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.

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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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3
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8
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D
A
N
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4
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4
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7
S
† 4
3
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6
7
3
.6
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6
−0
.4
2
1
1
.3
8
1
0
.9
9
4
S
† 4
4
0
.1
6
7
1
.0
0
0
0
.6
6
9
T
R
E
S
4
5
0
.8
7
0
0
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1
2
1
.0
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0
1
.0
2
2
T
R
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4
6
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52
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53




































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































































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