2. Consider the ultimatum game, where player I makes a take-it-or-leave-it offer to player 2, but suppose that player 2 is one of two types: with probability p, player 2 is a "standard" type, who maximizes his utility as usual; - with probability 1 - p, player 2 is a "crazy" type, who accepts an offer if and only if the offer gives player 2 at least fraction q of the surplus. Player 1 does not know the type of player 2. In this game, a strategy profile must specify strategies for both player 1 and the standard type of player 2 (note that the strategy of the crazy type of player 2 is given). For every pair (p,q) = (0,1] x (0,1], find all strategy profiles where player 1 and the standard type of player 2 are both maximizing their expected utility, given their opponent's strategy. Hint: First, figure out the best responses for the standard type of player 2. Then, to find player 1's best response to these strategies for various values of (p,q), start by writing down the possible expected utilities for player 1 as a function of the offer, p, and q.
2. Consider the ultimatum game, where player I makes a take-it-or-leave-it offer to player 2, but suppose that player 2 is one of two types: with probability p, player 2 is a "standard" type, who maximizes his utility as usual; - with probability 1 - p, player 2 is a "crazy" type, who accepts an offer if and only if the offer gives player 2 at least fraction q of the surplus. Player 1 does not know the type of player 2. In this game, a strategy profile must specify strategies for both player 1 and the standard type of player 2 (note that the strategy of the crazy type of player 2 is given). For every pair (p,q) = (0,1] x (0,1], find all strategy profiles where player 1 and the standard type of player 2 are both maximizing their expected utility, given their opponent's strategy. Hint: First, figure out the best responses for the standard type of player 2. Then, to find player 1's best response to these strategies for various values of (p,q), start by writing down the possible expected utilities for player 1 as a function of the offer, p, and q.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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![2. Consider the ultimatum game, where player 1 makes a take-it-or-leave-it offer to player 2,
but suppose that player 2 is one of two types:
with probability p, player 2 is a "standard" type, who maximizes his utility as usual;
with probability 1 - p, player 2 is a "crazy" type, who accepts an offer if and only if
the offer gives player 2 at least fraction q of the surplus.
Player 1 does not know the type of player 2.
In this game, a strategy profile must specify strategies for both player 1 and the standard type
of player 2 (note that the strategy of the crazy type of player 2 is given).
For every pair (p, q) E (0,1] x (0,1], find all strategy profiles where player 1 and the
standard type of player 2 are both maximizing their expected utility, given their opponent's
strategy.
Hint: First, figure out the best responses for the standard type of player 2. Then, to find
player l's best response to these strategies for various values of (p, q), start by writing down
the possible expected utilities for player 1 as a function of the offer, p, and q.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4454b7c5-cfde-458f-b3d2-867aac67c9f3%2Fb7e95449-f34a-42b3-8906-13ed0c192f37%2Fe9i2saj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. Consider the ultimatum game, where player 1 makes a take-it-or-leave-it offer to player 2,
but suppose that player 2 is one of two types:
with probability p, player 2 is a "standard" type, who maximizes his utility as usual;
with probability 1 - p, player 2 is a "crazy" type, who accepts an offer if and only if
the offer gives player 2 at least fraction q of the surplus.
Player 1 does not know the type of player 2.
In this game, a strategy profile must specify strategies for both player 1 and the standard type
of player 2 (note that the strategy of the crazy type of player 2 is given).
For every pair (p, q) E (0,1] x (0,1], find all strategy profiles where player 1 and the
standard type of player 2 are both maximizing their expected utility, given their opponent's
strategy.
Hint: First, figure out the best responses for the standard type of player 2. Then, to find
player l's best response to these strategies for various values of (p, q), start by writing down
the possible expected utilities for player 1 as a function of the offer, p, and q.
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