- Consider the infinitely repeated game of Prisoners Dilemma that has the following payoff structure. Each player has a discount rate r. Suppose both players are using the Tit-For-Tat strategy.
Column
Defect
Cooperate
Row
Defect
2, 2
10, 1
Cooperate
1, 10
6, 6
- (a) Under what condition does a player have an incentive to deviate once from cooperating and then revert to it? [4 points]
- (b) Under what condition does a player have an incentive to deviate from cooperating forever? [4 points]
- (c) Under what condition is it a Nash equilibrium that both players use the Tit-For-Tat strategy? [2 points]
- Consider the infinitely repeated game of Prisoners Dilemma that has the following payoff structure. Each player has a discount rate r. Suppose both players are using the Tit-For-Tat strategy.
|
|
Column |
||
|
|
Defect |
Cooperate |
|
|
Row |
Defect |
2, 2 |
10, 1 |
|
Cooperate |
1, 10 |
6, 6 |
|
- (a) Under what condition does a player have an incentive to deviate once from cooperating and then revert to it? [4 points]
- (b) Under what condition does a player have an incentive to deviate from cooperating forever? [4 points]
- (c) Under what condition is it a Nash equilibrium that both players use the Tit-For-Tat strategy? [2 points]
Related textbook solutions
Related questions
Problem 1
In each of the three games shown below, let p be the probability that player 1 plays cooperates (and 1- p the probability that player 1 defects), and let q be the probability that Player 2 plays cooperates (and 1- q the probability that player 2 defects).
Prisonerââ¬â¢s Dilemma
| Player 2 | |||
| Player 1 | cooperate | defect | |
| cooperate | 70,70 | 10,80 | |
| defect | 80,10 | 40,40 | |
Stag Hunt
| Player 2 | |||
| Player 1 | cooperate | defect | |
| cooperate | 70,70 | 5,40 | |
| defect | 40,5 | 40,40 | |
Chicken
| Player 2 | |||
| Player 1 | cooperate | defect | |
| cooperate | 70,70 | 50,80 | |
| defect | 80,50 | 40,40 | |
1. For each game, draw a graph with player 1ââ¬â¢s best response function (choice of p as a function of q), and player 2ââ¬â¢s best response function (choice of q as a function of p), with p on the horizontal axis and q on the vertical axis.
2. Using this graphs, find all the Nash equilibriums for the game, both pure and mixed strategy Nash equilibriums (if any). Label these equilibriums on the corresponding graph.
3. In those games that have multiple pure strategy Nash equilibriums, how do the expected payoffs from playing the mixed strategy Nash equilibrium compare with the payoffs from playing the pure strategy Nash equilibriums? Which type of strategy (mixed or pure) would players prefer to play in these games?
Problem 2
Two people are involved in a dispute. Player 1 does not know whether player 2 is strong or weak; she assigns probability ñ to player 2 being strong. Player 2 is fully informed. Each player can either fight or yield. Each player obtains a payoff of 0 is she yields (regardless of the other personââ¬â¢s action) and a payoff of 1 if she fights and her opponent yields. If both players fight, then their payoffs are (-1; 1) if player 2 is strong and (1;-1) if player 2 is weak. The Bayesian game is the following, depending on the type of player 2:
| Y | F | Y | F | ||||||
| Y | 0, 0 | 0, 1 | Y | 0, 0 | 0, 1 | ||||
| F | 1, 0 | -1, 1 | F | 1, 0 | 1, -1 | ||||
| Player 2 is strong (ñ) | Player 2 is weak (1-ñ) | Player 2 is strong (ñ) | |||||||
After writing all the strategies and payoffs in the same matrix, find the Bayesian Nash equilibriums, depending on the value of ñ (ñ ââ°Â¤ 1/2 or ñ ââ°Â¥1/2).
