The Prisoners' Dilemma
The prisoners' dilemma is a very popular example of a two-person game of strategic interaction, and it's a common introductory example in many game theory textbooks. The logic of the game is simple:
- The two players in the game have been accused of a crime and have been placed in separate rooms so that they cannot communicate with one another. (In other words, they can't collude or commit to cooperating.)
- Each player is asked independently whether he is going to confess to the crime or remain silent.
- Because each of the two players has two possible options (strategies), there are four possible outcomes to the game.
- If both players confess, they each get sent to jail, but for fewer years than if one of the players got ratted out by the other.
- If one player confesses and the other remains silent, the silent player gets punished severely while the player who confessed gets to go free.
- If both players remain silent, they each get a punishment that is less severe than if they both confess.
In the game itself, punishments (and rewards, where relevant) are represented by utility numbers. Positive numbers represent good outcomes, negative numbers represent bad outcomes, and one outcome is better than another if the number associated with it is greater. (Be careful, however, of how this works for negative numbers, since -5, for example, is greater than -20!)
In the table above, the first number in each box refers to the outcome for player 1 and the second number represents the outcome for player 2. These numbers represent just one of many sets of numbers that are consistent with the prisoners' dilemma setup.02of 04
Analyzing the Players' Options
Once a game is defined, the next step in analyzing the game is to assess the players' strategies and try to understand how the players are likely to behave. Economists make a few assumptions when they analyze games- first, they assume that both players are aware of the payoffs both for themselves and for the other player, and, second, they assume that both players are looking to rationally maximize their own payoff from the game.
One easy initial approach is to look for what are called dominant strategies- strategies that are best regardless of what strategy the other player chooses. In the example above, choosing to confess is a dominant strategy for both players:
- Confess is better for player 1 if player 2 chooses to confess since -6 is better than -10.
- Confess is better for player 1 if player 2 chooses to remain silent since 0 is better than -1.
- Confess is better for player 2 if player 1 chooses to confess since -6 is better than -10.
- Confess is better for player 2 if player 1 chooses to remain silent since 0 is better than -1.
Given that confessing is best for both players, it's not surprising that the outcome where both players confess is an equilibrium outcome of the game. That said, it's important to be a bit more precise with our definition.03of 04
The concept of a Nash Equilibrium was codified by mathematician and game theorist John Nash. Simply put, a Nash Equilibrium is a set of best-response strategies. For a two-player game, a Nash equilibrium is an outcome where player 2's strategy is the best response to player 1's strategy and player 1's strategy is the best response to player 2's strategy.
Finding the Nash equilibrium via this principle can be illustrated in the table of outcomes. In this example, player 2's best responses to player one are circled in green. If player 1 confesses, player 2's best response is to confess, since -6 is better than -10. If player 1 doesn't confess, player 2's best response is to confess, since 0 is better than -1. (Note that this reasoning is very similar to the reasoning used to identify dominant strategies.)
Player 1's best responses are circled in blue. If player 2 confesses, player 1's best response is to confess, since -6 is better than -10. If player 2 doesn't confess, player 1's best response is to confess, since 0 is better than -1.
The Nash equilibrium is the outcome where there is both a green circle and a blue circle since this represents a set of best response strategies for both players. In general, it is possible to have multiple Nash equilibria or none at all (at least in pure strategies as described here).04of 04
Efficiency of the Nash Equilibrium
You may have noticed that the Nash equilibrium in this example seems suboptimal in a way (specifically, in that it is not Pareto optimal) since it is possible for both players to get -1 rather than -6. This is a natural outcome of the interaction present in the game- in theory, not confessing would be an optimal strategy for the group collectively, but individual incentives prevent this outcome from being achieved. For example, if player 1 thought that player 2 would remain silent, he would have an incentive to rat him out rather than to stay silent, and vice versa.
For this reason, a Nash equilibrium can also be thought of as an outcome where no player has an incentive to unilaterally (i.e. by himself) deviate from the strategy that led to that outcome. In the example above, once the players choose to confess, neither player can do better by changing his mind by himself.