Normal-Form Payoff Matrix
The game is typically written with payoffs $(u_R, u_C)$ for row and column players:
| Cooperate | Defect | |
|---|---|---|
| Cooperate | $(R, R)$ | $(S, T)$ |
| Defect | $(T, S)$ | $(P, P)$ |
The dilemma requires $T > R > P > S$ (temptation, reward, punishment, sucker) and typically $2R > T + S$ so mutual cooperation exceeds the average of exploitation outcomes.
Dominant Strategy and Nash Equilibrium
Defection strictly dominates cooperation for each player: regardless of the opponent’s choice, $T > R$ and $P > S$. Iterated elimination of strictly dominated strategies leaves (Defect, Defect) as the unique survivor. This profile is the unique Nash equilibrium, yielding payoffs $(P, P)$.
The Pareto-optimal outcome $(R, R)$ is unreachable through rational self-interest in the one-shot game, illustrating Pareto inefficiency of equilibrium — a defining feature of social dilemmas.
Quantitative Example
A standard parameterisation sets $T=5, R=3, P=1, S=0$. The payoff matrix becomes:
| C | D | |
|---|---|---|
| C | (3, 3) | (0, 5) |
| D | (5, 0) | (1, 1) |
Social surplus under mutual cooperation is $6$; under mutual defection it is $2$. The efficiency loss ratio is $1 - 2/6 \approx 67\%$.
Mixed-Strategy Equilibrium
Although (Defect, Defect) is the unique pure-strategy Nash equilibrium, one can verify no profitable mixed deviation exists. If the column player cooperates with probability $p$, the row player’s expected gain from defecting over cooperating is $(T-R)p + (P-S)(1-p) > 0$ for all $p \in [0,1]$ under the dilemma constraints, confirming that defection is always a strict best response.