Although a large branch of game theory is devoted to the study of expected utility, we generally consider each player’s payoffs as a ranking of his most preferred outcome to his least preferred outcome. In the prisoner’s dilemma, we assumed that players only wanted to minimize their jail time. Game theory does not force players to have these preferences, as critics frequently claim. Instead, game theory analyzes what should happen given what players desire. So if players only want to minimize jail time, we could use the negative number of months spent in jail as their payoffs. This preserves their individual orderings over outcomes, as the most preferred outcome is worth 0, the least preferred outcome is -12, and everything else logically follows in between.
Interestingly, the cardinal values of the numbers are irrelevant to the outcome of the prisoner’s dilemma.
To be clear, this preference ordering exclusively over time spent in jail is just one way the players may interpret the situation. Suppose you and a friend were actually arrested and the interrogator offered you a similar deal. The results here do not generally tell you what to do in that situation, unless you and your friend only cared about jail time. Perhaps your friendship is strong, and both of you value it more than avoiding jail time. Since confessing might destroy the friendship, you could prefer to keep quiet if your partner kept quiet, which changes the ranking of your outcomes. Your preferences here are perfectly rational. However, we do not yet have the tools to solve the corresponding game. We will reconsider these alternative sets of preferences in Lesson 1.3.
Indeed, the possibility of alternative preferences highlights game theory’s role in making predictions about the world. In general, we take a three step approach:
- Make assumptions.
- Do some math.
- Draw conclusions.
For the given payoffs in the #prisoners_dilemma, we have seen an example of #strict_dominance. We say that a strategy x strictly dominates strategy y for a player if strategy x provides a greater payoff for that player than strategy y regardless of what the other players do. In this example, confessing strictly dominated keeping quiet for both players. Unsurprisingly, players never optimally select strictly dominated strategies—by definition, a better option always exists regardless of what the other players do.