Plate ii in print

The Minimax

von Neumann; the games of pure opposition, the saddle point, and the minimax theorem with which game theory was founded.

Even in a contest where one player's gain is exactly another's loss, the game holds a rational value that neither can be denied nor surpass; and the way to secure it is to play so unpredictably that no adversary, however able, can turn your own plan against you.

— the argument of this plate, after John von Neumann, 1928

The plates of this volume turn, for the most part, on mixed motives — on games in which the players' interests neither perfectly align nor perfectly clash but tangle together, some shared and some opposed, so the hard question is how cooperation holds among parties who are partly rivals. The subject did not begin in that tangled country. It began with a cleaner and more brutal case: the contest of pure opposition, in which one player's gain is exactly another's loss, nothing can be created, only a fixed sum divided, and cooperation is not difficult but impossible by construction. For these games John von Neumann proved, in 1928, a theorem so complete that it founded the discipline — that even in unrelieved conflict there is a rational way to play, and a definite value the play must yield. This plate sets out that case and theorem, the bedrock on which every later and subtler game is built.

A game of pure opposition carries a name in the theory: it is zero-sum, which means that across every outcome the players' gains and losses cancel exactly to nothing, so that whatever the one wins the other must precisely have lost.1 Picture two children at matching pennies, each concealing a coin and uncovering it at the same instant, one taking both whenever the faces agree and surrendering both when they differ; or two players contesting a settled pot across a card table. In neither is there anything to be made — no surplus that cleverness or cooperation might enlarge, no bargain that could leave both sides better off than before. There is only a fixed quantity to be divided, the players on opposite sides of every choice, each one's interest the precise negative of the other's. It is the starkest relation two parties can stand in, and for that reason the first that yielded to analysis.

1. The phrase is now used loosely for any bitter rivalry, but its technical sense is exact: the payoffs at every outcome sum to a constant, so one player's schedule of gains is the other's losses with the signs reversed. Most of economic life is emphatically not of this kind — exchange, the rest of the table insists, brings into being value that did not exist before — which is why this clean case, foundational as it is, remains a special and even peculiar one.

How, then, ought a player to choose? Von Neumann's reasoning starts from one severe assumption: in a game of pure opposition, treat your opponent as perfectly hostile and perfectly capable, since his interest is to do you the greatest harm the game permits. Granting that, weigh each of your strategies by its worst result — the most the opponent could strip from you, reading your intention and answering flawlessly — and settle on the one whose worst case is least bad of all.2 This is the maximin: you maximize your minimum, securing the highest floor you can stand upon whatever the other does. The number that floor guarantees is your security level, the value beneath which no adversary, however cunning, can drive you. It is a profoundly conservative way to play, in most of life too gloomy to be wise; but where the opponent's gain is your loss to the penny, his hostility is no worst case to be guessed at but a flat certainty, and the gloom is accurate.

2. The opponent, reasoning the same way from his side, computes his own maximin — the largest portion he can guarantee himself. Because the game is zero-sum, his guaranteeing himself a floor is the very same act as his holding you to a ceiling: the most he can lock away for himself is exactly the least he can confine you to. The two computations are mirror images, and the whole question is whether they meet.

There is a flaw in the floor as so far described, and it is fatal to any settled plan. Suppose that at matching pennies you resolve always to show heads: your opponent, anticipating it, shows tails, takes your coin every round, and your floor collapses into ruin. Any predictable rule can be found out and turned against its maker, so a fixed strategy's apparent security evaporates the instant the opponent learns it. Von Neumann's escape is to enlarge what may count as a strategy: let the player choose not a move but a lottery over moves — show heads or tails as an unseen coin of his own decides, with fixed odds but no fixed result.3 Such a mixed strategy cannot be exploited, for there is no pattern to read; the most the opponent can know is the odds, and against well-chosen odds his best reply earns nothing beyond the value. At matching pennies the right mixture is plainly even, heads and tails half the time apiece, and the player who randomizes so can be neither outguessed nor outdone. Unpredictability, the most intuitive defense, proves the rigorous one as well.

3. To randomize is not the same as to choose carelessly. The player fixes the odds with care and then hands the particular outcome to chance, so that he himself cannot know in advance what he will do — which is precisely why no one watching him can know it either. Using chance deliberately to deny an adversary information is among the theory's least obvious lessons, and its most useful in practice.
⁂

With mixed strategies admitted, von Neumann could prove the result that crowns the case. In any finite game of pure opposition, he showed, the largest value a player can guarantee himself through his best mixture is exactly equal to the smallest to which his opponent, through hers, can hold him — the maximin and the minimax, the floor and the ceiling, coincide at a single point.4 This is the minimax theorem, and its force lies in the closing of a gap that might have yawned wide: the distance between what you can secure and what your enemy can deny you shrinks, in every such game, to nothing at all. There follows a single number belonging not to either player but to the game itself — its value, the amount the first can always obtain and the second always keep him from exceeding. The contest, for all its opposition, has a determinate answer, a worth fixed by its structure that optimal play will exactly realize, from which neither party has reason to stray. What had looked like a duel without a resting place proves to have precisely one.

4. The theorem holds only because the lottery is permitted; in pure strategies alone the maximin and the minimax may genuinely differ, leaving a real gap and no settled value. It was admitting the mixed strategy, von Neumann's decisive stroke, that closed the gap in every case — and the proof, which turned on a property of convex sets, was one reason the young theory reached so readily for the heavier instruments of mathematics.

The point where the two best mixtures meet has a shape that earns it a name. Imagine the array of outcomes laid out as a landscape, one player's choices running east and the other's north: the solution sits at a saddle, the lowest point along one ridge and the highest along the ridge that crosses it, so that either player who wanders off it can only worsen his standing.5 This saddle point was the first equilibrium game theory possessed — the first exact sense in which a combination of strategies could be called a game's solution, stable because no one gains by moving from it. From the bare theorem of 1928 von Neumann built outward, and in the 1944 volume he wrote with Oskar Morgenstern the scattered insight grew into a discipline, with an apparatus for games of every kind. Economics has numbered him among its founders ever since, though he came to it sidelong, a mathematician by trade; the saddle point of a zero-sum game was the seed from which it all grew.

5. Morgenstern, an economist, brought the conviction that these ideas belonged to his field and not to mathematics alone; von Neumann brought the theorems. Their collaboration gave the subject both its name and its first systematic treatment, and though the zero-sum case was where the method's power showed first, the book reached well beyond it, toward the games of mixed interest that occupy the rest of this volume.

For all the completeness of the result, it governs only the special case — and the specialness is exactly the point. Pure opposition is rare; in most of the situations this volume studies, the players share some interest even while they contest the rest, and there the clean guarantee of the minimax quietly dissolves. It was to those wider games that John Nash carried the equilibrium two decades later, showing a resting point survives even when the players' interests are not perfectly opposed — the idea with which the volume opens, from which the prisoner's dilemma draws its sting. And it was past pure conflict altogether that Thomas Schelling would press, into the mixed-motive country of bargains, threats, and promises, where the decisive move is a commitment, not a calculation. The minimax stands behind all of it as the solved and settled base: the single kind of game whose answer is never in doubt, the foundation laid first precisely because it could be finished — before the subject turned to the harder country where opposition and common interest lie tangled, and cooperation becomes a question rather than a contradiction.