Plate i in print

The Dilemma

Nash; the prisoner's dilemma and the equilibrium of mutual betrayal, where each player, reasoning faultlessly, ends worse off than cooperation would have left him.

In any tangle of strategies there is a resting point — a choice for each player from which no one, acting alone, can do better; yet the resting point of the prisoner's dilemma is mutual betrayal, where both, reasoning faultlessly, end worse off than had they trusted, and neither dares to be the first to trust.

— the argument of this plate, after John Nash, 1950

The volume's subject is the strategic situation — the predicament in which what is best for me depends on what you do, and what is best for you depends on what I do. Ordinary choice theory imagines a person optimizing against fixed circumstances; but when the circumstances are other people, each reasoning about the others, the problem changes utterly, and a new mathematics is required. That mathematics is game theory, and its central object is the equilibrium, the resting point of mutual strategy, given its general form by John Nash. The volume opens with the equilibrium and with the single most consequential game of all, the prisoner's dilemma, because together they pose the question the rest of the volume must answer: if each player rationally pursues his own interest, why does cooperation, which would serve everyone, so often fail?

A game, in the technical sense, is any situation with players, the strategies available to each, and the payoffs that result — where each player's payoff depends not only on his own choice but on the choices of all the others. This interdependence is the whole difficulty: I cannot simply choose my best action, because my best action depends on yours, which depends on mine, in a regress with no obvious floor. Nash cut the knot with a definition of breathtaking simplicity. An equilibrium is a combination of strategies, one for each player, such that no player can do better by changing his own strategy alone, given what everyone else is doing. At an equilibrium everyone is playing a best response to everyone else; no one has any regret, and no one wishes to move. And Nash proved that every finite game possesses at least one such equilibrium. The strategic tangle always has a resting point, and with that proof game theory became a general science of interaction rather than a collection of puzzles.1

1. A game = players, the strategies open to each, and payoffs that depend on everyone's choices. The interdependence is the difficulty: my best action depends on yours and yours on mine. Nash's equilibrium cuts the knot: a combination of strategies from which no player can do better by deviating alone — everyone playing a best response to everyone, no one wishing to move. Nash proved every finite game has at least one. With that, game theory became a general science of interaction.

The most famous and consequential game is the prisoner's dilemma, and its lesson is unsettling. Picture two accomplices held separately, each offered the same deal: betray the other and, if the other stays silent, go free while the other takes the heavy sentence; but if both betray, both are punished heavily; and if both stay silent, both get off lightly. Now reason as one of the prisoners. If the other stays silent, I do better to betray, going free rather than taking even the light sentence. If the other betrays, I again do better to betray, taking the heavy sentence rather than the worst one. Whatever the other does, betrayal is my better move — it is, in the language of the theory, a dominant strategy. So both betray. Yet mutual betrayal is worse for both than mutual silence would have been. The unique equilibrium of the game is precisely the outcome that leaves everyone worse off, and there is no escaping it by clearer thinking, for it is the clear thinking that produces it.2

2. The prisoner's dilemma: two accomplices held apart, each able to betray or stay silent. Betray while the other is silent, go free; both betray, both punished heavily; both silent, both punished lightly. Reasoning it through: whatever the other does, betrayal pays more — it is a dominant strategy. So both betray, though both would be better off staying silent. The unique equilibrium is the outcome that makes everyone worse off, and clearer thinking only confirms it.
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Here is the heart of the matter, and the reason the prisoner's dilemma haunts the social sciences. The mutual-betrayal outcome is not a blunder. Each prisoner reasons flawlessly; each plays his genuinely best response; the equilibrium is the product of impeccable individual rationality. And yet the result is collectively irrational — both would be better off at the cooperative outcome, which individual rationality cannot reach. Individual rationality and collective rationality come apart, and the gap between them is not closed by anyone being smarter or more careful, for it is exactly the careful reasoning that opens it. The invisible hand, which elsewhere in the table turns private interest to public good, here turns it to public harm: each pursuing his own advantage, all are made worse off than they need be. The prisoner's dilemma is the invisible hand's dark mirror — the standing proof that self-interest and the common good, so often allies, can be flatly opposed.3

3. The unsettling core: mutual betrayal is not a blunder but the product of flawless individual rationality — each plays his best response, yet the result is collectively irrational, worse for both than cooperation. Individual and collective rationality come apart, and careful reasoning only widens the gap. The invisible hand, elsewhere turning private interest to public good, here turns it to public harm. The prisoner's dilemma is its dark mirror: self-interest and the common good, flatly opposed.

The dilemma's importance lies in how often the world wears its shape. Two firms would both profit by keeping prices high, but each is tempted to undercut the other, so both cut and both lose, in a price war. Two nations would both be safer spending less on arms, but each fears the other's buildup, so both arm and both are poorer and no safer, in an arms race. Fishers sharing a sea, herders sharing a pasture, factories sharing an airshed: each gains by taking more or fouling more, and the shared thing collapses — the tragedy of the commons, treated in its own volume, is a prisoner's dilemma played by many hands at once. Wherever individual incentive points toward defection while collective welfare requires cooperation, the dilemma is at work, and its grim equilibrium looms over the outcome. Much of the rest of this volume is the study of how, and whether, that equilibrium can be escaped.4

4. The dilemma is everywhere the world has its shape. Price wars: each firm undercuts, both lose. Arms races: each arms from fear, both poorer and no safer. Overfishing, overgrazing, pollution: each takes more, the shared resource collapses — the tragedy of the commons is a many-player prisoner's dilemma. Wherever private incentive points to defection while collective welfare needs cooperation, the dilemma operates. The volume's other plates ask how, and whether, its equilibrium can be escaped.

Taken alone, the prisoner's dilemma is a counsel of despair: rational, self-interested agents will not cooperate, however much cooperation would serve them, and the equilibrium of their meeting is mutual harm. Were this the last word, the prospects for cooperation among the self-interested — which is to say, for most of social and economic life — would be bleak indeed. But it is not the last word, and the volume traces the routes out. One can change the game by binding oneself, issuing commitments and threats that alter the payoffs, as the next plate shows with Schelling. One can ask which of those threats are even believable, as Selten does. One can let strategies compete not in a single play but across a whole population over time, as Maynard Smith does. And, above all, one can play the game again and again, so that the shadow of future encounters reaches back to discipline the present — the discovery, at the volume's flagship, that repetition can rescue cooperation from the dilemma itself. The escape begins by changing the game. The next plate is the commitment.5

5. Alone, the dilemma counsels despair — the self-interested will not cooperate, and their equilibrium is mutual harm. But it is not the last word. The routes out: change the game by binding oneself (Schelling); ask which threats are credible (Selten); let strategies compete across a population (Maynard Smith); and above all, play again and again, so the shadow of the future disciplines the present (Axelrod, the flagship). The escape begins by changing the game. Next: the commitment.

The volume opens with the strategic predicament and its central result. A game is any situation where each player's fortune turns on the others' choices, and Nash showed that every such tangle has an equilibrium — a resting point where no one can gain by moving alone. But the most important game, the prisoner's dilemma, delivers a hard verdict: when betrayal is each player's dominant strategy, both betray, and the unique equilibrium leaves everyone worse off than cooperation would have. This is not error but flawless individual rationality producing collective ruin — the invisible hand's dark mirror — and it recurs wherever private incentive points away from the common good, in price wars and arms races and the tragedy of the commons. Taken alone it is a counsel of despair. The rest of the volume is the search for the way out, which begins by changing the game itself. The next plate is the commitment.