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TI 2015-107/II

Tinbergen Institute Discussion Paper

Bubble Formation and (In)Efficient Markets in

Learning-to-Forecast and -optimise

Experiments

Te Baoa

Cars Hommesb

Tomasz Makarewiczb

a University of Groningen, the Netherlands;

b Faculty of Economics and Business, University of Amsterdam, and Tinbergen Institute, the

Netherlands.

Tinbergen Institute is the graduate school and research institute in economics of Erasmus University

Rotterdam, the University of Amsterdam and VU University Amsterdam.

More TI discussion papers can be downloaded at http://www.tinbergen.nl

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Bubble Formation and (In)Efficient Markets in

Learning-to-Forecast and -Optimise Experiments∗

Te Baoa Cars Hommesb Tomasz Makarewiczb

September 2015, Economic Journal forthcoming

a University of Groningen

b University of Amsterdam and Tinbergen Institute

Abstract. This experiment compares the price dynamics and bubble formation in an

asset market with a price adjustment rule in three treatments where subjects (1) submit

a price forecast only, (2) choose quantity to buy/sell and (3) perform both tasks. We

find deviation of the market price from the fundamental price in all treatments, but to

a larger degree in treatments (2) and (3). Mispricing is therefore a robust finding in

markets with positive expectation feedback. Some very large, recurring bubbles arise,

where the price is 3 times larger than the fundamental value, which were not seen in

former experiments.

JEL Classification: C91, C92, D53, D83, D84

Keywords: Financial Bubbles, Experimental Finance, Rational Expectations, Learning

to Forecast, Learning to Optimise

∗The authors are grateful to Alan Kirman for stimulating discussions and to the Editor An-

drea Galeotti and two anonymous referees for helpful and detailed comments. We also thank

participants at seminars at New York University, University of Goettingen, University of Ni-

jmegen, University of Namur, University of Groningen and conferences/workshops on “Com-

puting in Economics and Finance” 2013, Vancouver, “Experimental Finance” 2014, Zurich, in

particular our discussant David Schindler, “Expectations in Dynamic Macroeconomic Models”

2014, Bank of Finland, Helsinki and “Economic Dynamics and Quantitative Finance”, Decem-

ber 2014, Sydney. We gratefully acknowledge the financial support from NWO (Dutch Science

Foundation) Project No. 40611142 “Learning to Forecast with Evolutionary Models” and other

projects: the EUFP7 projects Complexity Research Initiative for Systemic Instabilities (CRISIS,

grant 288501) and Integrated Macro-Financial Modeling for Robust Policy Design (MACFIN-

ROBODS, grant 612796), and from the Institute of New Economic Thinking (INET) grant

project “Heterogeneous Expectations and Financial Crises” (INO 1200026). Email addresses:

T.Bao@rug.nl, C.H.Hommes@uva.nl, T.A.Makarewicz@uva.nl.

1

This paper investigates the price dynamics and bubble formation in an experi-

mental asset pricing market with a price adjustment rule. The purpose of the study

is to address a fundamental question about the origins of bubbles: do bubbles arise

because agents fail to form rational expectations or because they fail to optimise

their trading quantity given their expectations? Our experiment indicates that

both forces have a destabilising effect on the financial markets, which implies that

both deviations from rationality deserve more attention in future theoretical or

policy-oriented inquiries on bubble formation and market efficiency.

We design three experimental treatments: (1) subjects make a forecast only,

and are paid according to forecasting accuracy; (2) subjects make a quantity deci-

sion only, and are paid according to the profitability of their decision; (3) subjects

make both a forecast and a quantity decision, and are paid by their performance of

either of the tasks with equal probability. We design the payoff functions carefully

so that under the assumptions of perfect rationality and price taking behaviour,

these three tasks are equivalent in our experiment and should lead the subjects

to an equilibrium with a constant fundamental price. In contrast, we find none

of the experimental markets to show a reliable convergence to the fundamental

outcome. The market price is relatively most stable, with a slow upward trend, in

the treatment where the subjects make forecasts only. There are recurring bubbles

and crashes with high frequency and magnitude when the subjects submit both a

price forecast and a trading quantity decision.

Asset bubbles can be traced back to the very beginning of financial markets,

but have not been investigated extensively by modern economics and finance lit-

erature. One possible reason is that it contradicts the standard theory of rational

expectations (Muth, 1961; Lucas Jr., 1972) and efficient markets (Fama, 1970).

Recent finance literature however has shown growing interest in bounded rational-

ity (Farmer and Lo, 1999; Shiller, 2003) and ‘abnormal’ market movements such

as over- and under-reaction to changes in fundamentals (Bondt and Thaler, 2012)

and excess volatility (Campbell and Shiller, 1989). The recent financial crisis and

the preceding boom and bust in the US housing market highlight the importance

of understanding the mechanism of financial bubbles in order for policy makers to

design policies/institutions to enhance market stability.

It is usually difficult to identify bubbles using data from the field, since people

may substantially disagree about the underlying fundamental price of the asset

(see Hommes and in’t Veld, 2014, for a discussion about the S&P 500 example).

Laboratory experiments have an advantage in investigating this question by taking

full control over the underlying fundamental price. Smith et al. (1988) are among

the first authors to reliably reproduce price bubbles and crashes of asset prices in

2

a laboratory setting. They let the subjects trade an asset that pays a dividend

in each of 15 periods. The fundamental price at each period equals the sum of

the remaining expected dividends and follows a decreasing step function. The

authors find the price to go substantially above the fundamental price after the

initial periods before it crashes towards the end of the experiment. This approach

has been followed in many studies e.g. Lei et al. (2001); Noussair et al. (2001);

Dufwenberg et al. (2005); Haruvy and Noussair (2006); Akiyama et al. (2012);

Haruvy et al. (2013); Fu¨llbrunn et al. (2014).1 A typical result of these papers is

that the price boom and bust is a robust finding despite several major changes in

the experimental design.

Nevertheless, Kirchler et al. (2012); Huber and Kirchler (2012) argue that

the non-fundamental outcomes in these experiments are due to misunderstanding:

subjects may be simply confused by the declining fundamental price. They support

their argument by showing that no bubble appears when the fundamental price is

not declining or when the declining fundamental price is further illustrated by an

example of ‘stocks of a depletable gold mine’. Another potential concern about

these experiments, due to a typically short horizon (15 periods), is that one cannot

test whether financial crashes are likely to be followed by new bubbles. It is very

important to study the recurrence of boom-bust cycles in asset prices, for example

to understand the evolution of the asset prices between the dot-com bubble and

crash and the 2007/2008 financial crisis.

The Smith et al. (1988) experiment are categorised as ‘learning to optimise’

(henceforth LtO) experiments (see Duffy, 2008, for an extensive discussion). Be-

sides this approach, there is a complementary ‘learning to forecast’ (henceforth

LtF) experimental design introduced by Marimon et al. (1993) (see Hommes,

2011; Assenza et al., 2014, for comprehensive surveys). Hommes et al. (2005)

run an experiment where subjects act as professional advisers (forecasters) for a

pension fund: they submit a price forecasts, which is transformed into a quantity

decision of buying/selling by a computer program based on optimization over a

standard myopic mean-variance utility function. Subjects are paid according to

their forecasting accuracy. The fundamental price is defined as the rational expec-

tation equilibrium and remains constant over time. The results are twofold: (1)

the asset price fails to converge to the fundamental, but oscillates and forms bub-

bles in several markets; (2) instead of having rational expectations, most subjects

follow price trend extrapolation strategies (cf. Bostian and Holt, 2009). Heemei-

jer et al. (2009) and Bao et al. (2012) investigate whether the non-convergence

1For surveys of the literature, see Sunder (1995); Noussair and Tucker (2013).

3

result is driven by the positive expectation feedback nature of the experimental

market. Positive/negative expectation feedback means that the realised market

price increases/decreases when the average price expectation increases/decreases.

The results show that while negative feedback markets converge quickly to the

fundamental price, and adjust quickly to a new fundamental after a large shock,

positive feedback markets usually fail to converge, but under-react to the shocks

in the short run, and over-react in the long run.

The subjects in Hommes et al. (2005) and other ‘learning to forecast’ experi-

ments do not directly trade, but are assisted by a computer program to translate

their forecasts into optimal trading decisions. A natural question is what happens

if they submit explicit quantity decisions, i.e. if the experiment is based on the

‘learning to optimise’ design. Are the observed bubbles robust against the LtO

design or are they just an artifact of the computerised trading in the LtF design?

In this paper we design an experiment, in which the fundamental price is con-

stant over time (as in Hommes et al., 2005), but the subjects are asked to directly

indicate the amount of asset they want to buy/sell. Different from the double auc-

tion mechanism in the Smith et al. (1988) design, the price in our experiment is

determined by a price adjustment rule based on excess supply/demand (Beja and

Goldman, 1980; Campbell et al., 1997; LeBaron, 2006). Our experiment is help-

ful in testing financial theory based on such demand/supply market mechanisms.

Furthermore, our design allows us to have a longer time span of the experimental

sessions, which will enable a test for the recurrence of bubbles and crashes.

The main finding of our experiment is that the persistent deviation from the

fundamental price in Hommes et al. (2005) is a robust finding against task design.

Based on Relative Absolute Deviation (RAD) and Relative Deviation (RD) as

defined by Sto¨ckl et al. (2010), we find that the amplitude of the mis-pricing in

treatment (2) and (3) is much higher than in treatment (1). We also find larger

heterogeneity in traded quantities than individual price forecasts. These finding

suggest that learning to optimise is even harder than learning to forecast, and

therefore leads to even larger deviations from rationality and efficiency.

An important finding of our experiment is that in the Mixed, LtO and LtF

designs some very large and repeated price oscillations occur, where the price peaks

at more than 3 times the fundamental price. This was not observed in the former

experimental literature. Since bubbles in stock and housing prices reached similar

levels (the housing price index increases by 300% in several local markets before it

decreased by about 50% during the crisis), our experimental design may provide

a potentially better test bed for policies that deal with large recurrent bubbles.

Another contribution is that we provide an empirical micro foundation of ob-

4

served differences in aggregate macro behaviour across treatments. We estimate

individual forecasting and trading rules and find significant differences across treat-

ments. In the LtF treatment individual forecasting behaviour is more cautious in

the sense that subjects use a more conservative anchor (a weighted average of

last observed price and last forecast) in their trend-following rules, while in the

Mixed treatment almost all weight is given to the last observed price leading to a

more aggressive trend-following forecasting rule. Individual trading behaviour of

most subjects is characterized by extrapolation of past and/or expected returns.

Moreover, in the LtO and Mixed treatments the return extrapolation coefficients

are higher. These differences in individual behaviour explain the more unstable

aggregate behaviour with recurring booms and busts in the LtO and Mixed treat-

ments. We also perform a formal statistical test on individual heterogeneity in

trading strategies under the Mixed treatment. In particular, in some trading mar-

kets we observe a large degree of heterogeneity in the quantity decision even when

the price is rather stable. In the Mixed treatment most subjects fail to trade at

the conditionally optimal quantity given their own forecast. Learning to trade

optimally thus appears to be difficult.

Our paper is related to Bao et al. (2013) who run an experiment to compare

the LtF, LtO and Mixed designs in a cobweb economy. The main difference is

that they consider a negative feedback system, for which all markets converge

to the fundamental price, and find differences in the speed of convergence across

treatments.

The paper is organised as follows: Section 1 presents the experimental de-

sign and formulates testable hypotheses. Section 2 summarises the experimental

results and performs statistical tests of convergence to REE and for differences

across treatments based on aggregate variables as well as individual decision rules.

Finally, Section 3 concludes.

1 Experimental Design

In this section we explain the design of our experiment. We begin by defining

the treatments, followed by a discussion of the information given to the subjects.

Thereafter, we derive the micro foundations of the experimental economy, discuss

the implementation of the experiment, and specify hypotheses that will be tested

empirically.

5

1.1 Experimental Treatments

The experimental economy is based on a simple asset market with a constant

fundamental price. There are I = 6 subjects in each market, and each subject

plays an advisory role to a professional trading company. Subject task is either

to predict the asset price, suggest trading quantity or both, and subjects are re-

warded depending on their forecasting accuracy or trading profits. These decisions

generate an excess demand that determines the market price for the asset. The

experimental sessions last for T = 50 trading rounds. To present a quick overview

of the treatment designs, we only show the reduced form law of motion of the

price in each treatment in this section. The microfoundation of the experimental

economy and choice of parameters will be explained in detail in section 1.3.

Based on the nature of the task and the payoff scheme, there are three treat-

ments in the experiment. It is important to note here that the underlying market

structure is the same regardless of the subject task. We carefully choose the pa-

rameters of the model and payoff function so that under rational expectations,

these treatments are equivalent and lead to the same market equilibrium. The

treatments are specified as follows:

LtF Classical Learning-to-Forecast experiment. Subjects act as forecasting ad-

visers, namely they are asked for one-period ahead price predictions pei,t+1.

The subjects’ reward depends only on the prediction accuracy, defined by

(see also Table B.1 in Appendix B)

Payoffi,t+1 = max

{

0, (1300− 1300

49

(

pei,t+1 − pt+1

)2

)

}

, (1)

where pei,t+1 denotes the forecast of price at period t+1 formulated by subject

i and pt+1 is the realised asset price at period t+ 1.

The subject forecasts are automatically translated into excess demand for the

asset, yielding the following law of motion for the LtF treatment economy:

pt+1 = 66 +

20

21

(

p¯et+1 − 66

)

+ εt, (2)

where p∗ = pf = 66 is the fundamental price of the asset as well as the

unique Rational Expectations Equilibrium, p¯et+1 ≡ 16

∑6

i=1 p

e

i,t+1 denotes the

average price forecast of the six subjects and εt ∼ N(0, 1) is a small IID

shock to price pt+1.

For the price adjustment rule (2) the subjects’ payoff is maximised when all

predict the fundamental price, so that on average they make the smallest

prediction errors. Hence, in the LtF treatment it is optimal for all subjects

to predict pei,t+1 = 66.

6

LtO Classical Learning-to-Optimise experiment, where the subjects are asked to

decide on the asset quantity zi,t. Unlike the experiments in the spirit of

Smith et al. (1988), subjects in this treatment do not accumulate the asset

over periods. Instead, zi,t represents the final position of subject i in period

t. This position can be short with zi,t < 0 and is cleared once pt+1 is realised.

Subjects earn payoff based on the realised return ρt+1, which is defined as

a (constant) dividend y = 3.3 plus the capital gain over the constant gross

interest rate R = 1.05 of a secure bond:

ρt+1 ≡ pt+1 + y −Rpt = pt+1 + 3.3− 1.05pt. (3)

Subjects are not explicitly asked for a price prediction, but can use a built-in

calculator in the experimental program to compute the expected asset return

ρet+1 for any price forecast p

e

t+1 as in equation (3). Subjects are rewarded

according to

Payoffi,t+1 = max

{

0, 800 + 40(zi,t(pt+1 + 3.3− 1.05pt)− 3z2i,t)

}

. (4)

This payoff corresponds to a mean-variance utility function of the financial

firms in the underlying economy, as explained below. Expected payoff can be

computed by the subjects or read from a payoff table, depending on the cho-

sen quantity and the expected excess return (see Table B.2 in Appendix B).

Under the assumption of price-taking behaviour, i.e., when the subjects

ignore the impact of their own trading decisions on the realised market

price, the optimal demand for asset given one’s own price forecast pei,t+1

is z∗i,t =

pei,t+1+3.3−1.05pt

6

=

ρet+1

6

.

The law of motion of the LtO treatment is given by the price adjustment

rule based on the aggregate excess demand

pt+1 = pt +

20

21

6∑

i=1

zi,t + εt (5)

for the same set of IID shocks εt as in the LtF treatment. Under the as-

sumption of price-taking behaviour, the Rational Expectation Equilibrium

(REE) of the market is p∗ = pf = 66, and the associated optimal demand

for the asset is z∗ = 0 for each individual. Therefore, the optimal choices are

equivalent in the LtF and LtO treatments. For other cases in which subjects

deviate from price-taking behaviour, e.g. by taking their market power into

account and playing collusive or non-cooperative Nash strategies, a detailed

discussion is provided in Appendix C.

7

Mixed Each subject is asked first for his or her price forecast pei,t+1 and second for

the choice of the asset quantity zi,t. In order to avoid hedging, the reward for

the whole experiment is based on either the payoff in (1) or (4) with equal

probability (flip of a coin at the end of the session). The law of motion of

the Mixed treatment is given by (5), the same price adjustment rule as in

LtO and does not depend on the submitted price forecasts.

The points in each treatment are exchanged into Euro at the end of the experiment

with the conversion rate 3000 points = 1 Euro. We add a max function to the

forecasting and trading payoffs to avoid negative rewards.

1.2 Information to the Subjects

At the beginning of the experimental sessions, subjects were informed about their

task and payoff scheme, including the payoff functions (1) or (4) depending on the

treatment. We supplemented the subjects with payoff tables (see Appendix B).

Subjects from the LtF treatment were told that the asset price depends pos-

itively on the average price forecast, while subjects in the two other treatments

were informed that the price increases with the excess demand. In addition, in the

Mixed treatment we made it clear that the subject payoffs may be related to the

forecasting accuracy, but that the realized price itself depends exclusively on their

trades. Regardless of the treatment, we provided the subjects only with qualitative

information about the market, that is we did not explicate the respective laws of

motion (2) or (5).

Throughout the experiment, the subject could observe past market prices and

their individual decisions, in graphical and table form, but they could not see the

decisions, or an average decision, of the other participants. We did not mention the

fundamental price in the instructions at all, though we did provide the information

about the interest rate and the asset dividend in all the three treatments, which

could be used to compute the fundamental price p∗ = y¯/r = 66. Finally, the

subjects know the specification of their payoff function, i.e., the payoff is higher if

the prediction error (trading profit) is lower (higher) for the forecasters (traders).

1.3 Experimental Economy

This section provides some micro-foundations of our experimental economy. We

build our experimental economy upon an asset market with heterogeneous beliefs

as in Brock and Hommes (1998). There are I = 6 agents, who allocate investment

between a risky asset that pays a fixed dividend y and a risk-free bond that pays

8

a fixed gross return R = 1 + r.2 The wealth of agent i evolves according to

Wi,t+1 = RWi,t + zi,t(pt+1 + y −Rpt), (6)

where zi,t is the demand (in the sense of the final position) for the risky asset

by agent i in period t (positive sign for buying and negative sign for selling) and

pt and pt+1 are the prices of the risky asset in periods t and t + 1 respectively.

Let Ei,t and Vi,t denote the beliefs or forecasts of agent i about the conditional

expectation and the conditional variance based on publicly available information.

The agents are assumed to be simple myopic mean-variance maximizers of next

period’s wealth, i.e. they solve the myopic optimisation problem:

max

zi,t

{

Ei,tWi,t+1 − a

2

Vi,t(Wi,t+1)

}

≡ max

zi,t

{

zi,tEi,tρt+1 − a

2

z2i,tVi,t(ρt+1)

}

, (7)

where a is a parameter for risk aversion, and ρt+1 is the excess return as defined

in equation (3). In the experiment, we use an affine transformation of this utility

function as in (4) as a payoff for the trading task.

Optimal demand of agent i is given by3

z∗i,t =

Ei,t(ρt+1)

aVi,t(ρt+1)

=

pei,t+1 + y −Rpt

aσ2

, (8)

where pei,t+1 = Ei,tpt+1 is the individual forecast by agent i of the price in period

t+ 1. The market price is set by a market maker using a simple price adjustment

mechanism in response to excess demand (Beja and Goldman, 1980),4 given by

pt+1 = pt + λ

(

ZDt − ZSt

)

+ εt, (9)

2Fixed dividend allows for a constant fundamental price throughout the experiment. In a

more general model with the same demand functions and market equilibrium, y corresponds to

the mean of an (exogenous) IID stochastic dividend process yt; see Brock and Hommes (1998)

for a discussion.

3The last equality in (8) follows from a simplifying assumption made in Brock and Hommes

(1998) that all agents have homogeneous and constant beliefs about the conditional variance,

i.e. Vi,t(ρt+1) = σ

2. See Hommes (2013), Chapter 6, for a more detailed discussion.

4See e.g. Chiarella et al. (2009) for a survey on the abundant literature about the price

adjustment market mechanisms. We decided to use (9) instead of a market clearing mechanism

for two reasons: (i) market maker is a stylized description of a specialist driven market, a common

case for financial markets (e.g. NASDAQ); and (ii) the current one-period ahead design is much

simpler for the subjects than one based on a market clearing mechanism, which requires two-

period ahead trading and forecasting. In particular, the two-period ahead trading/forecasting

feature would lead to a 3-dimensional payoff table instead of the 2-dimensional payoff table in

Appendix B.2. The two-period ahead market clearing design results in much more volatile price

patterns in the LtF experiments (Hommes, 2011), which suggests that our main finding –that

the boundedly rational trading can be a destabilizing force in the financial markets– is likely to

be robust in a similar two-period ahead LtO experiment with a market clearing design.

9

where εt ∼ N(0, 1) is a small IID shock, λ > 0 is a scaling factor, ZSt is the

exogenous supply and ZDt is the total demand. This mechanism guarantees that

excess demand/supply increases/decreases the price.

For simplicity, the exogenous supply ZSt is normalised to 0 in all periods. In the

experiment, we take Rλ = 1, specifically R = 1 + r = 21/20, λ = 20/21, aσ2z = 6,

and y = 3.3. We chose these specific parameters mainly for simplicity of the law of

motion of the price. For example, by imposing aσ2z = 6, the total excess demand

coincide with the average expected excess return, and when Rλ = 1, this ensures

that the final law of motion of asset price in the LtF treatment only depends on

the average forecast p¯et+1, but does not contain pt. The price adjustment based on

aggregate individual demand thus takes the simple form

pt+1 = pt +

20

21

6∑

i=1

zi,t + εt, (10)

which constitutes the law of motion (5) for the LtO and Mixed treatments, in

which the subjects are asked to elicit their asset demands.

For an optimising agent and the chosen parameters, the individual optimal

demand (8) conditional on a price forecast pei,t+1 equals

z∗i,t =

ρei,t+1

aσ2

=

pei,t+1 + 3.3− 1.05pt

6

, (11)

with ρei,t+1 the forecast of excess return in period t+ 1 by agent i. Substituting it

back into (5) gives

pt+1 = 66 +

20

21

(

p¯et+1 − 66

)

+ εt, (12)

where p¯et+1 =

1

6

∑6

i=1 p

e

i,t+1 is the average prediction of the price pt+1 by six sub-

jects.5 This price is the temporary equilibrium with point-beliefs about prices and

represents the price adjustment process as a function of the average individual

forecast. It constitutes the law of motion (2) for the LtF treatment, in which the

subjects are asked to elicit their price expectations.

We note that from the optimal demand (11) it is clear that optimising the

(quadratic) mean-variance utility function (7) is equivalent to minimising the

quadratic penalty for forecasting errors as in the LtF payoff function (1). This

implies that the trading and forecasting tasks in the experiment are equivalent

under perfect rationality.

5Heemeijer et al. (2009) used a similar price adjustment rule in a learning to forecast exper-

iment that compares positive versus negative expectation feedback, but their fundamental price

is 60 instead of 66.

10

By imposing the rational expectations condition p¯et+1 = p

f = Et(pt+1), a simple

computation shows that pf = 66 is the unique Rational Expectation Equilibrium

(REE) of the system. This fundamental price equals the discounted sum of all

expected future dividends, i.e., pf = y/r . If all agents have rational expectations,

the realised price becomes pt = p

f + εt = 66 + εt, i.e. the fundamental price plus

(small) white noise and, on average, the price forecasts are self-fulfilling. When

the price is pf , the (expected) excess return of the risky asset in (3) equals 0 and

the optimal demand for the risky asset in (8) by each agent is also 0, that is excess

demand is equal to 0.

1.4 Liquidity Constraints

To limit the effect of extreme price forecasts or quantity decisions in the experi-

ment, we impose the following liquidity constraints on the subjects. For the LtF

treatment, price predictions such that pei,t+1 > pt+30 or p

e

i,t+1 < pt−30 are treated

as pei,t+1 = pt+30 and p

e

i,t+1 = pt−30 respectively. For the LtO treatment, quantity

decisions greater than 5 or smaller than −5 are treated as 5 and −5 respectively.

These two liquidity constraints are roughly the same, since the optimal asset de-

mand (11) is close to one sixth of the expected price difference. Nevertheless, the

liquidity constraint in the LtF treatment was never binding, while under the LtO

treatment subjects would sometimes trade at the edges of the allowed quantity

interval. We also imposed additional constraint that pt has to be non-negative

and not greater than 300. In the experiment, this constraint never had to be

implemented.

1.5 Number of Observations

The experiment was conducted on December 14, 17, 18 and 20, 2012 and June

6, 2014 at the CREED Laboratory, University of Amsterdam. 144 subjects were

recruited. The experiment employs a group design with 6 subjects in each ex-

perimental market. There are 24 markets in total and 8 for each treatment. No

subject participates in more than one session. The duration of the experiment

is typically about 1 hour for the LtF treatment, 1 hour and 15 minutes for the

LtO treatment, and 1 hour 45 minutes for the Mixed treatment. Experimental

instructions are shown in Appendix A.

11

1.6 Testable Hypotheses

The RE benchmark suggests that the subjects should learn to play the REE and

behave similarly in all treatments. In addition, a rational decision maker should

be able to find the optimal demand for the asset given his price forecast according

to Equation (11) in the Mixed treatment. These theoretical predictions can be

formulated into the following testable hypotheses:

HYPOTHESIS 1: The asset prices converge to the Rational Expectation Equi-

librium in all markets;

HYPOTHESIS 2: There is no systematic difference between the market prices

across the treatments;

HYPOTHESIS 3: Subjects’ earnings efficiency (defined as the ratio of the ex-

perimental payoff divided by the hypothetical payoff when all subjects play

the REE) are independent from the treatment;

HYPOTHESIS 4: In the Mixed treatment the quantity decisions by the sub-

jects are optimal conditional on their price expectations;

HYPOTHESIS 5: There is no systematic difference between the decision rules

used by the subjects for the same task across the treatments.

These hypotheses are further translated into rigorous statistical tests. To be

specific, we will use Relative (Absolute) Deviation (Sto¨ckl et al., 2010) to measure

price convergence, and test the difference of the distribution of this measure be-

tween the three treatments. (HYPOTHESIS 1 and 2). Relative earnings can be

compared with the Mann-Whitney-Wilcoxon rank-sum test (HYPOTHESIS 3).

Finally, we estimate individual behavioural rules for every subject: a simple re-

striction test will reveal whether HYPOTHESIS 4 is true, while the rank-

sum test can again be used to test the rule differences between the treatments

(HYPOTHESIS 5). Notice that HYPOTHESIS 1 is nested within HY-

POTHESIS 2, while HYPOTHESIS 4 is nested within HYPOTHESIS 5.

2 Experimental Results

2.1 Overview

Figure 1 (LtF treatment), Figure 2 (LtO treatment) and Figure 3 (Mixed treat-

ment) show plots of the market prices in each treatment. For most of the groups,

the prices and predictions remained in the interval [0, 100]. The exceptions are

markets 1, 4 and 8 (Figures 3a, 3d and 3h) in the Mixed treatment. In the first

12

two of these three groups, prices peaked at almost 150 (more than twice the funda-

mental price pf = 66) and in the last group, prices reached 225, almost 3.5 times

the fundamental price. Moreover, markets 4 and 8 of the Mixed treatment show

repeated booms and busts.

The figures suggest that the market price is the most stable in the LtF treat-

ment, and the most unstable in the Mixed treatment. In the LtF treatment, there

is little heterogeneity between the individual forecasts, shown by the green dashed

lines. In the LtO treatment, however, there is a high level of heterogeneity in the

quantity decisions shown by the blue dashed lines. In the Mixed treatment, it

is somewhat surprising that the low heterogeneity in price forecasts and the high

heterogeneity in quantity decisions coexist.6

It is noticeable that in two markets in the LtO and Mixed treatment, the

market price stabilises after a few periods, but stays at a non REE level. Market

2 in the LtO treatment stabilises around price 40, and Market 6 in the Mixed

treatment stabilises around price 50. In these two markets, the optimal demand

by each individual as implied by (11) should be about 0.2 (0.15) when the price

stabilises at 40 (50). However, the actual average demand in the experiment stays

very close to 0 in both cases. This is an indication of sub-optimal behaviour by

some subjects. It may be caused by two reasons: (1) the subjects mistakenly

ignored the role of dividend in the return function, and thought that buying is not

profitable unless the price change is strictly positive, or (2) some of them held a

pessimistic view about the market, and kept submitting a lower demand than the

optimal level as implied by their price forecast.

In general, convergence to the REE does not seem to occur in any of the

treatments. This suggests that the hypotheses based on the rational expectations

benchmark are likely to be rejected. Furthermore, the figures suggest clear differ-

ences between the treatments. In the remainder of this section, we will discuss the

statistical evidence for the hypotheses in detail.

6We compare the dispersion of individual decisions using the standard deviation of the (im-

plied) quantity decisions averaged over all periods in each market. A rank-sum test suggests that

there is no difference between dispersion of quantity decisions in the LtO versus Mixed treat-

ment (with p-value equal to 0.083 for dispersion over all periods and p-value equal to 0.161 for

dispersion over last 40 periods). The dispersions of the quantity decisions in the LtO and Mixed

treatments are indeed significantly larger than the dispersion of (implied) quantity decisions in

the LtF treatment, with p-values equal to 0 for both all and last 40 periods.

13

Figure 1: Price Dynamics in LtF Treatment

0

20

40

60

80

100

0 10 20 30 40 50

(a) Group 1

0

20

40

60

80

100

0 10 20 30 40 50

(b) Group 2

0

20

40

60

80

100

0 10 20 30 40 50

(c) Group 3

0

20

40

60

80

100

0 10 20 30 40 50

(d) Group 4

0

20

40

60

80

100

0 10 20 30 40 50

(e) Group 5

0

20

40

60

80

100

0 10 20 30 40 50

(f) Group 6

0

20

40

60

80

100

0 10 20 30 40 50

(g) Group 7

0

20

40

60

80

100

0 10 20 30 40 50

(h) Group 8

Notes. Groups 1-8 for the Learning to Forecast treatment. Straight line shows the

fundamental price pf = 66, solid black line denotes the realised price, while green dashed

lines denote individual forecasts.

14

Figure 2: Price Dynamics in LtO Treatment

0

20

40

60

80

100

0 10 20 30 40 50

Realised prices

-4

-2

0

2

4

0 10 20 30 40 50

Inividual quantity decisions

(a) Group 1

0

20

40

60

80

100

0 10 20 30 40 50

Realised prices

-4

-2

0

2

4

0 10 20 30 40 50

Inividual quantity decisions

(b) Group 2

0

20

40

60

80

100

0 10 20 30 40 50

Realised prices

-4

-2

0

2

4

0 10 20 30 40 50

Inividual quantity decisions

(c) Group 3

0

20

40

60

80

100

0 10 20 30 40 50

Realised prices

-4

-2

0

2

4

0 10 20 30 40 50

Inividual quantity decisions

(d) Group 4

0

20

40

60

80

100

0 10 20 30 40 50

Realised prices

-4

-2

0

2

4

0 10 20 30 40 50

Inividual quantity decisions

(e) Group 5

0

20

40

60

80

100

0 10 20 30 40 50

Realised prices

-4

-2

0

2

4

0 10 20 30 40 50

Inividual quantity decisions

(f) Group 6

0

20

40

60

80

100

120

140

0 10 20 30 40 50

Realised prices

-4

-2

0

2

4

0 10 20 30 40 50

Inividual quantity decisions

(g) Group 7

0

20

40

60

80

100

0 10 20 30 40 50

Realised prices

-4

-2

0

2

4

0 10 20 30 40 50

Inividual quantity decisions

(h) Group 8

Notes. Groups 1-8 for the Learning to Optimise treatment. Each group is presented

in two panels. The upper panel displays the fundamental price pf = 66 (straight line)

and the realised prices (solid black line), while the lower panel displays individual trades

(dashed blue lines) and average trade (solid red line). Notice the different y-axis scale

for group 7 (picture g).

15

Figure 3: Price Dynamics in Mixed Treatment

0

20

40

60

80

100

120

140

0 10 20 30 40 50

Realised prices

-4

-2

0

2

4

0 10 20 30 40 50

Inividual quantity decisions

(a) Group 1, price scale [0, 150]

0

20

40

60

80

100

0 10 20 30 40 50

Realised prices

-4

-2

0

2

4

0 10 20 30 40 50

Inividual quantity decisions

(b) Group 2

0

20

40

60

80

100

0 10 20 30 40 50

Realised prices

-4

-2

0

2

4

0 10 20 30 40 50

Inividual quantity decisions

(c) Group 3

0

20

40

60

80

100

120

140

0 10 20 30 40 50

Realised prices

-4

-2

0

2

4

0 10 20 30 40 50

Inividual quantity decisions

(d) Group 4, price scale [0, 150]

0

20

40

60

80

100

0 10 20 30 40 50

Realised prices

-4

-2

0

2

4

0 10 20 30 40 50

Inividual quantity decisions

(e) Group 5

0

20

40

60

80

100

0 10 20 30 40 50

Realised prices

-4

-2

0

2

4

0 10 20 30 40 50

Inividual quantity decisions

(f) Group 6

0

20

40

60

80

100

0 10 20 30 40 50

Realised prices

-4

-2

0

2

4

0 10 20 30 40 50

Inividual quantity decisions

(g) Group 7

0

50

100

150

200

250

0 10 20 30 40 50

Realised prices

-4

-2

0

2

4

0 10 20 30 40 50

Inividual quantity decisions

(h) Group 8, price scale [0, 250]

Notes. Groups 1-8 for the Mixed treatment with subject forecasting and trading. Each

group is presented in a picture with two panels. The upper panel displays the funda-

mental price pf = 66 (straight line), the realised prices (solid black line) and individual

predictions (green dashed lines), while the lower panel displays individual trades (dashed

blue lines) and average trade (solid red line). Notice the different y-axis scale for groups

1, 4 and 8 (pictures a, d and h respectively).

16

2.2 Quantifying the Bubbles

The term ‘bubble’ is informally used in the literature to describe, loosely speaking,

a prolonged spell of an asset price growth beyond its fundamental. In order to

capture this notion with a rigid statistic, we follow Sto¨ckl et al. (2010) and evaluate

the experimental mispricing with the Relative Absolute Deviation (RAD) and

Relative Deviation (RD). These two indices measure respectively the absolute and

relative deviation from the fundamental in a specific period t and are given by

RADg,t ≡ |p

g

t − pf |

pf

× 100%, (13)

RDg,t ≡ p

g

t − pf

pf

× 100%, (14)

where pf = 66 is the fundamental price and pgt is the realised asset price at period

t in the session of group g. The average RADg and RDg are defined as

RADg ≡ 1

50

50∑

t=1

RADg,t, (15)

RDg ≡ 1

50

50∑

t=1

RDg,t, (16)

RADg shows the average relative distance between the realised prices and the

fundamental in group g, while the average RDg focuses on the sign of this re-

lationship. Groups with average RDg close to zero could either converge to the

fundamental (in which case the RADg is also close to zero) or oscillate around the

fundamental (possibly with high RADg), while positive or negative average RDg

signals that the group typically over- or underpriced the asset.

It is difficult to come up with a formal criterion for a bubble in terms of these

measures. In particular, when bubbles are accompanied by a price plunge, or

“negative bubbles”, the RD may be very close to 0. Therefore, in this paper we

focus on the differences between the three treatments.7

The results for average RAD and RD measures for each treatment are pre-

sented in Table 1. They confirm that the LtF groups were the closest to, though

7As empirical benchmarks we computed these two measures for the US stock and housing

markets. The RAD (RD) is 40% (20.2%) for S&P500, based on quarterly data 1950Q1−2012Q4

and the fundamental computed by a standard Gordon present discounted value model; for the

same data set, using deviations from the Campbell-Cochrane consumption-habit fundamental

model the RAD (RD) is 19% (3.9%) (Hommes and in’t Veld, 2014). For US housing market

data in deviations from a benchmark fundamental based on housing rents the RAD (RD) are

7.7% (0.4%) for over 40 years of quarterly data 1970Q1− 2013Q1 and 9.7% (2.2%) for 20 years

of quarterly data 1993Q1− 2013Q1 (Bolt et al., 2014).

17

still quite far from, the REE (with an average RAD of about 9.5%), while Mixed

groups exhibited the largest price deviations with an average RAD of 36%. Inter-

estingly, LtO groups had significant oscillations (on average high RAD of 24.7%),

but centered close to the fundamental price (average RD of 1.4%, compared to av-

erage RD of −3% and 16.1% for the LtF and Mixed treatments respectively). LtF

groups on average are below the fundamental price and Mixed groups typically

overshoot it.

Table 1: RAD and RD

Treatment LtF LtO Mixed

Group RAD RD RAD RD RAD RD

#1 10.03∗∗∗ −7.011 18.26∗∗∗ −8.148∗ 38.65∗∗∗ 36.84∗

#2 17.98∗∗∗ −16.94∗ 34.52∗∗∗ −34.52∗ 7.27∗∗∗ −5.657∗

#3 8.019∗∗ −6.048 30.2∗∗∗ −12.95∗ 8.025∗∗∗ 4.014∗

#4 7.285∗∗ −5.196 20.63∗∗∗ 3.844∗ 42.86∗∗∗ 35.46∗

#5 8.366∗∗∗ 4.152∗ 16.55∗∗∗ 5.256∗ 14.98∗∗∗ 3.341∗

#6 14.52∗∗∗ 6.503∗ 17.51∗∗∗ 7.056∗ 23.08∗∗∗ −23.08∗

#7 4.222 1.104∗ 31.22∗∗∗ 23.82∗ 32.14∗∗∗ −18.71

#8 5.365 −0.2539∗ 28.48∗∗∗ 26.65∗ 120.7∗∗∗ 96.5∗

Average 9.473 −2.961 24.67 1.376 35.97 16.09

Notes. Relative Absolute Deviation (RAD) and Relative Deviation (RD) of the experi-

mental prices for the three treatments, in percentages. ∗∗∗ (∗∗) denotes groups for which

the average RAD from the last 40 periods is larger than 3% on 1% (5%) significance

level. ∗ denotes groups for which the average RD from the last 40 periods is outside

[−1.5%, 1.5%] interval on 5% significance level.

A simple t-test shows that for the LtO and Mixed treatment, as well as for 6 out

of 8 LtF groups (exceptions are Markets 7 and 8), the means of the groups’ RAD

measures (disregarding the initial 10 periods to allow for learning) are significantly

larger than 3%.8 Furthermore, for all groups in all three treatments, t-test on

any meaningful significance level rejects the null of the average price (for periods

11−50, i.e. the last 40 periods to allow for learning by the subjects) being equal to

the fundamental value. This result shows negative evidence on HYPOTHESIS 1:

none of the treatments converges to the REE.

There is no significant difference between the treatments in terms of RD ac-

83% RAD is approximately equivalent to a typical price deviation of 2 in absolute terms,

which corresponds to twice the standard deviations of the idiosyncratic supply shocks, i.e 95%

confidence bounds of the REE.

18

cording to the Mann-Whitney-Wilcoxon rank-sum test (p-value> 0.1 for each

pair of the treatments, z-statistic is −0.735,−0.735 and −0.420 for LtF, LtO

and Mixed respectively. The unit of observation is per market, i.e. 8 for each

treatment). However, the difference between the LtF treatment and each of the

other treatments in terms of RAD is significant at 5% according to the rank-

sum test (p-value= 0.002 and 0.003, and z-statistic is −3.151 and −2.205 for the

LtO and Mixed respectively, number of observations: 8 for each treatment), while

the difference between the LtO and Mixed is not significant (p-value= 0.753, z-

statistic= −0.135 number of observations: 8 for each treatment). This is strong

evidence against HYPOTHESIS 2, as it shows that trading and forecasting tasks

yield different market dynamics.

The RAD values in our paper are similar to those in Sto¨ckl et al. (2010) (see

specifically their Table 4 for the RAD/RD measures). Nevertheless, there are

some important differences. First, group 8 from the Mixed treatment (with RAD

equal to 120.7%) exhibits the largest price bubble in the experiment. Second,

the four experiments investigated by Sto¨ckl et al. (2010) have shorter spans (with

sessions of either 10 or 25 periods) and so typically witness one bubble. Our data

shows that the mispricing in experimental asset markets is a robust finding. The

crash of a bubble does not enforce the subjects to converge to the fundamental,

but instead marks the beginning of a ‘crisis’ until the market turns around and

a new bubble emerges. This succession of over- and under-pricing of the asset is

reflected in our RD measures, which are smaller than the typical ones reported by

Sto¨ckl et al. (2010), and can even be negative, despite high RAD.

In addition, our experiment yields measures resembling the above mentioned

benchmark stock and housing markets (see footnote 7). Indeed, the LtF, LtO and

Mixed experimental treatments yields boom/boost cycles of a realistic magnitude,

comparable to what has been observed in recent stock and housing market bubbles

and crashes.

RESULT 1. Among the three treatments, LtF incurs dynamics closest to the

REE. Nevertheless, the average price is still far from the rational expectations

equilibrium. Furthermore, in terms of aggregate dynamics LtF treatment is signif-

icantly different from the other two treatments, which are indistinguishable between

themselves. We conclude that HYPOTHESIS 1 and 2 are rejected.

2.3 Earnings Efficiency

Subjects’ earnings in the experiment are compared to the hypothetical case where

all subjects play according to the REE in all 50 periods. Subjects can earn 1300

19

points per period for the forecasting task when they play according to REE because

they make no prediction errors, and 800 points for the trading task when they play

according to the REE because the asset return is 0 and they should not buy or

sell. We use the ratio of actual against hypothetical REE payoffs as a measure

of payoff efficiency. This measure can be larger than 100% in treatments with

the LtO and Mixed Treatments, because the subjects can profit if they buy and

the price increases and vice versa. These earnings efficiency ratios, as reported in

Table D.1 in the appendix, are generally high (more than 75%).

The earnings efficiency for the forecasting task is higher in the LtF treatment

than in the Mixed treatment (rank-sum test for difference in distributions with

p-value=0.001). At the same time, the earnings efficiency for the trading task is

very similar in the LtO treatment and the Mixed treatment (rank-sum test with

p-value=0.753).

RESULT 2. Forecasting efficiency is significantly higher in the LtF than in the

Mixed treatment, while there is no significant difference in the trading efficiency

in treatments LtO and Mixed. HYPOTHESIS 3 is partially rejected.

2.4 Conditional Optimality of Forecast and Quantity Decision in

Mixed Treatment

In the Mixed treatment, each subject makes both a price forecast and a quantity

decision. It is therefore possible to investigate whether these two are consistent,

namely, whether the subjects’ quantity choices are close to the optimal demand

conditional on the price forecast as in Eq. (11) (the optimal quantity is 1/6 of

the corresponding expected asset return). Figure 4 shows the scatter plot of the

quantity decision against the implied predicted return ρei,t+1 = p

e

i,t+1 +3.3−1.05pt,

for each subject and each period separately.9 If all individuals made consistent

decisions, these points should lie on the (blue) line with slope 1/6.

Figure 4a illustrates two interesting observations. First, subjects have some

degree of ‘digit preference’, in the sense that the trading quantities are typically

round numbers or contain only one digit after the decimal. Second, the quantity

choices are far from being consistent with the price expectations. In fact, the

subjects sometimes sold (bought) the asset even though they believed its return

will be substantially positive (negative).

9Sometimes the subjects submit extremely high price predictions, which in most cases seem to

be typos. The scatter plot excludes these outliers, by restricting the horizontal scale of predicted

returns on the asset between −60 and 60.

20

Figure 4: Conditional Optimality of Quantity Decisions

-4

-2

0

2

4

-60 -40 -20 0 20 40 60

(a) Expected return vs trade

-2

-1.5

-1

-0.5

0

0.5

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6

(b) Trade rule (17): slope vs constant

Notes. ML estimation for trading rule (17) in the Mixed treatment. Panel (a) is the

scatter plot of the traded quantity (vertical axis) against the implied expected return

(horizontal axis). Each point represents one decision of one subject in one period from

one group. Panel (b) is the scatter plot of the estimated trading rules (17) slope (reaction

to expected return; horizontal axis) against constant (trading bias; vertical axis). Each

point represents one subject from one group. Solid line (left panel)/triangle (right panel)

denotes the optimal trade rule (zi,t = ρ

e

i,t/6). Dashed line (left panel)/circle (right panel)

denotes the estimated rule under restriction of homogeneity (zi,t = c+ θρ

e

i,t).

To further evaluate this finding, we run a series of Maximum Likelihood (ML)

regressions based on the trading rule

zi,t = ci + θiρ

e

i,t+1 + ηi,t, (17)

with ηi,t ∼ NID(0, σ2η,i). The estimated coefficients for all subjects are shown in

the scatter plot of Figure 4b. This model has a straightforward interpretation: it

takes the quantity choice of subject i in period t as a linear function of the implied

(by the price forecast) expected return on the asset. It has two important special

cases: homogeneity and optimality (nested within homogeneity). To be specific,

subject homogeneity (heterogeneity) corresponds to an insignificant (significant)

variation in the slope θi = θj (θi 6= θj) for any (some) pair of subjects i and j.

The constant ci shows the ‘irrational’ optimism/pessimism bias of subject i. Op-

timality of individual quantity decisions implies homogeneity with the additional

restrictions that θi = θj = 1/6 and ci = cj = 0 (no agent has a decision bias).

The assumptions of homogeneity and perfect optimisation are tested by esti-

mation of equation (17) with the restrictions on the parameters ci and θi.

10 These

10We use ML since the optimality constraint does not exclude heterogeneity of the idiosyncratic

shocks ηi,t. We exclude outliers defined as observations when a subject predicts an asset return

21

regressions are compared with an unrestricted regression (with θi 6= θj and ci 6= cj)

via a Likelihood Ratio (LR) test. The result of the LR test shows that both the

assumption of homogeneity and perfect optimisation are rejected (with p-values

below 0.001). Furthermore, we explicitly tested for zi,t = ρ

e

i,t/6 when estimating

individual rules. Estimations identified 11 subjects (23%) as consistent optimal

traders (see footnote 13 for a detailed discussion). In sum, we find evidence for

heterogeneity of individual trading rules. The majority of the subjects are unable

to learn the optimal solution.

This result has important implications for economic modelling. The RE hy-

pothesis is built on homogeneous and model consistent expectations, which the

agents in turn use to optimise their decisions. Many economists find the first ele-

ment of RE unrealistic: it is difficult for the agents to form rational expectations

due to limited understanding of the structure of the economy. But the second

part of RE is often taken as a good approximation: agents are assumed to make

an optimal decision conditional on what they think about the economy, even if

their forecast is wrong. Our subjects were endowed with as much information as

possible, including an asset return calculator, a table for profits based on the pre-

dicted asset return and chosen quantity and the explicit formula for profits; and

yet many failed to behave optimally in forecasting as well as choosing quantities.

The simplest explanation is that individuals in general lack the computational

capacity to make perfect mathematical optimisations.

RESULT 3. The subjects’ quantity decisions are not conditionally optimal given

their price forecasts in the Mixed treatment. We conclude that HYPOTHESIS 4

is rejected for 77% (37 out of 48) of the subjects.

2.5 Estimation of Individual Behavioural Rules

In this subsection we estimate individual forecasting and trading rules and inves-

tigate whether there are significant differences between treatments. Prior exper-

imental work (Heemeijer et al., 2009) suggests that in LtF experiments, subjects

use heterogeneous forecasting rules which nevertheless typically are well described

by a simple linear First-Order Rule

pei,t = αipt−1 + βip

e

i,t−1 + γi(pt−1 − pt−2). (18)

higher than 60 in absolute terms. To account for an initial learning phase, we exclude the first ten

periods from the sample. We also drop subjects 4 and 5 from group 6, since they would always

pass zi,t = 0 for t > 10. Interestingly, these two subjects had non-constant price predictions,

which suggests that they were not optimisers.

22

This rule may be viewed as an anchor and adjustment rule (Tversky and Kah-

neman, 1974), as it extrapolates a price change (the last term) from an anchor

(the first two terms). Two important special cases of (18) are the pure trend

following rule with αi = 1 and βi = 0, yielding

pei,t = pt−1 + γi(pt−1 − pt−2), (19)

and adaptive expectations with γi = 0 and αi + βi = 1, namely

pei,t = αipt−1 + (1− αi)pei,t−1. (20)

The pure trend-following rule (19) uses an anchor giving all weight to the last

observed price (pt−1), while in the general rule (18) the anchor gives weight to

the last observed price (pt−1) as well as the last forecast (pei,t−1). In this sense

the general rule (18) is more cautious and extrapolates the trend from a more

gradually evolving anchor, while the pure trend-following rule is more aggressive

extrapolating the trend from the last price observation.

To explain the trading behaviour of the subjects from the LtO and Mixed

treatments, we estimate a general trading strategy in the following specifications:

zi,t = ci + χizi,t−1 + φiρt, (LtO) (21a)

zi,t = ci + χizi,t−1 + φiρt + ζiρei,t+1. (Mixed) (21b)

This rule captures the most relevant and most recent possible elements of indi-

vidual trading. Notice however, that the trading rule (21a) in the LtO treatment

only contains a past return (ρt) term, while the trading rule (21b) in the Mixed

treatment contains an additional term for expected excess return (ρet+1), which

is not observable in the LtO treatment because subjects did not give price fore-

casts. Both trading rules have two interesting special cases. First, what we call

persistent demand (φi = ζi = 0) is characterised by a simple AR(1) process:

zi,t = ci + χizi,t−1. (22)

A second special case is a return extrapolation rule (with χi = 0):

zi,t = ci + φiρt (LtO), (23a)

zi,t = ci + φiρt + ζiρ

e

i,t+1 (Mixed). (23b)

For the LtF and LtO treatments, for each subject we estimate her behavioural

heuristic starting with the general forecasting rule (18) or the general trading

23

rule (21a) respectively. To allow for learning, all estimations are based on the

last 40 periods. Testing for special cases of the estimated rules is straightforward:

insignificant variables are dropped until all the remaining coefficients are significant

at 5% level.11

A similar approach is used for the Mixed treatment (now also allowing for

the expected return coefficient ζi).

12 Equations (18) and (21b) are estimated

simultaneously. One potential concern for the estimation is that the contemporary

idiosyncratic errors in these two equations are correlated, given that the trade

decision depends on the contemporary expected forecast (if ζi 6= 0). Since the

contemporary trade does not appear in the forecasting rule, the forecast based

on the rule (18) can be estimated independently in the first step. The potential

endogeneity only affects the trading heuristic (23b), and can be solved with a

simple instrumental variable approach. The first step is to estimate the forecasting

rule (18), which yields fitted price forecasts of each subject. In the second step,

the trading rule (23b) is estimated with both the fitted forecasts as instruments,

and directly with the reported forecasts. Endogeneity can be tested by comparing

the two estimators using the Hausman test. Finally, the special cases of (21a–21b)

are tested based on reported or fitted price forecasts according to the Hausman

test.13

The estimation results can be found in Appendix E, in Tables E.1, E.2 and E.3

respectively for the LtF, LtO and Mixed treatments. In order to quantify whether

agents use different decision rules in different treatments, we test the differences

of the coefficients in the decision rules with the rank-sum test.

2.5.1 Forecasting rules in LtF versus Mixed

The LtF treatment can be directly compared to the Mixed treatment by com-

paring the estimated forecasting rules (18). We observe that rules with a trend

extrapolation term γi are popular in both treatments (respectively 39 in LtF and

25 in Mixed out of 48). A few other subjects use a pure adaptive rule (20) (none

in LtF and 3 in Mixed treatments respectively). A few others use a rule defined by

11Adaptive expectations (20) impose a restriction α ∈ [0, 1] (with α = 1 − β), so we follow

here a simple ML approach. If αi > 1 (αi < 0) maximises the likelihood for (20), we use the

relevant corner solution αi = 1 (αi = 0) instead. We check the relevance of the two constrained

models (trend and adaptive) with the Likelihood Ratio test against the likelihood of (18).

12See footnote 9.

13Whenever the estimations indicated that a subject from the Mixed treatment used a return

extrapolation rule of the form zi,t = ζiρ

e

i,t+1, that is a rule in which only the implied expected

return was significant, we directly tested ζi = 1/6. This restriction implies optimal trading

consistently with the price forecast, which we could not reject for 11 out of 48 subjects.

24

(18) where γi = 0, but αi + βi 6= 1. There were no subjects in the LtF treatment

and only 2 in the Mixed treatment, for whom we could not identify a significant

forecasting rule. The average trend coefficients in both treatments are close to

γ¯ ≈ 0.4, and not significantly different in terms of distribution (with p-value of

the rank-sum test equal to 0.736). The difference between the two treatments lies

in the anchor of the forecasting rule. For the LtF treatment the average coefficients

are α¯ = 0.45 and β¯ = 0.56, while in the Mixed treatment these are α¯ = 0.84 and

β¯ = 0.06 (the differences are significant according to the rank-sum test, with both

p-values close to zero). This suggests that subjects in the LtF treatment are more

cautious in revising their expectations, with a gradually evolving anchor that puts

equal weight on past price and their previous forecast. In contrast, in the Mixed

treatments subjects use an anchor that puts almost all weight on the last price

observation and are thus closer to using a pure trend-following rule extrapolating

a trend from the last price observation.

2.5.2 Trading rules in LtO versus Mixed

The LtO and Mixed treatments can be compared by the estimated trading rules.

Recall however, that the trading rule (21a) in the LtO treatment only contains a

past return term (ρt) with coefficient φi, while the trading rule (21b) in the Mixed

treatment contains an additional term for expected excess return (ρet+1) with a

coefficient ζi. In both treatments we find that the rules with a term on past or

expected return is the dominating rule (33 in the LtO and 32 in the Mixed treat-

ment). There are only 12 subjects using a significant AR1 coefficient χi in the LtO

treatment, and 8 in the Mixed treatment. This shows that in both the LtO and

Mixed treatments the majority of subjects tried to extrapolate realized and/or

expected asset returns, which leads to relatively strong trend chasing behaviour.

Nevertheless, there are 11 subjects in the LtO treatment and 8 in the Mixed treat-

ment for whom we can not identify a trading rule within this simple class. The

average demand persistence was χ¯ = 0.07 and χ¯ = 0.006, and the average trend

extrapolation was φ¯ = 0.09 and φ+ ζ = 0.06 in the LtO and Mixed treatment re-

spectively.14 The distributions of the two coefficients are not significantly different

across the treatments according to the rank-sum test, with p-values of 0.425 and

14The trading rules (21a) and (21b) are not directly comparable, since (21b) is a function of

both the past and the expected asset return, and the latter is unobservable in the LtO treatment.

For the sake of comparability, we look at what we interpret as an individual reaction to asset

return dynamics: φi in LtO treatment and φi + ζi in the Mixed treatment. As a robustness

check, we also estimated the simplest trading rule (21a) for both the LtO and Mixed treatments

(ignoring expected asset returns) and found no significant difference between treatments.

25

0.885 for χi and φi/φi+ζi respectively. Hence, based upon individual trading rules

we do not find significant differences between the LtO and Mixed treatments.

2.5.3 Implied trading rules in LtF versus LtO

It is more difficult to compare the LtF and LtO treatments based upon individual

decision rules, since there was no trading in the LtF and no forecasting in the

LtO treatment. We can however use the estimated individual forecasting rules to

obtain the implied optimal trading rules (8) in the LtF treatment and compare

these to the general trading rule (21a) in the LtO treatment. A straightforward

computation shows that for a forecasting rule (18) with coefficients (αi, βi, γi), the

implied optimal trading rule has coefficients χi = βi and φi = (αi + γi − R)/6.15

Hence, for the LtF and LtO treatments we can compare the coefficients for the

adaptive term, i.e. the weight given to the last trade, and the return extrapolation

coefficients. The averages of the first coefficient are β¯ = 0.56 and χ¯ = 0.07 for

the LtF and LtO treatments respectively, and it is significantly higher in the LtF

treatment (rank-sum test p-value close to zero). Moreover the second coefficient,

the implied reaction to the past asset return, is weaker in the LtF treatment

(average implied φ¯ = −0.03) than in the LtO treatment (average φ¯ = 0.09), and

this difference is again significant (rank-sum test p-value close to zero). Hence,

these results on the individual (implied) trading rules show differences between

the LtO and LtF treatments. The LtO treatment is more unstable than the LtF

treatment because subjects are less cautious in the sense that they give less weight

to their previous trade and they give more weight to extrapolating past returns.

We summarise the results on estimated individual behavioural rules as follows:

RESULT 4. Most subjects, regardless of the treatment, follow an anchor and

adjustment rule. In forecasting, LtF subjects were more cautious, using an an-

chor that puts more weight on their previous forecast, while the Mixed treatments

subjects use an anchor with almost all weight on recent prices. In trading, most

subjects extrapolate past returns and/or expected returns. In the LtO subjects give

more weight to past return extrapolation compared to the implied trading behaviour

in the LtF. These individual rules explain more unstable aggregate dynamics in the

LtO and Mixed treatments. We conclude that HYPOTHESIS 5 is rejected.

15The implied trading rule (8) however cannot exactly be rewritten in the form (21a), but

has one additional term pt−1 with coefficient [R(βi + αi + γi − R) − γi]/(aσ2). This coefficient

typically is small however, since γi is small and αi + βi is close to 1. The mean estimated

coefficient over 48 subjects is very close to zero (−0.00229), and with a simple t-test we can not

reject the hypothesis that the mean coefficient is 0 (p-value 0.15).

26

3 Conclusions

The origin of asset price bubbles is an important topic for both researchers and

policy makers. This paper investigates the price dynamics and bubble formation

in an experimental asset pricing market with a price adjustment rule. We find that

the mispricing is largest in the treatment where subjects do both forecasting and

trading, and smallest when subjects only make a prediction. Our result suggests

that price instability is the result of both inaccurate forecasting and imperfect

optimisation. There has been empirical work quantifying forecast biases by house-

holds and finance professionals in real markets, and theoretical works that start to

incorporate the stylized facts into modelling of expectations in macroeconomics.

Our result suggests it may be equally important to collect evidence on failure in

making optimal decisions conditional on one’s own belief by market participants,

and incorporate this behavioural bias into modelling of simple heuristics as an

alternative to perfectly optimal individual decisions.

Which behavioural biases can explain the differences in the individual decisions

and aggregate market outcomes in the learning to forecast and learning to optmise

markets? A first possibility is that the quantity decision task is more cognitive

demanding than the forecasting task, when the subjects in the LtF treatment are

assisted by a computer program. Following Rubinstein (2007), we use decision time

as a proxy for cognitive load and compare the average decision time in each treat-

ment. It turns out that while subjects take significantly longer time in the Mixed

treatment than the other two treatments according to Mann-Whitney-Wilcoxon

test, there is no significant difference between the LtF and LtO treatments. It

helps to explain why the markets are particularly volatile in the Mixed treat-

ment, but does not explain why the LtO treatment is more unstable than the LtF

treatment. Second, in the LtF treatment, the subjects’ goal is to find an accu-

rate forecast. Only the size of the prediction error matters, while the sign does not

matter. Conversely, in a LtO market it is in a way more important for the subjects

to predict the direction of the price movement right, and the size of the prediction

error is important only to a secondary degree. For example, if a subject predicts

the return will be high and decided to buy, he can still make a profit if the price

goes up far more than he expected, and his prediction error is large. Therefore,

the subjects may have a natural tendency to pay more attention to price changes

or follow the “wisdom of crowd”, which leads to assigning more weight to past or

expected returns. Furthermore, for price forecasting past individual behaviour can

be directly compared to observed market prices. If an individual forecasting strat-

egy fits well with observed price behaviour, more weight may be given to past own

27

individual behaviour. In contrast, individual trading decisions cannot be directly

compared or anchored against trading volume or other aggregate information. It

then becomes more natural to evaluate and anchor individual trading by giving

more weight to recent past prices and/or recent past returns.

Asset mispricing and financial bubbles can cause serious market inefficiencies,

and may become a threat to the overall economic stability, as shown by the 2007

financial-economic crisis. Proponents of rational expectations often claim that se-

rious asset bubbles cannot arise, because rational economic agents would efficiently

arbitrage against it and quickly push the ‘irrational’ (non-fundamental) investors

out of the market. Our experiment suggests otherwise: people exhibit heteroge-

neous and not necessarily optimal behaviour. Because they are trend-followers,

their non-fundamental beliefs are correlated. This is reinforced by the positive

feedback between expectations and realised prices in asset markets, as stressed

e.g. in Hommes (2013). Therefore, price oscillations cannot be mitigated by more

rational market investors. As a result, waves of optimism and pessimism can arise

despite the fundamentals being relatively stable.

Our experiment can be extended in several ways. For example, the subjects in

our experiment can short-sell the assets, which may not be feasible in real mar-

kets. An interesting topic for future research is the case where agents face short

selling constraints (Anufriev and Tuinstra, 2013). Another possible extension is

to impose a network structure among the traders, i.e. one trader can only trade

with some, but not all the other traders; or traders need to pay a cost in order to

be connected to other traders. This design can help us to examine the mechanism

of bubble formation in financial networks (Gale and Kariv, 2007), and network

games (Galeotti et al., 2010) in general. There has been a pioneering experimen-

tal literature by Gale and Kariv (2009) and Choi et al. (2014) that study how

network structure influences market efficiency when subjects act as intermediaries

between sellers and buyers. Our experimental setup can be extended to study how

network structure influences market efficiency and stability when subjects act as

traders of financial assets in the over the counter (OTC) market.

University of Amsterdam and Tinbergen Institute

University of Groningen

28

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32

A Experimental Instructions

(Not For Publication)

A.1 LtF treatment

General information

In this experiment you participate in a market. Your role in the market is a profes-

sional Forecaster for a large firm, and the firm is a major trading company of an

asset in the market. In each period the firm asks you to make a prediction of the

market price of the asset. The price should be predicted one period ahead. Based

on your prediction, your firm makes a decision about the quantity of the asset the

firm should buy or sell in this market. Your forecast is the only information the

firm has on the future market price. The more accurate your prediction is, the

better the quality of your firm’s decision will be. You will get a payoff based on

the accuracy of your prediction. You are going to advise the firm for 50 successive

time periods.

About the price determination

The price is determined by the following price adjustment rule: when there is more

demand (firm’s willingness to buy) of the asset, the price goes up; when there is

more supply (firm’s willingness to sell), the price will go down.

There are several large trading companies on this market and each of them is ad-

vised by a forecaster like you. Usually, higher price predictions make a firm to buy

more or sell less, which increases the demand and vice versa. Total demand and

supply is largely determined by the sum of the individual demand of these firms.

About your job

Your only task in this experiment is to predict the market price in each time period

as accurately as possible. Your prediction in period 1 should lie between 0

and 100. At the beginning of the experiment you are asked to give a prediction

for the price in period 1. When all forecasters have submitted their predictions for

the first period, the firms will determine the quantity to demand, and the market

price for period 1 will be determined and made public to all forecasters. Based on

the accuracy of your prediction in period 1, your earnings will be calculated.

Subsequently, you are asked to enter your prediction for period 2. When all par-

ticipants have submitted their prediction and demand decisions for the second

period, the market price for that period, will be made public and your earnings

will be calculated, and so on, for all 50 consecutive periods. The information you

can refer to at period t consists of all past prices, your predictions and earnings.

33

Please note that due to liquidity constraint, your firm can only buy and sell up

to a maximum amount of assets in each period. This means although you can

submit any prediction for period 2 and all periods after period 2, if the price in

last period is pt−1, and you prediction is pet : the firm’s trading decision is con-

strained by pet ∈ [pt−1 − 30, pt−1 + 30]. More precisely, the firm will trade as

if pet = pt−1 +30 if p

e

t > pt−1 +30, and trade as if p

e

t = pt−1−30 if pet < pt−1−30.

About your payoff

Your earnings depend only on the accuracy of your predictions. The earnings

shown on the computer screen will be in terms of points. If your prediction is pet

and the price turns out to be pt in period t, your earnings are determined by the

following equation:

Payoff = max

[

1300− 1300

49

(pet − pt)2 , 0

]

.

The maximum possible points you can earn for each period (if you make no pre-

diction error) is 1300, and the larger your prediction error is, the fewer points you

can make. You will earn 0 points if your prediction error is larger than 7. There is

a Payoff Table on your table, which shows the points you can earn for different

prediction errors.

We will pay you in cash at the end of the experiment based on the points you

earned. You earn 1 euro for each 2600 points you make.

A.2 LtO treatment

General information

In this experiment you participate in a market. Your role in the market is a Trader

of a large firm, and the firm is a major trading company of an asset. In each period

the firm asks you to make a trading decision on the quantity Dt your firm will

BUY to the market. (You can also decide to sell, in that case you just submit a

negative quantity.) You are going to play this role for 50 successive time periods.

The better the quality of your decision is, the better your firm can achieve her

target. The target of your firm is to maximize the expected asset value minus the

variance of the asset value, which is also the measure by the firm concerning your

performance:

(1) pit = Wt − 1

2

V ar (Wt)

2

The total asset value Wt equals the return of the per unit asset multiplied by the

number of unit you buy Dt. The return of the asset is pt + y − Rpt−1, where

34

R is the gross interest rate which equals 1.05, pt is the asset price at period t,

therefore pt − Rpt−1 is the capital gain of the asset, and y = 3.3 is the dividend

paid by the asset. We assume the variance of the price of a unit of the asset is

σ2 = 6, therefore the expected variance of the asset value is 6D2t . Therefore we

can rewrite the performance measure in the following way

(2) pit = (pt + y −Rpt−1)Dt − 3D2t

The asset price in the next period pt is not observable in the current period. You

can make a forecast pet on it. There is an asset return calculator in the

experimental interface that gives the asset return for each price forecast pet you

make. Your own payoff is a function of the value of target function of the firm:

(3) Payofft = 800 + 40 ∗ pit

This function means you get 800 points (experimental currency) as basic salary,

and 40 points for each 1 unit of performance (target function of the firm) you

make. If your trades will be unsuccessful, you may lose points and earn less than

your basic salary, down to 0. Based on the asset return, you can look up your

payoff for each quantity decision you make in the payoff table.

You can of course also calculate your payoff for each given forecast and quantity

using equation (2) and (3) directly. In that situation you can ask us for a calculator.

About the price determination

The price is determined by the following price adjustment rule: when there is more

demand than supply of the asset (namely, more traders want to buy), the price

will go up; and when there is more supply than demand of the asset (namely, more

people want to sell), the price will go down.

About your job

Your only task in this experiment is to decide the quantity the firm will buy/sell.

At the beginning of period 1 you determine the quantity to buy or sell (submitting

a positive number means you want to buy, and negative number means you want

to sell) for period 1. After all traders submit their quantity decisions, the market

price for period 1 will be determined and made public to all traders. Based on

the value of the target function of your firm in period 1, your earnings in the first

period will be calculated.

Subsequently, you make trading decisions for the second period, the market price

for that period will be made public and your earnings will be calculated, and so

on, for all 50 consecutive periods. The information you can refer to at period t

35

consists of all previous prices, your quantity decisions and earnings.

Please notice that due to the liquidity constraint of the firm, the amount of asset

you buy or sell cannot be more than 5 units. Which means you quantity decision

should be between −5 and 5. The numbers on the payoff table are just examples.

You can use any other number such as 0.01, −1.3, 2.15 etc., as long as they are

within [−5, 5]. if When you want to submit numbers with a decimal point, please

write a “.”, NOT a “,”.

About your payoff

In each period you are paid according to equation (3). The earnings shown on the

computer screen will be in terms of points. We will pay you in cash at the end

of the experiment based on the points you earned. You earn 1 euro for each 2600

points you make.

A.3 Mixed treatment

General information

In this experiment you participate in a market. Your role in the market is a Trader

of a large firm, and the firm is a major trading company of an asset. In each period

the firm asks you to make a trading decision on the quantity Dt your firm will

BUY to the market. (You can also decide to sell, in that case you just submit a

negative quantity.) You are going to play this role for 50 successive time periods.

The better the quality of your decision is, the better your firm can achieve her

target. The target of your firm is to maximize the expected asset value minus the

variance of the asset value, which is also the measure by the firm concerning your

performance:

(1) pit = Wt − 1

2

V ar (Wt)

2

The total asset value Wt equals the return of the per unit asset multiplied by the

number of unit you buy Dt. The return of the asset is pt + y − Rpt−1, where

R is the gross interest rate which equals 1.05, pt is the asset price at period t,

therefore pt − Rpt−1 is the capital gain of the asset, and y = 3.3 is the dividend

paid by the asset. We assume the variance of the price of a unit of the asset is

σ2 = 6, therefore the expected variance of the asset value is 6D2t . Therefore we

can rewrite the performance measure in the following way

(2) pit = (pt + y −Rpt−1)Dt − 3D2t

The asset price in the next period pt is not observable in the current period. You

can make a forecast pet on it. There is an asset return calculator in the

36

experimental interface that gives the asset return for each price forecast pet you

make. Your own payoff is a function of the value of target function of the firm:

(3) Payofft = 800 + 40 ∗ pit

This function means you get 800 points (experimental currency) as basic salary,

and 40 points for each 1 unit of performance (target function of the firm) you

make. If your trades will be unsuccessful, you may lose points and earn less than

your basic salary, down to 0. Based on the asset return, you can look up your

payoff for each quantity decision you make in the payoff table.

You can of course also calculate your payoff for each given forecast and quantity

using equation (2) and (3) directly. In that situation you can ask us for a calcu-

lator.

The payoff for the forecasting task is simply a decreasing function of your fore-

casting error (the distance between your forecast and the realized price). When

your forecasting error is larger than 7, you earn 0 points.

(4) Payoffforecasting = max

[

1300− 1300

49

(pet − pt)2 , 0

]

About the price determination

The price is determined by the following price adjustment rule: when there is more

demand than supply of the asset (namely, more traders want to buy), the price

will go up; and when there is more supply than demand of the asset (namely, more

people want to sell), the price will go down.

About your job

Your task in this experiment consists of two parts: (1) to make a price forecast;

(2) to decide the quantity the firm will buy/sell. At the beginning of period

1 you submit your price forecast between 0 and 100, and then determine

the quantity to buy or sell (submitting a positive number means you want to buy,

and negative number means you want to sell) for period 1, and the market price

for period 1 will be determined and made public to all traders. Based on your

forecasting error and performance measure for the trading task, in period 1, your

earnings in the first period will be calculated.

Subsequently, you make forecasting and trading decisions for the second period,

the market price for that period will be made public and your earnings will be

calculated, and so on, for all 50 consecutive periods. The information you can

refer to at period t consists of all previous prices, your past forecasts, quantity

decisions and earnings.

37

Please notice that due to the liquidity constraint of the firm, the amount of asset

you buy or sell cannot be more than 5 units. Which means you quantity decision

should always be between −5 and 5. The numbers on the payoff table are just

examples. You can use any other numbers such as 0.01, −1.3, 2.15 etc. as long

as they are within [−5, 5].

About your payoff

In each period you are paid for the forecasting task according to equation (4)

and trading task according to equation (3). The earnings shown on the computer

screen will be in terms of points. We will pay you in cash at the end of the experi-

ment based on the points you earned for either the forecasting task or the trading

task. Which task will be paid will be determined randomly (we will invite one of

the participants to toss a coin). That is, depending on the coin toss, your

earnings will be calculated either based on equation (3) or equation (4).

You earn 1 euro for each 2600 points you make.

38

B Payoff Tables

(Not For Publication)

Table B.1: Payoff Table for Forecasting Task

Payoff Table for Forecasting Task

Your Payoff=max[1300− 1300

49

(Your Prediction Error)2, 0]

3000 points equal 1 euro

error points error points error points error points

0 1300 1.85 1209 3.7 937 5.55 483

0.05 1300 1.9 1204 3.75 927 5.6 468

0.1 1300 1.95 1199 3.8 917 5.65 453

0.15 1299 2 1194 3.85 907 5.7 438

0.2 1299 2.05 1189 3.9 896 5.75 423

0.25 1298 2.1 1183 3.95 886 5.8 408

0.3 1298 2.15 1177 4 876 5.85 392

0.35 1297 2.2 1172 4.05 865 5.9 376

0.4 1296 2.25 1166 4.1 854 5.95 361

0.45 1295 2.3 1160 4.15 843 6 345

0.5 1293 2.35 1153 4.2 832 6.05 329

0.55 1292 2.4 1147 4.25 821 6.1 313

0.6 1290 2.45 1141 4.3 809 6.15 297

0.65 1289 2.5 1134 4.35 798 6.2 280

0.7 1287 2.55 1127 4.4 786 6.25 264

0.75 1285 2.6 1121 4.45 775 6.3 247

0.8 1283 2.65 1114 4.5 763 6.35 230

0.85 1281 2.7 1107 4.55 751 6.4 213

0.9 1279 2.75 1099 4.6 739 6.45 196

0.95 1276 2.8 1092 4.65 726 6.5 179

1 1273 2.85 1085 4.7 714 6.55 162

1.05 1271 2.9 1077 4.75 701 6.6 144

1.1 1268 2.95 1069 4.8 689 6.65 127

1.15 1265 3 1061 4.85 676 6.7 109

1.2 1262 3.05 1053 4.9 663 6.75 91

1.25 1259 3.1 1045 4.95 650 6.8 73

1.3 1255 3.15 1037 5 637 6.85 55

1.35 1252 3.2 1028 5.05 623 6.9 37

1.4 1248 3.25 1020 5.1 610 6.95 19

1.45 1244 3.3 1011 5.15 596 error ≥ 0

1.5 1240 3.35 1002 5.2 583

1.55 1236 3.4 993 5.25 569

1.6 1232 3.45 984 5.3 555

1.65 1228 3.5 975 5.35 541

1.7 1223 3.55 966 5.4 526

1.75 1219 3.6 956 5.45 512

1.8 1214 3.65 947 5.5 497

39

Table B.2: Payoff Table for Trading Task

Y

o

u

r

p

ro

fi

t

A

ss

e

t

q

u

a

n

ti

ty

:

p

o

si

ti

v

e

n

u

m

b

e

r

m

e

a

n

s

to

b

u

y,

n

e

g

a

ti

v

e

to

se

ll

−5

−4

.5

−4

−3

.5

−3

−2

.5

−2

−1

.5

−1

−0

.5

0

0

.5

1

1

.5

2

2

.5

3

3

.5

4

4

.5

5

A s s e t r e t u r n

−1

5

80

0

10

70

12

80

14

30

15

20

15

50

15

20

1

4

3

0

1

2

8

0

1

0

7

0

8

0

0

4

7

0

8

0

0

0

0

0

0

0

0

0

−1

4

60

0

89

0

11

20

12

90

14

00

14

50

14

40

1

3

7

0

1

2

4

0

1

0

5

0

8

0

0

4

9

0

1

2

0

0

0

0

0

0

0

0

0

−1

3

40

0

71

0

96

0

11

50

12

80

13

50

13

60

1

3

1

0

1

2

0

0

1

0

3

0

8

0

0

5

1

0

1

6

0

0

0

0

0

0

0

0

0

−1

2

20

0

53

0

80

0

10

10

11

60

12

50

12

80

1

2

5

0

1

1

6

0

1

0

1

0

8

0

0

5

3

0

2

0

0

0

0

0

0

0

0

0

0

−1

1

0

35

0

64

0

87

0

10

40

11

50

12

00

1

1

9

0

1

1

2

0

9

9

0

8

0

0

5

5

0

2

4

0

0

0

0

0

0

0

0

0

−1

0

0

17

0

48

0

73

0

92

0

10

50

11

20

1

1

3

0

1

0

8

0

9

7

0

8

0

0

5

7

0

2

8

0

0

0

0

0

0

0

0

0

−9

0

0

32

0

59

0

80

0

95

0

10

40

1

0

7

0

1

0

4

0

9

5

0

8

0

0

5

9

0

3

2

0

0

0

0

0

0

0

0

0

−8

0

0

16

0

45

0

68

0

85

0

96

0

1

0

1

0

1

0

0

0

9

3

0

8

0

0

6

1

0

3

6

0

5

0

0

0

0

0

0

0

0

−7

0

0

0

31

0

56

0

75

0

88

0

9

5

0

9

6

0

9

1

0

8

0

0

6

3

0

4

0

0

1

1

0

0

0

0

0

0

0

0

−6

0

0

0

17

0

44

0

65

0

80

0

8

9

0

9

2

0

8

9

0

8

0

0

6

5

0

4

4

0

1

7

0

0

0

0

0

0

0

0

−5

0

0

0

30

32

0

55

0

72

0

8

3

0

8

8

0

8

7

0

8

0

0

6

7

0

4

8

0

2

3

0

0

0

0

0

0

0

0

−4

0

0

0

0

20

0

45

0

64

0

7

7

0

8

4

0

8

5

0

8

0

0

6

9

0

5

2

0

2

9

0

0

0

0

0

0

0

0

−3

0

0

0

0

80

35

0

56

0

7

1

0

8

0

0

8

3

0

8

0

0

7

1

0

5

6

0

3

5

0

8

0

0

0

0

0

0

0

−2

0

0

0

0

0

25

0

48

0

6

5

0

7

6

0

8

1

0

8

0

0

7

3

0

6

0

0

4

1

0

1

6

0

0

0

0

0

0

0

−1

0

0

0

0

0

15

0

40

0

5

9

0

7

2

0

7

9

0

8

0

0

7

5

0

6

4

0

4

7

0

2

4

0

0

0

0

0

0

0

0

0

0

0

0

0

50

32

0

5

3

0

6

8

0

7

7

0

8

0

0

7

7

0

6

8

0

5

3

0

3

2

0

5

0

0

0

0

0

0

1

0

0

0

0

0

0

24

0

4

7

0

6

4

0

7

5

0

8

0

0

7

9

0

7

2

0

5

9

0

4

0

0

1

5

0

0

0

0

0

0

2

0

0

0

0

0

0

16

0

4

1

0

6

0

0

7

3

0

8

0

0

8

1

0

7

6

0

6

5

0

4

8

0

2

5

0

0

0

0

0

0

3

0

0

0

0

0

0

80

3

5

0

5

6

0

7

1

0

8

0

0

8

3

0

8

0

0

7

1

0

5

6

0

3

5

0

8

0

0

0

0

0

4

0

0

0

0

0

0

0

2

9

0

5

2

0

6

9

0

8

0

0

8

5

0

8

4

0

7

7

0

6

4

0

4

5

0

2

0

0

0

0

0

0

5

0

0

0

0

0

0

0

2

3

0

4

8

0

6

7

0

8

0

0

8

7

0

8

8

0

8

3

0

7

2

0

5

5

0

3

2

0

3

0

0

0

0

6

0

0

0

0

0

0

0

1

7

0

4

4

0

6

5

0

8

0

0

8

9

0

9

2

0

8

9

0

8

0

0

6

5

0

4

4

0

1

7

0

0

0

0

7

0

0

0

0

0

0

0

1

1

0

4

0

0

6

3

0

8

0

0

9

1

0

9

6

0

9

5

0

8

8

0

7

5

0

5

6

0

3

1

0

0

0

0

8

0

0

0

0

0

0

0

5

0

3

6

0

6

1

0

8

0

0

9

3

0

1

0

0

0

1

0

1

0

9

6

0

8

5

0

6

8

0

4

5

0

1

6

0

0

0

9

0

0

0

0

0

0

0

0

3

2

0

5

9

0

8

0

0

9

5

0

1

0

4

0

1

0

7

0

1

0

4

0

9

5

0

8

0

0

5

9

0

3

2

0

0

0

1

0

0

0

0

0

0

0

0

0

2

8

0

5

7

0

8

0

0

9

7

0

1

0

8

0

1

1

3

0

1

1

2

0

1

0

5

0

9

2

0

7

3

0

4

8

0

1

7

0

0

1

1

0

0

0

0

0

0

0

0

2

4

0

5

5

0

8

0

0

9

9

0

1

1

2

0

1

1

9

0

1

2

0

0

1

1

5

0

1

0

4

0

8

7

0

6

4

0

3

5

0

0

1

2

0

0

0

0

0

0

0

0

2

0

0

5

3

0

8

0

0

1

0

1

0

1

1

6

0

1

2

5

0

1

2

8

0

1

2

5

0

1

1

6

0

1

0

1

0

8

0

0

5

3

0

2

0

0

1

3

0

0

0

0

0

0

0

0

1

6

0

5

1

0

8

0

0

1

0

3

0

1

2

0

0

1

3

1

0

1

3

6

0

1

3

5

0

1

2

8

0

1

1

5

0

9

6

0

7

1

0

4

0

0

1

4

0

0

0

0

0

0

0

0

1

2

0

4

9

0

8

0

0

1

0

5

0

1

2

4

0

1

3

7

0

1

4

4

0

1

4

5

0

1

4

0

0

1

2

9

0

1

1

2

0

8

9

0

6

0

0

1

5

0

0

0

0

0

0

0

0

8

0

4

7

0

8

0

0

1

0

7

0

1

2

8

0

1

4

3

0

1

5

2

0

1

5

5

0

1

5

2

0

1

4

3

0

1

2

8

0

1

0

7

0

8

0

0

N

ot

e

th

at

30

00

p

oi

n

ts

of

yo

u

r

p

ro

fi

t

co

rr

es

p

on

d

s

to

e

1.

C Rational Strategic Behaviour

(Not For Publication)

Our experimental results are clearly different from the predictions of the rational

expectation equilibrium (REE). However, we also found that subjects typically

earn high payoffs, implying some sort of profit seeking behaviour.

In this appendix, we discuss whether rational strategic behaviour can explain

our experimental results, in particular in the LtO and Mixed treatments. Based on

the assumption on the subjects’ perception of the game and information structure,

three cases are discussed: (1) agents are price takers; (2) agents know their market

power and coordinate on monopolistic behaviour; (3) agents know their market

power but play non-cooperatively. We show that under price-taking behaviour, the

LtF and LtO treatments are equivalent. If the subjects behave strategically or try

to collude, the economy can have alternative equilibria, where the subjects collec-

tively ‘ride a bubble’, or jump around the fundamental price. Nevertheless, these

rational equilibria predict different outcomes than the individual and aggregate

behaviour observed in the experiment.

Without loss of generality, we focus on the one-shot game version of the ex-

perimental market to derive our results. More precisely, we look at the optimal

quantity decisions zi,t that the agents in period t (knowing prices and individual

traded quantity until and including period t) have to formulate to maximise the

expected utlity in period t + 1. This is supported by two observations. First,

by definition agents are myopic and their payoff in t + 1 depends only on the

realised profit from that period, and not on the stream of future profits from pe-

riod t+ 2 onward. Second, the experiment is a repeated game with finitely many

repetitions, and subjects knew it would end after 50 periods. Using the standard

backward induction reasoning, one can easily show that a sequence of one-period

game equilibria forms a rational equilibrium of the finitely repeated game as well.

C.1 Price takers

Realised utility of investors in the LtO treatment is given by (4) and is equivalent

to the following form:

Ui,t(zi,t) = zi,t (pt+1 + y −Rpt)− aσ

2

2

z2i,t, (C.1)

where zi,t is the traded quantity and Ui,t is a quadratic function of the traded

quantity. As discussed in Section 2, assuming the agent is a price taker, the

41

optimal traded quantity conditional on the expected price pei,t+1 is given by

z∗i,t = arg max

zi,t

Ui,t =

pei,t+1 + y −Rpt

aσ2

. (C.2)

Note that this result relies on the assumption that the subjects do not know

the price generating function. We argue that the subjects also have an incentive

to minimise their forecasting error when they choose the quantity and are paid

according to the risk adjusted profit. To see that, suppose that the realised market

price in the next period is pt+1, and the subject makes a prediction error of , i.e.

her prediction is pei,t+1 = pt+1 + . The payoff function can be rewritten as:

Ui,t(zi,t) = zi,t (pt+1 + y −Rpt)− aσ

2

2

z2i,t

=

(pt+1 + + y −Rpt)(pt+1 + y −Rpt)

aσ2

− (pt+1 + + y −Rpt)

2

2aσ2

=

(pt+1 + y −Rpt)2

2aσ2

−

2

2aσ2

. (C.3)

This shows that utility is maximised when = 0, namely, when all subjects have

correct belief. Assuming perfect rationality and price taking behaviour (perfect

competition), the task of finding the optimal trade coincides with the task of

minimizing the forecast error. Subjects have incentives to search for and play the

REE also when they choose the quantity. We summarise this finding below.

FINDING 1. When the subjects act as price takers, the utility function in the

Learning to Optimise treatment is a quadratic function of the prediction error, the

same (up to a monotonic transformation) as in the Learning to Forecast treat-

ment. The subjects’ payoff is maximised when they play the Rational Expectation

Equilibrium regardless of the design: the REE of LtF and LtO treatments are

equivalent.

C.2 Collusive outcome

Consider now the case when agents realise how their predictions/trading quanti-

ties influence the price and are able to coordinate on a common strategy. This

resembles a collusive (oligopoly) market, e.g. similar to a cobweb economy in which

the sellers can coordinate their production.

In the collusive case, all agents behave as a monopoly that maximises joint

(unweighted) utility; thus the solution is symmetric, that is for each agent i,

zi,t = zt. In our experiment the price determination function is:

pt+1 = pt + 6λzt, (C.4)

42

and so the monopoly maximises

Ut =

6∑

i=1

Ui,t(zt) = 6

[

zt(pt+1 + y −Rpt)− aσ

2

2

z2t

]

= 6

[

z2t

(

6λ− aσ

2

2

)

+ zt(y − rpt)

]

. (C.5)

Here we assume that a rational agent has perfect knowledge about the pricing

function (C.4). Notice that when λ = 20/21, aσ2 = 6, as in the experiment, the

coefficient before z2t is positive, 6λ− aσ

2

2

= 19

7

> 0, and thus the profit function is U

shaped, instead of inversely U shaped.16 This means that a finite global maximum

does not exist (utility goes to +∞ when zt goes to either +∞ or −∞). The global

minimum is obtained when zi,t =

7

38

(rpt − y) = 7r38(pt − pf ).

In our experiment, the subjects are constrainted to choose a quantity from

[−5,+5] and the price is bound to the interval [0, 300]. Collusive equilibrium

in the one-shot game implies that the subjects coordinate on zi,t = 5 or zi,t =

−5, depending on which is further away from 7(rpt−y)

38

(as (C.5) is a symmetric

parabola). Since 7(rpt−y)

38

> 0 when the price is above the fundamental (pf = y/r),

we can see that the agents coordinate on −5 if the price is higher than the REE

(pt > y/r). Similarly, rational agents coordinate on +5 if the price is lower than

the REE (pt < y/r). If the price is exactly at the fundamental, rational agents

are indifferent between −5 and 5. Notice that in such a case trading the REE

quantity (zi,t = 0) gives the global minimum for the monopoly.

As a consequence, the collusive outcome predicts that the subjects will ‘jump

up and down’ around the fundamental. When the initial price is below (above)

the fundamental, the monopoly will buy (sell) the asset until the price overshoots

(undershoots) the fundamental, and so forth. Then the subjects start to ‘jump up

and down’ as described before.

FINDING 2. When the subjects know the price determination function and are

able to form a coalition, the collusive profit function in the LtO treatment is U

shaped. Subjects would buy under-priced and sell an over-priced asset. In the long

run rational collusive subjects will alternate their trading quantities between −5

and 5 and so the price will alternate around the equilibrium.

16If 6λ− aσ22 < 0, this objective function is inversely U shaped. The maximum point is achieved

when zi,t =

rpt−y

12λ−aσ2 . This means when pt = y/r, namely when the price is at the REE, the

optimal quantity under collusive equilibrium is still 0. When the price is higher or lower than

the REE, the optimal quantity increases with the difference between the price and the REE.

This means there is a continuum of equilibria when the economy does not start at the REE.

43

Such alternating dynamics would resemble coordination on contrarian type of

behaviour, but has not been observed in any of the experimental groups. In-

stead, our subjects coordinated on trend-following trading rules, which resulted

in smooth, gradual price oscillations. Moreover, quantity decisions equal to 5 or

−5 happened rarely in the experiment (7 times in the LtO and 44 times in the

Mixed treatment). Typical subject behaviour was much more conservative: 97%

and 91% traded quantities in the LtO and Mixed treatments respectively were

confined in the interval [−2.5, 2.5].

C.3 Perfect information non-cooperative game

Consider a scenario, in which the subjects realise the experimental price deter-

mination mechanism, but cannot coordinate their actions and play a symmetric

Nash equilibrium (NE) instead of the collusive one. There is a positive external-

ity of the subjects’ decisions: when one subject buys the asset, it pushes up the

price and also the benefits of all the other subjects. The collusive equilibrium

internalises this externality, while the non-cooperative NE does not. What will

rational subjects do in this situation?

In the case of a non-cooperative one-shot game, we again focus on a symmetric

solution. Consider agent i, who optimises her quantity choice believing that all

other agents will choose zot . This means that the price at t+ 1 becomes

pt+1 = pt + 5λz

o

t + λzi,t. (C.6)

Agent i maximises therefore

Ui,t = zi,t (λzi,t + 5λz

o

t + y − rpt)−

aσ2

2

z2i,t

= z2it

2λ− aσ2

2

+ zi,t(5λz

o

t + y − rpt). (C.7)

Notice that 2λ− aσ2 = −86/21 < 0. This is an inversely U shaped parabola with

the unique maximum given by the best response function

z∗i,t(z

o

t ) =

5λzot + y − rpt

aσ2 − 2λ . (C.8)

A symmetric solution requires z∗i,t(z

o

t ) = z

o

t , which implies

z∗t =

rpt − y

7λ− aσ2 =

3

2

(rpt − y). (C.9)

Furthermore the reaction function z∗i,t(z

o

t ) is linear with respect to z

o

t , with a slope

5λ

aσ2−2λ =

100

86

> 1. Thus, zot > z

∗

t (< and =) implies z

∗

i,t > z

o

t (< and =), or in

44

words, if agent i believes that the other players will buy (sell) the asset, she has an

incentive to buy (sell) even more. Then as a best response, the other agents should

further increase/decrease their demand, and this is limited only by the liquidity

constraints. The strategy (C.9) thus defines the threshold point between the two

corner strategies, i.e. the full NE best response strategy is defined as

zNEi,t =

5 if zot > z

∗

t

z∗t if z

o

t = z

∗

t

−5 if zot < z∗t .

(C.10)

The boundary strategies can be infeasible if the previous price is too close to zero

or 300.17 To sum up, as long as the price pt is sufficiently far from the edges of the

allowed interval [0, 300], there are three NE of the one-shot non-cooperative game,

which are defined as fixed points of (C.10), namely all players playing zi,t = −5,

zi,t = z

∗

t and zi,t = +5 for all i ∈ {1, . . . , 6}.

A simple interpretation is that, given the parametrization, our model is an

example of a (Nash) coordination game. As long as 5λ

aσ2−2λ > 1, the best response

(C.8) is to amplify the average trade of the other players. This is not a surprising

result, as it merely exhibits the strength of the positive feedback present in this

market18.

If the agents coordinate on the strategy zi,t = z

∗

t , the price evolves according

to the following law of motion:

pt+1 =

10pt − 60y

7

. (C.11)

In contrast to the collusive game, in the non-cooperative game the fundamental

price is a possible steady state, but only if it is an outcome in the initial period.

Additional equilibrium refinements may further exclude it as a rational outcome,

since it is the least profitable one. Recall that the subjects earn 0 when they play

z∗t with price at the fundamental (because there is no trade). On the other hand,

they may earn a positive profit by coordinating on −5 or 5. For example, when

all of them buy 5 units of asset, the utility for each of them will be (pt−1 + y +

6λzi,t − (1 + r)pt−1)zi,t − ασ22 z2i,t = (33.3 − 0.05pt−1) ∗ 5 − 75. This equals 76.5

when pt−1 = 60, 16.5 when pt−1 = 300 and 75 when the previous price is equal

to the fundamental, pt−1 = 66. This explains why the payoff efficiency (average

17Notice that we can interpret zot as the average quantity traded by all other agents, besides

agent i, and the reasoning for NE strategy (C.10) remains intact. This implies that NE has to

be symmetric.

18In practice, such an equilibrium could not be sustained in the long run, since then the market

maker would incur accumulating losses every period.

45

experimental payoff divided by payoff under REE) is larger than 100% in some

markets in the LtO or Mixed treatments where prices have large oscillations.

Notice that the linear equation (C.11) is unstable, so the NE of the one-shot

game leads to unstable price dynamics in the repeated game even if the agents

coordinate on zi,t = z

∗

t , as long as the initial price is different from the fundamental

price. Indeed, if the initial price is 67 or 65 (fundamental price plus or minus

one), the price will go to the upper cap of 300 or the lower cap of 0 respectively.

Furthermore the agents can switch at any moment between the three one-shot

game NE defined by (C.10). This implies that in the repeated non-cooperative

game, many rational price paths are possible. This includes many price paths

where agents coordinate on 5 or −5, including the alternating collusive equilibrium

discussed in the last section.

FINDING 3. In the non-cooperative game with perfect information, there are

two possible types of NE. The fundamental outcome is a possible outcome only

if the initial price is equal to the fundamental price. Otherwise, the agents will

coordinate on unstable, possibly oscillatory price dynamics, with traded quantities

of −5 or 5. When they coordinate on a non-zero quantity, their payoff can be

higher than their payoff under the REE under the price-taking beliefs.

C.4 Summary

To conclude, the perfectly rational agents can coordinate on price boom-bust cycles

and earn positive profit19. However, this would require even stronger assumptions

than the fundamental equilibrium, namely that the agents perfectly understand

the underlying price determination mechanism.

Furthermore, such rational equilibria with price oscillations predict that the

subjects coordinate on homogeneous trades at the edge of the liquidity constraints.

The subjects from the LtO and Mixed treatments behaved differently. Their traded

quantities were highly heterogeneous, and rarely reached the liquidity constraints.

Therefore, the alternative rational equilibrium from the perfect information,

non-cooperative games provide some useful insights on why subjects “ride the

bubbles” in the LtO and Mixed treatment. However, since the rational solution

19Note that the subjects earn more in collusive and non-cooperative Nash setting because we

pay them according to the book value of the asset, and the taˆtonnement process ensures the

price movement is relatively smooth. In real life, people may not be able to realize the full book

value of their asset holdings because the asset price will fall when a large fraction of them start

to sell, and without the marker maker in the taˆtonnements process absorbing all these losses,

they may suffer huge losses when the asset price declines sharply.

46

cannot explain the heterogeneity of the individual decisions and non-boundary

trading quantities, the mispricing in the experimental data is more likely a result

of the joint forces of rational (profit seeking) and boundedly rational behaviour

with some coordination on trend-following buy and hold and short sell strategies.

D Earnings Ratios

(Not For Publication)

Table D.1: Earnings Efficiency

Treatment LtF LtO Mixed Forecasting Mixed Trading

Market 1 96.35% 102.54% 87.62% 100.89%

Market 2 94.47% 95.25% 67.27% 87.33%

Market 3 96.03% 98.21% 75.63% 79.61%

Market 4 96.18% 100.43% 77.41% 114.63%

Market 5 95.15% 97.39% 87.07% 99.03%

Market 6 94.06% 99.64% 91.94% 97.24%

Market 7 96.18% 98.58% 81.20% 94.55%

Market 8 96.54% 98.41% 60.80% 132.01%

Average 95.62% 98.81% 78.62% 100.66%

Notes. Earnings efficiency for each market. The efficiency is defined as the average

experimental payoff divided by the payoff under REE, which is 26.67 euro for the fore-

casting task, and 18.33 euro for the trading task.

47

E Estimation Of Individual Forecasting Rules

(Not For Publication)

Rule coefficients

Subject cons. Past price AR(1) Past trend R2 Type

Group 1

1 0.288 0.756 0.680 0.995

2 −1.952 1.090 0.448 0.996

3 1.000 0.744 0.734 TRE

4 −1.349 0.982 0.427 0.998

5 −2.080 0.307 0.725 0.362 0.997

6 1.000 0.770 0.648 TRE

Group 2

1 1.014 0.998

2 0.626 0.347 0.519 0.998

3 −2.110 0.346 0.697 0.996

4 1.013 0.992

5 1.013 0.997

6 −1.857 0.475 0.561 0.391 0.996

Group 3

1 0.463 0.522 0.707 0.993

2 0.513 0.495 0.655 0.994

3 0.476 0.660 0.395 0.993

4 1.000 0.302 0.310 TRE

5 1.000 0.364 0.390 TRE

6 0.471 0.544 0.579 0.998

Group 4

1 0.596 0.568 0.482 0.988

2 1.000 0.679 0.320 TRE

3 1.000 0.161 0.025 TRE

4 −2.553 0.418 0.621 0.405 0.992

5 0.389 0.608 0.539 0.996

6 1.000 0.341 0.385 TRE

Table E.1: Estimated individual rules for the LtF treatment.

48

Rule coefficients

Subject cons. Past price AR(1) Past trend R2 Type

Group 5

1 0.260 0.715 0.729 0.990

2 1.021 0.895

3 −53.068 −0.369 2.125 −1.591 0.655

4 0.178 0.902 0.836 0.980

5 0.452 0.587 0.791 0.993

6 0.281 0.719 1.245 0.985

Group 6

1 0.993 0.880

2 1.000 0.921 0.507 TRE

3 1.000 0.712 0.761 TRE

4 1.000 0.827 0.804 TRE

5 0.452 0.411 0.977 0.986

6 1.000 0.804 0.809 TRE

Group 7

1 6.914 0.902 0.910

2 1.010 0.998

3 0.926 0.924

4 0.359 0.590 0.399 0.966

5 0.990 0.973

6 0.308 0.536 0.545 0.960

Group 8

1 1.000 0.451 0.293 TRE

2 1.000 0.370 0.502 TRE

3 2.778 0.822 0.470 0.984

4 7.958 0.884 0.783 0.911

5 0.316 0.701 0.471 0.992

6 1.000 0.342 0.081 TRE

Table E.1: (continued) Estimated individual rules for the LtF treatment.

49

Rule coefficients R2 rule stability

Subject cons. AR(1) past return

Group 1

1 −0.447 0.203 0.904 mixed S

2 0.175 0.819 return U

3 0.167 0.804 return U

4 0.111 0.856 return S

5 −0.125 0.168 0.833 return U

6 0.159 0.854 return S

Group 2

1 0.0451 random S

2 0.168 random S

3 0.00997 random S

4 0.106 random S

5 0.478 −0.0473 0.24 mixed U

6 0.0473 random S

Group 3

1 −0.188 −0.291 0.221 0.836 mixed U

2 0.16 0.272 return S

3 −0.26 0.16 0.645 return S

4 0.0781 0.124 return S

5 0.283 0.105 0.676 mixed S

6 0.152 0.879 return S

Group 4

1 0.811 0.677 AR(1) N

2 0.174 0.549 return U

3 0.113 0.69 return S

4 0.14 0.824 return S

5 0.174 0.798 return U

6 0.119 0.346 return S

Table E.2: Estimated individual rules for the LtO treatment.

50

Rule coefficients R2 rule stability

Subject cons. AR(1) past return

Group 5

1 0.0975 random S

2 0.0695 random S

3 0.579 0.333 AR(1) N

4 0.00356 random S

5 0.0238 random S

6 0.0487 0.183 return S

Group 6

1 0.0496 random S

2 0.135 0.588 return S

3 0.125 0.854 return S

4 0.566 0.663 AR(1) N

5 0.108 0.468 return S

6 0.148 0.595 return S

Group 7

1 0.29 0.0795 0.741 mixed S

2 0.743 0.551 AR(1) N

3 −0.3 0.177 0.759 mixed S

4 0.44 0.0893 0.675 mixed S

5 0.136 0.269 0.0521 0.59 mixed S

6 0.156 0.884 return S

Group 8

1 0.2 0.258 return U

2 0.118 0.439 return S

3 0.118 0.207 0.757 return U

4 0.0522 0.0482 0.546 return S

5 0.131 random S

6 0.143 0.703 return S

Table E.2: (continued) Estimated individual rules for the LtO treatment.

51

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53

学霸联盟

Tinbergen Institute Discussion Paper

Bubble Formation and (In)Efficient Markets in

Learning-to-Forecast and -optimise

Experiments

Te Baoa

Cars Hommesb

Tomasz Makarewiczb

a University of Groningen, the Netherlands;

b Faculty of Economics and Business, University of Amsterdam, and Tinbergen Institute, the

Netherlands.

Tinbergen Institute is the graduate school and research institute in economics of Erasmus University

Rotterdam, the University of Amsterdam and VU University Amsterdam.

More TI discussion papers can be downloaded at http://www.tinbergen.nl

Tinbergen Institute has two locations:

Tinbergen Institute Amsterdam

Gustav Mahlerplein 117

1082 MS Amsterdam

The Netherlands

Tel.: +31(0)20 525 1600

Tinbergen Institute Rotterdam

Burg. Oudlaan 50

3062 PA Rotterdam

The Netherlands

Tel.: +31(0)10 408 8900

Fax: +31(0)10 408 9031

Bubble Formation and (In)Efficient Markets in

Learning-to-Forecast and -Optimise Experiments∗

Te Baoa Cars Hommesb Tomasz Makarewiczb

September 2015, Economic Journal forthcoming

a University of Groningen

b University of Amsterdam and Tinbergen Institute

Abstract. This experiment compares the price dynamics and bubble formation in an

asset market with a price adjustment rule in three treatments where subjects (1) submit

a price forecast only, (2) choose quantity to buy/sell and (3) perform both tasks. We

find deviation of the market price from the fundamental price in all treatments, but to

a larger degree in treatments (2) and (3). Mispricing is therefore a robust finding in

markets with positive expectation feedback. Some very large, recurring bubbles arise,

where the price is 3 times larger than the fundamental value, which were not seen in

former experiments.

JEL Classification: C91, C92, D53, D83, D84

Keywords: Financial Bubbles, Experimental Finance, Rational Expectations, Learning

to Forecast, Learning to Optimise

∗The authors are grateful to Alan Kirman for stimulating discussions and to the Editor An-

drea Galeotti and two anonymous referees for helpful and detailed comments. We also thank

participants at seminars at New York University, University of Goettingen, University of Ni-

jmegen, University of Namur, University of Groningen and conferences/workshops on “Com-

puting in Economics and Finance” 2013, Vancouver, “Experimental Finance” 2014, Zurich, in

particular our discussant David Schindler, “Expectations in Dynamic Macroeconomic Models”

2014, Bank of Finland, Helsinki and “Economic Dynamics and Quantitative Finance”, Decem-

ber 2014, Sydney. We gratefully acknowledge the financial support from NWO (Dutch Science

Foundation) Project No. 40611142 “Learning to Forecast with Evolutionary Models” and other

projects: the EUFP7 projects Complexity Research Initiative for Systemic Instabilities (CRISIS,

grant 288501) and Integrated Macro-Financial Modeling for Robust Policy Design (MACFIN-

ROBODS, grant 612796), and from the Institute of New Economic Thinking (INET) grant

project “Heterogeneous Expectations and Financial Crises” (INO 1200026). Email addresses:

T.Bao@rug.nl, C.H.Hommes@uva.nl, T.A.Makarewicz@uva.nl.

1

This paper investigates the price dynamics and bubble formation in an experi-

mental asset pricing market with a price adjustment rule. The purpose of the study

is to address a fundamental question about the origins of bubbles: do bubbles arise

because agents fail to form rational expectations or because they fail to optimise

their trading quantity given their expectations? Our experiment indicates that

both forces have a destabilising effect on the financial markets, which implies that

both deviations from rationality deserve more attention in future theoretical or

policy-oriented inquiries on bubble formation and market efficiency.

We design three experimental treatments: (1) subjects make a forecast only,

and are paid according to forecasting accuracy; (2) subjects make a quantity deci-

sion only, and are paid according to the profitability of their decision; (3) subjects

make both a forecast and a quantity decision, and are paid by their performance of

either of the tasks with equal probability. We design the payoff functions carefully

so that under the assumptions of perfect rationality and price taking behaviour,

these three tasks are equivalent in our experiment and should lead the subjects

to an equilibrium with a constant fundamental price. In contrast, we find none

of the experimental markets to show a reliable convergence to the fundamental

outcome. The market price is relatively most stable, with a slow upward trend, in

the treatment where the subjects make forecasts only. There are recurring bubbles

and crashes with high frequency and magnitude when the subjects submit both a

price forecast and a trading quantity decision.

Asset bubbles can be traced back to the very beginning of financial markets,

but have not been investigated extensively by modern economics and finance lit-

erature. One possible reason is that it contradicts the standard theory of rational

expectations (Muth, 1961; Lucas Jr., 1972) and efficient markets (Fama, 1970).

Recent finance literature however has shown growing interest in bounded rational-

ity (Farmer and Lo, 1999; Shiller, 2003) and ‘abnormal’ market movements such

as over- and under-reaction to changes in fundamentals (Bondt and Thaler, 2012)

and excess volatility (Campbell and Shiller, 1989). The recent financial crisis and

the preceding boom and bust in the US housing market highlight the importance

of understanding the mechanism of financial bubbles in order for policy makers to

design policies/institutions to enhance market stability.

It is usually difficult to identify bubbles using data from the field, since people

may substantially disagree about the underlying fundamental price of the asset

(see Hommes and in’t Veld, 2014, for a discussion about the S&P 500 example).

Laboratory experiments have an advantage in investigating this question by taking

full control over the underlying fundamental price. Smith et al. (1988) are among

the first authors to reliably reproduce price bubbles and crashes of asset prices in

2

a laboratory setting. They let the subjects trade an asset that pays a dividend

in each of 15 periods. The fundamental price at each period equals the sum of

the remaining expected dividends and follows a decreasing step function. The

authors find the price to go substantially above the fundamental price after the

initial periods before it crashes towards the end of the experiment. This approach

has been followed in many studies e.g. Lei et al. (2001); Noussair et al. (2001);

Dufwenberg et al. (2005); Haruvy and Noussair (2006); Akiyama et al. (2012);

Haruvy et al. (2013); Fu¨llbrunn et al. (2014).1 A typical result of these papers is

that the price boom and bust is a robust finding despite several major changes in

the experimental design.

Nevertheless, Kirchler et al. (2012); Huber and Kirchler (2012) argue that

the non-fundamental outcomes in these experiments are due to misunderstanding:

subjects may be simply confused by the declining fundamental price. They support

their argument by showing that no bubble appears when the fundamental price is

not declining or when the declining fundamental price is further illustrated by an

example of ‘stocks of a depletable gold mine’. Another potential concern about

these experiments, due to a typically short horizon (15 periods), is that one cannot

test whether financial crashes are likely to be followed by new bubbles. It is very

important to study the recurrence of boom-bust cycles in asset prices, for example

to understand the evolution of the asset prices between the dot-com bubble and

crash and the 2007/2008 financial crisis.

The Smith et al. (1988) experiment are categorised as ‘learning to optimise’

(henceforth LtO) experiments (see Duffy, 2008, for an extensive discussion). Be-

sides this approach, there is a complementary ‘learning to forecast’ (henceforth

LtF) experimental design introduced by Marimon et al. (1993) (see Hommes,

2011; Assenza et al., 2014, for comprehensive surveys). Hommes et al. (2005)

run an experiment where subjects act as professional advisers (forecasters) for a

pension fund: they submit a price forecasts, which is transformed into a quantity

decision of buying/selling by a computer program based on optimization over a

standard myopic mean-variance utility function. Subjects are paid according to

their forecasting accuracy. The fundamental price is defined as the rational expec-

tation equilibrium and remains constant over time. The results are twofold: (1)

the asset price fails to converge to the fundamental, but oscillates and forms bub-

bles in several markets; (2) instead of having rational expectations, most subjects

follow price trend extrapolation strategies (cf. Bostian and Holt, 2009). Heemei-

jer et al. (2009) and Bao et al. (2012) investigate whether the non-convergence

1For surveys of the literature, see Sunder (1995); Noussair and Tucker (2013).

3

result is driven by the positive expectation feedback nature of the experimental

market. Positive/negative expectation feedback means that the realised market

price increases/decreases when the average price expectation increases/decreases.

The results show that while negative feedback markets converge quickly to the

fundamental price, and adjust quickly to a new fundamental after a large shock,

positive feedback markets usually fail to converge, but under-react to the shocks

in the short run, and over-react in the long run.

The subjects in Hommes et al. (2005) and other ‘learning to forecast’ experi-

ments do not directly trade, but are assisted by a computer program to translate

their forecasts into optimal trading decisions. A natural question is what happens

if they submit explicit quantity decisions, i.e. if the experiment is based on the

‘learning to optimise’ design. Are the observed bubbles robust against the LtO

design or are they just an artifact of the computerised trading in the LtF design?

In this paper we design an experiment, in which the fundamental price is con-

stant over time (as in Hommes et al., 2005), but the subjects are asked to directly

indicate the amount of asset they want to buy/sell. Different from the double auc-

tion mechanism in the Smith et al. (1988) design, the price in our experiment is

determined by a price adjustment rule based on excess supply/demand (Beja and

Goldman, 1980; Campbell et al., 1997; LeBaron, 2006). Our experiment is help-

ful in testing financial theory based on such demand/supply market mechanisms.

Furthermore, our design allows us to have a longer time span of the experimental

sessions, which will enable a test for the recurrence of bubbles and crashes.

The main finding of our experiment is that the persistent deviation from the

fundamental price in Hommes et al. (2005) is a robust finding against task design.

Based on Relative Absolute Deviation (RAD) and Relative Deviation (RD) as

defined by Sto¨ckl et al. (2010), we find that the amplitude of the mis-pricing in

treatment (2) and (3) is much higher than in treatment (1). We also find larger

heterogeneity in traded quantities than individual price forecasts. These finding

suggest that learning to optimise is even harder than learning to forecast, and

therefore leads to even larger deviations from rationality and efficiency.

An important finding of our experiment is that in the Mixed, LtO and LtF

designs some very large and repeated price oscillations occur, where the price peaks

at more than 3 times the fundamental price. This was not observed in the former

experimental literature. Since bubbles in stock and housing prices reached similar

levels (the housing price index increases by 300% in several local markets before it

decreased by about 50% during the crisis), our experimental design may provide

a potentially better test bed for policies that deal with large recurrent bubbles.

Another contribution is that we provide an empirical micro foundation of ob-

4

served differences in aggregate macro behaviour across treatments. We estimate

individual forecasting and trading rules and find significant differences across treat-

ments. In the LtF treatment individual forecasting behaviour is more cautious in

the sense that subjects use a more conservative anchor (a weighted average of

last observed price and last forecast) in their trend-following rules, while in the

Mixed treatment almost all weight is given to the last observed price leading to a

more aggressive trend-following forecasting rule. Individual trading behaviour of

most subjects is characterized by extrapolation of past and/or expected returns.

Moreover, in the LtO and Mixed treatments the return extrapolation coefficients

are higher. These differences in individual behaviour explain the more unstable

aggregate behaviour with recurring booms and busts in the LtO and Mixed treat-

ments. We also perform a formal statistical test on individual heterogeneity in

trading strategies under the Mixed treatment. In particular, in some trading mar-

kets we observe a large degree of heterogeneity in the quantity decision even when

the price is rather stable. In the Mixed treatment most subjects fail to trade at

the conditionally optimal quantity given their own forecast. Learning to trade

optimally thus appears to be difficult.

Our paper is related to Bao et al. (2013) who run an experiment to compare

the LtF, LtO and Mixed designs in a cobweb economy. The main difference is

that they consider a negative feedback system, for which all markets converge

to the fundamental price, and find differences in the speed of convergence across

treatments.

The paper is organised as follows: Section 1 presents the experimental de-

sign and formulates testable hypotheses. Section 2 summarises the experimental

results and performs statistical tests of convergence to REE and for differences

across treatments based on aggregate variables as well as individual decision rules.

Finally, Section 3 concludes.

1 Experimental Design

In this section we explain the design of our experiment. We begin by defining

the treatments, followed by a discussion of the information given to the subjects.

Thereafter, we derive the micro foundations of the experimental economy, discuss

the implementation of the experiment, and specify hypotheses that will be tested

empirically.

5

1.1 Experimental Treatments

The experimental economy is based on a simple asset market with a constant

fundamental price. There are I = 6 subjects in each market, and each subject

plays an advisory role to a professional trading company. Subject task is either

to predict the asset price, suggest trading quantity or both, and subjects are re-

warded depending on their forecasting accuracy or trading profits. These decisions

generate an excess demand that determines the market price for the asset. The

experimental sessions last for T = 50 trading rounds. To present a quick overview

of the treatment designs, we only show the reduced form law of motion of the

price in each treatment in this section. The microfoundation of the experimental

economy and choice of parameters will be explained in detail in section 1.3.

Based on the nature of the task and the payoff scheme, there are three treat-

ments in the experiment. It is important to note here that the underlying market

structure is the same regardless of the subject task. We carefully choose the pa-

rameters of the model and payoff function so that under rational expectations,

these treatments are equivalent and lead to the same market equilibrium. The

treatments are specified as follows:

LtF Classical Learning-to-Forecast experiment. Subjects act as forecasting ad-

visers, namely they are asked for one-period ahead price predictions pei,t+1.

The subjects’ reward depends only on the prediction accuracy, defined by

(see also Table B.1 in Appendix B)

Payoffi,t+1 = max

{

0, (1300− 1300

49

(

pei,t+1 − pt+1

)2

)

}

, (1)

where pei,t+1 denotes the forecast of price at period t+1 formulated by subject

i and pt+1 is the realised asset price at period t+ 1.

The subject forecasts are automatically translated into excess demand for the

asset, yielding the following law of motion for the LtF treatment economy:

pt+1 = 66 +

20

21

(

p¯et+1 − 66

)

+ εt, (2)

where p∗ = pf = 66 is the fundamental price of the asset as well as the

unique Rational Expectations Equilibrium, p¯et+1 ≡ 16

∑6

i=1 p

e

i,t+1 denotes the

average price forecast of the six subjects and εt ∼ N(0, 1) is a small IID

shock to price pt+1.

For the price adjustment rule (2) the subjects’ payoff is maximised when all

predict the fundamental price, so that on average they make the smallest

prediction errors. Hence, in the LtF treatment it is optimal for all subjects

to predict pei,t+1 = 66.

6

LtO Classical Learning-to-Optimise experiment, where the subjects are asked to

decide on the asset quantity zi,t. Unlike the experiments in the spirit of

Smith et al. (1988), subjects in this treatment do not accumulate the asset

over periods. Instead, zi,t represents the final position of subject i in period

t. This position can be short with zi,t < 0 and is cleared once pt+1 is realised.

Subjects earn payoff based on the realised return ρt+1, which is defined as

a (constant) dividend y = 3.3 plus the capital gain over the constant gross

interest rate R = 1.05 of a secure bond:

ρt+1 ≡ pt+1 + y −Rpt = pt+1 + 3.3− 1.05pt. (3)

Subjects are not explicitly asked for a price prediction, but can use a built-in

calculator in the experimental program to compute the expected asset return

ρet+1 for any price forecast p

e

t+1 as in equation (3). Subjects are rewarded

according to

Payoffi,t+1 = max

{

0, 800 + 40(zi,t(pt+1 + 3.3− 1.05pt)− 3z2i,t)

}

. (4)

This payoff corresponds to a mean-variance utility function of the financial

firms in the underlying economy, as explained below. Expected payoff can be

computed by the subjects or read from a payoff table, depending on the cho-

sen quantity and the expected excess return (see Table B.2 in Appendix B).

Under the assumption of price-taking behaviour, i.e., when the subjects

ignore the impact of their own trading decisions on the realised market

price, the optimal demand for asset given one’s own price forecast pei,t+1

is z∗i,t =

pei,t+1+3.3−1.05pt

6

=

ρet+1

6

.

The law of motion of the LtO treatment is given by the price adjustment

rule based on the aggregate excess demand

pt+1 = pt +

20

21

6∑

i=1

zi,t + εt (5)

for the same set of IID shocks εt as in the LtF treatment. Under the as-

sumption of price-taking behaviour, the Rational Expectation Equilibrium

(REE) of the market is p∗ = pf = 66, and the associated optimal demand

for the asset is z∗ = 0 for each individual. Therefore, the optimal choices are

equivalent in the LtF and LtO treatments. For other cases in which subjects

deviate from price-taking behaviour, e.g. by taking their market power into

account and playing collusive or non-cooperative Nash strategies, a detailed

discussion is provided in Appendix C.

7

Mixed Each subject is asked first for his or her price forecast pei,t+1 and second for

the choice of the asset quantity zi,t. In order to avoid hedging, the reward for

the whole experiment is based on either the payoff in (1) or (4) with equal

probability (flip of a coin at the end of the session). The law of motion of

the Mixed treatment is given by (5), the same price adjustment rule as in

LtO and does not depend on the submitted price forecasts.

The points in each treatment are exchanged into Euro at the end of the experiment

with the conversion rate 3000 points = 1 Euro. We add a max function to the

forecasting and trading payoffs to avoid negative rewards.

1.2 Information to the Subjects

At the beginning of the experimental sessions, subjects were informed about their

task and payoff scheme, including the payoff functions (1) or (4) depending on the

treatment. We supplemented the subjects with payoff tables (see Appendix B).

Subjects from the LtF treatment were told that the asset price depends pos-

itively on the average price forecast, while subjects in the two other treatments

were informed that the price increases with the excess demand. In addition, in the

Mixed treatment we made it clear that the subject payoffs may be related to the

forecasting accuracy, but that the realized price itself depends exclusively on their

trades. Regardless of the treatment, we provided the subjects only with qualitative

information about the market, that is we did not explicate the respective laws of

motion (2) or (5).

Throughout the experiment, the subject could observe past market prices and

their individual decisions, in graphical and table form, but they could not see the

decisions, or an average decision, of the other participants. We did not mention the

fundamental price in the instructions at all, though we did provide the information

about the interest rate and the asset dividend in all the three treatments, which

could be used to compute the fundamental price p∗ = y¯/r = 66. Finally, the

subjects know the specification of their payoff function, i.e., the payoff is higher if

the prediction error (trading profit) is lower (higher) for the forecasters (traders).

1.3 Experimental Economy

This section provides some micro-foundations of our experimental economy. We

build our experimental economy upon an asset market with heterogeneous beliefs

as in Brock and Hommes (1998). There are I = 6 agents, who allocate investment

between a risky asset that pays a fixed dividend y and a risk-free bond that pays

8

a fixed gross return R = 1 + r.2 The wealth of agent i evolves according to

Wi,t+1 = RWi,t + zi,t(pt+1 + y −Rpt), (6)

where zi,t is the demand (in the sense of the final position) for the risky asset

by agent i in period t (positive sign for buying and negative sign for selling) and

pt and pt+1 are the prices of the risky asset in periods t and t + 1 respectively.

Let Ei,t and Vi,t denote the beliefs or forecasts of agent i about the conditional

expectation and the conditional variance based on publicly available information.

The agents are assumed to be simple myopic mean-variance maximizers of next

period’s wealth, i.e. they solve the myopic optimisation problem:

max

zi,t

{

Ei,tWi,t+1 − a

2

Vi,t(Wi,t+1)

}

≡ max

zi,t

{

zi,tEi,tρt+1 − a

2

z2i,tVi,t(ρt+1)

}

, (7)

where a is a parameter for risk aversion, and ρt+1 is the excess return as defined

in equation (3). In the experiment, we use an affine transformation of this utility

function as in (4) as a payoff for the trading task.

Optimal demand of agent i is given by3

z∗i,t =

Ei,t(ρt+1)

aVi,t(ρt+1)

=

pei,t+1 + y −Rpt

aσ2

, (8)

where pei,t+1 = Ei,tpt+1 is the individual forecast by agent i of the price in period

t+ 1. The market price is set by a market maker using a simple price adjustment

mechanism in response to excess demand (Beja and Goldman, 1980),4 given by

pt+1 = pt + λ

(

ZDt − ZSt

)

+ εt, (9)

2Fixed dividend allows for a constant fundamental price throughout the experiment. In a

more general model with the same demand functions and market equilibrium, y corresponds to

the mean of an (exogenous) IID stochastic dividend process yt; see Brock and Hommes (1998)

for a discussion.

3The last equality in (8) follows from a simplifying assumption made in Brock and Hommes

(1998) that all agents have homogeneous and constant beliefs about the conditional variance,

i.e. Vi,t(ρt+1) = σ

2. See Hommes (2013), Chapter 6, for a more detailed discussion.

4See e.g. Chiarella et al. (2009) for a survey on the abundant literature about the price

adjustment market mechanisms. We decided to use (9) instead of a market clearing mechanism

for two reasons: (i) market maker is a stylized description of a specialist driven market, a common

case for financial markets (e.g. NASDAQ); and (ii) the current one-period ahead design is much

simpler for the subjects than one based on a market clearing mechanism, which requires two-

period ahead trading and forecasting. In particular, the two-period ahead trading/forecasting

feature would lead to a 3-dimensional payoff table instead of the 2-dimensional payoff table in

Appendix B.2. The two-period ahead market clearing design results in much more volatile price

patterns in the LtF experiments (Hommes, 2011), which suggests that our main finding –that

the boundedly rational trading can be a destabilizing force in the financial markets– is likely to

be robust in a similar two-period ahead LtO experiment with a market clearing design.

9

where εt ∼ N(0, 1) is a small IID shock, λ > 0 is a scaling factor, ZSt is the

exogenous supply and ZDt is the total demand. This mechanism guarantees that

excess demand/supply increases/decreases the price.

For simplicity, the exogenous supply ZSt is normalised to 0 in all periods. In the

experiment, we take Rλ = 1, specifically R = 1 + r = 21/20, λ = 20/21, aσ2z = 6,

and y = 3.3. We chose these specific parameters mainly for simplicity of the law of

motion of the price. For example, by imposing aσ2z = 6, the total excess demand

coincide with the average expected excess return, and when Rλ = 1, this ensures

that the final law of motion of asset price in the LtF treatment only depends on

the average forecast p¯et+1, but does not contain pt. The price adjustment based on

aggregate individual demand thus takes the simple form

pt+1 = pt +

20

21

6∑

i=1

zi,t + εt, (10)

which constitutes the law of motion (5) for the LtO and Mixed treatments, in

which the subjects are asked to elicit their asset demands.

For an optimising agent and the chosen parameters, the individual optimal

demand (8) conditional on a price forecast pei,t+1 equals

z∗i,t =

ρei,t+1

aσ2

=

pei,t+1 + 3.3− 1.05pt

6

, (11)

with ρei,t+1 the forecast of excess return in period t+ 1 by agent i. Substituting it

back into (5) gives

pt+1 = 66 +

20

21

(

p¯et+1 − 66

)

+ εt, (12)

where p¯et+1 =

1

6

∑6

i=1 p

e

i,t+1 is the average prediction of the price pt+1 by six sub-

jects.5 This price is the temporary equilibrium with point-beliefs about prices and

represents the price adjustment process as a function of the average individual

forecast. It constitutes the law of motion (2) for the LtF treatment, in which the

subjects are asked to elicit their price expectations.

We note that from the optimal demand (11) it is clear that optimising the

(quadratic) mean-variance utility function (7) is equivalent to minimising the

quadratic penalty for forecasting errors as in the LtF payoff function (1). This

implies that the trading and forecasting tasks in the experiment are equivalent

under perfect rationality.

5Heemeijer et al. (2009) used a similar price adjustment rule in a learning to forecast exper-

iment that compares positive versus negative expectation feedback, but their fundamental price

is 60 instead of 66.

10

By imposing the rational expectations condition p¯et+1 = p

f = Et(pt+1), a simple

computation shows that pf = 66 is the unique Rational Expectation Equilibrium

(REE) of the system. This fundamental price equals the discounted sum of all

expected future dividends, i.e., pf = y/r . If all agents have rational expectations,

the realised price becomes pt = p

f + εt = 66 + εt, i.e. the fundamental price plus

(small) white noise and, on average, the price forecasts are self-fulfilling. When

the price is pf , the (expected) excess return of the risky asset in (3) equals 0 and

the optimal demand for the risky asset in (8) by each agent is also 0, that is excess

demand is equal to 0.

1.4 Liquidity Constraints

To limit the effect of extreme price forecasts or quantity decisions in the experi-

ment, we impose the following liquidity constraints on the subjects. For the LtF

treatment, price predictions such that pei,t+1 > pt+30 or p

e

i,t+1 < pt−30 are treated

as pei,t+1 = pt+30 and p

e

i,t+1 = pt−30 respectively. For the LtO treatment, quantity

decisions greater than 5 or smaller than −5 are treated as 5 and −5 respectively.

These two liquidity constraints are roughly the same, since the optimal asset de-

mand (11) is close to one sixth of the expected price difference. Nevertheless, the

liquidity constraint in the LtF treatment was never binding, while under the LtO

treatment subjects would sometimes trade at the edges of the allowed quantity

interval. We also imposed additional constraint that pt has to be non-negative

and not greater than 300. In the experiment, this constraint never had to be

implemented.

1.5 Number of Observations

The experiment was conducted on December 14, 17, 18 and 20, 2012 and June

6, 2014 at the CREED Laboratory, University of Amsterdam. 144 subjects were

recruited. The experiment employs a group design with 6 subjects in each ex-

perimental market. There are 24 markets in total and 8 for each treatment. No

subject participates in more than one session. The duration of the experiment

is typically about 1 hour for the LtF treatment, 1 hour and 15 minutes for the

LtO treatment, and 1 hour 45 minutes for the Mixed treatment. Experimental

instructions are shown in Appendix A.

11

1.6 Testable Hypotheses

The RE benchmark suggests that the subjects should learn to play the REE and

behave similarly in all treatments. In addition, a rational decision maker should

be able to find the optimal demand for the asset given his price forecast according

to Equation (11) in the Mixed treatment. These theoretical predictions can be

formulated into the following testable hypotheses:

HYPOTHESIS 1: The asset prices converge to the Rational Expectation Equi-

librium in all markets;

HYPOTHESIS 2: There is no systematic difference between the market prices

across the treatments;

HYPOTHESIS 3: Subjects’ earnings efficiency (defined as the ratio of the ex-

perimental payoff divided by the hypothetical payoff when all subjects play

the REE) are independent from the treatment;

HYPOTHESIS 4: In the Mixed treatment the quantity decisions by the sub-

jects are optimal conditional on their price expectations;

HYPOTHESIS 5: There is no systematic difference between the decision rules

used by the subjects for the same task across the treatments.

These hypotheses are further translated into rigorous statistical tests. To be

specific, we will use Relative (Absolute) Deviation (Sto¨ckl et al., 2010) to measure

price convergence, and test the difference of the distribution of this measure be-

tween the three treatments. (HYPOTHESIS 1 and 2). Relative earnings can be

compared with the Mann-Whitney-Wilcoxon rank-sum test (HYPOTHESIS 3).

Finally, we estimate individual behavioural rules for every subject: a simple re-

striction test will reveal whether HYPOTHESIS 4 is true, while the rank-

sum test can again be used to test the rule differences between the treatments

(HYPOTHESIS 5). Notice that HYPOTHESIS 1 is nested within HY-

POTHESIS 2, while HYPOTHESIS 4 is nested within HYPOTHESIS 5.

2 Experimental Results

2.1 Overview

Figure 1 (LtF treatment), Figure 2 (LtO treatment) and Figure 3 (Mixed treat-

ment) show plots of the market prices in each treatment. For most of the groups,

the prices and predictions remained in the interval [0, 100]. The exceptions are

markets 1, 4 and 8 (Figures 3a, 3d and 3h) in the Mixed treatment. In the first

12

two of these three groups, prices peaked at almost 150 (more than twice the funda-

mental price pf = 66) and in the last group, prices reached 225, almost 3.5 times

the fundamental price. Moreover, markets 4 and 8 of the Mixed treatment show

repeated booms and busts.

The figures suggest that the market price is the most stable in the LtF treat-

ment, and the most unstable in the Mixed treatment. In the LtF treatment, there

is little heterogeneity between the individual forecasts, shown by the green dashed

lines. In the LtO treatment, however, there is a high level of heterogeneity in the

quantity decisions shown by the blue dashed lines. In the Mixed treatment, it

is somewhat surprising that the low heterogeneity in price forecasts and the high

heterogeneity in quantity decisions coexist.6

It is noticeable that in two markets in the LtO and Mixed treatment, the

market price stabilises after a few periods, but stays at a non REE level. Market

2 in the LtO treatment stabilises around price 40, and Market 6 in the Mixed

treatment stabilises around price 50. In these two markets, the optimal demand

by each individual as implied by (11) should be about 0.2 (0.15) when the price

stabilises at 40 (50). However, the actual average demand in the experiment stays

very close to 0 in both cases. This is an indication of sub-optimal behaviour by

some subjects. It may be caused by two reasons: (1) the subjects mistakenly

ignored the role of dividend in the return function, and thought that buying is not

profitable unless the price change is strictly positive, or (2) some of them held a

pessimistic view about the market, and kept submitting a lower demand than the

optimal level as implied by their price forecast.

In general, convergence to the REE does not seem to occur in any of the

treatments. This suggests that the hypotheses based on the rational expectations

benchmark are likely to be rejected. Furthermore, the figures suggest clear differ-

ences between the treatments. In the remainder of this section, we will discuss the

statistical evidence for the hypotheses in detail.

6We compare the dispersion of individual decisions using the standard deviation of the (im-

plied) quantity decisions averaged over all periods in each market. A rank-sum test suggests that

there is no difference between dispersion of quantity decisions in the LtO versus Mixed treat-

ment (with p-value equal to 0.083 for dispersion over all periods and p-value equal to 0.161 for

dispersion over last 40 periods). The dispersions of the quantity decisions in the LtO and Mixed

treatments are indeed significantly larger than the dispersion of (implied) quantity decisions in

the LtF treatment, with p-values equal to 0 for both all and last 40 periods.

13

Figure 1: Price Dynamics in LtF Treatment

0

20

40

60

80

100

0 10 20 30 40 50

(a) Group 1

0

20

40

60

80

100

0 10 20 30 40 50

(b) Group 2

0

20

40

60

80

100

0 10 20 30 40 50

(c) Group 3

0

20

40

60

80

100

0 10 20 30 40 50

(d) Group 4

0

20

40

60

80

100

0 10 20 30 40 50

(e) Group 5

0

20

40

60

80

100

0 10 20 30 40 50

(f) Group 6

0

20

40

60

80

100

0 10 20 30 40 50

(g) Group 7

0

20

40

60

80

100

0 10 20 30 40 50

(h) Group 8

Notes. Groups 1-8 for the Learning to Forecast treatment. Straight line shows the

fundamental price pf = 66, solid black line denotes the realised price, while green dashed

lines denote individual forecasts.

14

Figure 2: Price Dynamics in LtO Treatment

0

20

40

60

80

100

0 10 20 30 40 50

Realised prices

-4

-2

0

2

4

0 10 20 30 40 50

Inividual quantity decisions

(a) Group 1

0

20

40

60

80

100

0 10 20 30 40 50

Realised prices

-4

-2

0

2

4

0 10 20 30 40 50

Inividual quantity decisions

(b) Group 2

0

20

40

60

80

100

0 10 20 30 40 50

Realised prices

-4

-2

0

2

4

0 10 20 30 40 50

Inividual quantity decisions

(c) Group 3

0

20

40

60

80

100

0 10 20 30 40 50

Realised prices

-4

-2

0

2

4

0 10 20 30 40 50

Inividual quantity decisions

(d) Group 4

0

20

40

60

80

100

0 10 20 30 40 50

Realised prices

-4

-2

0

2

4

0 10 20 30 40 50

Inividual quantity decisions

(e) Group 5

0

20

40

60

80

100

0 10 20 30 40 50

Realised prices

-4

-2

0

2

4

0 10 20 30 40 50

Inividual quantity decisions

(f) Group 6

0

20

40

60

80

100

120

140

0 10 20 30 40 50

Realised prices

-4

-2

0

2

4

0 10 20 30 40 50

Inividual quantity decisions

(g) Group 7

0

20

40

60

80

100

0 10 20 30 40 50

Realised prices

-4

-2

0

2

4

0 10 20 30 40 50

Inividual quantity decisions

(h) Group 8

Notes. Groups 1-8 for the Learning to Optimise treatment. Each group is presented

in two panels. The upper panel displays the fundamental price pf = 66 (straight line)

and the realised prices (solid black line), while the lower panel displays individual trades

(dashed blue lines) and average trade (solid red line). Notice the different y-axis scale

for group 7 (picture g).

15

Figure 3: Price Dynamics in Mixed Treatment

0

20

40

60

80

100

120

140

0 10 20 30 40 50

Realised prices

-4

-2

0

2

4

0 10 20 30 40 50

Inividual quantity decisions

(a) Group 1, price scale [0, 150]

0

20

40

60

80

100

0 10 20 30 40 50

Realised prices

-4

-2

0

2

4

0 10 20 30 40 50

Inividual quantity decisions

(b) Group 2

0

20

40

60

80

100

0 10 20 30 40 50

Realised prices

-4

-2

0

2

4

0 10 20 30 40 50

Inividual quantity decisions

(c) Group 3

0

20

40

60

80

100

120

140

0 10 20 30 40 50

Realised prices

-4

-2

0

2

4

0 10 20 30 40 50

Inividual quantity decisions

(d) Group 4, price scale [0, 150]

0

20

40

60

80

100

0 10 20 30 40 50

Realised prices

-4

-2

0

2

4

0 10 20 30 40 50

Inividual quantity decisions

(e) Group 5

0

20

40

60

80

100

0 10 20 30 40 50

Realised prices

-4

-2

0

2

4

0 10 20 30 40 50

Inividual quantity decisions

(f) Group 6

0

20

40

60

80

100

0 10 20 30 40 50

Realised prices

-4

-2

0

2

4

0 10 20 30 40 50

Inividual quantity decisions

(g) Group 7

0

50

100

150

200

250

0 10 20 30 40 50

Realised prices

-4

-2

0

2

4

0 10 20 30 40 50

Inividual quantity decisions

(h) Group 8, price scale [0, 250]

Notes. Groups 1-8 for the Mixed treatment with subject forecasting and trading. Each

group is presented in a picture with two panels. The upper panel displays the funda-

mental price pf = 66 (straight line), the realised prices (solid black line) and individual

predictions (green dashed lines), while the lower panel displays individual trades (dashed

blue lines) and average trade (solid red line). Notice the different y-axis scale for groups

1, 4 and 8 (pictures a, d and h respectively).

16

2.2 Quantifying the Bubbles

The term ‘bubble’ is informally used in the literature to describe, loosely speaking,

a prolonged spell of an asset price growth beyond its fundamental. In order to

capture this notion with a rigid statistic, we follow Sto¨ckl et al. (2010) and evaluate

the experimental mispricing with the Relative Absolute Deviation (RAD) and

Relative Deviation (RD). These two indices measure respectively the absolute and

relative deviation from the fundamental in a specific period t and are given by

RADg,t ≡ |p

g

t − pf |

pf

× 100%, (13)

RDg,t ≡ p

g

t − pf

pf

× 100%, (14)

where pf = 66 is the fundamental price and pgt is the realised asset price at period

t in the session of group g. The average RADg and RDg are defined as

RADg ≡ 1

50

50∑

t=1

RADg,t, (15)

RDg ≡ 1

50

50∑

t=1

RDg,t, (16)

RADg shows the average relative distance between the realised prices and the

fundamental in group g, while the average RDg focuses on the sign of this re-

lationship. Groups with average RDg close to zero could either converge to the

fundamental (in which case the RADg is also close to zero) or oscillate around the

fundamental (possibly with high RADg), while positive or negative average RDg

signals that the group typically over- or underpriced the asset.

It is difficult to come up with a formal criterion for a bubble in terms of these

measures. In particular, when bubbles are accompanied by a price plunge, or

“negative bubbles”, the RD may be very close to 0. Therefore, in this paper we

focus on the differences between the three treatments.7

The results for average RAD and RD measures for each treatment are pre-

sented in Table 1. They confirm that the LtF groups were the closest to, though

7As empirical benchmarks we computed these two measures for the US stock and housing

markets. The RAD (RD) is 40% (20.2%) for S&P500, based on quarterly data 1950Q1−2012Q4

and the fundamental computed by a standard Gordon present discounted value model; for the

same data set, using deviations from the Campbell-Cochrane consumption-habit fundamental

model the RAD (RD) is 19% (3.9%) (Hommes and in’t Veld, 2014). For US housing market

data in deviations from a benchmark fundamental based on housing rents the RAD (RD) are

7.7% (0.4%) for over 40 years of quarterly data 1970Q1− 2013Q1 and 9.7% (2.2%) for 20 years

of quarterly data 1993Q1− 2013Q1 (Bolt et al., 2014).

17

still quite far from, the REE (with an average RAD of about 9.5%), while Mixed

groups exhibited the largest price deviations with an average RAD of 36%. Inter-

estingly, LtO groups had significant oscillations (on average high RAD of 24.7%),

but centered close to the fundamental price (average RD of 1.4%, compared to av-

erage RD of −3% and 16.1% for the LtF and Mixed treatments respectively). LtF

groups on average are below the fundamental price and Mixed groups typically

overshoot it.

Table 1: RAD and RD

Treatment LtF LtO Mixed

Group RAD RD RAD RD RAD RD

#1 10.03∗∗∗ −7.011 18.26∗∗∗ −8.148∗ 38.65∗∗∗ 36.84∗

#2 17.98∗∗∗ −16.94∗ 34.52∗∗∗ −34.52∗ 7.27∗∗∗ −5.657∗

#3 8.019∗∗ −6.048 30.2∗∗∗ −12.95∗ 8.025∗∗∗ 4.014∗

#4 7.285∗∗ −5.196 20.63∗∗∗ 3.844∗ 42.86∗∗∗ 35.46∗

#5 8.366∗∗∗ 4.152∗ 16.55∗∗∗ 5.256∗ 14.98∗∗∗ 3.341∗

#6 14.52∗∗∗ 6.503∗ 17.51∗∗∗ 7.056∗ 23.08∗∗∗ −23.08∗

#7 4.222 1.104∗ 31.22∗∗∗ 23.82∗ 32.14∗∗∗ −18.71

#8 5.365 −0.2539∗ 28.48∗∗∗ 26.65∗ 120.7∗∗∗ 96.5∗

Average 9.473 −2.961 24.67 1.376 35.97 16.09

Notes. Relative Absolute Deviation (RAD) and Relative Deviation (RD) of the experi-

mental prices for the three treatments, in percentages. ∗∗∗ (∗∗) denotes groups for which

the average RAD from the last 40 periods is larger than 3% on 1% (5%) significance

level. ∗ denotes groups for which the average RD from the last 40 periods is outside

[−1.5%, 1.5%] interval on 5% significance level.

A simple t-test shows that for the LtO and Mixed treatment, as well as for 6 out

of 8 LtF groups (exceptions are Markets 7 and 8), the means of the groups’ RAD

measures (disregarding the initial 10 periods to allow for learning) are significantly

larger than 3%.8 Furthermore, for all groups in all three treatments, t-test on

any meaningful significance level rejects the null of the average price (for periods

11−50, i.e. the last 40 periods to allow for learning by the subjects) being equal to

the fundamental value. This result shows negative evidence on HYPOTHESIS 1:

none of the treatments converges to the REE.

There is no significant difference between the treatments in terms of RD ac-

83% RAD is approximately equivalent to a typical price deviation of 2 in absolute terms,

which corresponds to twice the standard deviations of the idiosyncratic supply shocks, i.e 95%

confidence bounds of the REE.

18

cording to the Mann-Whitney-Wilcoxon rank-sum test (p-value> 0.1 for each

pair of the treatments, z-statistic is −0.735,−0.735 and −0.420 for LtF, LtO

and Mixed respectively. The unit of observation is per market, i.e. 8 for each

treatment). However, the difference between the LtF treatment and each of the

other treatments in terms of RAD is significant at 5% according to the rank-

sum test (p-value= 0.002 and 0.003, and z-statistic is −3.151 and −2.205 for the

LtO and Mixed respectively, number of observations: 8 for each treatment), while

the difference between the LtO and Mixed is not significant (p-value= 0.753, z-

statistic= −0.135 number of observations: 8 for each treatment). This is strong

evidence against HYPOTHESIS 2, as it shows that trading and forecasting tasks

yield different market dynamics.

The RAD values in our paper are similar to those in Sto¨ckl et al. (2010) (see

specifically their Table 4 for the RAD/RD measures). Nevertheless, there are

some important differences. First, group 8 from the Mixed treatment (with RAD

equal to 120.7%) exhibits the largest price bubble in the experiment. Second,

the four experiments investigated by Sto¨ckl et al. (2010) have shorter spans (with

sessions of either 10 or 25 periods) and so typically witness one bubble. Our data

shows that the mispricing in experimental asset markets is a robust finding. The

crash of a bubble does not enforce the subjects to converge to the fundamental,

but instead marks the beginning of a ‘crisis’ until the market turns around and

a new bubble emerges. This succession of over- and under-pricing of the asset is

reflected in our RD measures, which are smaller than the typical ones reported by

Sto¨ckl et al. (2010), and can even be negative, despite high RAD.

In addition, our experiment yields measures resembling the above mentioned

benchmark stock and housing markets (see footnote 7). Indeed, the LtF, LtO and

Mixed experimental treatments yields boom/boost cycles of a realistic magnitude,

comparable to what has been observed in recent stock and housing market bubbles

and crashes.

RESULT 1. Among the three treatments, LtF incurs dynamics closest to the

REE. Nevertheless, the average price is still far from the rational expectations

equilibrium. Furthermore, in terms of aggregate dynamics LtF treatment is signif-

icantly different from the other two treatments, which are indistinguishable between

themselves. We conclude that HYPOTHESIS 1 and 2 are rejected.

2.3 Earnings Efficiency

Subjects’ earnings in the experiment are compared to the hypothetical case where

all subjects play according to the REE in all 50 periods. Subjects can earn 1300

19

points per period for the forecasting task when they play according to REE because

they make no prediction errors, and 800 points for the trading task when they play

according to the REE because the asset return is 0 and they should not buy or

sell. We use the ratio of actual against hypothetical REE payoffs as a measure

of payoff efficiency. This measure can be larger than 100% in treatments with

the LtO and Mixed Treatments, because the subjects can profit if they buy and

the price increases and vice versa. These earnings efficiency ratios, as reported in

Table D.1 in the appendix, are generally high (more than 75%).

The earnings efficiency for the forecasting task is higher in the LtF treatment

than in the Mixed treatment (rank-sum test for difference in distributions with

p-value=0.001). At the same time, the earnings efficiency for the trading task is

very similar in the LtO treatment and the Mixed treatment (rank-sum test with

p-value=0.753).

RESULT 2. Forecasting efficiency is significantly higher in the LtF than in the

Mixed treatment, while there is no significant difference in the trading efficiency

in treatments LtO and Mixed. HYPOTHESIS 3 is partially rejected.

2.4 Conditional Optimality of Forecast and Quantity Decision in

Mixed Treatment

In the Mixed treatment, each subject makes both a price forecast and a quantity

decision. It is therefore possible to investigate whether these two are consistent,

namely, whether the subjects’ quantity choices are close to the optimal demand

conditional on the price forecast as in Eq. (11) (the optimal quantity is 1/6 of

the corresponding expected asset return). Figure 4 shows the scatter plot of the

quantity decision against the implied predicted return ρei,t+1 = p

e

i,t+1 +3.3−1.05pt,

for each subject and each period separately.9 If all individuals made consistent

decisions, these points should lie on the (blue) line with slope 1/6.

Figure 4a illustrates two interesting observations. First, subjects have some

degree of ‘digit preference’, in the sense that the trading quantities are typically

round numbers or contain only one digit after the decimal. Second, the quantity

choices are far from being consistent with the price expectations. In fact, the

subjects sometimes sold (bought) the asset even though they believed its return

will be substantially positive (negative).

9Sometimes the subjects submit extremely high price predictions, which in most cases seem to

be typos. The scatter plot excludes these outliers, by restricting the horizontal scale of predicted

returns on the asset between −60 and 60.

20

Figure 4: Conditional Optimality of Quantity Decisions

-4

-2

0

2

4

-60 -40 -20 0 20 40 60

(a) Expected return vs trade

-2

-1.5

-1

-0.5

0

0.5

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6

(b) Trade rule (17): slope vs constant

Notes. ML estimation for trading rule (17) in the Mixed treatment. Panel (a) is the

scatter plot of the traded quantity (vertical axis) against the implied expected return

(horizontal axis). Each point represents one decision of one subject in one period from

one group. Panel (b) is the scatter plot of the estimated trading rules (17) slope (reaction

to expected return; horizontal axis) against constant (trading bias; vertical axis). Each

point represents one subject from one group. Solid line (left panel)/triangle (right panel)

denotes the optimal trade rule (zi,t = ρ

e

i,t/6). Dashed line (left panel)/circle (right panel)

denotes the estimated rule under restriction of homogeneity (zi,t = c+ θρ

e

i,t).

To further evaluate this finding, we run a series of Maximum Likelihood (ML)

regressions based on the trading rule

zi,t = ci + θiρ

e

i,t+1 + ηi,t, (17)

with ηi,t ∼ NID(0, σ2η,i). The estimated coefficients for all subjects are shown in

the scatter plot of Figure 4b. This model has a straightforward interpretation: it

takes the quantity choice of subject i in period t as a linear function of the implied

(by the price forecast) expected return on the asset. It has two important special

cases: homogeneity and optimality (nested within homogeneity). To be specific,

subject homogeneity (heterogeneity) corresponds to an insignificant (significant)

variation in the slope θi = θj (θi 6= θj) for any (some) pair of subjects i and j.

The constant ci shows the ‘irrational’ optimism/pessimism bias of subject i. Op-

timality of individual quantity decisions implies homogeneity with the additional

restrictions that θi = θj = 1/6 and ci = cj = 0 (no agent has a decision bias).

The assumptions of homogeneity and perfect optimisation are tested by esti-

mation of equation (17) with the restrictions on the parameters ci and θi.

10 These

10We use ML since the optimality constraint does not exclude heterogeneity of the idiosyncratic

shocks ηi,t. We exclude outliers defined as observations when a subject predicts an asset return

21

regressions are compared with an unrestricted regression (with θi 6= θj and ci 6= cj)

via a Likelihood Ratio (LR) test. The result of the LR test shows that both the

assumption of homogeneity and perfect optimisation are rejected (with p-values

below 0.001). Furthermore, we explicitly tested for zi,t = ρ

e

i,t/6 when estimating

individual rules. Estimations identified 11 subjects (23%) as consistent optimal

traders (see footnote 13 for a detailed discussion). In sum, we find evidence for

heterogeneity of individual trading rules. The majority of the subjects are unable

to learn the optimal solution.

This result has important implications for economic modelling. The RE hy-

pothesis is built on homogeneous and model consistent expectations, which the

agents in turn use to optimise their decisions. Many economists find the first ele-

ment of RE unrealistic: it is difficult for the agents to form rational expectations

due to limited understanding of the structure of the economy. But the second

part of RE is often taken as a good approximation: agents are assumed to make

an optimal decision conditional on what they think about the economy, even if

their forecast is wrong. Our subjects were endowed with as much information as

possible, including an asset return calculator, a table for profits based on the pre-

dicted asset return and chosen quantity and the explicit formula for profits; and

yet many failed to behave optimally in forecasting as well as choosing quantities.

The simplest explanation is that individuals in general lack the computational

capacity to make perfect mathematical optimisations.

RESULT 3. The subjects’ quantity decisions are not conditionally optimal given

their price forecasts in the Mixed treatment. We conclude that HYPOTHESIS 4

is rejected for 77% (37 out of 48) of the subjects.

2.5 Estimation of Individual Behavioural Rules

In this subsection we estimate individual forecasting and trading rules and inves-

tigate whether there are significant differences between treatments. Prior exper-

imental work (Heemeijer et al., 2009) suggests that in LtF experiments, subjects

use heterogeneous forecasting rules which nevertheless typically are well described

by a simple linear First-Order Rule

pei,t = αipt−1 + βip

e

i,t−1 + γi(pt−1 − pt−2). (18)

higher than 60 in absolute terms. To account for an initial learning phase, we exclude the first ten

periods from the sample. We also drop subjects 4 and 5 from group 6, since they would always

pass zi,t = 0 for t > 10. Interestingly, these two subjects had non-constant price predictions,

which suggests that they were not optimisers.

22

This rule may be viewed as an anchor and adjustment rule (Tversky and Kah-

neman, 1974), as it extrapolates a price change (the last term) from an anchor

(the first two terms). Two important special cases of (18) are the pure trend

following rule with αi = 1 and βi = 0, yielding

pei,t = pt−1 + γi(pt−1 − pt−2), (19)

and adaptive expectations with γi = 0 and αi + βi = 1, namely

pei,t = αipt−1 + (1− αi)pei,t−1. (20)

The pure trend-following rule (19) uses an anchor giving all weight to the last

observed price (pt−1), while in the general rule (18) the anchor gives weight to

the last observed price (pt−1) as well as the last forecast (pei,t−1). In this sense

the general rule (18) is more cautious and extrapolates the trend from a more

gradually evolving anchor, while the pure trend-following rule is more aggressive

extrapolating the trend from the last price observation.

To explain the trading behaviour of the subjects from the LtO and Mixed

treatments, we estimate a general trading strategy in the following specifications:

zi,t = ci + χizi,t−1 + φiρt, (LtO) (21a)

zi,t = ci + χizi,t−1 + φiρt + ζiρei,t+1. (Mixed) (21b)

This rule captures the most relevant and most recent possible elements of indi-

vidual trading. Notice however, that the trading rule (21a) in the LtO treatment

only contains a past return (ρt) term, while the trading rule (21b) in the Mixed

treatment contains an additional term for expected excess return (ρet+1), which

is not observable in the LtO treatment because subjects did not give price fore-

casts. Both trading rules have two interesting special cases. First, what we call

persistent demand (φi = ζi = 0) is characterised by a simple AR(1) process:

zi,t = ci + χizi,t−1. (22)

A second special case is a return extrapolation rule (with χi = 0):

zi,t = ci + φiρt (LtO), (23a)

zi,t = ci + φiρt + ζiρ

e

i,t+1 (Mixed). (23b)

For the LtF and LtO treatments, for each subject we estimate her behavioural

heuristic starting with the general forecasting rule (18) or the general trading

23

rule (21a) respectively. To allow for learning, all estimations are based on the

last 40 periods. Testing for special cases of the estimated rules is straightforward:

insignificant variables are dropped until all the remaining coefficients are significant

at 5% level.11

A similar approach is used for the Mixed treatment (now also allowing for

the expected return coefficient ζi).

12 Equations (18) and (21b) are estimated

simultaneously. One potential concern for the estimation is that the contemporary

idiosyncratic errors in these two equations are correlated, given that the trade

decision depends on the contemporary expected forecast (if ζi 6= 0). Since the

contemporary trade does not appear in the forecasting rule, the forecast based

on the rule (18) can be estimated independently in the first step. The potential

endogeneity only affects the trading heuristic (23b), and can be solved with a

simple instrumental variable approach. The first step is to estimate the forecasting

rule (18), which yields fitted price forecasts of each subject. In the second step,

the trading rule (23b) is estimated with both the fitted forecasts as instruments,

and directly with the reported forecasts. Endogeneity can be tested by comparing

the two estimators using the Hausman test. Finally, the special cases of (21a–21b)

are tested based on reported or fitted price forecasts according to the Hausman

test.13

The estimation results can be found in Appendix E, in Tables E.1, E.2 and E.3

respectively for the LtF, LtO and Mixed treatments. In order to quantify whether

agents use different decision rules in different treatments, we test the differences

of the coefficients in the decision rules with the rank-sum test.

2.5.1 Forecasting rules in LtF versus Mixed

The LtF treatment can be directly compared to the Mixed treatment by com-

paring the estimated forecasting rules (18). We observe that rules with a trend

extrapolation term γi are popular in both treatments (respectively 39 in LtF and

25 in Mixed out of 48). A few other subjects use a pure adaptive rule (20) (none

in LtF and 3 in Mixed treatments respectively). A few others use a rule defined by

11Adaptive expectations (20) impose a restriction α ∈ [0, 1] (with α = 1 − β), so we follow

here a simple ML approach. If αi > 1 (αi < 0) maximises the likelihood for (20), we use the

relevant corner solution αi = 1 (αi = 0) instead. We check the relevance of the two constrained

models (trend and adaptive) with the Likelihood Ratio test against the likelihood of (18).

12See footnote 9.

13Whenever the estimations indicated that a subject from the Mixed treatment used a return

extrapolation rule of the form zi,t = ζiρ

e

i,t+1, that is a rule in which only the implied expected

return was significant, we directly tested ζi = 1/6. This restriction implies optimal trading

consistently with the price forecast, which we could not reject for 11 out of 48 subjects.

24

(18) where γi = 0, but αi + βi 6= 1. There were no subjects in the LtF treatment

and only 2 in the Mixed treatment, for whom we could not identify a significant

forecasting rule. The average trend coefficients in both treatments are close to

γ¯ ≈ 0.4, and not significantly different in terms of distribution (with p-value of

the rank-sum test equal to 0.736). The difference between the two treatments lies

in the anchor of the forecasting rule. For the LtF treatment the average coefficients

are α¯ = 0.45 and β¯ = 0.56, while in the Mixed treatment these are α¯ = 0.84 and

β¯ = 0.06 (the differences are significant according to the rank-sum test, with both

p-values close to zero). This suggests that subjects in the LtF treatment are more

cautious in revising their expectations, with a gradually evolving anchor that puts

equal weight on past price and their previous forecast. In contrast, in the Mixed

treatments subjects use an anchor that puts almost all weight on the last price

observation and are thus closer to using a pure trend-following rule extrapolating

a trend from the last price observation.

2.5.2 Trading rules in LtO versus Mixed

The LtO and Mixed treatments can be compared by the estimated trading rules.

Recall however, that the trading rule (21a) in the LtO treatment only contains a

past return term (ρt) with coefficient φi, while the trading rule (21b) in the Mixed

treatment contains an additional term for expected excess return (ρet+1) with a

coefficient ζi. In both treatments we find that the rules with a term on past or

expected return is the dominating rule (33 in the LtO and 32 in the Mixed treat-

ment). There are only 12 subjects using a significant AR1 coefficient χi in the LtO

treatment, and 8 in the Mixed treatment. This shows that in both the LtO and

Mixed treatments the majority of subjects tried to extrapolate realized and/or

expected asset returns, which leads to relatively strong trend chasing behaviour.

Nevertheless, there are 11 subjects in the LtO treatment and 8 in the Mixed treat-

ment for whom we can not identify a trading rule within this simple class. The

average demand persistence was χ¯ = 0.07 and χ¯ = 0.006, and the average trend

extrapolation was φ¯ = 0.09 and φ+ ζ = 0.06 in the LtO and Mixed treatment re-

spectively.14 The distributions of the two coefficients are not significantly different

across the treatments according to the rank-sum test, with p-values of 0.425 and

14The trading rules (21a) and (21b) are not directly comparable, since (21b) is a function of

both the past and the expected asset return, and the latter is unobservable in the LtO treatment.

For the sake of comparability, we look at what we interpret as an individual reaction to asset

return dynamics: φi in LtO treatment and φi + ζi in the Mixed treatment. As a robustness

check, we also estimated the simplest trading rule (21a) for both the LtO and Mixed treatments

(ignoring expected asset returns) and found no significant difference between treatments.

25

0.885 for χi and φi/φi+ζi respectively. Hence, based upon individual trading rules

we do not find significant differences between the LtO and Mixed treatments.

2.5.3 Implied trading rules in LtF versus LtO

It is more difficult to compare the LtF and LtO treatments based upon individual

decision rules, since there was no trading in the LtF and no forecasting in the

LtO treatment. We can however use the estimated individual forecasting rules to

obtain the implied optimal trading rules (8) in the LtF treatment and compare

these to the general trading rule (21a) in the LtO treatment. A straightforward

computation shows that for a forecasting rule (18) with coefficients (αi, βi, γi), the

implied optimal trading rule has coefficients χi = βi and φi = (αi + γi − R)/6.15

Hence, for the LtF and LtO treatments we can compare the coefficients for the

adaptive term, i.e. the weight given to the last trade, and the return extrapolation

coefficients. The averages of the first coefficient are β¯ = 0.56 and χ¯ = 0.07 for

the LtF and LtO treatments respectively, and it is significantly higher in the LtF

treatment (rank-sum test p-value close to zero). Moreover the second coefficient,

the implied reaction to the past asset return, is weaker in the LtF treatment

(average implied φ¯ = −0.03) than in the LtO treatment (average φ¯ = 0.09), and

this difference is again significant (rank-sum test p-value close to zero). Hence,

these results on the individual (implied) trading rules show differences between

the LtO and LtF treatments. The LtO treatment is more unstable than the LtF

treatment because subjects are less cautious in the sense that they give less weight

to their previous trade and they give more weight to extrapolating past returns.

We summarise the results on estimated individual behavioural rules as follows:

RESULT 4. Most subjects, regardless of the treatment, follow an anchor and

adjustment rule. In forecasting, LtF subjects were more cautious, using an an-

chor that puts more weight on their previous forecast, while the Mixed treatments

subjects use an anchor with almost all weight on recent prices. In trading, most

subjects extrapolate past returns and/or expected returns. In the LtO subjects give

more weight to past return extrapolation compared to the implied trading behaviour

in the LtF. These individual rules explain more unstable aggregate dynamics in the

LtO and Mixed treatments. We conclude that HYPOTHESIS 5 is rejected.

15The implied trading rule (8) however cannot exactly be rewritten in the form (21a), but

has one additional term pt−1 with coefficient [R(βi + αi + γi − R) − γi]/(aσ2). This coefficient

typically is small however, since γi is small and αi + βi is close to 1. The mean estimated

coefficient over 48 subjects is very close to zero (−0.00229), and with a simple t-test we can not

reject the hypothesis that the mean coefficient is 0 (p-value 0.15).

26

3 Conclusions

The origin of asset price bubbles is an important topic for both researchers and

policy makers. This paper investigates the price dynamics and bubble formation

in an experimental asset pricing market with a price adjustment rule. We find that

the mispricing is largest in the treatment where subjects do both forecasting and

trading, and smallest when subjects only make a prediction. Our result suggests

that price instability is the result of both inaccurate forecasting and imperfect

optimisation. There has been empirical work quantifying forecast biases by house-

holds and finance professionals in real markets, and theoretical works that start to

incorporate the stylized facts into modelling of expectations in macroeconomics.

Our result suggests it may be equally important to collect evidence on failure in

making optimal decisions conditional on one’s own belief by market participants,

and incorporate this behavioural bias into modelling of simple heuristics as an

alternative to perfectly optimal individual decisions.

Which behavioural biases can explain the differences in the individual decisions

and aggregate market outcomes in the learning to forecast and learning to optmise

markets? A first possibility is that the quantity decision task is more cognitive

demanding than the forecasting task, when the subjects in the LtF treatment are

assisted by a computer program. Following Rubinstein (2007), we use decision time

as a proxy for cognitive load and compare the average decision time in each treat-

ment. It turns out that while subjects take significantly longer time in the Mixed

treatment than the other two treatments according to Mann-Whitney-Wilcoxon

test, there is no significant difference between the LtF and LtO treatments. It

helps to explain why the markets are particularly volatile in the Mixed treat-

ment, but does not explain why the LtO treatment is more unstable than the LtF

treatment. Second, in the LtF treatment, the subjects’ goal is to find an accu-

rate forecast. Only the size of the prediction error matters, while the sign does not

matter. Conversely, in a LtO market it is in a way more important for the subjects

to predict the direction of the price movement right, and the size of the prediction

error is important only to a secondary degree. For example, if a subject predicts

the return will be high and decided to buy, he can still make a profit if the price

goes up far more than he expected, and his prediction error is large. Therefore,

the subjects may have a natural tendency to pay more attention to price changes

or follow the “wisdom of crowd”, which leads to assigning more weight to past or

expected returns. Furthermore, for price forecasting past individual behaviour can

be directly compared to observed market prices. If an individual forecasting strat-

egy fits well with observed price behaviour, more weight may be given to past own

27

individual behaviour. In contrast, individual trading decisions cannot be directly

compared or anchored against trading volume or other aggregate information. It

then becomes more natural to evaluate and anchor individual trading by giving

more weight to recent past prices and/or recent past returns.

Asset mispricing and financial bubbles can cause serious market inefficiencies,

and may become a threat to the overall economic stability, as shown by the 2007

financial-economic crisis. Proponents of rational expectations often claim that se-

rious asset bubbles cannot arise, because rational economic agents would efficiently

arbitrage against it and quickly push the ‘irrational’ (non-fundamental) investors

out of the market. Our experiment suggests otherwise: people exhibit heteroge-

neous and not necessarily optimal behaviour. Because they are trend-followers,

their non-fundamental beliefs are correlated. This is reinforced by the positive

feedback between expectations and realised prices in asset markets, as stressed

e.g. in Hommes (2013). Therefore, price oscillations cannot be mitigated by more

rational market investors. As a result, waves of optimism and pessimism can arise

despite the fundamentals being relatively stable.

Our experiment can be extended in several ways. For example, the subjects in

our experiment can short-sell the assets, which may not be feasible in real mar-

kets. An interesting topic for future research is the case where agents face short

selling constraints (Anufriev and Tuinstra, 2013). Another possible extension is

to impose a network structure among the traders, i.e. one trader can only trade

with some, but not all the other traders; or traders need to pay a cost in order to

be connected to other traders. This design can help us to examine the mechanism

of bubble formation in financial networks (Gale and Kariv, 2007), and network

games (Galeotti et al., 2010) in general. There has been a pioneering experimen-

tal literature by Gale and Kariv (2009) and Choi et al. (2014) that study how

network structure influences market efficiency when subjects act as intermediaries

between sellers and buyers. Our experimental setup can be extended to study how

network structure influences market efficiency and stability when subjects act as

traders of financial assets in the over the counter (OTC) market.

University of Amsterdam and Tinbergen Institute

University of Groningen

28

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32

A Experimental Instructions

(Not For Publication)

A.1 LtF treatment

General information

In this experiment you participate in a market. Your role in the market is a profes-

sional Forecaster for a large firm, and the firm is a major trading company of an

asset in the market. In each period the firm asks you to make a prediction of the

market price of the asset. The price should be predicted one period ahead. Based

on your prediction, your firm makes a decision about the quantity of the asset the

firm should buy or sell in this market. Your forecast is the only information the

firm has on the future market price. The more accurate your prediction is, the

better the quality of your firm’s decision will be. You will get a payoff based on

the accuracy of your prediction. You are going to advise the firm for 50 successive

time periods.

About the price determination

The price is determined by the following price adjustment rule: when there is more

demand (firm’s willingness to buy) of the asset, the price goes up; when there is

more supply (firm’s willingness to sell), the price will go down.

There are several large trading companies on this market and each of them is ad-

vised by a forecaster like you. Usually, higher price predictions make a firm to buy

more or sell less, which increases the demand and vice versa. Total demand and

supply is largely determined by the sum of the individual demand of these firms.

About your job

Your only task in this experiment is to predict the market price in each time period

as accurately as possible. Your prediction in period 1 should lie between 0

and 100. At the beginning of the experiment you are asked to give a prediction

for the price in period 1. When all forecasters have submitted their predictions for

the first period, the firms will determine the quantity to demand, and the market

price for period 1 will be determined and made public to all forecasters. Based on

the accuracy of your prediction in period 1, your earnings will be calculated.

Subsequently, you are asked to enter your prediction for period 2. When all par-

ticipants have submitted their prediction and demand decisions for the second

period, the market price for that period, will be made public and your earnings

will be calculated, and so on, for all 50 consecutive periods. The information you

can refer to at period t consists of all past prices, your predictions and earnings.

33

Please note that due to liquidity constraint, your firm can only buy and sell up

to a maximum amount of assets in each period. This means although you can

submit any prediction for period 2 and all periods after period 2, if the price in

last period is pt−1, and you prediction is pet : the firm’s trading decision is con-

strained by pet ∈ [pt−1 − 30, pt−1 + 30]. More precisely, the firm will trade as

if pet = pt−1 +30 if p

e

t > pt−1 +30, and trade as if p

e

t = pt−1−30 if pet < pt−1−30.

About your payoff

Your earnings depend only on the accuracy of your predictions. The earnings

shown on the computer screen will be in terms of points. If your prediction is pet

and the price turns out to be pt in period t, your earnings are determined by the

following equation:

Payoff = max

[

1300− 1300

49

(pet − pt)2 , 0

]

.

The maximum possible points you can earn for each period (if you make no pre-

diction error) is 1300, and the larger your prediction error is, the fewer points you

can make. You will earn 0 points if your prediction error is larger than 7. There is

a Payoff Table on your table, which shows the points you can earn for different

prediction errors.

We will pay you in cash at the end of the experiment based on the points you

earned. You earn 1 euro for each 2600 points you make.

A.2 LtO treatment

General information

In this experiment you participate in a market. Your role in the market is a Trader

of a large firm, and the firm is a major trading company of an asset. In each period

the firm asks you to make a trading decision on the quantity Dt your firm will

BUY to the market. (You can also decide to sell, in that case you just submit a

negative quantity.) You are going to play this role for 50 successive time periods.

The better the quality of your decision is, the better your firm can achieve her

target. The target of your firm is to maximize the expected asset value minus the

variance of the asset value, which is also the measure by the firm concerning your

performance:

(1) pit = Wt − 1

2

V ar (Wt)

2

The total asset value Wt equals the return of the per unit asset multiplied by the

number of unit you buy Dt. The return of the asset is pt + y − Rpt−1, where

34

R is the gross interest rate which equals 1.05, pt is the asset price at period t,

therefore pt − Rpt−1 is the capital gain of the asset, and y = 3.3 is the dividend

paid by the asset. We assume the variance of the price of a unit of the asset is

σ2 = 6, therefore the expected variance of the asset value is 6D2t . Therefore we

can rewrite the performance measure in the following way

(2) pit = (pt + y −Rpt−1)Dt − 3D2t

The asset price in the next period pt is not observable in the current period. You

can make a forecast pet on it. There is an asset return calculator in the

experimental interface that gives the asset return for each price forecast pet you

make. Your own payoff is a function of the value of target function of the firm:

(3) Payofft = 800 + 40 ∗ pit

This function means you get 800 points (experimental currency) as basic salary,

and 40 points for each 1 unit of performance (target function of the firm) you

make. If your trades will be unsuccessful, you may lose points and earn less than

your basic salary, down to 0. Based on the asset return, you can look up your

payoff for each quantity decision you make in the payoff table.

You can of course also calculate your payoff for each given forecast and quantity

using equation (2) and (3) directly. In that situation you can ask us for a calculator.

About the price determination

The price is determined by the following price adjustment rule: when there is more

demand than supply of the asset (namely, more traders want to buy), the price

will go up; and when there is more supply than demand of the asset (namely, more

people want to sell), the price will go down.

About your job

Your only task in this experiment is to decide the quantity the firm will buy/sell.

At the beginning of period 1 you determine the quantity to buy or sell (submitting

a positive number means you want to buy, and negative number means you want

to sell) for period 1. After all traders submit their quantity decisions, the market

price for period 1 will be determined and made public to all traders. Based on

the value of the target function of your firm in period 1, your earnings in the first

period will be calculated.

Subsequently, you make trading decisions for the second period, the market price

for that period will be made public and your earnings will be calculated, and so

on, for all 50 consecutive periods. The information you can refer to at period t

35

consists of all previous prices, your quantity decisions and earnings.

Please notice that due to the liquidity constraint of the firm, the amount of asset

you buy or sell cannot be more than 5 units. Which means you quantity decision

should be between −5 and 5. The numbers on the payoff table are just examples.

You can use any other number such as 0.01, −1.3, 2.15 etc., as long as they are

within [−5, 5]. if When you want to submit numbers with a decimal point, please

write a “.”, NOT a “,”.

About your payoff

In each period you are paid according to equation (3). The earnings shown on the

computer screen will be in terms of points. We will pay you in cash at the end

of the experiment based on the points you earned. You earn 1 euro for each 2600

points you make.

A.3 Mixed treatment

General information

In this experiment you participate in a market. Your role in the market is a Trader

of a large firm, and the firm is a major trading company of an asset. In each period

the firm asks you to make a trading decision on the quantity Dt your firm will

BUY to the market. (You can also decide to sell, in that case you just submit a

negative quantity.) You are going to play this role for 50 successive time periods.

The better the quality of your decision is, the better your firm can achieve her

target. The target of your firm is to maximize the expected asset value minus the

variance of the asset value, which is also the measure by the firm concerning your

performance:

(1) pit = Wt − 1

2

V ar (Wt)

2

The total asset value Wt equals the return of the per unit asset multiplied by the

number of unit you buy Dt. The return of the asset is pt + y − Rpt−1, where

R is the gross interest rate which equals 1.05, pt is the asset price at period t,

therefore pt − Rpt−1 is the capital gain of the asset, and y = 3.3 is the dividend

paid by the asset. We assume the variance of the price of a unit of the asset is

σ2 = 6, therefore the expected variance of the asset value is 6D2t . Therefore we

can rewrite the performance measure in the following way

(2) pit = (pt + y −Rpt−1)Dt − 3D2t

The asset price in the next period pt is not observable in the current period. You

can make a forecast pet on it. There is an asset return calculator in the

36

experimental interface that gives the asset return for each price forecast pet you

make. Your own payoff is a function of the value of target function of the firm:

(3) Payofft = 800 + 40 ∗ pit

This function means you get 800 points (experimental currency) as basic salary,

and 40 points for each 1 unit of performance (target function of the firm) you

make. If your trades will be unsuccessful, you may lose points and earn less than

your basic salary, down to 0. Based on the asset return, you can look up your

payoff for each quantity decision you make in the payoff table.

You can of course also calculate your payoff for each given forecast and quantity

using equation (2) and (3) directly. In that situation you can ask us for a calcu-

lator.

The payoff for the forecasting task is simply a decreasing function of your fore-

casting error (the distance between your forecast and the realized price). When

your forecasting error is larger than 7, you earn 0 points.

(4) Payoffforecasting = max

[

1300− 1300

49

(pet − pt)2 , 0

]

About the price determination

The price is determined by the following price adjustment rule: when there is more

demand than supply of the asset (namely, more traders want to buy), the price

will go up; and when there is more supply than demand of the asset (namely, more

people want to sell), the price will go down.

About your job

Your task in this experiment consists of two parts: (1) to make a price forecast;

(2) to decide the quantity the firm will buy/sell. At the beginning of period

1 you submit your price forecast between 0 and 100, and then determine

the quantity to buy or sell (submitting a positive number means you want to buy,

and negative number means you want to sell) for period 1, and the market price

for period 1 will be determined and made public to all traders. Based on your

forecasting error and performance measure for the trading task, in period 1, your

earnings in the first period will be calculated.

Subsequently, you make forecasting and trading decisions for the second period,

the market price for that period will be made public and your earnings will be

calculated, and so on, for all 50 consecutive periods. The information you can

refer to at period t consists of all previous prices, your past forecasts, quantity

decisions and earnings.

37

Please notice that due to the liquidity constraint of the firm, the amount of asset

you buy or sell cannot be more than 5 units. Which means you quantity decision

should always be between −5 and 5. The numbers on the payoff table are just

examples. You can use any other numbers such as 0.01, −1.3, 2.15 etc. as long

as they are within [−5, 5].

About your payoff

In each period you are paid for the forecasting task according to equation (4)

and trading task according to equation (3). The earnings shown on the computer

screen will be in terms of points. We will pay you in cash at the end of the experi-

ment based on the points you earned for either the forecasting task or the trading

task. Which task will be paid will be determined randomly (we will invite one of

the participants to toss a coin). That is, depending on the coin toss, your

earnings will be calculated either based on equation (3) or equation (4).

You earn 1 euro for each 2600 points you make.

38

B Payoff Tables

(Not For Publication)

Table B.1: Payoff Table for Forecasting Task

Payoff Table for Forecasting Task

Your Payoff=max[1300− 1300

49

(Your Prediction Error)2, 0]

3000 points equal 1 euro

error points error points error points error points

0 1300 1.85 1209 3.7 937 5.55 483

0.05 1300 1.9 1204 3.75 927 5.6 468

0.1 1300 1.95 1199 3.8 917 5.65 453

0.15 1299 2 1194 3.85 907 5.7 438

0.2 1299 2.05 1189 3.9 896 5.75 423

0.25 1298 2.1 1183 3.95 886 5.8 408

0.3 1298 2.15 1177 4 876 5.85 392

0.35 1297 2.2 1172 4.05 865 5.9 376

0.4 1296 2.25 1166 4.1 854 5.95 361

0.45 1295 2.3 1160 4.15 843 6 345

0.5 1293 2.35 1153 4.2 832 6.05 329

0.55 1292 2.4 1147 4.25 821 6.1 313

0.6 1290 2.45 1141 4.3 809 6.15 297

0.65 1289 2.5 1134 4.35 798 6.2 280

0.7 1287 2.55 1127 4.4 786 6.25 264

0.75 1285 2.6 1121 4.45 775 6.3 247

0.8 1283 2.65 1114 4.5 763 6.35 230

0.85 1281 2.7 1107 4.55 751 6.4 213

0.9 1279 2.75 1099 4.6 739 6.45 196

0.95 1276 2.8 1092 4.65 726 6.5 179

1 1273 2.85 1085 4.7 714 6.55 162

1.05 1271 2.9 1077 4.75 701 6.6 144

1.1 1268 2.95 1069 4.8 689 6.65 127

1.15 1265 3 1061 4.85 676 6.7 109

1.2 1262 3.05 1053 4.9 663 6.75 91

1.25 1259 3.1 1045 4.95 650 6.8 73

1.3 1255 3.15 1037 5 637 6.85 55

1.35 1252 3.2 1028 5.05 623 6.9 37

1.4 1248 3.25 1020 5.1 610 6.95 19

1.45 1244 3.3 1011 5.15 596 error ≥ 0

1.5 1240 3.35 1002 5.2 583

1.55 1236 3.4 993 5.25 569

1.6 1232 3.45 984 5.3 555

1.65 1228 3.5 975 5.35 541

1.7 1223 3.55 966 5.4 526

1.75 1219 3.6 956 5.45 512

1.8 1214 3.65 947 5.5 497

39

Table B.2: Payoff Table for Trading Task

Y

o

u

r

p

ro

fi

t

A

ss

e

t

q

u

a

n

ti

ty

:

p

o

si

ti

v

e

n

u

m

b

e

r

m

e

a

n

s

to

b

u

y,

n

e

g

a

ti

v

e

to

se

ll

−5

−4

.5

−4

−3

.5

−3

−2

.5

−2

−1

.5

−1

−0

.5

0

0

.5

1

1

.5

2

2

.5

3

3

.5

4

4

.5

5

A s s e t r e t u r n

−1

5

80

0

10

70

12

80

14

30

15

20

15

50

15

20

1

4

3

0

1

2

8

0

1

0

7

0

8

0

0

4

7

0

8

0

0

0

0

0

0

0

0

0

−1

4

60

0

89

0

11

20

12

90

14

00

14

50

14

40

1

3

7

0

1

2

4

0

1

0

5

0

8

0

0

4

9

0

1

2

0

0

0

0

0

0

0

0

0

−1

3

40

0

71

0

96

0

11

50

12

80

13

50

13

60

1

3

1

0

1

2

0

0

1

0

3

0

8

0

0

5

1

0

1

6

0

0

0

0

0

0

0

0

0

−1

2

20

0

53

0

80

0

10

10

11

60

12

50

12

80

1

2

5

0

1

1

6

0

1

0

1

0

8

0

0

5

3

0

2

0

0

0

0

0

0

0

0

0

0

−1

1

0

35

0

64

0

87

0

10

40

11

50

12

00

1

1

9

0

1

1

2

0

9

9

0

8

0

0

5

5

0

2

4

0

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1.

C Rational Strategic Behaviour

(Not For Publication)

Our experimental results are clearly different from the predictions of the rational

expectation equilibrium (REE). However, we also found that subjects typically

earn high payoffs, implying some sort of profit seeking behaviour.

In this appendix, we discuss whether rational strategic behaviour can explain

our experimental results, in particular in the LtO and Mixed treatments. Based on

the assumption on the subjects’ perception of the game and information structure,

three cases are discussed: (1) agents are price takers; (2) agents know their market

power and coordinate on monopolistic behaviour; (3) agents know their market

power but play non-cooperatively. We show that under price-taking behaviour, the

LtF and LtO treatments are equivalent. If the subjects behave strategically or try

to collude, the economy can have alternative equilibria, where the subjects collec-

tively ‘ride a bubble’, or jump around the fundamental price. Nevertheless, these

rational equilibria predict different outcomes than the individual and aggregate

behaviour observed in the experiment.

Without loss of generality, we focus on the one-shot game version of the ex-

perimental market to derive our results. More precisely, we look at the optimal

quantity decisions zi,t that the agents in period t (knowing prices and individual

traded quantity until and including period t) have to formulate to maximise the

expected utlity in period t + 1. This is supported by two observations. First,

by definition agents are myopic and their payoff in t + 1 depends only on the

realised profit from that period, and not on the stream of future profits from pe-

riod t+ 2 onward. Second, the experiment is a repeated game with finitely many

repetitions, and subjects knew it would end after 50 periods. Using the standard

backward induction reasoning, one can easily show that a sequence of one-period

game equilibria forms a rational equilibrium of the finitely repeated game as well.

C.1 Price takers

Realised utility of investors in the LtO treatment is given by (4) and is equivalent

to the following form:

Ui,t(zi,t) = zi,t (pt+1 + y −Rpt)− aσ

2

2

z2i,t, (C.1)

where zi,t is the traded quantity and Ui,t is a quadratic function of the traded

quantity. As discussed in Section 2, assuming the agent is a price taker, the

41

optimal traded quantity conditional on the expected price pei,t+1 is given by

z∗i,t = arg max

zi,t

Ui,t =

pei,t+1 + y −Rpt

aσ2

. (C.2)

Note that this result relies on the assumption that the subjects do not know

the price generating function. We argue that the subjects also have an incentive

to minimise their forecasting error when they choose the quantity and are paid

according to the risk adjusted profit. To see that, suppose that the realised market

price in the next period is pt+1, and the subject makes a prediction error of , i.e.

her prediction is pei,t+1 = pt+1 + . The payoff function can be rewritten as:

Ui,t(zi,t) = zi,t (pt+1 + y −Rpt)− aσ

2

2

z2i,t

=

(pt+1 + + y −Rpt)(pt+1 + y −Rpt)

aσ2

− (pt+1 + + y −Rpt)

2

2aσ2

=

(pt+1 + y −Rpt)2

2aσ2

−

2

2aσ2

. (C.3)

This shows that utility is maximised when = 0, namely, when all subjects have

correct belief. Assuming perfect rationality and price taking behaviour (perfect

competition), the task of finding the optimal trade coincides with the task of

minimizing the forecast error. Subjects have incentives to search for and play the

REE also when they choose the quantity. We summarise this finding below.

FINDING 1. When the subjects act as price takers, the utility function in the

Learning to Optimise treatment is a quadratic function of the prediction error, the

same (up to a monotonic transformation) as in the Learning to Forecast treat-

ment. The subjects’ payoff is maximised when they play the Rational Expectation

Equilibrium regardless of the design: the REE of LtF and LtO treatments are

equivalent.

C.2 Collusive outcome

Consider now the case when agents realise how their predictions/trading quanti-

ties influence the price and are able to coordinate on a common strategy. This

resembles a collusive (oligopoly) market, e.g. similar to a cobweb economy in which

the sellers can coordinate their production.

In the collusive case, all agents behave as a monopoly that maximises joint

(unweighted) utility; thus the solution is symmetric, that is for each agent i,

zi,t = zt. In our experiment the price determination function is:

pt+1 = pt + 6λzt, (C.4)

42

and so the monopoly maximises

Ut =

6∑

i=1

Ui,t(zt) = 6

[

zt(pt+1 + y −Rpt)− aσ

2

2

z2t

]

= 6

[

z2t

(

6λ− aσ

2

2

)

+ zt(y − rpt)

]

. (C.5)

Here we assume that a rational agent has perfect knowledge about the pricing

function (C.4). Notice that when λ = 20/21, aσ2 = 6, as in the experiment, the

coefficient before z2t is positive, 6λ− aσ

2

2

= 19

7

> 0, and thus the profit function is U

shaped, instead of inversely U shaped.16 This means that a finite global maximum

does not exist (utility goes to +∞ when zt goes to either +∞ or −∞). The global

minimum is obtained when zi,t =

7

38

(rpt − y) = 7r38(pt − pf ).

In our experiment, the subjects are constrainted to choose a quantity from

[−5,+5] and the price is bound to the interval [0, 300]. Collusive equilibrium

in the one-shot game implies that the subjects coordinate on zi,t = 5 or zi,t =

−5, depending on which is further away from 7(rpt−y)

38

(as (C.5) is a symmetric

parabola). Since 7(rpt−y)

38

> 0 when the price is above the fundamental (pf = y/r),

we can see that the agents coordinate on −5 if the price is higher than the REE

(pt > y/r). Similarly, rational agents coordinate on +5 if the price is lower than

the REE (pt < y/r). If the price is exactly at the fundamental, rational agents

are indifferent between −5 and 5. Notice that in such a case trading the REE

quantity (zi,t = 0) gives the global minimum for the monopoly.

As a consequence, the collusive outcome predicts that the subjects will ‘jump

up and down’ around the fundamental. When the initial price is below (above)

the fundamental, the monopoly will buy (sell) the asset until the price overshoots

(undershoots) the fundamental, and so forth. Then the subjects start to ‘jump up

and down’ as described before.

FINDING 2. When the subjects know the price determination function and are

able to form a coalition, the collusive profit function in the LtO treatment is U

shaped. Subjects would buy under-priced and sell an over-priced asset. In the long

run rational collusive subjects will alternate their trading quantities between −5

and 5 and so the price will alternate around the equilibrium.

16If 6λ− aσ22 < 0, this objective function is inversely U shaped. The maximum point is achieved

when zi,t =

rpt−y

12λ−aσ2 . This means when pt = y/r, namely when the price is at the REE, the

optimal quantity under collusive equilibrium is still 0. When the price is higher or lower than

the REE, the optimal quantity increases with the difference between the price and the REE.

This means there is a continuum of equilibria when the economy does not start at the REE.

43

Such alternating dynamics would resemble coordination on contrarian type of

behaviour, but has not been observed in any of the experimental groups. In-

stead, our subjects coordinated on trend-following trading rules, which resulted

in smooth, gradual price oscillations. Moreover, quantity decisions equal to 5 or

−5 happened rarely in the experiment (7 times in the LtO and 44 times in the

Mixed treatment). Typical subject behaviour was much more conservative: 97%

and 91% traded quantities in the LtO and Mixed treatments respectively were

confined in the interval [−2.5, 2.5].

C.3 Perfect information non-cooperative game

Consider a scenario, in which the subjects realise the experimental price deter-

mination mechanism, but cannot coordinate their actions and play a symmetric

Nash equilibrium (NE) instead of the collusive one. There is a positive external-

ity of the subjects’ decisions: when one subject buys the asset, it pushes up the

price and also the benefits of all the other subjects. The collusive equilibrium

internalises this externality, while the non-cooperative NE does not. What will

rational subjects do in this situation?

In the case of a non-cooperative one-shot game, we again focus on a symmetric

solution. Consider agent i, who optimises her quantity choice believing that all

other agents will choose zot . This means that the price at t+ 1 becomes

pt+1 = pt + 5λz

o

t + λzi,t. (C.6)

Agent i maximises therefore

Ui,t = zi,t (λzi,t + 5λz

o

t + y − rpt)−

aσ2

2

z2i,t

= z2it

2λ− aσ2

2

+ zi,t(5λz

o

t + y − rpt). (C.7)

Notice that 2λ− aσ2 = −86/21 < 0. This is an inversely U shaped parabola with

the unique maximum given by the best response function

z∗i,t(z

o

t ) =

5λzot + y − rpt

aσ2 − 2λ . (C.8)

A symmetric solution requires z∗i,t(z

o

t ) = z

o

t , which implies

z∗t =

rpt − y

7λ− aσ2 =

3

2

(rpt − y). (C.9)

Furthermore the reaction function z∗i,t(z

o

t ) is linear with respect to z

o

t , with a slope

5λ

aσ2−2λ =

100

86

> 1. Thus, zot > z

∗

t (< and =) implies z

∗

i,t > z

o

t (< and =), or in

44

words, if agent i believes that the other players will buy (sell) the asset, she has an

incentive to buy (sell) even more. Then as a best response, the other agents should

further increase/decrease their demand, and this is limited only by the liquidity

constraints. The strategy (C.9) thus defines the threshold point between the two

corner strategies, i.e. the full NE best response strategy is defined as

zNEi,t =

5 if zot > z

∗

t

z∗t if z

o

t = z

∗

t

−5 if zot < z∗t .

(C.10)

The boundary strategies can be infeasible if the previous price is too close to zero

or 300.17 To sum up, as long as the price pt is sufficiently far from the edges of the

allowed interval [0, 300], there are three NE of the one-shot non-cooperative game,

which are defined as fixed points of (C.10), namely all players playing zi,t = −5,

zi,t = z

∗

t and zi,t = +5 for all i ∈ {1, . . . , 6}.

A simple interpretation is that, given the parametrization, our model is an

example of a (Nash) coordination game. As long as 5λ

aσ2−2λ > 1, the best response

(C.8) is to amplify the average trade of the other players. This is not a surprising

result, as it merely exhibits the strength of the positive feedback present in this

market18.

If the agents coordinate on the strategy zi,t = z

∗

t , the price evolves according

to the following law of motion:

pt+1 =

10pt − 60y

7

. (C.11)

In contrast to the collusive game, in the non-cooperative game the fundamental

price is a possible steady state, but only if it is an outcome in the initial period.

Additional equilibrium refinements may further exclude it as a rational outcome,

since it is the least profitable one. Recall that the subjects earn 0 when they play

z∗t with price at the fundamental (because there is no trade). On the other hand,

they may earn a positive profit by coordinating on −5 or 5. For example, when

all of them buy 5 units of asset, the utility for each of them will be (pt−1 + y +

6λzi,t − (1 + r)pt−1)zi,t − ασ22 z2i,t = (33.3 − 0.05pt−1) ∗ 5 − 75. This equals 76.5

when pt−1 = 60, 16.5 when pt−1 = 300 and 75 when the previous price is equal

to the fundamental, pt−1 = 66. This explains why the payoff efficiency (average

17Notice that we can interpret zot as the average quantity traded by all other agents, besides

agent i, and the reasoning for NE strategy (C.10) remains intact. This implies that NE has to

be symmetric.

18In practice, such an equilibrium could not be sustained in the long run, since then the market

maker would incur accumulating losses every period.

45

experimental payoff divided by payoff under REE) is larger than 100% in some

markets in the LtO or Mixed treatments where prices have large oscillations.

Notice that the linear equation (C.11) is unstable, so the NE of the one-shot

game leads to unstable price dynamics in the repeated game even if the agents

coordinate on zi,t = z

∗

t , as long as the initial price is different from the fundamental

price. Indeed, if the initial price is 67 or 65 (fundamental price plus or minus

one), the price will go to the upper cap of 300 or the lower cap of 0 respectively.

Furthermore the agents can switch at any moment between the three one-shot

game NE defined by (C.10). This implies that in the repeated non-cooperative

game, many rational price paths are possible. This includes many price paths

where agents coordinate on 5 or −5, including the alternating collusive equilibrium

discussed in the last section.

FINDING 3. In the non-cooperative game with perfect information, there are

two possible types of NE. The fundamental outcome is a possible outcome only

if the initial price is equal to the fundamental price. Otherwise, the agents will

coordinate on unstable, possibly oscillatory price dynamics, with traded quantities

of −5 or 5. When they coordinate on a non-zero quantity, their payoff can be

higher than their payoff under the REE under the price-taking beliefs.

C.4 Summary

To conclude, the perfectly rational agents can coordinate on price boom-bust cycles

and earn positive profit19. However, this would require even stronger assumptions

than the fundamental equilibrium, namely that the agents perfectly understand

the underlying price determination mechanism.

Furthermore, such rational equilibria with price oscillations predict that the

subjects coordinate on homogeneous trades at the edge of the liquidity constraints.

The subjects from the LtO and Mixed treatments behaved differently. Their traded

quantities were highly heterogeneous, and rarely reached the liquidity constraints.

Therefore, the alternative rational equilibrium from the perfect information,

non-cooperative games provide some useful insights on why subjects “ride the

bubbles” in the LtO and Mixed treatment. However, since the rational solution

19Note that the subjects earn more in collusive and non-cooperative Nash setting because we

pay them according to the book value of the asset, and the taˆtonnement process ensures the

price movement is relatively smooth. In real life, people may not be able to realize the full book

value of their asset holdings because the asset price will fall when a large fraction of them start

to sell, and without the marker maker in the taˆtonnements process absorbing all these losses,

they may suffer huge losses when the asset price declines sharply.

46

cannot explain the heterogeneity of the individual decisions and non-boundary

trading quantities, the mispricing in the experimental data is more likely a result

of the joint forces of rational (profit seeking) and boundedly rational behaviour

with some coordination on trend-following buy and hold and short sell strategies.

D Earnings Ratios

(Not For Publication)

Table D.1: Earnings Efficiency

Treatment LtF LtO Mixed Forecasting Mixed Trading

Market 1 96.35% 102.54% 87.62% 100.89%

Market 2 94.47% 95.25% 67.27% 87.33%

Market 3 96.03% 98.21% 75.63% 79.61%

Market 4 96.18% 100.43% 77.41% 114.63%

Market 5 95.15% 97.39% 87.07% 99.03%

Market 6 94.06% 99.64% 91.94% 97.24%

Market 7 96.18% 98.58% 81.20% 94.55%

Market 8 96.54% 98.41% 60.80% 132.01%

Average 95.62% 98.81% 78.62% 100.66%

Notes. Earnings efficiency for each market. The efficiency is defined as the average

experimental payoff divided by the payoff under REE, which is 26.67 euro for the fore-

casting task, and 18.33 euro for the trading task.

47

E Estimation Of Individual Forecasting Rules

(Not For Publication)

Rule coefficients

Subject cons. Past price AR(1) Past trend R2 Type

Group 1

1 0.288 0.756 0.680 0.995

2 −1.952 1.090 0.448 0.996

3 1.000 0.744 0.734 TRE

4 −1.349 0.982 0.427 0.998

5 −2.080 0.307 0.725 0.362 0.997

6 1.000 0.770 0.648 TRE

Group 2

1 1.014 0.998

2 0.626 0.347 0.519 0.998

3 −2.110 0.346 0.697 0.996

4 1.013 0.992

5 1.013 0.997

6 −1.857 0.475 0.561 0.391 0.996

Group 3

1 0.463 0.522 0.707 0.993

2 0.513 0.495 0.655 0.994

3 0.476 0.660 0.395 0.993

4 1.000 0.302 0.310 TRE

5 1.000 0.364 0.390 TRE

6 0.471 0.544 0.579 0.998

Group 4

1 0.596 0.568 0.482 0.988

2 1.000 0.679 0.320 TRE

3 1.000 0.161 0.025 TRE

4 −2.553 0.418 0.621 0.405 0.992

5 0.389 0.608 0.539 0.996

6 1.000 0.341 0.385 TRE

Table E.1: Estimated individual rules for the LtF treatment.

48

Rule coefficients

Subject cons. Past price AR(1) Past trend R2 Type

Group 5

1 0.260 0.715 0.729 0.990

2 1.021 0.895

3 −53.068 −0.369 2.125 −1.591 0.655

4 0.178 0.902 0.836 0.980

5 0.452 0.587 0.791 0.993

6 0.281 0.719 1.245 0.985

Group 6

1 0.993 0.880

2 1.000 0.921 0.507 TRE

3 1.000 0.712 0.761 TRE

4 1.000 0.827 0.804 TRE

5 0.452 0.411 0.977 0.986

6 1.000 0.804 0.809 TRE

Group 7

1 6.914 0.902 0.910

2 1.010 0.998

3 0.926 0.924

4 0.359 0.590 0.399 0.966

5 0.990 0.973

6 0.308 0.536 0.545 0.960

Group 8

1 1.000 0.451 0.293 TRE

2 1.000 0.370 0.502 TRE

3 2.778 0.822 0.470 0.984

4 7.958 0.884 0.783 0.911

5 0.316 0.701 0.471 0.992

6 1.000 0.342 0.081 TRE

Table E.1: (continued) Estimated individual rules for the LtF treatment.

49

Rule coefficients R2 rule stability

Subject cons. AR(1) past return

Group 1

1 −0.447 0.203 0.904 mixed S

2 0.175 0.819 return U

3 0.167 0.804 return U

4 0.111 0.856 return S

5 −0.125 0.168 0.833 return U

6 0.159 0.854 return S

Group 2

1 0.0451 random S

2 0.168 random S

3 0.00997 random S

4 0.106 random S

5 0.478 −0.0473 0.24 mixed U

6 0.0473 random S

Group 3

1 −0.188 −0.291 0.221 0.836 mixed U

2 0.16 0.272 return S

3 −0.26 0.16 0.645 return S

4 0.0781 0.124 return S

5 0.283 0.105 0.676 mixed S

6 0.152 0.879 return S

Group 4

1 0.811 0.677 AR(1) N

2 0.174 0.549 return U

3 0.113 0.69 return S

4 0.14 0.824 return S

5 0.174 0.798 return U

6 0.119 0.346 return S

Table E.2: Estimated individual rules for the LtO treatment.

50

Rule coefficients R2 rule stability

Subject cons. AR(1) past return

Group 5

1 0.0975 random S

2 0.0695 random S

3 0.579 0.333 AR(1) N

4 0.00356 random S

5 0.0238 random S

6 0.0487 0.183 return S

Group 6

1 0.0496 random S

2 0.135 0.588 return S

3 0.125 0.854 return S

4 0.566 0.663 AR(1) N

5 0.108 0.468 return S

6 0.148 0.595 return S

Group 7

1 0.29 0.0795 0.741 mixed S

2 0.743 0.551 AR(1) N

3 −0.3 0.177 0.759 mixed S

4 0.44 0.0893 0.675 mixed S

5 0.136 0.269 0.0521 0.59 mixed S

6 0.156 0.884 return S

Group 8

1 0.2 0.258 return U

2 0.118 0.439 return S

3 0.118 0.207 0.757 return U

4 0.0522 0.0482 0.546 return S

5 0.131 random S

6 0.143 0.703 return S

Table E.2: (continued) Estimated individual rules for the LtO treatment.

51

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