Game Theory · Classic Problems
The Prisoner’s Dilemma
Why two perfectly rational individuals can end up worse off by acting in their own self-interest — the most famous problem in all of game theory.
Imagine you and a friend are arrested for a crime. The police separate you and offer each of you the same deal: betray your partner and go free while they serve 10 years — or stay silent. If both stay silent, you each serve 1 year on a minor charge. If both betray, you each serve 5 years. You cannot communicate with each other. What do you do? This is the Prisoner’s Dilemma — a parable that reveals one of the deepest tensions in economics: individual rationality can produce collectively irrational outcomes.
📘 Key Term
The Prisoner’s Dilemma is a simultaneous game in which two players, each acting in their own rational self-interest, choose a strategy that leads to a mutually suboptimal outcome. The dominant strategy for both players is to defect (betray), even though mutual cooperation yields a better result for both.
The Payoff Matrix
| Player B: Cooperate (Silent) | Player B: Defect (Betray) | |
|---|---|---|
| Player A: Cooperate (Silent) | (-1, -1) ← Best collective | (-10, 0) |
| Player A: Defect (Betray) | (0, -10) | (-5, -5) ← Nash Equilibrium |
Why Defection is the Dominant Strategy
A dominant strategy is one that gives a better payoff regardless of what the other player does. Consider Player A’s reasoning:
• If B cooperates: A gets -1 (cooperate) vs 0 (defect) → A prefers to defect
• If B defects: A gets -10 (cooperate) vs -5 (defect) → A prefers to defect
Defection dominates cooperation in every scenario. By symmetry, B also defects. Both end up with (-5, -5) — worse than (-1, -1) from mutual cooperation. This is the tragedy of the Prisoner’s Dilemma.
💡 Key Insight
The Prisoner’s Dilemma shows that markets and individual rationality do not always produce efficient outcomes. This is why economists invoke the Prisoner’s Dilemma to justify regulation of cartels, environmental agreements, and arms control treaties — situations where cooperation benefits everyone but defection tempts each party.
Repeated Games and the Shadow of the Future
The one-shot Prisoner’s Dilemma always ends in defection. But what if the game is played repeatedly? In a repeated game, players can reward past cooperation and punish defection — which creates an incentive to cooperate.
Robert Axelrod’s famous tournaments in the 1980s found that Tit-for-Tat — cooperate on the first move, then copy your opponent’s last move — consistently outperformed all other strategies. The lesson: cooperation can emerge from self-interest when players interact repeatedly and value the future.
⚠️ Common Error
Students often write that the Nash Equilibrium in the Prisoner’s Dilemma is (Cooperate, Cooperate). It is not — it is (Defect, Defect). Mutual cooperation is the socially optimal outcome, but it is not a Nash Equilibrium because either player can improve by defecting unilaterally. Always distinguish between the socially optimal outcome and the Nash Equilibrium.
Real-World Examples
🌍 Climate Change: Each country benefits from others reducing emissions while not reducing its own. The Nash Equilibrium is low cooperation — the Prisoner’s Dilemma explains why climate agreements are so hard to enforce.
🏭 Price Wars: Two supermarkets both lowering prices is the Prisoner’s Dilemma — both earn lower profits than if they had held prices steady.
💊 Antibiotic Resistance: Each patient taking antibiotics unnecessarily is individually rational but collectively catastrophic — a public health version of the Prisoner’s Dilemma.
Q1. Using a payoff matrix, explain why the Prisoner’s Dilemma results in a Nash Equilibrium that is not Pareto optimal. [8 marks]
Answer: In the Prisoner’s Dilemma, each player has a dominant strategy to defect regardless of the other’s choice. The Nash Equilibrium is (Defect, Defect) yielding payoffs of (-5, -5). This is not Pareto optimal because both players could be made better off by moving to (Cooperate, Cooperate) yielding (-1, -1) — no one would be worse off and both would be better off. However, (Cooperate, Cooperate) is not a Nash Equilibrium because each player has an incentive to deviate and defect, earning 0 rather than -1. The conflict between individual rationality and collective optimality is the defining feature of the Prisoner’s Dilemma.
References
1. Tucker, A.W. (1950) ‘A Two-Person Dilemma’, unpublished notes, Stanford University.
2. Axelrod, R. (1984) The Evolution of Cooperation. Basic Books.
3. Rapoport, A. and Chammah, A.M. (1965) Prisoner’s Dilemma. University of Michigan Press.
4. Dixit, A. and Nalebuff, B. (1991) Thinking Strategically. W.W. Norton.
5. Fudenberg, D. and Tirole, J. (1991) Game Theory. MIT Press.