Solution to Prisoner's Dilemma
Solution to Prisoner's Dilemma
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.
Proof
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 $16.2$