Problem Set 5: Due on July 4
Advanced Microeconomics II (Spring, 2nd, 2013)
1. Question 1 (6 points) See the following game tree.
(a) How many information sets (including singleton sets) does this game have? (b) How many subgames (including the entire game) does this game have?
(c) Find all (pure-strategy) subgame perfect equilibria. 2. Question 2 (3 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] is player 1’s own share. 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.
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3. Question 3 (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?
4. Question 4 (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.
5. Question 5 (6 points)
Two players, 1 and 2, each own a house. Each player i values her own house at vi
and this is private information. The value of player i’s house to the other player j(6= i) is 3
2vi. The values vi are drawn independently from the interval [0, 1] with uniform distribution. Suppose players announce simultaneously whether they want to exchange (E) their house of not (N). If both players agree to an exchange, the exchange takes place. Otherwise no exchange occurs.
(a) Find a Bayesian Nash equilibrium of the game in pure strategies in which each player i accepts an exchange if and only if the value vi does not exceed some threshold θi
(b) How would your answer to (a) change if the value of player i’s house to the other player j becomes 5
2vi?
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