Ann and Bob are trying to divide a dollar, which they can consume only when they divide. If they agree on a division that gives a € [0, 1] to player i in period t, then i's payoff is d'r, where & € (0, 1) is the discount factor. The periods are t = 0, 1, 2,..... If the players do not agree by the end of period T, then the game ends and each player gets 0. Think of T as the horizon the game. At each period t≤ T, the negotiation between Ann and Bob proceeds as follows:
Ann and Bob are trying to divide a dollar, which they can consume only when they divide. If they agree on a division that gives a € [0, 1] to player i in period t, then i's payoff is d'r, where & € (0, 1) is the discount factor. The periods are t = 0, 1, 2,..... If the players do not agree by the end of period T, then the game ends and each player gets 0. Think of T as the horizon the game. At each period t≤ T, the negotiation between Ann and Bob proceeds as follows:
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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![Ann and Bob are trying to divide a dollar, which they can consume only when they
divide. If they agree on a division that gives r E [0, 1] to player i in period t, then i's
payoff is 8'r, where d e (0, 1) is the discount factor. The periods are t = 0, 1, 2,... If
the players do not agree by the end of period T, then the game ends and each player
gets 0. Think of T as the horizon the game.
At each period t<T, the negotiation between Ann and Bob proceeds as follows:
• One player is selected to be the proposer. Say that An is the proposer when t
is even (i.e., when t = 0,2,...), while Bob is the proposer when t is odd (i.e.,
when t = 1,3,...).
• The proposer offers r to the other player who (after having observed the offer)
can accept or reject it.
• If the offer is accepted, then the proposer gets 1-r and the opponent gets r.
• If the offer is rejected, the game proceeds to the next period.
Notice that this is a game with perfect information (all past moves are observed).
The horizon is finite if T < o, infinite if T = 00.
(a) Find a SPE for T = 0, that is, the game lasts only one period.
(b) Find a SPE for T = 1, that is, the game lasts for two periods.
(c) Find a SPE for T = co, that is, the game has infinite horizon.
Hint. Look for an "stationary" equilibrium in which the proposer always offers
the same quantity r* to the other player, and the respondent accepts any offer
that is more generous than r*, rejects otherwise. Find the value of r* that
makes this strategy profile a SPE.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe40f9647-7e3d-44de-bb78-c93fb275dbdc%2F33ff2608-4de1-4ad2-ac88-c727d4304233%2F0wq7lz9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Ann and Bob are trying to divide a dollar, which they can consume only when they
divide. If they agree on a division that gives r E [0, 1] to player i in period t, then i's
payoff is 8'r, where d e (0, 1) is the discount factor. The periods are t = 0, 1, 2,... If
the players do not agree by the end of period T, then the game ends and each player
gets 0. Think of T as the horizon the game.
At each period t<T, the negotiation between Ann and Bob proceeds as follows:
• One player is selected to be the proposer. Say that An is the proposer when t
is even (i.e., when t = 0,2,...), while Bob is the proposer when t is odd (i.e.,
when t = 1,3,...).
• The proposer offers r to the other player who (after having observed the offer)
can accept or reject it.
• If the offer is accepted, then the proposer gets 1-r and the opponent gets r.
• If the offer is rejected, the game proceeds to the next period.
Notice that this is a game with perfect information (all past moves are observed).
The horizon is finite if T < o, infinite if T = 00.
(a) Find a SPE for T = 0, that is, the game lasts only one period.
(b) Find a SPE for T = 1, that is, the game lasts for two periods.
(c) Find a SPE for T = co, that is, the game has infinite horizon.
Hint. Look for an "stationary" equilibrium in which the proposer always offers
the same quantity r* to the other player, and the respondent accepts any offer
that is more generous than r*, rejects otherwise. Find the value of r* that
makes this strategy profile a SPE.
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