Date: April 4, 2011
Subject: Game Theory (ECO290E)
Please complete your answer in either one of doc, pdf, or text file, and send it (via attached file) to the following e‐mail address by 23:59pm (Tokyo time) on April 18. Yosuke YASUDA <email@example.com>
Remark: This assignment counts for 50 points, the same weight as the final exam. You should work alone, i.e., NOT allowed to work together with somebody. Good luck!
1. Bargaining (5 points)
Players 1, 2 and 3 are bargaining over how to split the ice‐cream of size 1. In the first period, player 1 proposes a share (α, β, 1‐α‐β) to players 2 and 3 where each share must be between 0 and 1. Players 2 and 3 can decide whether accepting the offer or reject it. If both players accept, then the game finishes and each player gets the proposed share. If either one of them rejects, the game moves to the second period in which the size of the ice‐cream becomes 80% of the original size due to melting. In the second stage, the ice‐cream is randomly assigned to player 2 with probability p and 3 with probability 1‐p (by using some randomization devise). Suppose that each player maximizes expected size of the ice‐cream that she can get. Derive a subgame perfect Nash equilibrium of this game.
Hint: You can focus on the equilibrium in which both players 2 and 3 accept the offer in the first period. For such equilibrium, solve α and β as a function of p.
2. Repeated Game (10 points)
Consider the following two persons 2 x 2 game.
1 / 2 L R
U 4, 4 0, 7
D 6, 0 1, 2
A) Find all pure‐strategy Nash equilibria.
B) Consider the two‐period repeated game in which the above stage game will be played twice. Suppose that the payoff for each player is simply the sum of the payoffs in the stage games. Then, can (U, L) be sustained as a subgame perfect Nash equilibrium? If yes, derive the equilibrium. If not, explain why.
C) Now suppose that the game will be played infinitely many times, and each player tries to maximize the discounted sum of payoffs with the discount factor δ (< 1), which is common across players. For what value of δ, can (U, L) be sustained as a subgame perfect Nash equilibrium?
Hint: You can focus on the trigger strategy, i.e., start playing (U, L) and switch to one shot Nash equilibrium forever once somebody deviates. Consider both players incentive constraints, since the game is not symmetric.
3. Incomplete Information (10 points)
Consider a game of election with asymmetric information among voters (citizens). Whether candidate A or candidate B is elected depends on the votes of two citizens. The social situation may be in one of two states, a and b. The citizens agree that candidate A is best if the state is a, and candidate B is best if the state is b. The payoff for each citizen is symmetric and given as follows: 1 if the best candidate wins, 0 if the other candidate wins, and 0.5 if the candidates tie. Suppose that citizen 1 knows the true state, whereas citizen 2 believes that the state is a with probability 0.8 and b with probability 0.2. Each citizen takes either one of the three actions: vote for candidate A, vote for candidate B, and not vote.
A) Consider the corresponding Bayesian game: the nature first chooses the true state which is informed only to citizen 1. Then, what is the strategy for each player?
B) Derive the pure strategy Bayesian Nash equilibria.
Hint: There are two equilibria, one of which involves weakly dominated strategy.
C) What does happen if each player has only two actions, vote for A and vote for B? Explain why or how the introduction of “not vote” can improve efficiency.
4. Application of Game Theory (25 points)
On March 11, the huge earthquake and tsunami hit the north‐east coast of Japan. This catastrophic natural disaster brought down noteworthy social behaviors, e.g., panic buying, evacuations, herding, etc, which are not often observed in the usual situation. Although these unusual behaviors look difficult to analyze theoretically, some intuitive explanation might well be obtained by game theoretical models. In this question, I ask you to 1) find a specific social behavior related to natural disaster (it doesn’t necessarily related to the Japan’s disaster), 2) construct the game which capture the behavior, 3) solve it by finding corresponding solutions such as Nash equilibria, and 4) provide the explanation based on your model. The game can be of any form: you can consider either static game, dynamic game, or incomplete information game (Bayesian game).
Remark: I will give you partial credits even if the connection between your model and the actual social behavior looks very weak, as long as your model is mathematically correct and you can successfully derive its solutions.