Amir and Beatrice play the following game. Amir offers an amount of money z € [0, 1] to Beatrice. Beatrice can either accept or reject. If Beatrice accepts, then Amir receives 1 - z
Amir and Beatrice play the following game. Amir offers an amount of money z € [0, 1] to Beatrice. Beatrice can either accept or reject. If Beatrice accepts, then Amir receives 1 - z
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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![Amir and Beatrice play the following game. Amir offers an amount of money z € [0, 1] to
Beatrice. Beatrice can either accept or reject. If Beatrice accepts, then Amir receives 1 - z
dollars and Beatrice receives a dollars. If Beatrice rejects, then both receive no money. As in
the Ultimatum Game, Amir cares only about maximizing the amount of money he receives.
Beatrice, on the other hand, detests Amir, and therefore cares about the amount of money
that both receive: if Amir receives y dollars and Beatrice receives a dollars, then Beatrice's
payoff is a-ay where a > 0.
(a) Find all pure strategy Nash equilibria of the game in which the two players choose
simultaneously (thus Beatrice accepts or rejects without seeing Amir's offer).
Solution: The NE are the strategy profiles in which Beatrice rejects and Amir's offer a
satisfies 2-a(1-x) ≤0, i.e. r ≤a/(1+a).
(b) Find all subgame perfect equilibria of the sequential game in which Amir first makes the
offer and Beatrice observes the offer before choosing whether to accept or reject.
Solution: There is a unique SPE given by Amir offering r* = a/(1+ a), and Beatrice
accepting any offer x ≥a/(1+a) and rejecting any lower offer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F898cbd98-1de8-493f-9103-82a631e26b34%2F51ed8a64-0879-416c-b0c9-413e7fc05711%2Frq4gula_processed.png&w=3840&q=75)
Transcribed Image Text:Amir and Beatrice play the following game. Amir offers an amount of money z € [0, 1] to
Beatrice. Beatrice can either accept or reject. If Beatrice accepts, then Amir receives 1 - z
dollars and Beatrice receives a dollars. If Beatrice rejects, then both receive no money. As in
the Ultimatum Game, Amir cares only about maximizing the amount of money he receives.
Beatrice, on the other hand, detests Amir, and therefore cares about the amount of money
that both receive: if Amir receives y dollars and Beatrice receives a dollars, then Beatrice's
payoff is a-ay where a > 0.
(a) Find all pure strategy Nash equilibria of the game in which the two players choose
simultaneously (thus Beatrice accepts or rejects without seeing Amir's offer).
Solution: The NE are the strategy profiles in which Beatrice rejects and Amir's offer a
satisfies 2-a(1-x) ≤0, i.e. r ≤a/(1+a).
(b) Find all subgame perfect equilibria of the sequential game in which Amir first makes the
offer and Beatrice observes the offer before choosing whether to accept or reject.
Solution: There is a unique SPE given by Amir offering r* = a/(1+ a), and Beatrice
accepting any offer x ≥a/(1+a) and rejecting any lower offer.
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