Consider an application of the Prisoner’s Dilemma game where two countries in a cartel decide whether to Cooperate or Defect. If both choose to follow the cartel agreement and continues cooperating with each other, each receives a payoff of 2. If both decide to defect on each other, they both get -2. If one cooperates and the other defects, the first one loses 10 (she receives -10) while the other gets 0. Denote the probability of cooperating for Player 1 as p and for Player 2 as q.How much does Player 1 earn in the mixed-strategy Nash equilibrium of this game?
Question
Consider an application of the Prisoner’s Dilemma game where two countries in a cartel decide whether to Cooperate or Defect. If both choose to follow the cartel agreement and continues cooperating with each other, each receives a payoff of 2. If both decide to defect on each other, they both get -2. If one cooperates and the other defects, the first one loses 10 (she receives -10) while the other gets 0. Denote the probability of cooperating for Player 1 as p and for Player 2 as q.How much does Player 1 earn in the mixed-strategy Nash equilibrium of this game?
Solution
To find the mixed-strategy Nash equilibrium, we first need to set up the payoff matrix for the game.
| Cooperate (q) | Defect (1-q) | |
|---|---|---|
| Cooperate (p) | 2,2 | -10,0 |
| Defect (1-p) | 0,-10 | -2,-2 |
In a mixed-strategy Nash equilibrium, each player is indifferent between their strategies given the strategy of the other player. This means that the expected payoff from cooperating must equal the expected payoff from defecting for both players.
For Player 1, the expected payoff from cooperating is 2q - 10(1-q) and the expected payoff from defecting is -2q. Setting these equal to each other gives:
2q - 10 + 10q = -2q 12q = 10 q = 10/12 = 5/6
So in the mixed-strategy Nash equilibrium, Player 1 cooperates with probability 5/6.
The expected payoff for Player 1 in the mixed-strategy Nash equilibrium is then the expected payoff from either cooperating or defecting, which are now equal, so we can just calculate one of them:
E[Payoff] = 2(5/6) - 10(1 - 5/6) = 10/3 - 10/3 = 0
So Player 1 earns 0 in the mixed-strategy Nash equilibrium of this game.
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