## Prisoners Dilemma

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**The Prisoner's Dilemma**

The game called The Prisoner's Dilemma is probably the most well known example of game theory. It is also an example of a normal form game (or strategic form game), which means that the players choose simultaneously. (Formally, a normal form game is a game that can be defined by only specifying the players, the strategies, and the payoffs.) One way to construct the game is the following. Two players, A and B, have been arrested (somewhere where the rule of law is somewhat substandard) and are kept in isolation. A prosecutor suggests A the following:

- If you confess and B does not, you will be set free as a sign of our
- If the both of you confess, you each get 2 years in prison.
- If B confesses and you do not, you get 10 years in prison while B is set free.
- If none of you confesses, we will frame you for a petty crime and you will each have to pay a small fine. At the same time, B gets the same suggestion. The two players cannot communicate with each other and therefore must consider a solution in solitude. Let us now identify the different elements that make this a game, in the game theoretical sense of the word.
- The players; Individuals A and B.
- Actions; For A: choose "Confess" or "Do not confess"; and similarly for B: choose "Confess" or "Do not confess.”
- Information; Both A and B know that the other has received the same offer, but they do not know how the other chooses.
- Strategies. Both A and B can only choose one of two different actions. Possible strategies for A are then "choose confess" or "choose not to confess", and similarly for B.
- Payoffs; Here we need to know the two players’ preferences. For simplicity, we assume that they have the same preferences and that they are as follows: 10 years in prison (-10), 2 years in prison (-2), a small fine (-1), and freedom (+1).

Many normal form games can be represented with a so-called payoff matrix, where one player’s strategies are displayed in the vertical direction and the other’s strategies in the horizontal direction. Their respective payoffs that correspond to certain strategy pair are then indicated in the squares. If we do this for the present game, we get the payoff matrix in Figure 13.1. Note that player A’s payoffs are to the left in the squares and player B’s are to the right.

Let us first look at the game from the perspective of player A. She does not know how player B will choose, but she does know that player B will choose either “Confess” or “Do not confess.” Say that player B would choose “Do not confess.” Then, obviously, the best thing player A can do is to choose “Confess,” since she will then get a utility of +1 (freedom) instead of -1 (a small fine). Now, say that player B chooses “Confess” instead. Then the best thing player A can do is still to choose “Confess,” since she will then get a utility of -2 (2 years in prison) instead of -10 (10 years in prison). Consequently, player A has a strategy that is the best one, independently of what player B chooses.

Such a strategy is called a **dominant strategy.** Player B’s problem is the same as player A’s, and hence it is a dominant strategy for player B as well to choose “Confess.” As a result, they both choose “Confess” and get two years in prison. This is so, even though it is possible for them both to get away with a small fine (if they both choose “Do not confess”). This is the dilemma. For both player A and B it is individually rational to confess, but acting that way they achieve an outcome worse than what is “collectively” possible. If they had been able to cooperate, they would both have been able to reach a higher utility level. Games that have properties such as this one are called Prisoner’s Dilemmagames. It could just as well be two countries deciding on whether to wage war on each other, two firms deciding on whether to start a price war or not, or two fishers deciding on whether to restrict their fishing or take the risk that the fish will go extinct. The players are kept from a rather good solution, because they choose their own individual best.

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