Prisoner's Dilemma

Canonical 2x2 game where individually rational defection leads to a Pareto-inferior outcome.

Ask the Game theory assistant 1 min read · Updated September 9, 2026

Definition

A symmetric $2\times 2$ game with payoffs satisfying $T>R>P>S$ and $2R>T+S$ , where $C$ and $D$ denote Cooperate and Defect.

A standard payoff matrix is $\begin{pmatrix}(R,R)&(S,T)\\(T,S)&(P,P)\end{pmatrix}$ with e.g. $(T,R,P,S)=(5,3,1,0)$ .

Intuition

Each player strictly prefers to defect regardless of the opponent, yet mutual cooperation is strictly better than mutual defection.

It exposes the tension between individual incentives and collective welfare — rationality yields $(P,P)$ rather than $(R,R)$ .

Worked example

With $(T,R,P,S)=(5,3,1,0)$ , defecting yields $5$ vs cooperating $3$ if the other cooperates, and $1$ vs $0$ if the other defects, so $D$ strictly dominates $C$ .

Real-world analogs: arms races, cartel cheating, doping in sports, and climate-change free-riding.

The math

In the one-shot game $D$ is strictly dominant; the unique Nash equilibrium is $(D,D)$ . In infinitely repeated play with discount factor $\delta\ge (T-R)/(T-P)$ , cooperation can be sustained by trigger strategies (Folk Theorem).

Axelrod's tournaments showed that simple reciprocal strategies like Tit-for-Tat often prevail in iterated play.

Where it is used

Models public-goods provision, oligopoly collusion stability, and international treaty enforcement.

It motivates mechanism design, reputation, and long-run relationships.

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