Problem Set 5: Due on July 5
Advanced Microeconomics II (Spring, 2nd, 2012)
1. Question 1 (4 points)
Players 1 (proposer) and 2 (receiver) are bargaining over how to split the ice-cream of size 1. In the first stage, player 1 proposes a share {x, 1 − x} to player 2 where x ∈ [0, 1]. Player 2 can decide whether accept the offer or reject it. If player 2 accepts, then the game finishes and players get their shares. If player 2 rejects, the game move to the second stage, in which the size of the ice-cream becomes δ(∈ (0, 1)) of the original size due to melting. In the second stage, by flipping a coin, the ice-cream is randomly assigned to one of the players. Suppose each player maximizes expected size of the ice-cream that she can get. Derive a subgame perfect Nash equilibrium of this game.
2. Question 2 (5 points)
Players 1 and 2 are forming a firm. The value of their relationship depends on the effort that each expends. Suppose that player i’s utility from the relationship is x2j+ xj− xixj, where xi is i’s effort and xj is the effort of the other player. Assume x1, x2 ≥ 0.
(a) Find the Nash equilibrium of this game. Is it Pareto efficient?
(b) Suppose that the players interact over time, which we model with the infinitely repeated version of the game. Let δ denote the (common) discount factor of the players. Under what conditions can the players sustain some positive effort level k = x1 = x2 > 0 over time?
3. Question 3 (5 points)
Consider a game of election with asymmetric information among voters. Whether candidate A or candidate B is elected depends on the votes of two citizens (denoted by 1 and 2). The economy may be in one of two states, α and β. The citizens agree that candidate A is best if the state is α and candidate B is best if the state is β. The payoff for each citizen is symmetric and given as follows: 1 if the best candidate wins, 0 if the other candidate wins, and 1/2 if the candidates tie. Suppose that citizen 1 is informed of the true state, whereas citizen 2 believes it is α with probability 0.9 and β with probability 0.1. Each citizen may either vote for candidate A, vote for candidate B, or not vote.
(a) Formulate this situation as a Bayesian game.
(b) Show that the game has exactly two pure strategy Nash equilibria, the one of which involves weakly dominated strategy.
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4. Question 4 (6 points)
There are 10 envelopes and each of them contains a number 1 through 10. That is, one envelope contains 1, another envelope contains 2, and so on; these numbers cannot be observable from outside. Suppose there are two individuals. Each of them randomly receives one envelope and observes the number inside of her/his own envelope. Then, they are given an option to exchange the envelope to the other person; exchange occurs if and only if both individuals wish to exchange. Finally, individuals receive prize ($) equal to the number, i.e., she receives $X if the number is X. Assume that both individuals are risk-neutral so that they maximize expected value of prizes.
(a) Consider the above situation as a Bayesian game. Then, what is the individ- ual’s strategy?
(b) If an individual receives the envelope with $10, will she have an incentive to exchange or not? Explain why.
(c) Solve for the Bayesian Nash equilibrium. Can the exchange happen (with positive probability) in equilibrium?
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