Numéraire
Volume the Twenty-fifth · of the table of twenty-six
The Behaviors · Group VI · № 25 · Gm

The Game

strategy among the self-interested

The game is the second of the behaviors, and it concerns the most intricate of them: what happens when each person's best move depends on what everyone else does, so that no one can simply choose well in isolation. Choice studied the single mind weighing its options; the game studies minds entangled, each reasoning about the others who are reasoning about it. The volume opens with the predicament in its starkest form — the prisoner's dilemma, where two parties, each acting with flawless self-interest, reach an outcome worse for both than cooperation would have been, and where the resting point of their strategy, the Nash equilibrium, is mutual betrayal. That very equilibrium was the widening of an older and narrower one, for the games of pure opposition — where one player's gain is exactly another's loss — John von Neumann had already proved to hold a rational value, won if need be by playing unpredictably, the theorem with which the whole subject began. That dilemma poses the question the volume must answer, the question the author set it: when each best move depends on every other, how does cooperation hold among the self-interested? The plates trace the routes out. One may change the game by binding one's own hands, for the power to commit can be a source of strength (Schelling); one must then ask which threats and promises are credible, which a clear-eyed opponent should believe (Selten); one may let strategies compete not by reason but by survival, across a whole population over time (Maynard Smith). And the flagship gives the deepest answer: that when the dilemma is played not once but again and again, between parties who expect to meet in the future, cooperation can hold among the purely self-interested — sustained by no altruism and no authority, but by the long shadow that tomorrow casts back upon today (Axelrod). The game is the connective tissue of the whole table, the strategic logic that runs beneath markets and firms and contracts and the commons; and it shows that the self-interested, meeting again and again, can build out of the dilemma itself a working peace.

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The argument of the volume

Begin where the volume begins, with the predicament itself. A game, in the technical sense, is any situation in which each player's payoff depends on the choices of all the others, so that no one can choose well without reasoning about how the rest will choose; and John Nash gave such situations their organizing idea, the equilibrium — a combination of strategies from which no player can do better by deviating alone — proving that every finite game has one. This widened a result John von Neumann had reached two decades earlier for the special case of pure opposition — the zero-sum games, in which one player's gain is exactly the other's loss — where a rational value, and an optimal way of securing it, always exist; Nash's achievement was to show that an equilibrium persists even when the players' interests are not perfectly opposed. But the most consequential game, the prisoner's dilemma, delivers a hard verdict: when betrayal is each player's dominant strategy, both betray, and the unique equilibrium of their flawless self-interest is the outcome that leaves them both worse off than cooperation would have. This is the invisible hand's dark mirror, individual rationality producing collective ruin, and it recurs wherever private incentive points away from the common good — in price wars, arms races, and the tragedy of the commons. The first route out, Thomas Schelling showed, is to change the game by binding oneself: in a contest of strategy, surrendering one's options can be a source of power, for by committing irrevocably to a course of action — burning the bridge, tying the bargainer's hands — one alters what the other expects, and so what the other does. A threat or a promise works only when it is believed, and commitment is what makes it believable.

But a commitment must be credible, and Reinhard Selten asked which threats deserve belief. A Nash equilibrium, in a game played out over time, can rest on a threat so hollow that no clear-eyed opponent should fear it — the incumbent who vows a ruinous price war against any entrant, though waging it would ruin him too. Selten's subgame-perfect equilibrium strips such empty menaces away, by demanding that each player act optimally not merely at the start of the game but at every point within it, so that an equilibrium survives only if its every threat is one the player would actually choose to keep; and the method that finds it, backward induction, solves the game from its last move toward its first, each step fixed by the rational play to follow. Then John Maynard Smith, a biologist, dispensed with rational players altogether. Reinterpret a game in the terms of evolution — strategies as inherited programs, players as organisms, payoffs as reproductive fitness — and equilibrium still appears, reached not by calculation but by selection: the evolutionarily stable strategy, one that, once common, no rare mutant can invade. The hawk-dove game shows such an equilibrium to be a stable mixture of aggression and restraint, and explains why real conflict is so often ritualized rather than lethal. Equilibrium, it turns out, has two roots — reason and survival — and the deeper of them needs no mind at all.

The flagship gives the volume its answer. Robert Axelrod set the prisoner's dilemma to be played not once but again and again, between parties who expect to meet in the future, and found that this single change transforms it. When the game repeats, the present acquires a shadow, for what I do to you now will shape what you do to me later; defection's one-time gain is dwarfed by the retaliation it invites, and cooperation's small cost is repaid by the cooperation it earns. In computer tournaments Axelrod found the winning strategy to be the simplest, tit-for-tat — cooperate first, then answer like with like, and forgive — and from it drew the deep result: that cooperation among the self-interested requires neither altruism, nor a central authority, nor even rationality, but only repetition, memory, and a long enough horizon. The achievement is fragile, resting on an open horizon and modest stakes, and the folk theorem warns that repetition makes cooperation possible without making it unique; but it is real. And with it the volume reveals that the whole table has been playing games all along — the commons a many-player dilemma, oligopoly a repeated game, the relational contract a reciprocity, the central bank's credibility a subgame-perfect discipline, the wage a bargain — so that strategy is the connective tissue of economic life, and trust is self-interest that has learned to count the future.

The argument remains open. What follows is an attempt to fix its exact edges.

The twelve plates of this volume

  1. i The Dilemmain print 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. Read the plate →
  2. ii The Minimaxin print von Neumann; the games of pure opposition, the saddle point, and the minimax theorem with which game theory was founded. Read the plate →
  3. iii The Commitmentin print Schelling; the strategy of conflict, and the paradox that binding one's own hands — burning the bridge, throwing away the choice — can be a source of power. Read the plate →
  4. iv The Credibilityin print Selten; subgame perfection and backward induction, and why a threat the threatener would never carry out fails to bind at all. Read the plate →
  5. v The Populationin print Maynard Smith; the evolutionarily stable strategy, where equilibrium is reached not by reason but by survival, and conflict proves self-limiting. Read the plate →
  6. vi The Coordinationin print coordination games and focal points; when several outcomes would suit everyone, how players settle, without a word, on the same one. Read the plate →
  7. vii The Fogin print incomplete information; Harsanyi's games of types, and how one plays against an opponent whose very aims cannot be seen. Read the plate →
  8. viii The Reciprocityin print the capstone: played again and again, the dilemma yields to cooperation — held among the self-interested by no altruism or authority, but by the long shadow of the future. Read the plate →
  9. ix The Brinkin print brinkmanship and deterrence; the threat that leaves the outcome partly to chance, and how one makes credible a threat one would never wish to carry out. Read the plate →
  10. x The Designin print mechanism design and the revelation principle; whether the rules of a game can be built so that honesty becomes each player's best move. Read the plate →
  11. xi The Coalitionin print cooperative game theory, the core and the Shapley value; which alliances hold together when cooperation pays, and how its gains are divided. Read the plate →
  12. xii The Huntin print the stag hunt and the assurance game; the great prize that needs every hand, and the mutual trust required before anyone dares reach for it. Read the plate →

In closing

This volume took up the game, the second of the behaviors and the most intricate of them — the situation in which each player's best move depends on what every other player does, so that strategy must reason about strategy. Its subject was game theory: the equilibrium and the prisoner's dilemma, commitment and credibility, the strategies that survive, and above all the question of how cooperation holds among self-interested parties. And the question the author set it was this: when each best move depends on every other, how does cooperation hold?

The volume traced the routes from the dilemma to its answer. John Nash gave strategy its organizing idea, the equilibrium from which no player can profitably deviate, and the prisoner's dilemma gave the hard case, where that equilibrium is mutual betrayal and individual rationality yields collective ruin. Behind that idea stood an older one: John von Neumann, founding the subject, had shown that the games of pure opposition — where one side's gain is the other's loss — always possess a rational value, won if need be by playing unpredictably; Nash's equilibrium widened von Neumann's to the games where interest and opposition mix. Thomas Schelling found the first way out, the paradox that committing oneself — throwing away one's own options — can be a source of power, since it changes what the other expects and so what he does. Reinhard Selten sharpened it into a discipline of credibility, his subgame-perfect equilibrium ruling out the threats a player would never trouble to carry out, found by reasoning backward from the game's last move. And John Maynard Smith dispensed with reason entirely, showing that equilibrium can be reached by survival rather than calculation — the evolutionarily stable strategy that no rare mutant can invade, with conflict proving self-limiting, as in the contest of hawk and dove.

The flagship gave the answer. Robert Axelrod set the dilemma to be played again and again, between parties who expect to meet in the future, and found that repetition transforms it: with a long enough shadow of the future, cooperation holds among the purely self-interested, sustained by no altruism and no authority but by the simple fact that today's betrayal invites tomorrow's revenge. The plain strategy of tit-for-tat — open with trust, answer like with like, and forgive — calls cooperation forth and keeps it, though the achievement is fragile, and the folk theorem warns that repetition opens many doors and not the cooperative one alone. So the question is answered: cooperation among the self-interested needs neither virtue nor a sovereign, only repetition, memory, and a horizon long enough that the future can discipline the present. And with the game the volume reveals that the whole table has been playing games throughout — the commons a many-player dilemma, oligopoly a repeated game, the relational contract a reciprocity, the central bank's credibility a subgame-perfect discipline, the wage a Nash bargain — so that strategy is the connective tissue of economic life. The behaviors, of which the game is the deepest, are the foundation on which the rest of the table stands; and with the game joining the choice in print, the family of the behaviors is begun, and the table nears its close.

Cooperation among the self-interested is not, in the end, a miracle or a sentiment but an equilibrium — fragile, conditional, and hard-won. Played once, the game of self-interest ends in mutual defection; played again and again, between parties who expect to meet tomorrow, it can yield a stable peace, for the shadow of the future reaches back to make today's cooperation today's best move. Trust, then, is self-interest that has learned to count the future, and the institutions that lengthen that future — lasting communities, enduring firms and reputations, the expectation of a shared tomorrow — are the quiet machinery of human cooperation. So the question the game leaves open is not whether the self-interested can cooperate, for they can, but how long a shadow their meeting casts: when each best move depends on every other, cooperation holds exactly so far as the future is allowed to matter.