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Game Theory

The Prisoner's Dilemma

The Prisoner's Dilemma is a game with two strategies available to players: cooperate or defect. Each player has a dominant strategy to defect, and the Nash equilibrium produces a worse outcome for both players than if they had cooperated with one another.

The classic example in game theory is the prisoner's dilemma, a game that demonstrates why two rational players might not cooperate even if it appears to be in their best interest. Consider the example of two men, Walter and Jesse, who are driving on the highway with two unregistered weapons in their car when they are pulled over by police. Although the police can charge the men for having unregistered weapons and a conviction is guaranteed, they believe that Walter and Jesse are guilty of something much more serious: being manufacturers and sellers of large quantities of narcotics. Unfortunately, the police lack the evidence to make the narcotics charges stick. They need one of the two to betray the other. Therefore, they arrest Walter and Jesse, put them in separate interrogation rooms, and offer each the same deal:

We know you are both manufacturing and selling large quantities of narcotics. A conviction on drug charges will get you a sentence of 25 years. But we're not sure we have enough evidence to put you away for that crime. We are, however, confident we can convict you on a weapons charge that will land you in jail for three years. So, we are willing to make you a deal. If you confess to manufacturing and selling narcotics and testify against your partner, we will give you only one year in jail, and your partner will receive a sentence of 25 years. But if you deny the crime and your partner confesses, he will get only one year while you receive a sentence of 25 years. If you both confess and save us the cost of a trial, each of you will get a sentence of only 10 years.
The possible outcomes for this game can be shown by using a payoff matrix showing the prisoners' two choices: to deny committing the crime (and not testify against the other person) or to confess to the crimes (and take the deal offered).
In the prisoner's dilemma game, the payoffs, or consequences of a strategy, are arranged so that each player has the greatest incentive to confess (shown by the green check marks) and receive the shortest prison sentence. This situation leads to a Nash Equilibrium in which both players receive a worse outcome than if they had cooperated and not confessed.
If both Jesse and Walter remain quiet and deny any participation in manufacturing and selling drugs, they both will only be charged with having unregistered weapons and will each go to jail for three years. Thus, in the upper-left quadrant of the payoff matrix (where the row is “Deny” and the column is “Deny”), both get a payoff of three years. Conversely, if both confess to being involved in the drug trade, both receive a sentence of 10 years as seen on the bottom right quadrant of the matrix (where the row is “Confess” and the column is “Confess”).

The other two quadrants in the payoff matrix represent the cases where one of the players confesses while the other remains silent. In the upper right, Jesse (column) confesses and Walter (row) denies involvement. Walter receives 25 years and Jesse walks away after only one year. The opposite is the case in the lower left of the payoff matrix; Walter (row) confesses and gets a one-year sentence, while Jesse (column) remains quiet and gets sent away for 25 years.

Solving a Game

A game can be solved by looking for each player's best response, given the other player's choice.

The easiest way to determine a strategy for playing a game is to look for a player's best response to each option his or her opponent may select. The player's best response is the strategy that maximizes the player's well-being, given the other player's choice. For example, using Jesse and Walter's Prisoner's Dilemma, consider what Jesse will do if he believes that Walter will remain silent and deny their involvement in manufacturing and selling drugs. If Walter chooses to deny, Jesse can also deny and receive a three-year sentence for the weapons charge. Alternatively, he can confess, testify against Walter, and receive only a one-year sentence in exchange for his testimony. Jesse's best choice is to confess, because this reduces his prison sentence.

Now, suppose Jesse instead believes that Walter will confess to manufacturing and selling drugs. Jesse can deny any involvement and receive a sentence of 25 years, or he can also confess and receive 10 years. In this case, as in the last case, his best response is to confess.

There are two important things to note here. First, this analysis considers only the options listed in the payoff matrix and assumes that, in this case, the players only care about the length of sentence they receive. The payoff matrix must contain all of the relevant information about payoffs to be useful in determining which choice makes each player better or worse off. If each cared about their reputations, for example, that would have to be added into the game through the payoff matrix.

Second, in this particular game the payoffs are symmetric, meaning the payoffs are the same for both players if the choices are reversed. In this example, the result is that the payoffs to Jesse when Walter denies or confesses are exactly the same as those to Walter when Jesse denies or confesses. With both of these assumptions, Walter will choose to confess when he believes that Jesse will deny, and he will also choose to confess if he thinks that Jesse will also confess. Therefore, the choice to confess is Walter's best response in both cases.

Dominant Strategies

A dominant strategy is a strategy that is the best choice regardless of the option chosen by the player's opponent.

In the prisoner's dilemma, the best response is for Jesse to confess regardless of whether Walter denies involvement in the drug industry or confesses to it. This means that for Jesse, "Confess" is a dominant strategy. A dominant strategy is the best response for a player regardless of what the other player chooses. Because "Confess" is the best option for Jesse whether Walter chooses "Confess" or "Deny," "Confess" is a dominant strategy for Jesse.

Because the game is symmetric, "Confess" is a dominant strategy for Walter as well. When Jesse chooses to deny, Walter is better off confessing. When Jesse opts to confess, Walter receives a lower sentence from confessing. So, no matter what Jesse does, Walter should confess.

In the prisoner's dilemma, both players will have a dominant strategy. However, not all games have dominant strategies. In some games, only one of the players has a dominant strategy. In other games, neither player has a dominant strategy. Thus, it is important to follow the "best response" method to determine how each player will act.

Nash Equilibrium

A Nash equilibrium is a game outcome in which no participant can gain by a change of strategy if the strategies of others remain unchanged.

Given that both Jesse and Walter have dominant strategies to confess, it is reasonable to expect the outcome of the game will be both players choosing the dominant strategy. This is a Nash equilibrium: a game outcome in which no participant can gain by a change of strategy if the strategies of others remain unchanged. Both players are playing their best strategies, given the choice made by their opponent. This equilibrium is named after John Nash, who won the Nobel Prize in Economics for his work studying game theory.

At this Nash equilibrium, neither player has an incentive to change his mind. If Jesse and Walter both confess, neither has a reason to regret his choice, given the actions of the other player. Because Walter is confessing, Jesse cannot receive a lighter sentence than he receives from also confessing. Therefore, "Confess" is his best choice, given Walter's decision. The same holds for Walter. If Jesse confesses, Walter will receive the lightest sentence by confessing as well. In order to determine if a game has a Nash equilibrium, check marks can be placed in the box of each player's best strategy on the payoff matrix. If a box has two check marks in it, this indicates that each player is playing his best option, given the strategy chosen by his opponent. A payoff matrix for this example would have two check marks in the "Confess, Confess" box, showing that it is the Nash equilibrium.

This game is referred to as a dilemma because, despite both players having a dominant strategy in which neither could do better by unilaterally changing strategy, the dominant strategy results in a worse outcome for both than if they had both selected the other option. By both confessing, Walter and Jesse each serve 10 years in jail, but if they had both denied the charge, they would have served only three years each. The dilemma is that the best outcome is for both prisoners to deny the charges, even though the rational decision is to confess when each prisoner considers his own self-interest.