Prisoner's Dilemma — Definition, Formula & Examples

The Prisoner's Dilemma is a two-player game in which each player does better by betraying the other, no matter what the other does. Yet if both betray, both end up worse off than if both had cooperated.

A Prisoner's Dilemma is a symmetric two-player, simultaneous-move game with strategies Cooperate (C) and Defect (D). The payoffs are T (temptation, for defecting against a cooperator), R (reward, for mutual cooperation), P (punishment, for mutual defection), and S (sucker's payoff, for cooperating against a defector). They satisfy $T > R > P > S$ . Defect is then a strictly dominant strategy, and the unique Nash equilibrium $(D, D)$ is Pareto-dominated by $(C, C)$ . For the repeated game, the extra condition $2R > T + S$ is usually imposed. It ensures that mutual cooperation beats taking turns exploiting each other.

Key Formula

T > R > P > S \qquad \text{and} \qquad 2R > T + S

Where:

$T$ = Temptation payoff: you defect while the other player cooperates
$R$ = Reward payoff: both players cooperate
$P$ = Punishment payoff: both players defect
$S$ = Sucker's payoff: you cooperate while the other player defects

How It Works

You write the game as a payoff matrix. Rows are Player 1's choices, columns are Player 2's choices, and each cell lists (Player 1's payoff, Player 2's payoff). To analyze it, fix the opponent's choice and ask which of your own choices pays more. Then repeat for the other opponent choice. If one strategy wins in every case, it is dominant. When both players use their dominant strategies, you get the equilibrium, and you can compare it with the outcome that is best for the group.

Worked Example

Problem: Two players each choose Cooperate (C) or Defect (D) at the same time. The payoffs, with Player 1's listed first, are (C, C) = (3, 3), (C, D) = (0, 5), (D, C) = (5, 0), (D, D) = (1, 1). Find each player's best strategy and the Nash equilibrium. Then compare the equilibrium with mutual cooperation.

Step 1: Set up the matrix. Suppose Player 2 cooperates. Player 1 gets 3 from C and 5 from D. Since 5 > 3, Player 1 should defect.

\begin{array}{c|cc} & C & D \\ \hline C & (3,3) & (0,5) \\ D & (5,0) & (1,1) \end{array}

Step 2: Now suppose Player 2 defects. Player 1 gets 0 from C and 1 from D. Since 1 > 0, Player 1 should defect again.

u_1(D, D) = 1 > 0 = u_1(C, D)

Step 3: Defect is better for Player 1 in both cases, so it is a strictly dominant strategy. The game is symmetric, so the same holds for Player 2. The unique Nash equilibrium is (D, D), with payoffs (1, 1).

Step 4: Compare total payoffs. At the equilibrium, the players earn 2 in total. Mutual cooperation would give them 6, and each player would earn more.

1 + 1 = 2 \quad < \quad 3 + 3 = 6

Answer: Both players defect, so the Nash equilibrium is (D, D) with payoffs (1, 1). Each player would earn 3 instead of 1 if both cooperated. Rational individual choices produce a worse result for both.

Another Example

Problem: Two suspects are questioned separately. If both stay silent, each serves 1 year. If both confess, each serves 5 years. If one confesses and the other stays silent, the confessor goes free and the silent one serves 10 years. Is this a Prisoner's Dilemma?

Step 1: Fewer years is better, so write each payoff as negative years. Staying silent is cooperating with your partner.

T = 0, \quad R = -1, \quad P = -5, \quad S = -10

Step 2: Check the ordering condition.

0 > -1 > -5 > -10

Step 3: Check the repeated-game condition.

2R = -2 > -10 = T + S

Answer: Yes. Both conditions hold, so this is a Prisoner's Dilemma. Confessing is each suspect's dominant strategy.

Why It Matters

The Prisoner's Dilemma is the standard opening example in intermediate microeconomics and game theory courses. It is used to model price wars between firms, arms races, and overfishing of shared resources. Evolutionary biologists also use the repeated version to study how cooperation can arise among self-interested organisms.

Common Mistakes

Mistake: Assuming the Nash equilibrium is the best outcome for the players.

Correction: A Nash equilibrium only means that no player can gain by changing strategy alone. In the Prisoner's Dilemma, the equilibrium (D, D) is worse for both players than (C, C). That gap is the whole point of the dilemma.

Mistake: Picking the larger number when payoffs are given as years in prison.

Correction: Years in prison are costs, so smaller is better. Convert them to negative numbers, or minimize instead of maximize, before you compare strategies.

Related Terms

Game Theory — The field that studies strategic games like this
Fair Game — Another game-theory concept, defined by expected payoffs
Optimization — Each player optimizes their own payoff individually
Graph Theory — Models repeated dilemmas played across social networks