1. Consider a variant of the ultimatum game we studied in class in which players have fairness considerations. The timing of the game is as usual. First, player 1 proposes the split (100 - x, x) of a hundred dollars to player 2, where x = [0, 100]. Player 2 observes the split and decides whether to accept (in which case they receive money according to the proposed split) or reject (in which case they both get 0 dollars). Now suppose that each player's utility from an outcome equals the amount of money she gets minus the "unfairness disutility" which is proportional to the squared difference in the monetary outcomes. That is, if an offer of x is accepted by P2, then the "unfairness disutility” of player i equals ẞi(x - (100 - x))², where ẞi is a parameters of the game indicating how strongly player i cares about fairness. Note that if an offer is rejected, they both get $0 and so the disutility term (and, hence, the final utility as well) equals zero. Note also that the case we considered in class corresponds to ẞ₁ = ẞ₂ = 0. (a) Represent this game in extensive form. (b) Let B₁ = 0,2 = 0. Which offers will player 2 definitely accept? reject? Describe all sequentially rational strategies of player 2. (c) Let ẞ₁ = 1,2 = 0. For each sequentially rational strategy of player 2 you identified in (c), either describe which proposal maximizes player 1's continuation value or explain why it does not exist. (d) Let B₁ = 1, B2 = 0. Describe all SPNE of the game. (e) Let ẞ10, ẞ₂ = player 2. 1 = 10° Which offers will player 2 definitely accept? reject? Describe all sequentially rational strategies of
1. Consider a variant of the ultimatum game we studied in class in which players have fairness considerations. The timing of the game is as usual. First, player 1 proposes the split (100 - x, x) of a hundred dollars to player 2, where x = [0, 100]. Player 2 observes the split and decides whether to accept (in which case they receive money according to the proposed split) or reject (in which case they both get 0 dollars). Now suppose that each player's utility from an outcome equals the amount of money she gets minus the "unfairness disutility" which is proportional to the squared difference in the monetary outcomes. That is, if an offer of x is accepted by P2, then the "unfairness disutility” of player i equals ẞi(x - (100 - x))², where ẞi is a parameters of the game indicating how strongly player i cares about fairness. Note that if an offer is rejected, they both get $0 and so the disutility term (and, hence, the final utility as well) equals zero. Note also that the case we considered in class corresponds to ẞ₁ = ẞ₂ = 0. (a) Represent this game in extensive form. (b) Let B₁ = 0,2 = 0. Which offers will player 2 definitely accept? reject? Describe all sequentially rational strategies of player 2. (c) Let ẞ₁ = 1,2 = 0. For each sequentially rational strategy of player 2 you identified in (c), either describe which proposal maximizes player 1's continuation value or explain why it does not exist. (d) Let B₁ = 1, B2 = 0. Describe all SPNE of the game. (e) Let ẞ10, ẞ₂ = player 2. 1 = 10° Which offers will player 2 definitely accept? reject? Describe all sequentially rational strategies of
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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![1. Consider a variant of the ultimatum game we studied in class in which players have fairness considerations. The timing of
the game is as usual. First, player 1 proposes the split (100 - x, x) of a hundred dollars to player 2, where x = [0, 100]. Player 2
observes the split and decides whether to accept (in which case they receive money according to the proposed split) or reject
(in which case they both get 0 dollars).
Now suppose that each player's utility from an outcome equals the amount of money she gets minus the "unfairness disutility"
which is proportional to the squared difference in the monetary outcomes. That is, if an offer of x is accepted by P2, then
the "unfairness disutility” of player i equals ẞi(x - (100 - x))², where ẞi is a parameters of the game indicating how strongly
player i cares about fairness. Note that if an offer is rejected, they both get $0 and so the disutility term (and, hence, the final
utility as well) equals zero. Note also that the case we considered in class corresponds to ẞ₁ = ẞ₂ = 0.
(a) Represent this game in extensive form.
(b) Let B₁ = 0,2 = 0. Which offers will player 2 definitely accept? reject? Describe all sequentially rational strategies of
player 2.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1858a710-c111-4513-9004-6c10bf7d1d6e%2F61bf2504-2f53-4147-a879-6c9a991fdfbd%2F63oda9r_processed.png&w=3840&q=75)
Transcribed Image Text:1. Consider a variant of the ultimatum game we studied in class in which players have fairness considerations. The timing of
the game is as usual. First, player 1 proposes the split (100 - x, x) of a hundred dollars to player 2, where x = [0, 100]. Player 2
observes the split and decides whether to accept (in which case they receive money according to the proposed split) or reject
(in which case they both get 0 dollars).
Now suppose that each player's utility from an outcome equals the amount of money she gets minus the "unfairness disutility"
which is proportional to the squared difference in the monetary outcomes. That is, if an offer of x is accepted by P2, then
the "unfairness disutility” of player i equals ẞi(x - (100 - x))², where ẞi is a parameters of the game indicating how strongly
player i cares about fairness. Note that if an offer is rejected, they both get $0 and so the disutility term (and, hence, the final
utility as well) equals zero. Note also that the case we considered in class corresponds to ẞ₁ = ẞ₂ = 0.
(a) Represent this game in extensive form.
(b) Let B₁ = 0,2 = 0. Which offers will player 2 definitely accept? reject? Describe all sequentially rational strategies of
player 2.
Transcribed Image Text:(c) Let ẞ₁ = 1,2 = 0. For each sequentially rational strategy of player 2 you identified in (c), either describe which proposal
maximizes player 1's continuation value or explain why it does not exist.
(d) Let B₁ = 1, B2 = 0. Describe all SPNE of the game.
(e) Let ẞ10, ẞ₂ =
player 2.
1
=
10°
Which offers will player 2 definitely accept? reject? Describe all sequentially rational strategies of
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