# 11.3 The Prisoner’s Dilemma

Because of the complexity of oligopoly, which is the result of mutual interdependence among firms, there is no single, generally-accepted theory of how oligopolies behave, in the same way that we have theories for all the other market structures. Instead, economists use game theory, a branch of mathematics that analyzes situations in which players must make decisions and then receive payoffs based on what other players decide to do. Game theory has found widespread applications in the social sciences, as well as in business, law, and military strategy.

All games have three basic elements: players, strategies and payoffs.

• The players of the game are the agents actively participating in the game and who will experience outcomes based on the play of all players.
• The strategies are all of the possible strategic choices available to each player, they can be the same for all players or different for each player.
• The payoffs are the outcomes associated with every possible strategic combination, for each player.

There are different types of games: Games can also be single-shot or repeated.

• Single-shot games are played once and then the game is over.
• Repeated games are simultaneous move games played repeatedly by the same players. Below we see examples of single-shot games. We will discuss sequential games in section 11.7.

The prisoner’s dilemma is a scenario in which the gains from cooperation are larger than the rewards from pursuing self-interest. It applies well to oligopoly. The story behind the prisoner’s dilemma goes like this: There are two prisoners (players of the game), who have two strategies: confess and deny the crime, and serve a jail sentence (payoffs)

### The Prisoner’s Dilemma

Two co-conspiratorial criminals are arrested. When they are taken to the police station, they refuse to say anything and are put in separate interrogation rooms. Eventually, a police officer enters the room where Prisoner A is being held and says: “You know what? Your partner in the other room is confessing. Your partner is going to get a light prison sentence of just one year, and because you’re remaining silent, the judge is going to stick you with eight years in prison. Why don’t you get smart? If you confess, too, we’ll cut your jail time down to five years, and your partner will get five years, also.” Over in the next room, another police officer is giving exactly the same speech to Prisoner B. What the police officers do not say is that if both prisoners remain silent, the evidence against them is not especially strong, and the prisoners will end up with only two years in jail each.

The game theory situation facing the two prisoners is in Fig 11.1 below.

 Prisoner B Prisoner A Remain Silent (cooperate with other prisoner) Confess (do not cooperate with other prisoner) Remain Silent (cooperate with other prisoner) A gets 2 years, B gets 2 years A gets 8 years, B gets 1 year Confess (do not cooperate with other prisoner A gets 1 year, B gets 8 years A gets 5 years, B gets 5 years

## Understanding the situation from the figure: Dominant Strategy and Nash Equilibrium

First consider the choices from Prisoner A’s point of view. If A believes that B will confess, then A should confess, too, so as to not get stuck with the eight years in prison. However, if A believes that B will not confess, then A will be tempted to act selfishly and confess, so as to serve only one year. The key point is that A has an incentive to confess regardless of what choice B makes! B faces the same set of choices, and thus will have an incentive to confess regardless of what choice A makes. To confess is called the dominant strategy. It is the strategy an individual (or firm) will pursue regardless of the other individual’s (or firm’s) decision. The result is that if prisoners pursue their own self-interest, both are likely to confess, and end up doing a total of 10 years of jail time between them.

The game is called a dilemma because if the two prisoners had cooperated by both remaining silent, they would only have had to serve a total of four years of jail time between them. If the two prisoners can work out some way of cooperating so that neither one will confess, they will both be better off than if they each follow their own individual self-interest, which in this case leads straight into longer jail terms.

The solution concept most commonly used in game theory is the Nash Equilibrium concept. A Nash Equilibrium is an outcome where, given the strategy choices of the other players, no individual player can obtain a higher payoff by altering their strategy choice. An equivalent way to think about Nash Equilibrium is that it is an outcome of a game where all players are simultaneously playing a best response to the others’ strategy choices. The Nash Equilibrium of the game is that both players confess and serve a five-year sentence.