Consider a modified version of the ultimatum game, with altruism. There is a pie of size 50 to be split between Proposer (P) and Responder (R). In the first stage, P offers a share 0 ≤ x ≤ 1 of the pie to R. In the second stage, R accepts or rejects P's offer. If R accepts, the pie is split according to the proposed division, with P receiving (1 - x) × 50 and R receiving x x 50. If R rejects, both players receive none of the pie. But players' payoffs are not equal to how much of the pie they receive. In particular, payoffs in the event of an accepted offer are: Up (1-x) x 50+ bx × 50 UR = x×50+b(1 − x) × 50 (Payoffs in the event of a rejected offer are 0 for both players.) Here, the parameter b≥ 0 measures how altruistic players are, i.e. how much "happier" a player is when his opponent receives more pie. a) Suppose b = 0.5. How much pie does P receive in the subgame perfect Nash equilibrium of the game? (specify the total amount of pie, not the share). b) Suppose b = 0.5. What is P's payoff in the subgame perfect Nash equilibrium of the game? c) Suppose b = 1.2. How much pie does P receive in the subgame perfect Nash equilibrium of the game? (specify the total amount of pie, not the share). d) Suppose b = 1.2. What is P's payoff in the subgame perfect Nash equilibrium of the game?
Consider a modified version of the ultimatum game, with altruism. There is a pie of size 50 to be split between Proposer (P) and Responder (R). In the first stage, P offers a share 0 ≤ x ≤ 1 of the pie to R. In the second stage, R accepts or rejects P's offer. If R accepts, the pie is split according to the proposed division, with P receiving (1 - x) × 50 and R receiving x x 50. If R rejects, both players receive none of the pie. But players' payoffs are not equal to how much of the pie they receive. In particular, payoffs in the event of an accepted offer are: Up (1-x) x 50+ bx × 50 UR = x×50+b(1 − x) × 50 (Payoffs in the event of a rejected offer are 0 for both players.) Here, the parameter b≥ 0 measures how altruistic players are, i.e. how much "happier" a player is when his opponent receives more pie. a) Suppose b = 0.5. How much pie does P receive in the subgame perfect Nash equilibrium of the game? (specify the total amount of pie, not the share). b) Suppose b = 0.5. What is P's payoff in the subgame perfect Nash equilibrium of the game? c) Suppose b = 1.2. How much pie does P receive in the subgame perfect Nash equilibrium of the game? (specify the total amount of pie, not the share). d) Suppose b = 1.2. What is P's payoff in the subgame perfect Nash equilibrium of the game?
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Transcribed Image Text:Consider a modified version of the ultimatum game, with altruism. There is a pie of size 50 to be split
between Proposer (P) and Responder (R). In the first stage, P offers a share 0 ≤ x ≤ 1 of the pie to R. In the
second stage, R accepts or rejects P's offer. If R accepts, the pie is split according to the proposed division,
with P receiving (1 - x) × 50 and R receiving x x 50. If R rejects, both players receive none of the pie.
But players' payoffs are not equal to how much of the pie they receive. In particular, payoffs in the event of an
accepted offer are:
Up (1-x) x 50+ bx × 50
UR = x×50+b(1 − x) × 50
(Payoffs in the event of a rejected offer are 0 for both players.)
Here, the parameter b≥ 0 measures how altruistic players are, i.e. how much "happier" a player is when his
opponent receives more pie.
a) Suppose b = 0.5. How much pie does P receive in the subgame perfect Nash equilibrium of the game?
(specify the total amount of pie, not the share).
b) Suppose b
=
0.5. What is P's payoff in the subgame perfect Nash equilibrium of the game?
c) Suppose b = 1.2. How much pie does P receive in the subgame perfect Nash equilibrium of the game?
(specify the total amount of pie, not the share).
d) Suppose b
= 1.2. What is P's payoff in the subgame perfect Nash equilibrium of the game?
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