Prisoner's Dilemma
The game where defecting pays whatever the other player does, so the only equilibrium is mutual defection, though both would be better off cooperating. The course calls it the battle line.
Prisoner's Dilemma
Two players each choose to cooperate (C: contribute, hold back) or defect (D). Payoffs are ordinal, 4 best to 1 worst, written (row, column).
| Column C | Column D | |
|---|---|---|
| Row C | 3, 3 | 1, 4 |
| Row D | 4, 1 | 2, 2 |
The ordering is temptation > reward > punishment > sucker (T > R > P > S). In the terms of Greed and fear, it has both: if you cooperate I gain by defecting (greed), and if you defect I lose by cooperating (fear).
Equilibria
Defecting is best whatever the other player does, so it dominates. There is one equilibrium, both defect, and no mixed one. Both cooperating is better for both than the equilibrium, but neither can get there alone.
the game has a unique Nash equilibrium (Confess, Confess).
— Martin J. Osborne and Ariel Rubinstein, A Course in Game Theory (1994), p. 16
Osborne and Rubinstein describe each player as having an incentive to be a "free rider" (p. 16).
Names
Free-rider problem, n-person prisoner's dilemma. The course calls it the battle line: breaking ranks pays whatever the rest of the line does. The n-person versions are the Tragedy of the commons on the taking side and the Public goods game on the building side (Taking and building). More aliases: Names of collective action problems.
Typical failure and fix
- Failure: universal shirking. The good isn't provided, or the resource is run down.
- Fix: change the payoffs for everyone: sanctions, rewards, monitoring, repetition. In the indefinitely repeated game, cooperation becomes one equilibrium among very many, which leaves a coordination problem (Osborne and Rubinstein p. 134, pp. 143–144). Finite repetition doesn't help (p. 134). Sanctions only work if they are accurate enough: in Ostrom's herders example a regulator who sanctions correctly 70% of the time leaves the herders in a prisoner's dilemma (Ostrom, Governing the Commons, 1990, p. 11).
- What doesn't help: assurance. Knowing others will cooperate doesn't change anything: defecting still pays more.
Cases
- Open-access fisheries, such as the Newfoundland cod. Ostrom (p. 48) says open-access appropriation "frequently can be characterized as a PD game"; limited-access cases often aren't.
- The public-goods experiment: a pot doubled and shared equally.
- Arms races fit here and as a Stag hunt, depending on the payoffs.
- Ozone depletion looks like a battle line, but Sandler reads the Montreal Protocol as a move towards a privileged group: twelve countries caused over 78% of emissions, substitutes existed, and for the United States cutting CFCs was a dominant strategy, so "the major emitters could 'privilege' the rest of the world" (Sandler, Global Collective Action, 2004, pp. 217–218). Many of the cutbacks would have happened without the treaty (p. 218).
Not every collective action problem is a prisoner's dilemma. Michael Taylor (The Possibility of Cooperation, 1987, p. 18) warns against equating the two. Dixit, Skeath and Reiley (Games of Strategy, 4th ed., 2015, p. 430) add that in large groups public-good provision "almost always" takes this form. See Patterns of collective action.
