Prisoner’s Dilemma

Definition

Prisoner's Dilemma (PD)

A symmetric two-player game in which each player chooses Cooperate (C) or Defect (D), with row-player payoffs $R$ (both cooperate), $S$ (I cooperate, partner defects), $T$ (I defect, partner cooperates) and $P$ (both defect), satisfying $T>R>P>S$

The names: Reward for mutual cooperation, Sucker’s payoff, Temptation to defect, Punishment for mutual defection.

Intuition

Self-interest destroys cooperation

Whatever my partner does, I do better by defecting: against a cooperator I get $T$ instead of $R$, against a defector $P$ instead of $S$. My partner reasons the same way. So we both defect and get $P$ each, although we both prefer $R$. Each player’s reasoning is individually sound; the collective result is bad. That gap between individual rationality and collective welfare is what makes it a dilemma of cooperation.

Two motives push towards defection:

• Greed: $T>R$, defecting pays more when the partner cooperates.
• Fear: $P>S$, defecting pays more when the partner defects.

The PD has both, which makes D a dominant strategy. Its problem is one of incentives: even an agent certain its partner will cooperate is tempted to defect. Compare the Stag Hunt, which has fear but no greed, and where the problem is trust.

Two pricing bots

	partner C	partner D
C	3, 3	0, 5
D	5, 0	1, 1

Two autonomous agents owned by different companies (pricing bots in one market, scheduling agents sharing a cluster) each maximise their owner’s payoff. Game theory predicts (D,D), with payoff 1 each, where (C,C) would have given 3 each.

Mathematical Formulation

Against a partner who cooperates with probability $p$ (or, in a population, a fraction $x$ of cooperators):

$f_C-f_D=p(R-T)+(1-p)(S-P)$

where:

• $f_C=pR+(1-p)S$: expected payoff of cooperating
• $f_D=pT+(1-p)P$: expected payoff of defecting
• $R-T<0$ by greed, and $S-P<0$ by fear

Both terms are negative, so $f_C-f_D<0$ for every $p$: defection always earns more. In the Replicator Dynamics this means the fraction of cooperators falls from any starting point. $x=1$ (all cooperate) is a rest point but unstable, since a single defector earns more and spreads; $x=0$ (all defect) is the stable rest point and corresponds to the Nash equilibrium (D,D).

Key Properties / Variants

• Donation game form (AGAI L02): $R=b-c$, $S=-c$, $T=b$, $P=0$ with $b>c>0$ (cooperating costs $c$ and gives the partner $b$).
• Unique Nash equilibrium (D,D), in dominant strategies. See Nash Equilibrium.
• (D,D) is the only outcome that is not Pareto optimal; (C,C) is the social optimum. See Pareto Optimality.
• One-shot versus repeated. Played once, defection is the rational choice. Played repeatedly with memory, reciprocal strategies such as Tit-for-Tat and Win-Stay Lose-Shift can make cooperation pay, and Axelrod’s 1980 tournament was won by Tit-for-Tat. See Repeated Game.
• Extra condition for repeated play (background beyond the course material): many definitions also require $2R>T+S$, so that taking turns exploiting each other pays less than steady mutual cooperation.
• Populations. Without memory or reciprocity, adaptive populations (imitation, simple reinforcement learning) drift to all-defect, matching the rational prediction.
• Why it matters for AI agents. Self-interested agents deployed by different owners can end up in PD-like situations: competing bots, negotiating LLM agents, shared resources.

Connections

• Solution concepts: Nash Equilibrium, Pareto Optimality
• Contrast: Stag Hunt (fear without greed; coordination problem)
• Escapes from the dilemma: Repeated Game (memory, reciprocity)
• Population view: Replicator Dynamics
• Setting: Multi-Agent System, Rational Agent

Appears In

• AGAI-L01 - Agents, Multiagent Systems and Agentic AI
• AGAI-Lab01 - Simple Agents, Complex Dynamics
• AGAI-L02 - Game Theory and Population Dynamics
• AGAI-L03 - Intro to Multi-Agent Reinforcement Learning (iterated PD as a stochastic game)