How to solve a prisoner's dilemma game in game theory?

To solve a prisoner's dilemma game in game theory, we use the concept of dominant strategies.

In a prisoner's dilemma game, two players have the option to either cooperate or defect. The payoff matrix for the game is as follows:

| | Player 2 Cooperates | Player 2 Defects |
|----------|-------------------|------------------|
| Player 1 Cooperates | (3,3) | (0,5) |
| Player 1 Defects | (5,0) | (1,1) |

To find the dominant strategy for each player, we compare the payoffs for each strategy regardless of what the other player chooses.

For Player 1, if Player 2 cooperates, Player 1's payoff is higher if they defect (5 > 3). If Player 2 defects, Player 1's payoff is higher if they defect (1 > 0). Therefore, Player 1's dominant strategy is to defect.

For Player 2, if Player 1 cooperates, Player 2's payoff is higher if they defect (5 > 3). If Player 1 defects, Player 2's payoff is higher if they defect (1 > 0). Therefore, Player 2's dominant strategy is to defect.

When both players play their dominant strategy, the outcome is (1,1), which is worse for both players than if they had both cooperated (3,3). This is the dilemma of the game.

In conclusion, to solve a prisoner's dilemma game in game theory, we use the concept of dominant strategies to determine the best strategy for each player. However, the outcome of playing the dominant strategy may not be the best outcome for both players, leading to the dilemma of the game.