№ XXV
The twelve plates
The chooser of the previous volume stood alone, ranking options against a world that held still. Step into almost any real decision and the world stops holding still: it is made, in good part, of other people, each choosing too, each watching you as you watch them. The firm's best price depends on its rival's; the rival's on its own. This volume is the study of such entangled choice — of the games whose outcomes no single player commands, and of the equilibria, settled by mutual anticipation, that those games leave behind.
when each player's best move depends on every other's, what decides where they come to rest — and might they have rested elsewhere?
The chooser of the volume before this one stood, in the end, alone. The options were laid before him, the prices posted, the odds fixed; his task was to rank what was offered and take the best, and the world against which he chose sat obligingly still. Step into almost any decision that matters, though, and the world refuses to sit still — for it is made, in large part, of other people, each of them deciding too, and each watching the watcher. The merchant who sets his price must guess what his rival will charge, while the rival guesses back; the driver choosing his side of the road is choosing what he trusts the oncoming driver to choose; the general, the bidder, the suitor, the negotiator are each trying to fare well in a situation whose outcome turns not on their own move alone but on the interlocking moves of others who are straining to outguess them. This is the strategic predicament, and its study — austere, powerful, and barely a century old — is the theory of games. It is the subject of this volume: choice when the world that the chooser faces is itself made of choosers.
The theory begins by stripping the predicament to three bare things. A game has players — the parties who decide; strategies — the moves available to each; and payoffs — what each player receives for every possible combination of moves. What makes it a game, and not the solitary weighing of the previous volume, is the single fact of interdependence: your payoff hangs not on your move alone but on the whole array of moves, yours and everyone's together, so that to choose well you must anticipate the others — who are, at the very same instant, anticipating you, in a hall of facing mirrors that recedes without end. The earliest games the theory mastered were the zero-sum kind, in which one player's gain is exactly another's loss and there is nothing to share — the pure conflicts for which von Neumann proved his celebrated minimax theorem. But the games that matter most to economic life are not zero-sum at all: in them the players may rise together or fall together, and the question is never merely who wins but how much is made between them and how it is split. Von Neumann and Oskar Morgenstern set the whole study down with the first real rigor in 1944, and gave it its founding picture — the matrix of payoffs, a grid with a cell for every combination of choices, each cell holding what each player carries off should the play fall there.
What, in such a tangle, will the players do? The answer that organized the entire science was given by John Nash, and it is the idea of an equilibrium: a combination of strategies, one belonging to each player, with the property that no player, taking the others' moves as given, could do any better by changing his own. It is a resting point — stable not because anyone has commanded it or agreed to it, but because, once there, no one has any reason to move away. The equilibrium is the game's verdict, the outcome on which the players' interlocking expectations come to settle. And in just this lies the theory's bearing on everything these volumes have argued. For an equilibrium, once it has settled, wears the perfect look of a fact: a stable state of affairs, the way the situation simply and obviously is. It is nothing of the kind. It is the settled outcome of a contest of strategies, held in its place by the threats and the expectations of the players, and a different balance of those forces would have held a different outcome in place instead. A price is a verdict on a hidden contest, mistaken for a fact of nature; an equilibrium is that verdict in its most general form. Nash proved that such a resting point always exists, at least once one allows the mixed and random strategies that enlarge the field; what he could not prove was that it is unique, and the embarrassment of riches — many equilibria, with nothing in the bare logic to choose among them — will shadow the plates that follow.
The most celebrated result of the theory is also its most unsettling, for it proves that reason, faithfully followed, can march a body of rational people straight into an outcome that every one of them would do anything to avoid. In the situation called the prisoner's dilemma, each of two parties has a choice — to cooperate or to defect — so arranged that defection serves him better whatever the other does; and so both defect, and both end far worse off than had both cooperated. What makes the result so bleak is that the failure is structural, not a lapse of virtue or of wit to be cured by exhortation: each player is behaving exactly as cold reason demands, and to urge him to cooperate is to urge him to act against his own interest — so that nothing short of altering the payoffs themselves, by repetition or reputation or a rule imposed from without, can mend it. The shape recurs through the whole of economic life. Two firms that would both grow rich keeping prices high instead undercut one another into a war that beggars both; nations that would all be safer disarmed each arm against the rest; herdsmen who would all prosper by grazing less each graze more until the common is destroyed. The price war, the arms race, the ruined commons are one structure in three disguises, and each teaches the same hard lesson — that a contest can deliver a verdict that no party to it ever wanted.
Yet the dilemma is not fate, and a great part of the theory is given to the ways its grim verdict is escaped, evaded, or designed away. The first escape is repetition. Players who know they will meet again, with the shadow of the future falling over each encounter, can hold one another to a cooperation that a single round would never permit — present restraint bought by the threat of future retaliation, the logic that Axelrod's patient tit-for-tat made famous; though the rescue is fragile, and can unravel if the players know just when the game will end, the last round's defection working its way backward to poison the first. The second escape is commitment, and it wears a paradox on its face: a player can sometimes gain by destroying his own freedom to act, binding himself so visibly to a course that the others must believe he will hold it — the strange potency, as Thomas Schelling showed, of the tied hand and the burned bridge. And where the players cannot save themselves, the rules may be rewritten to save them: the discipline of mechanism design runs the whole theory backward, engineering the payoffs of a game so that the pursuit of private advantage is made to yield a public good — the quiet craft behind the well-built auction, and the dream behind every well-made law. Strategy, in short, is not only a diagnosis of conflict but a kit of instruments for its resolution.
So the volume arrives, by the road of strategy, at the conviction that governs the whole table. A price is a verdict on a hidden contest, mistaken for a fact of nature — and the theory of games is the anatomy of the contest itself. Wherever interests are entangled, which is to say very nearly everywhere, the outcome that prevails is an equilibrium: the settlement of a strategic contest conducted in the currency of anticipation, threat, and commitment. The price two rivals charge, the wage a firm and a union reach, the treaty two powers sign, the convention a whole people keeps without thinking — each is an equilibrium, held in place by what the players expect of one another, and each might have been a different equilibrium, for a game commonly has many. We meet the arrangement that happens to prevail and we take it for the order of nature; it is the resting point of a contest that could have come to rest elsewhere, and which of the possible equilibria we inhabit is itself the trace of a hidden struggle — over who could commit, who moved first, what each was led to expect — that singled it out from the rest. The twelve plates follow the game from the move and the standoff that settles it, through the dilemma, the coordination problem, the threat, and the repetition that rescues cooperation from reason, to the rivals of the marketplace, the bluff of hidden knowledge, the offer of the bargaining table, the mechanism that builds the rules, the anomalies that flesh-and-blood players display, and the regress of mutual knowledge on which the whole structure trembles. Where the volume on choice gave us the chooser alone, this one gives us choosers entangled, each fate knotted into the others'. The argument remains open.