Definition:Prisoner's Dilemma

Definition

The Prisoner's Dilemma is an instance of a class of games whose mechanics are as follows:

Two suspects in a crime are interviewed separately.

If they both confess, they will be sentenced to $3$ years in prison.

If only one of them confesses, he will be freed and used as a witness against the other, who will then receive a sentence of $4$ years.

If neither one confesses, they will both be sentenced for lesser crime, and each will spend $1$ year in prison.

Payoff Table

The payoff table of the prisoner's dilemma is as follows:

		$\text B$
$\text A$		$\begin{array} {r {{|}} c {{|}} } & \text{Don't Confess} & \text{Confess} \\ \hline \text{Don't Confess} & -1, -1 & 0, -4 \\ \hline \text{Confess} & -4, 0 & -3, -3 \\ \hline \end{array}$

Analysis

Analysis of Prisoner's Dilemma

Solution

From the payoff table:

		$\text B$
$\text A$		$\begin{array} {r {{|}} c {{|}} } & \text{Don't Confess} & \text{Confess} \\ \hline \text{Don't Confess} & -1, -1 & 0, -4 \\ \hline \text{Confess} & -4, 0 & -3, -3 \\ \hline \end{array}$

Each player gains by cooperating, and the best outcome is for neither player to confess.

However, because each player then has the opportunity to improve his position by changing his strategy to confessing, $\left({\text{Don't Confess}, \text{Don't Confess} }\right)$ is not a Nash equilibrium.

Thus there is a single Nash equilibrium:

$\left({\text{Confess}, \text{Confess} }\right)$

Sources

• 1994: Martin J. Osborne and Ariel Rubinstein: A Course in Game Theory ... (previous) ... (next): $2.3$: Examples: Example $16.2$
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): tit-for-tat
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): prisoner's dilemma (in game theory)