Econ 504: Game Theory with Economic and Financial Applications Problem Set 2
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Problem Set 2
Econ 504: Game Theory with Economic and Financial Applications.
November 6, 2022
1. PUBLIC GOOD GAME
There are two players, i = 1, 2. In each period, t = 0, 1, players decide simultaneously whether to contribute to the period-t public good and contributing is a 0 · 1 decision. In a given period, each player derive a benefit of 1 if at least one of them provides the public good and 0 if none does; player i’s cost of contributing in a period is ci and is the same in both periods. Per-period payoffs are depicted below:
|
|
Contribute |
Don\t |
|
Contribute |
1 · c1 , 1 · c2 |
1 · c1 , 1 |
|
Don\t |
1, 1 · c2 |
0, 0 |
We assume that payoffs are discounted so that a player’s objective function is the sum of his first-period payoff plus δ times his second-period payoff, where 0 < δ < 1. Though the benefits of the public good– 1 each– are common knowledge, each player’s cost is known only to that player. However both players believe that the ci are drawn independently from the uniform distribution on [0, 
], where 
> 1.
1. Assume first that the game is played over a single period. Find the Bayesian Nash equilibrium of this static game of incomplete information? (hint: at equilibrium there exists a cutoff cost for each player for which the player contributes if his cost is less than the threshold).
2. Show that we can find this equilibrium using iterated elimination of strictly dominated strategies.
For the reminder of the exercise assume a repeated version of the game. A strategy for player i is a pair consisting of σi(0)(1|ci ) ( player i’s probability of contributing in the first period when his cost is ci ) and σi(1)(1|h1 , ci ) (the probability that player i contributes in the second period when his cost is ci and when the history is h1 e (00, 01, 10, 11}, where history 01 means player 1 didn’t contribute in the first period and player 2 contributed). Note that each player contribution cost remains the same in period 1 and period 2.
1. Assume that there exist a Perfect Bayesian Equilibrium (PBE) with a cutoff cost cˆi for each player i such that player i contributes in the first period if and only if ci < cˆi . Show that 0 < cˆi < 1.
2. Find the symmetric PBE, where 
1 = 
2 = 
.
2. COURT
There are two players, a plaintiff and a defendant in a civil suit.. The plaintiff knows whether or not he will win the case if it goes to trial, but the defendant does not have this information. The defendant knows the plaintiff knows who would win; and the defendant has prior beliefs that there is a probability 1/4 that the plaintiff will win; these prior beliefs are common knowledge. If the plaintiff wins, his payoff is 4 and the defendant’s payoff is -5; if the plaintiff loses his payoff is -1 and the defendant’s is 0. The plaintiff has two possible actions: He can ask for either a low settlement of m = 1 or a high settlement of m = 2. If the defendant accepts a settlement offer of m, the plaintiff payoff’s is m and the defendant’s is ·m. If the defendant rejects the settlement offer, the case goes to court.
1. Represent the game in extensive form i.e. draw a game tree.
2. List all the pure-strategy Perfect Bayesian Equilibrium strategy profiles. For each such profile, specify the posterior beliefs of the defendant (about the probability that the plaintiff wins the case) as a function of m and verify that the combination of these beliefs and the profile is in fact a PBE.
2022-11-08