A problem that previously would have taken three people three months to solve could now be worked out by the same three people in ten hours. The research on the H-bomb was, thanks to MANIAC, lifted out of its slump.
Through the late 1940s and early 1950s, von Neumann made frequent trips to RAND. John Williams, head of RAND’s math division, adored Johnny von Neumann, and was thrilled when, in December 1947, he convinced von Neumann to join RAND as a part-time consultant. Williams would frequently try out immensely difficult math problems on von Neumann during his visits, but never stumped him. Von Neumann had the sort of mind that could routinely solve in his head the most elaborate calculations to the second or third decimal point.
In 1928, when he was twenty-four, von Neumann had sat in on a fateful poker game that set in motion a remarkable train of logical observations. First, he noted that a player’s winnings and losses depended not only on his own moves, but also on the moves of the other players. In devising a strategy, he had to take into account the strategies of the other players, assuming that they too were rational; that, therefore, the essence of the good strategy was to win the game, regardless of what the other players did, even though, paradoxically, what the other players do determines, in part, the playing of the game.
Von Neumann then realized that the game of poker was fundamentally similar to the economic marketplace. Economists had been attempting to impose mathematical models on classical economic theory, but with no success. The reason for their failure, von Neumann reflected, was that the theory assumed an independent consumer trying to maximize his gains and independent sellers trying to maximize theirs—whereas, in fact, just as in the game of poker, the consumer and the seller formed a unit, competing but interdependent, and their moves could not be systematically analyzed or strategically planned except in the context of each other’s moves. So it goes with any situation in which two or more players have a conflict of interest and in which a good deal of uncertainty is involved.
Von Neumann developed what ultimately came to be called “game theory” as a mathematically precise method of determining rational strategies in the face of critical uncertainties. The classical case of game theory is the Prisoners’ Dilemma. Two prisoners, arrested on suspicion for the same crime, are kept in separate cells with no chance to communicate. They are separately approached by guards and given the following proposition: if neither squeals on the other, they will both serve brief sentences; if Prisoner A tells on B, but B keeps quiet, then A will be let free and B will serve maximum sentence; likewise, if B talks but A remains silent, then B will be freed and A forced to serve full sentence; if both A and B rat on each other, then they will serve half sentences. On the surface, it seems that it would serve both their interests to remain silent. However, there is a great deal of uncertainty. Prisoner A worries that Prisoner B might feel compelled to talk, since it would be to B’s advantage to do so; if, under such circumstances, A does not talk, A serves a full jail sentence. Prisoner B is, of course, thinking similar thoughts about Prisoner A’s possible moves. Therefore, both prisoners will talk and both will serve half jail sentences, even though both would have been better off keeping quiet.
According to game theory, moreover, both prisoners would be perfectly rational if they did talk. Both have to assume that the other prisoner, the other player, will play his best move; thus, each has to play the move that would be best for himself given the best move of the other player. That is the essence of game theory: find out your opponent’s best strategy and act accordingly. Such a strategy may not get you the maximum gain, but it will prevent you from taking the maximum loss.
Von Neumann proved his theory with highly complex mathematics, which have never been disputed, and also illustrated his proof with some lower-level matrixes. The Prisoners’ Dilemma can be outlined in a rather simple diagram.
The rows represent A’s possible choices; the columns represent B’s. The numbers in the boxes represent the values that they each place on the possible choices, with A’s value cited first. If they both talk (-.5,-.5), they both serve half sentences; if one talks and the other remains silent (1,-1; -1,1), one goes free and the other serves full sentence; if both remain silent (.7,.7), they serve brief sentences.
If Player A is rational, he will scan his possible strategies and pick the one that gives him the least-worst payoff, no matter what strategy B picks. Talking could give him -.5, but remaining silent might give him -1, if B decides to talk. So A will go with the strategy of talking. Similarly, B’s silence might give him a negative payoff of -1, whereas talking will reduce that loss by half. He too will talk. Von Neumann called that the “minimax solution”—the lowest maximum, the highest minimum.
Von Neumann wrote a scholarly paper on game theory in 1928 and created a minor sensation in the scientific and mathematical communities of Europe. The sensation exploded in 1944 when he and a Princeton economist named Oskar Morgenstern collaborated to write an enormous volume in English called Theory of Games and Economic Behavior, laying out the theory systematically, offering the mathematical proofs, suggesting applications of the theory to economics and the entire spectrum of social conflict.
It was a conservative theory and pessimistic as well. It said that it was irrational behavior for one to take a leap, do what is best for both parties and trust that one’s opponent might do the same. In this sense, game theory was the perfect intellectual rationale for the Cold War, the vehicle through which many intellectuals bought on to its assumptions. It was possible to apply the Prisoners’ Dilemma, for instance, to the Soviet-American arms race—substituting “Build More” for “Talk” and “Stop Building” for “Silence.” It made sense for both sides to stop building, but neither could have the confidence to agree to a treaty to stop building arms (.7,.7), suspecting that the other might cheat, build more and go on to win (-1,1). Distrust and the fostering of international tensions could be elevated to the status of an intellectual construct, a mathematical axiom.
Game theory caught on in a very big way at RAND in the late 1940s. John Williams was particularly entranced with it, and wrote a very lively compendium of dozens of situations and anecdotes in which game theory could play a valuable role in guiding decisions. But there was a major limitation to game theory. For it to be used precisely, as a science, the analyst had to have some way of calculating what numbers should go in the matrixes. This would be particularly true in games not just of two-by-two matrixes—such as Prisoners’ Dilemma—but three-by-three or five-by-five or ten-by-ten or any number of combinations. And then what about those games that involve not just two players but three or four or more? Then there were games where both players did not settle on the same box in the matrix, or had no logical reason to do so—where certain moves might be optimal 60 percent of the time, but other moves 40 percent of the time. In these cases, the players would have to play according to a mixture of random selection and the laws of probability, just as a good poker player bluffs systematically but randomly, so that his strategy is not discovered.