The Game

Begin with the difference that defines the subject. Through most of this table the chooser has faced a fixed world: the firm minimizing cost against given prices of inputs, the saver weighing a rate it cannot move, the consumer optimizing against the numbers on the shelf. In each case the environment is a backdrop — uncertain, perhaps, but indifferent, neither watching the chooser nor altering its behavior in response to his. A great deal can be understood by treating the world this way, and the table has understood it so. But the most important environment any economic actor faces is not a backdrop at all. It is other actors, who are choosing too, and whose choices turn on his as his turn on theirs. This is strategic interdependence, and where it is present, the placid logic of optimizing against a given world breaks down.

The instances are everywhere once one looks. The price a firm can charge depends on the price its rival sets; the wage a worker can hold out for depends on what his fellows will accept; a cartel's profits depend on whether each member trusts the rest to restrain output; a bank's survival depends on whether each depositor believes the others will keep their money in. In none of these is the relevant environment a fixed curve or a settled fact. It is a web of mutually dependent choices, and to model it as a backdrop is to miss the very thing that drives the outcome. The study of such webs is what von Neumann and Morgenstern, in the founding work of 1944, called the theory of games — a name whose lightness belies its reach, for the "games" in question are the most serious situations in economic life.

A game in this precise sense has three parts. There are the players, who choose; the strategies, the courses of action available to each; and the payoffs, the outcomes that result from every combination of the players' choices. What lifts a game above a mere catalogue of these is the kind of thinking it forces upon the players. To choose well, each must anticipate what the others will do; but each of the others is likewise anticipating him; so that he must anticipate their anticipation of him, and they his anticipation of their anticipation, in a regress of mutual expectation that mirrors back upon itself without any natural floor. This hall of mirrors is the signature of strategic thought, and it is what makes a game hard in a way that a decision against nature is not. Taming the regress — finding within it a configuration that is consistent with itself, from which the reasoning does not flee — is the problem the next plate takes up.

Seeing the economy as a tissue of games changes how its older categories look. The impersonal market forces of the earlier volumes — the price that seems to descend from nowhere, the wage that seems set by the air — resolve, on inspection, into the aggregate of strategic choices: the price is what emerges when firms contend, the wage what emerges when parties bargain, the rate what emerges when a central bank and a public anticipate each other. Game theory looks inside the impersonal force and finds reasoning agents, each responding to the others. It thereby brings the table's founding image into focus, for if every price is the verdict of a hidden contest, then the contest is, formally, a game, and the price its equilibrium — a claim the final plate of this volume will make good.

The hall of mirrors left us without a floor: each player anticipating the others, the others anticipating him, the reasoning chasing itself with no place to stop. John Nash, in a doctoral thesis of a few pages, gave the field its floor. A configuration of strategies is an equilibrium, he proposed, if no player can do better by changing his own strategy alone, given what all the others are doing. At such a configuration every player is already playing the best reply to the rest; none has any reason to move; and so the configuration sustains itself. The regress is tamed not by following the mirrors to a bottom they do not have, but by finding the point at which the reflections agree — the configuration that, once each expects it, each confirms.

The figure shows the idea in its smallest form. Two players each choose between A and B, and the four cells give the payoffs, the first number to the row player and the second to the column. Here the reasoning is easy, for A is the best reply to whatever the other does — three beats one if the other plays A, two beats nothing if the other plays B — so each plays A, and the cell A-A is the equilibrium, ringed in the figure. Neither can improve by deviating; each is doing the best he can against the other; the point holds. In this particular game the equilibrium is also happy, the best outcome available to both, and one might be lulled into thinking equilibria generally are. The next plate will dispel that comfort.

The reach of the concept comes from this kinship with the self-enforcing. Everything the table has called an institution or a norm — the convention that fixes which side of the road we drive on, the practice that holds a cartel together, the expectation that sustains a currency, the custom that governs a commons — can be understood as an equilibrium of some game, a configuration that persists because no one party can gain by departing from it alone. Game theory thus supplies the precise meaning of a quality the table has invoked throughout under the looser names of trust, custom, and the done thing: to be self-enforcing is to be a Nash equilibrium. And Nash's existence theorem, that every finite game has at least one equilibrium, assures us that the search for such a resting point is never in vain, though it may lead, as the next plates show, to a point in randomized play or to one among several.

For the concept carries two warnings, and they govern all that follows. The first is that a game may have many equilibria, with the theory silent as to which will prevail — an indeterminacy that the plate on coordination must confront, and that only a focal point or a convention can resolve. The second, and graver, is that an equilibrium need not be good. It is merely a configuration no one can improve upon by moving alone; whether it is good for the players together, the definition neither asks nor answers. The comfortable game in the figure, whose equilibrium is the best outcome for both, is the exception that flatters. The general case includes equilibria that are bad for everyone caught in them, and it is to the most important of these that we now turn.

Of all the games, one matters more than the rest, for it overturns the central comfort of economics in a single four-celled table. It is the prisoner's dilemma, named for the parable of two suspects questioned apart, each offered leniency to betray the other, and each reasoning his way to a confession that leaves both worse off than silence would have. Strip away the parable and the structure stands bare. Two parties each choose to cooperate or to defect. The figure gives the payoffs, and the cruelty is in the arithmetic. If the other cooperates, you do better to defect — five beats three. If the other defects, you again do better to defect — one beats nothing. Whatever the other does, defection is your best reply; defection is, in the language of the last plate, a dominant strategy. So both defect, and both receive one, when both might have had three.

The table above is fixed and annotated. Here are the same four cells, made operable — so the cruelty can be checked by hand rather than taken on faith. Set your move and the other’s, or simply click a cell, and read off who receives what. Then try to find a single case in which cooperating serves you better than defecting: hold the other at cooperation, and your best reply is to defect; hold the other at defection, and your best reply is, again, to defect. There is no escape inside the table — and that is the whole of the dilemma, for the one cell in which neither can profitably change, where you both defect, pays each of you less than the cell you abandoned to reach it, where you both cooperated.

This is the invisible hand run backward. The early volumes watched self-interest, in a competitive market, promote an end that was no part of anyone's intention — the public good served by private greed. The dilemma exhibits the mirror image: private reason producing public ruin, an outcome worse for all than was available, reached precisely because each pursued his own advantage. And the structure is not a curiosity. It is the formal skeleton of nearly every failure the third family of this table set out: the herdsmen who each overgraze the common until it is barren; the firms that each strip a fishery; the nations that each build arms they would all be safer without; the polluters who each foul a sky they share; the citizens who each decline to fund a good they would all enjoy. Each is a prisoner's dilemma, often with many players rather than two, and each ends where the dilemma ends — at a ruin that individual rationality built.

One feature deserves emphasis because it is so often missed: the dilemma cannot be talked away. It is not a failure to coordinate, not a confusion that better communication would cure. Let the two parties confer, swear cooperation, shake hands — and each still does better to defect when the moment comes, for the handshake changed no payoff. A promise to cooperate in a one-shot dilemma is not credible, because keeping it is against the interest of the one who made it. This is what distinguishes the dilemma from the coordination problems a later plate treats, where merely agreeing on which equilibrium to play suffices. In the dilemma, agreement is cheap and worthless; only a change in the incentives themselves can help.

And a change in the incentives is exactly what the rest of the table supplies. There are two ways to escape a dilemma, and the table has devoted whole families to each. One may change the game — rewrite the payoffs so that cooperation becomes a dominant strategy or at least an equilibrium — which is the work of institutions: the state that enforces the contract and so makes the broken promise costly, the regulator that taxes the pollution and so makes fouling the sky expensive, the authority that compels the contribution to the public good. This is why the fourth family of this table, the institutions, exists at all: they are devices for converting dilemmas into games with happier equilibria. The other escape is to repeat the game, letting the shadow of the future do what no single encounter can; and that is the subject to which we now turn.

The dilemma of the last plate was a tragedy of a single encounter. But few of the encounters that matter are single. The buyer returns to the supplier, the firm faces the worker again next month, the members of a trade deal with one another for years, the neighbors share a boundary for life. When the same parties meet repeatedly, the game is no longer the one-shot dilemma but a repeated game, and repetition changes everything — for now today's defection can be answered by tomorrow's, and the prospect of that answer reaches back to alter today's choice.

This is the shadow of the future, and the volume on the contract met it already in the guise of relational dealing. The logic is a weighing of present against future. To defect today is to seize the temptation now — the five rather than the three — but to forfeit the cooperation of every tomorrow, as the other party, seeing the betrayal, withholds his own cooperation in retaliation. If the future matters enough, the stream of forfeited cooperation outweighs the one-time gain from cheating, and the rational course flips: it becomes better to cooperate, not from goodness but from prudence, because the shadow of the future has made betrayal unprofitable. A simple strategy enforces this — cooperate so long as the other does, and punish defection with defection — and the best known of these is the tit-for-tat that Robert Axelrod found triumphant in his tournaments, and that the volume on the norm met as reciprocity. Nice, retaliatory, forgiving, and clear, it sustains cooperation by making each party's future treatment hinge on his present conduct.

The condition on which all this depends can be stated precisely: cooperation can be sustained if the players weigh the future heavily enough — if they are patient, and expect the relationship to continue. Above a critical weight on the future, mutual cooperation is an equilibrium of the repeated game, self-enforcing in the exact sense of the second plate, for no one can gain by defecting once the loss of future cooperation is reckoned in. Below it — when the parties are impatient, or the relationship is unlikely to last — the future is too faint to discipline the present, and the game collapses back into the one-shot dilemma. The celebrated folk theorem records the upshot and its excess: with patient players and an open-ended horizon, not only cooperation but a vast range of outcomes can be sustained as equilibria, so many that the theorem predicts less what will happen than what can. Cooperation among the self-interested is shown to be possible; it is not shown to be inevitable.

Here, then, is the formal account of much that the table has described in the warmer language of trust and custom. The handshake of the contract holds because the parties expect to deal again; the norm of reciprocity endures because the community plays its games repeatedly; the cooperation that the third family's dilemmas seemed to forbid is recovered, in the repeated setting, as the rational equilibrium of patient parties with a future. But the same mechanism is neutral as to what it sustains, and its dark application must be named: a cartel holds its prices high by exactly this logic, each member refraining from undercutting for fear of the price war that retaliation would bring, so that the shadow of the future which underwrites the honest dealing of a trade also underwrites the collusion that fleeces its customers. Cooperation among firms is conspiracy against the public — the same equilibrium, viewed from the other side. The repeated game gives and takes away.

So far the players have chosen at once, in ignorance of one another, and the matrix has sufficed to hold the game. But many games unfold in sequence, one party moving while the other watches and then replying, and these are drawn not as a grid but as a tree, branching from a first decision down through the responses it invites to the outcomes at its tips. Such games are solved by reasoning backward: one asks what the last player will do at each point he might reach, then what the player before him will do anticipating that, and so on up the tree to the opening move. Backward induction, this is called, and it has a way of dissolving threats that look fearsome from the front.

The figure draws the classic case. An entrant considers whether to challenge an incumbent who holds a market. If the entrant stays out, the incumbent keeps his monopoly and its handsome profit. If the entrant comes in, the incumbent must choose: fight, with a price war that bloodies them both, or accommodate, sharing the market for a smaller but positive profit each. The incumbent thunders that he will fight any entry. But reason backward. Should entry occur, the incumbent compares fighting, which loses him money, against accommodating, which earns him some; he will accommodate. His threat to fight is therefore not credible — it asks him to do, in the event, what is against his interest — and the entrant, seeing this, enters without fear. The equilibrium, traced in bold, is entry met by accommodation, and the incumbent's fearsome threat counts for nothing.

The lesson generalizes into one of the deepest in the subject, and it was Thomas Schelling who drew it out. The power of a threat or a promise lies not in its making but in its credibility — in whether the party would actually carry it out when the moment came. And the way to make a threat or promise credible is, paradoxically, to surrender one's own freedom: to commit, by some prior and irreversible move, so that carrying it out becomes one's interest, or refusing it becomes impossible. The general who burns the bridge behind his army cannot retreat, and the enemy, knowing this, may yield; the incumbent who saddles himself with capacity he must run will fight any entrant, and so faces none; the negotiator who stakes his public honor on a position cannot concede it without ruin, and so the other side must move. In each case a player improves his position by destroying an option — by making himself less free, and therefore more believable.

This inverts the comfortable assumption that more options are always better, and it illuminates a great deal the table has met before. The volume on the rate found a central bank unable to keep inflation low because it could not credibly promise not to inflate: the public, expecting the broken promise, built the inflation in, and only inflation resulted. The cure was commitment — rules in place of discretion, an independent bank, a published target — a deliberate tying of the monetary hands so that the promise of stability would be believed. The same logic underwrites the contract that binds, the constitution that cannot be lightly amended, the treaty whose breach brings automatic penalty. Each is a device for foreclosing a future discretion that, if kept, would render some present promise empty. The capacity to commit is itself a thing that institutions supply, and it is among the most valuable things they supply.

Not every game is a contest of opposed interests. In a large and important class the players want, above all, to do the same thing as one another, and their difficulty is not conflict but coordination. Which side of the road shall we drive on? Either will do, provided we all choose alike; left-for-everyone and right-for-everyone are both equilibria, equally good, and the only catastrophe is to disagree. Which technical standard, which language of trade, which currency, which meeting place — a thousand such questions have many acceptable answers and only one requirement, that we converge upon a single one. These are coordination games, and they pose a problem the dilemma did not: not how to make cooperation an equilibrium, for it already is, but how to select, among several equilibria, the one we shall actually play.

Pure game theory is strikingly unhelpful here. Its central concept tells us that all-left and all-right are equilibria; it does not tell us which will prevail, for the payoffs are symmetric and the mathematics has no way to break the tie. The indeterminacy that the second plate flagged becomes acute. What resolves it lies outside the formal game, in what Schelling called a focal point: an equilibrium that stands out from the others for some reason the bare payoffs do not capture — its prominence, its tradition, its naturalness, the mere fact that everyone expects everyone to expect it. Asked to meet a stranger in a great city with no chance to confer, people converge in surprising numbers on the famous landmark at noon, not because it pays more but because it is the obvious place, the place each guesses the other will guess. The salience does the work that the incentives cannot.

And the great social supplier of focal points is the convention — which is why this plate is the near kin of the one on convention in the volume on the norm. A convention is a coordination game already solved, a standing answer to the question of which equilibrium, transmitted so that each new entrant inherits the expectation rather than having to forge it. Drive on the right; adopt the metric units; speak the language of the room; measure in the customary gauge. These are not laws of nature and not, mostly, laws of the state; they are equilibria selected by history and held in place by shared expectation, each self-enforcing in the exact sense of the second plate. Where the dilemma needed institutions to change its incentives, the coordination game needs only a convention to settle its expectations — a cheaper fix, and one that explains why so much of social order is convention rather than command.

Between the pure coordination game and the dilemma sits a third structure that deserves its own name, for it captures a cooperation problem of a distinct and consequential kind. It is the stag hunt of Rousseau's parable: a band may cooperate to take a stag, which feeds them all but requires every hunter to hold his post, or each may break off to catch a hare, which is meager but sure and needs no one's help. Hunting the stag together is an equilibrium — once all are committed, none wishes to defect, for the stag is the better prize. But hunting hare is also an equilibrium, the safe and defensive one, and a hunter will choose it the moment he doubts his fellows. Here cooperation is not forbidden, as in the dilemma; it is available, an equilibrium ready to be occupied. What stands between the band and the stag is not a perverse incentive but a failure of mutual confidence — each will cooperate if sure the others will, and each defects only from the fear that they will not.

Some games refuse to come to rest in any definite combination of moves. The figure shows the purest case, the game of matching pennies, in which two players each show heads or tails and one wins if the faces match, the other if they differ. Try to settle it. Wherever the players land, the loser of that cell wishes to switch; his switch makes a new loser, who wishes to switch in turn; and the best replies chase one another endlessly around the four cells, never alighting. There is no equilibrium in definite play — no pair of fixed choices from which neither would deviate — and the cycle in the figure has no exit.

The way out is to stop choosing a move and start choosing the odds. The equilibrium of matching pennies is in mixed strategies: each player randomizes, showing heads half the time and tails half the time, by some private device no opponent can foresee. At this point neither can gain from any change, because neither can be read; the unpredictability is itself the shield. This was the content of von Neumann's minimax theorem, the founding result of the whole field, proved years before Nash: in any finite game of pure conflict — any zero-sum game, where one player's gain is exactly the other's loss — there exist optimal strategies, generally randomized, and a definite value each player can secure by guarding against the worst the other can inflict. The poker player lives by this theorem. He must bluff sometimes and not others, betting strength and weakness in a calculated mixture, for a player who never bluffed would be folded against whenever he bet, and one who always bluffed would be called; only the unpredictable mixture cannot be exploited.

The same logic governs a class of economic games more sober than poker. The tax authority cannot audit every return, for auditing is dear; the taxpayer cannot safely cheat on every return, for the audit, though rare, is ruinous. So each randomizes — the authority audits with some probability, the citizen complies with some probability — and the system rests at a mixed equilibrium in which neither auditing-always nor cheating-always would pay. The regulator inspecting a firm, the employer monitoring a worker who might shirk, the customs officer searching a fraction of the cargo: all are inspection games of this shape, all without equilibrium in definite behavior, all resolved by the deliberate use of chance. Randomization, the bluff made rigorous, turns out to be a standard instrument of enforcement.

But a warning must close the plate, for the games of this section are games of pure conflict, and those are not the games that most of economic life consists of. In a zero-sum game there is nothing to share and nothing to negotiate; the whole of skill is the minimax, the guarding against an opponent whose interests are exactly opposed to one's own. Matching pennies is such a game, and poker, and very nearly war. The economic interactions that matter most are not. They are mixed-motive games, in which the parties' interests neither wholly align nor wholly oppose, and in which — crucially — outcomes exist that are better for everyone. To carry the habits of the zero-sum game into these is the error the edge below describes, and it is among the costliest a strategic actor can make.

Consider the most ordinary of strategic situations: two parties who can create a surplus together, but only by agreeing, and who must then divide what they have made. A buyer values a house above what the seller will part with it for, so a sale would benefit both — but at what price? An employer and a worker can together produce more than either alone, a surplus to be shared — but in what proportion of wage to profit? The structure is mixed in its motives. Both prefer agreement to none, for disagreement destroys the surplus and leaves each with nothing but his fallback; yet each prefers a larger share of what agreement creates, and over that division their interests are opposed. This is the bargaining problem, and its answer determines a great many of the prices the table has treated as if they fell from the sky.

The figure lays out its geometry. Every way of splitting the whole surplus lies on the frontier — the downward line along which one party's gain is the other's loss, since the pie is of fixed size once made. Rational parties will reach some point on this frontier, for to settle inside it would waste surplus they could divide; the question is which point. Anchoring the answer is the threat point, the marked outcome each would fall back to if no agreement were struck — the wage of the next-best job, the price from the next-best buyer, the misery of no deal at all. Neither party will accept a split that leaves him below his threat, so the bargaining range is the stretch of the frontier above both, and the line they draw must fall somewhere along it.

What, then, fixes the split within that range? Nash gave one answer by axioms, naming the point that maximizes the product of the parties' gains above their threats — under symmetry, the even division, the splitting of the difference that intuition expects. Rubinstein gave a deeper answer by modeling the haggling as a game in its own right, a sequence of alternating offers in which delay is costly because the future is discounted or the surplus decays. The equilibrium of that game yields a precise division, and its lesson is stark: patience is power. The party who discounts the future less, who can endure the delay of refusal while the other cannot, walks away with the larger share. To this the earlier plates add their determinants — the outside option that improves one's threat point, the commitment that makes a take-it-or-leave-it credible, the private information that lets one party feign greater resolve. Bargaining power is assembled from these: the power to wait, the power to walk, the power to refuse to move.

And here the volume rejoins the table's oldest concern. For if the wage is the split of the surplus a job creates, then the wage is a bargain, and its level turns not on some impartial marginal product alone but on the relative strength of worker and employer — on the worker's outside options, his ability to wait, his power to combine with others, against the employer's. If the price that divides the gains from trade falls within a bargaining range, then it too reflects the balance of power between buyer and seller, not merely the scarcities the early volumes invoked. The division of the surplus, which is to say the distribution of income, is in large measure a matter of who holds the stronger hand in the bargain — and that is the subject the next plates, on the auction and the mechanism, approach from the side of design, before the volume turns at last to the market itself.

An auction is a game stripped to its essentials, and it is also the place where the table's thesis is least in need of argument, for here the price is plainly the outcome of a contest. Bidders compete; each chooses what to offer, weighing the chance of winning against the cost of victory; and the rules determine who takes the prize and what he pays. Four classic formats recur: the ascending auction of open outcry, where the price climbs until one bidder is left; its mirror, the descending auction, where the price falls until someone accepts; the sealed first-price auction, where the highest bid wins and pays itself; and the sealed second-price auction of William Vickrey, where the highest bid wins but pays only the second. These are not mere procedures. Each is a different game, inducing a different strategy, and the differences repay study.

Vickrey's second-price rule has a small marvel hidden in it. Because the winner pays not his own bid but the next one down, he has no reason to shade his offer below what the prize is worth to him: bidding his true value is his dominant strategy, optimal whatever the others do. The auction thus coaxes the truth out of its bidders as a matter of self-interest — a property the next plate will name incentive compatibility and prize. The first-price auction, by contrast, sets a genuine strategic puzzle, for there bidding one's full value would leave no profit, and each must shade his bid below his value by an amount that trades the chance of winning against the gain from winning. And a deep theorem of Vickrey's holds that, under the right conditions, all four formats raise the same revenue on average — so that the seller's choice among them turns not on revenue but on subtler virtues of robustness and simplicity.

The most instructive complication arises when the prize has a value that is common but unknown — an oil tract, a company, a band of radio spectrum, worth roughly the same to all but estimated differently by each. The figure shows what then befalls the winner.

That auction theory should be the showcase of economics as an applied science is, given its abstraction, a pleasant surprise. When governments came to sell the radio spectrum on which the wireless age would run, they turned to economists to design the auctions, and the designs — intricate ascending formats that let bidders learn and recombine as the sale proceeded — raised enormous sums while placing the licenses in the hands that valued them most. The same theory governs the auctions by which treasuries borrow, by which electricity is bought and sold, and by which the advertising beside a search result is allocated in the instant before a page loads, billions of times a day, through mechanisms descended from Vickrey's. Here the abstract study of strategic bidding became a kind of engineering, and the price it produced was visibly and admittedly the verdict of a contest its designers had built.

The auction thus makes the table's thesis concrete and adds a twist the later volumes will need. Concrete, because nowhere is it plainer that the price is the outcome of a strategic contest rather than a fact of nature: change the rules of the auction and you change the price. The twist is the winner's curse, which is the first crack in the perfect rationality this whole volume has assumed, and the edge below follows it toward the volumes still to come.

Every plate so far has read game theory forward, taking the rules as given and asking what self-interested players will do within them. The tenth plate turns the telescope around. Mechanism design — reverse game theory, its founder Leonid Hurwicz called it — begins from the outcome one wants and asks what rules would induce self-interested players to produce it. The shift is from the scientist who predicts a game's result to the architect who builds a game to a purpose: a sale that places the goods with those who value them most, a vote that reflects the electorate's will, a division that all parties will call fair, an allocation that wastes nothing. Given that the players will pursue their own ends whatever rules we set, how shall we set the rules so that, in pursuing their ends, they serve ours?

The difficulty is that the players know things the designer does not — their own values, costs, and intentions, the private information on which the volume on asymmetry turned — and will reveal it only if revealing it pays. The cardinal virtue of a mechanism is therefore incentive compatibility: that honesty, or whatever good behavior the design requires, be in each player's own interest. Vickrey's second-price auction was the model, eliciting true values because the rules made truth-telling a dominant strategy. And a remarkable theorem of Roger Myerson, the revelation principle, shows that the search for good mechanisms can without loss be confined to such honest ones: whatever any mechanism can achieve, some mechanism can achieve while making truthful revelation optimal. The designer may thus assume the players will tell the truth, provided he builds the rules so that truth is what their interest counsels.

But design has limits as firm as its powers, and a mature theory states them as impossibility theorems. Some objectives cannot be reached by any mechanism whatever. When two parties with private valuations must trade voluntarily, no mechanism can guarantee that every beneficial trade will occur; some surplus is irreducibly lost to the parties' incentive to misrepresent. When a society must choose among three or more options, no voting rule that is not a dictatorship can be wholly immune to strategic manipulation — a result that rhymes with the impossibility the volume on the rule met in Arrow. These theorems are not failures of ingenuity but boundaries of the possible, and a good designer respects them, seeking the best mechanism within the constraints rather than a perfection the mathematics forbids.

Within those boundaries the field has flowered into a practical art that the work of Alvin Roth best exemplifies. Where markets cannot use prices — because money is unseemly, or forbidden, or simply absent — they can still be designed. Roth and his colleagues rebuilt the system that matches medical graduates to hospitals, redesigned the assignment of children to city schools, and, most strikingly, constructed the exchanges by which incompatible kidney donors and patients are linked into chains and cycles, so that a willing donor whose kidney suits a stranger sets in motion a sequence that gives his own loved one a kidney in return. A market in organs is rightly prohibited; a designed mechanism that arranges compatible gifts is not, and it has saved many thousands of lives. Here mechanism design rejoins the volume on the rule, for an institution simply is a mechanism — a standing set of rules that channels strategic behavior — and to design a mechanism well is to design a good institution. The rules of the game, this whole table has insisted, are made by someone; this plate shows that they can be made on purpose, and made well.

It remains to turn the apparatus upon the market itself, for the market is a game, and one this volume has been circling from the start. The perfectly competitive market of the early volumes — a swarm of sellers so numerous that none could move the price, each a passive taker of a number the multitude set — is the limiting case, valuable as an idealization but rare in fact. Most markets are peopled by few enough firms that each must reckon with the others: the price one can charge depends on what rivals charge, the quantity one should make depends on what rivals make, and the placid price-taking of the textbook gives way to strategic contention. This is oligopoly, the general case of which perfect competition and monopoly are the two extremes.

Augustin Cournot drew the first picture of it in 1838, more than a century before the field had a name, and the figure reproduces his insight. Two firms choose quantities; the combined output sets the price through demand; and each firm's best quantity depends on the other's, since the more the rival floods the market the less it pays to add. Each firm's reaction curve traces its best output against every output of the other, and where the two curves cross lies the equilibrium — each producing exactly its best reply to the other, neither wishing to change. It is a Nash equilibrium in all but name, found by Cournot a hundred and twelve years before Nash. And it sits between the extremes: the price it yields stands above the marginal cost that pure competition would enforce, for the firms retain some power, yet below the monopoly price, for they suffer some rivalry — and as the number of firms grows, the equilibrium slides toward the competitive floor.

That the outcome turns on the rules, and not on the scarcities alone, the comparison with Joseph Bertrand makes vivid. Bertrand objected that firms set prices, not quantities, and worked the model again on that assumption — with a startling result. If two firms sell the same good and compete by price, each undercutting the other to capture the whole market, the price is driven all the way down to marginal cost, the competitive outcome, by two firms alone. Quantity competition leaves a markup; price competition annihilates it; and nothing has changed but the variable over which the firms contend. The lesson is the volume's thesis in miniature: the price is not dictated by cost and demand alone but by the form of the competition, the strategic structure of the game the firms are playing. And that structure admits a third possibility the fourth plate prepared — collusion, in which the firms, playing a repeated game, hold their prices high under the threat of mutual punishment, lifting the price from the Bertrand floor toward the monopoly ceiling. Between marginal cost and monopoly the price may rest anywhere, and where it rests is a fact about the game, not about nature.

So the volume arrives, by way of the market, at the claim it was built to make. The price that the first family of this table examined one kind at a time — the wage, the rent, the rate, the premium, the spread — is in each case the equilibrium of a strategic game: the configuration at which the contest among the interested parties comes to rest. It is not read from the meeting of supply and demand as from a dial; it is the resting point of a game whose structure determines it, and whose structure is shaped by the institutions that count the firms, define what they may compete on, guard or open the gates to entry, and permit or forbid their collusion. The competitive price of the early volumes is revealed as the equilibrium of one game among many — the game of many sellers contending freely — and where that game is not the one in play, the price rests elsewhere. The final plate gathers what this means for the whole table.

This table announced at its outset that every price is a verdict on a hidden contest, and asked, volume upon volume, what each contest was and who was party to it. The present volume has supplied the form of the answer. The contest is a game — a situation of strategic interdependence, in which the outcome for each depends on the choices of all — and the price is its equilibrium, the configuration at which the contest comes to rest. The wage is the resting point of a game between worker and employer; the rate, of a game between lender, borrower, and the bank that anticipates them both; the price of a good, of a game among the firms that make it and the buyers who want it. What the metaphor of the contest expressed in figure, the theory of games states in form: the price is the Nash equilibrium of a strategic interaction, and to ask why a price is what it is, is to ask what game produced it.

Seen so, the strategic skeleton stands revealed beneath the whole table, and the families fall into place upon it. The failures of the third family are games gone wrong — prisoner's dilemmas, in which each party, acting rationally, helps to bring about a ruin none of them wanted: the exhausted common, the fouled sky, the unfunded public good, the arms race no one can stop. The institutions of the fourth family are devices for changing those games — for rewriting the payoffs by enforcement, regulation, and tax so that cooperation, impossible as an equilibrium of the original game, becomes the equilibrium of the altered one; mechanism design is the formal study of how this is done. The norms of the sixth family, beside which this volume sits, are equilibria of repeated games — the informal, self-enforcing configurations that the shadow of the future and the discipline of reciprocity hold in place without any enforcer at all. And the prices of the first family are, as we have just seen, the equilibria of strategic contests among the interested. The game is what all of these have in common; it is the substructure of which they are the aspects.

Upon this substructure the deepest tension of the table is at last resolved — not dissolved, but located. From the first volume two visions have contended: the harmony of the invisible hand, in which self-interest is turned to the common good, and the tragedy of the dilemma, in which self-interest is turned to common ruin. The theory of games shows that neither is the truth of markets as such, because which of them obtains is not a fact of nature but a property of the game's structure. A competitive market without external effects is a game whose equilibrium is benign, and there the hand is real. A prisoner's dilemma is a game whose equilibrium is malign, and there the tragedy is real. The very same self-interested agents, no better and no worse, produce harmony in the one game and ruin in the other; what differs is not the men but the rules under which they meet. Self-interest, the table may now conclude, is neither the benefactor the early economists hoped nor the villain their critics feared. It is a force whose issue depends entirely on the structure into which it is channeled — and the structure is something we build.

For the rules are made. This has been the steady insistence of the whole work, and the theory of games gives it a final and exact sense. The payoffs that decide whether a game is a harmony or a dilemma are set by the institutions of property, contract, and liability; the repetition that lets cooperation hold is sustained by the relationships and reputations a society maintains; the number of players, the variable they compete on, the barriers to their entry, and the collusion they may or may not practice are fixed by the law of competition; the very capacity to commit, on which credible promising depends, is supplied by institutions built for the purpose. Not one of these is given by nature, and every one of them is a human arrangement, contested and revisable. The market is therefore not a natural object obeying immutable laws but a game whose outcome follows from its rules — and the price, being the equilibrium of that game, is not a verdict of nature or of justice but the resting point of a contest whose terms we have set, and might set otherwise. A different structure would yield a different equilibrium: a different price, a different division of the surplus, a different distribution of the whole. The equilibrium is real, but it is not fate.

Honesty requires that the powers of this beautiful theory be matched against its limits, for it has both in full measure. Its powers are not in doubt: it is the common language of strategic life, the apparatus that unified the failures, the institutions, the norms, and the prices under a single account, and it has descended from the seminar into the world to design the auctions that sell the spectrum, the markets that match the willing, and the regulations that discipline the few. But its assumptions are strong, and at their edges they fray. It supposes the players perfectly rational, flawless best-responders reasoning without error — and the volumes on choice and bias, still to come, will show how far real minds depart from that ideal, as the ultimatum game of the norm and the winner's curse of the auction have already hinted. It supposes their rationality common knowledge, each knowing the others rational and knowing that they know, in a tower of mutual awareness that real people do not build. And it suffers a chronic indeterminacy, for many games have many equilibria and the theory cannot say which will prevail, but must borrow from outside itself — from focal points, conventions, norms, and history — the selection it cannot supply. Game theory is thus a lens of great power and a poor oracle: it reveals the strategic structure of a situation and the dependence of its outcome on the rules, but it rarely foretells the outcome, and it crowns no equilibrium as just. It shows us the game; it does not play it for us, nor tell us which game we ought to build.

This volume stands second in the last family of the table, beside the norm, and the two are twins. What the norm described in the warm language of custom, reciprocity, and trust, the game restates in the cool language of equilibrium, repetition, and best reply; the convention of the one is the coordination equilibrium of the other, the reciprocity of the one is the repeated-game strategy of the other, the self-enforcing rule of the one is the Nash equilibrium of the other. Together they give cooperation its two faces, the institutional and the formal, and show it to be at once a social achievement and a strategic possibility. And both lean upon a foundation they do not themselves examine — the rationality of the agent who chooses and reasons and reckons — which the remaining volumes of this family, on the choice and the bias, are written to inspect. The game has assumed that agent flawless; the choice will model him; the bias will catch him erring. The Behaviors complete themselves only when the strategist of this volume is set beside the chooser of the next and the erring mind of the one after.

So we may say, at the close, what an economy is. It is a vast and intricate web of interlocking games — strategic contests whose equilibria are the prices, wages, rents, and rates this table has examined one by one; whose structures decide, game by game, whether self-interest issues in the harmony of mutual gain or the tragedy of mutual ruin; and whose rules are written, amended, and fought over by the human beings who must then play them. The price is the verdict of a contest, the contest is a game, and the game has rules that are ours to make. A numéraire of honesty, toward which this whole work has reached, would begin from exactly this admission: that the price is not the impartial judgment of nature upon the worth of things but the equilibrium of a contest we have structured, and that to take responsibility for the price is to take responsibility for the rules. What those rules should be, the theory does not decide. It hands the question back to us, where it belongs. The argument remains open.