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无代写-ECON3124

时间：2021-04-21

Behavioural Economics ECON3124

Problem Set 4

Due 6pm (AEST), 27 April 2021

1. Consider the following game based on the Trust Game (which is a commonly used game

to measure the level of trust in a society). An “investor” (player 1) can decide whether to

send some portion of $10 to an “entrepreneur” (player 2). If she keeps all of the money,

the game ends with the investor getting $10 and the entrepreneur getting $0. If the

investor sends any money, it gets tripled (e.g., sending $10 gets tripled to $30), and the

entrepreneur decides whether to keep all the money to herself, or to send some amount

back to the investor.

(a) [2pts] What is the equilibrium of this game if both players are purely self-interested?

(b) [20pts] Suppose the two players have distributional preferences as introduced in class: if

x1 and x2 are the amounts of money players 1 and 2 end up with, respectively, then player

1's utility is

1(1, 2) = {

1

2

1 +

1

2

2, 1 ≤ 2

2

3

1 +

1

3

2, 1 ≥ 2

while player 2’s utility is

2(1, 2) = {

2

3

1 +

1

3

2, 2 ≥ 1

2, 2 ≤ 1

The players know each others’ utilities.

Show that a Nash equilibrium of this game is for the investor to send all $10 and the entrepreneur to

split the $30 equally.

(c) [5pts] Going back to looking only at the payoffs. Argue informally but carefully that the

above is NOT a fairness equilibrium in Rabin's intentions-based model of social preferences.

(d) Researchers find that entrepreneurs are often more generous in the Trust Game (if

investors send all $10) than in a dictator game in which they are asked to split $30. Show

whether this finding can be explained by (i) [3pts] a distributional model of social

preferences; and (ii) [10pts] Rabin's model of intentions-based preferences.

2. A large population of players are playing the following version of the beauty-contest

game. Each player guesses a number between 100 and 200 (inclusive), and the player

closest to p < 1 of the average wins a given prize. If multiple players are closest to p < 1

times the average, the prize is allocated randomly between them.

(a) [5pts] What guess(es) survive iterated elimination of dominated strategies?

From now on, suppose that a fraction 1 − of the players is rational, and a fraction of the

players is irrational. It is known that irrational players like high numbers, so they always

guess 200.

(b) [10pts] Solve for the rationals' equilibrium guess as a function of and .

(c) [5pts] What happens to the rationals' guess as approaches 1 from below? Explain the

intuition.

(d) [5pts] What happens to the rationals' guess as approaches 1? Explain the intuition.

3. Suppose that when allocating money between two random people, Xiaoyu is a rational

“expected-Rawlsian” in the sense that she will choose the allocation that maximizes the

expected value of min(1, 2), where 1 and 2 are the earnings that day for person 1

and person 2, respectively. For parts (a)-(d) below, suppose that the two people are not

going to get any more money that day than through the ways stated in the question.

(a) [5pts] Suppose Xiaoyu must make a decision of whether to give each of the two people

$4, or to give one of them $20 and the other $0, with the “winner” selected randomly with

equal probability. What will Xiaoyu choose?

(b) [10pts] Now suppose that Xiaoyu must make the above decision twice in a row. If she

chooses the $20/$0 allocation twice, the winners are selected independently (that is, a coin

is flipped on both occasions to determine who receives $20). What will Xiaoyu do now?

(c) [10pts] Now suppose again that Xiaoyu must make the above decision once, but she

knows that one of the two people has already received $10 that day, while the other one

has received nothing. What will she do?

(d) [5pts] Finally, suppose that Xiaoyu is not so rational, but instead “narrowly brackets” her

decision, for each choice acting as if that choice was going to determine the day's total

change in wealth for the two people. What are the answers for parts (b) and (c) in this case?

Explain the intuition for any differences between your answers to parts (b) and (c) and to

part (d).

(e) [5pts] Based on your answers to the previous parts, argue that it is often impossible to

infer a person's social preferences (e.g. whether she dislikes or does not mind inequality)

from her choices without knowing whether she brackets her choices narrowly.

学霸联盟

Problem Set 4

Due 6pm (AEST), 27 April 2021

1. Consider the following game based on the Trust Game (which is a commonly used game

to measure the level of trust in a society). An “investor” (player 1) can decide whether to

send some portion of $10 to an “entrepreneur” (player 2). If she keeps all of the money,

the game ends with the investor getting $10 and the entrepreneur getting $0. If the

investor sends any money, it gets tripled (e.g., sending $10 gets tripled to $30), and the

entrepreneur decides whether to keep all the money to herself, or to send some amount

back to the investor.

(a) [2pts] What is the equilibrium of this game if both players are purely self-interested?

(b) [20pts] Suppose the two players have distributional preferences as introduced in class: if

x1 and x2 are the amounts of money players 1 and 2 end up with, respectively, then player

1's utility is

1(1, 2) = {

1

2

1 +

1

2

2, 1 ≤ 2

2

3

1 +

1

3

2, 1 ≥ 2

while player 2’s utility is

2(1, 2) = {

2

3

1 +

1

3

2, 2 ≥ 1

2, 2 ≤ 1

The players know each others’ utilities.

Show that a Nash equilibrium of this game is for the investor to send all $10 and the entrepreneur to

split the $30 equally.

(c) [5pts] Going back to looking only at the payoffs. Argue informally but carefully that the

above is NOT a fairness equilibrium in Rabin's intentions-based model of social preferences.

(d) Researchers find that entrepreneurs are often more generous in the Trust Game (if

investors send all $10) than in a dictator game in which they are asked to split $30. Show

whether this finding can be explained by (i) [3pts] a distributional model of social

preferences; and (ii) [10pts] Rabin's model of intentions-based preferences.

2. A large population of players are playing the following version of the beauty-contest

game. Each player guesses a number between 100 and 200 (inclusive), and the player

closest to p < 1 of the average wins a given prize. If multiple players are closest to p < 1

times the average, the prize is allocated randomly between them.

(a) [5pts] What guess(es) survive iterated elimination of dominated strategies?

From now on, suppose that a fraction 1 − of the players is rational, and a fraction of the

players is irrational. It is known that irrational players like high numbers, so they always

guess 200.

(b) [10pts] Solve for the rationals' equilibrium guess as a function of and .

(c) [5pts] What happens to the rationals' guess as approaches 1 from below? Explain the

intuition.

(d) [5pts] What happens to the rationals' guess as approaches 1? Explain the intuition.

3. Suppose that when allocating money between two random people, Xiaoyu is a rational

“expected-Rawlsian” in the sense that she will choose the allocation that maximizes the

expected value of min(1, 2), where 1 and 2 are the earnings that day for person 1

and person 2, respectively. For parts (a)-(d) below, suppose that the two people are not

going to get any more money that day than through the ways stated in the question.

(a) [5pts] Suppose Xiaoyu must make a decision of whether to give each of the two people

$4, or to give one of them $20 and the other $0, with the “winner” selected randomly with

equal probability. What will Xiaoyu choose?

(b) [10pts] Now suppose that Xiaoyu must make the above decision twice in a row. If she

chooses the $20/$0 allocation twice, the winners are selected independently (that is, a coin

is flipped on both occasions to determine who receives $20). What will Xiaoyu do now?

(c) [10pts] Now suppose again that Xiaoyu must make the above decision once, but she

knows that one of the two people has already received $10 that day, while the other one

has received nothing. What will she do?

(d) [5pts] Finally, suppose that Xiaoyu is not so rational, but instead “narrowly brackets” her

decision, for each choice acting as if that choice was going to determine the day's total

change in wealth for the two people. What are the answers for parts (b) and (c) in this case?

Explain the intuition for any differences between your answers to parts (b) and (c) and to

part (d).

(e) [5pts] Based on your answers to the previous parts, argue that it is often impossible to

infer a person's social preferences (e.g. whether she dislikes or does not mind inequality)

from her choices without knowing whether she brackets her choices narrowly.

学霸联盟