We hope [the theory of games] will work, just as we hoped in 1942 that the atomic bomb would work. — ANONYMOUS PENTAGON SCIENTIST to Fortune, 1949
NASH’S NOVEL IDEA about games with many players had preceded him at RAND by several months. The first version of his elegant proof of the existence of equilibrium for games with many players — two skimpy pages in the November 1949 issue of the National Academy of Sciences proceedings — swept through the white stucco building at Fourth and Broadway like a California brushfire.1
The biggest appeal of the Nash equilibrium concept was its promise of liberation from the two-person zero-sum game. The mathematicians, military strategists, and economists at RAND had focused almost exclusively on games of total conflict — my win is your loss or vice versa — between two players. Shapley and Dresher’s 1949 review of game theory research at RAND refers to the organization’s “preoccupation with the zero-sum two person game.”2 That preoccupation was natural, given that these were games for which the von Neumann theory was both sound and reasonably complete. Zero-sum games also seemed to fit the problem — nuclear conflict between two superpowers — which absorbed most of RAND’s attention.
Only it really didn’t. At least some of the researchers at RAND were already chafing at the central assumption of a fixed payoff in such games, Arrow recalled.3 As weapons got ever more destructive, even all-out war had ceased to be a situation of pure conflict in which opponents had no common interest whatever. Inflicting the greatest amount of damage on an enemy — bombing him back to the Stone Age — no longer made any sense, as American strategists realized during the final phase of the campaign against Germany when they decided not to destroy the coal mines and industrial complexes of the Ruhr.4 As Thomas C. Schelling, one of RAND’s nuclear strategists, would put it a decade later,5
In international affairs, there is mutual dependence as well as opposition. Pure conflict, in which the interests of two antagonists are completely opposed, is a special case; it would arise in a war of complete extermination, otherwise not even in war. The possibility of mutual accommodation is as important and dramatic as the element of conflict. Concepts like deterrence, limited war, and disarmament, as well as negotiation, are concerned with the common interest and mutual dependence that can exist between participants in a conflict.
Schelling goes on to say why this is so: “These are games in which, though the element of conflict provides the dramatic interest, mutual dependence is part of the logical structure and demands some kind of collaboration or mutual accommodation—tacit, if not explicit — even if only in the avoidance of mutual disaster.”6
In 1950, at least the economists at RAND were aware that if game theory were to evolve into a descriptive theory that could be usefully applied to real-life military and economic conflicts, one had to focus on games that allowed for cooperation as well as conflict. “Everybody was already bothered by the zero-sum game,” Arrow recalled. “You’re trying to decide whether to go to war or not. You couldn’t say that the losses to the losers were gains to the winner. It was a troublesome thing.”7
• • •
Military strategists were the first to seize on the ideas of game theory. Most economists ignored The Theory of Games and Economic Behavior and the few that didn’t, like John Kenneth Galbraith writing in Fortune and Carl Kaysen, later director of the Institute for Advanced Study, turn out to have had significant contact with military strategists during the war.8 An article in Fortune in 1949 by John McDonald made it clear that the military hoped to use von Neumann’s theory of games to work out intelligence missions, bombing patterns, and nuclear defense strategy.9 On the lookout for new ideas and with plenty of money to spend, the Air Force embraced game theory with the same enthusiasm with which the Prussian military had embraced probability theory a couple hundred years earlier.10
Game theory had already made its debut in military planning rooms. It had been used during the war to develop antisubmarine tactics when German submarines were destroying American military transports. As McDonald reported in Fortune:11
The military application of “Games” was begun early in the last war, some time in fact before the publication of the complete theory, by ASWOEG (Anti-Submarine Warfare Operations Evaluation Group). Mathematicians in the group had got hold of von Neumann’s first paper on poker, published in 1928.
But von Neumann actually spent his frenetic visits to Santa Monica almost exclusively with the computer engineers and the nuclear scientists.12 His enormous prestige and Williams’s deft salesmanship led to a major concentration on game theory at RAND from 1947 into the 1950s. The hope was that game theory would provide the mathematical underpinning for a theory of human conflict and spread to disciplines other than mathematics. Williams convinced the Air Force to let RAND create two new divisions, economics and social science. By the time Nash arrived, a “trust” of game theory research had grown up at RAND including such game theorists as Lloyd S. Shapley, J. C. McKinsey, N. Dalkey, F. B. Thompson, and H. F. Bohnenblust, such pure mathematicians as John Milnor, statisticians David Blackwell, Sam Karlin, and Abraham Girschick, and economists Paul Samuelson, Kenneth Arrow, and Herbert Simon.13
Most of the RAND military applications of game theory concerned tactics. Air battles between fighters and bombers were modeled as duels.14 The strategic problem in a duel is one of timing. For each opponent, having the first shot maximizes the chance of a miss. But having the better shot also maximizes the chance for being hit. The question is when to fire. There’s a tradeoff. By waiting a little longer each opponent improves his own chance of scoring a hit, but also increases the risk of being shot down. Such duels can be both noisy and silent. With “silent guns,” the duelist doesn’t know the other has fired unless he is hit. Therefore, neither participant knows whether the other still has a bullet or has fired and missed and is now defenseless.
A report by Dresher and Shapley summarizing RAND’s game theory research between the fall of 1947 and the spring of 1949 gives the flavor.15 The mathematicians describe a problem of staggered attacks in a bombing mission:
Problem
A single intercepter base, having I fighters, is located on a base line. Each fighter has a given endurance. If a fighter, vectored out against a bomber attack, has not yet engaged his original target, then at the option of the ground controller he may be vectored back to engage a second attack.
The attacker has a stock of N bombers and A bombs. The attacker chooses two points to attack and sends N1 bombers including A1 bomb carriers on the first attack and t minutes later he sends N2 = N — N1 bombers including A2 = A — A 1 carriers on the second attack.
The payoff to the attacker is the number of bomb carriers that are not destroyed by the fighters.
Solution
Both players have pure optimal strategies. An optimal strategy of the attacker is to attack both targets simultaneously and distribute the A bomb carriers in proportion to the number of bombers in each attack. An optimum strategy of the defender is to dispatch interceptors in proportion to the number of attacking bombers and not to revector fighters. The value of the game to the attacker will be
V = max (0,A(1 — 1/Nk))
where k is the kill probability of the fighter
The game Nash had in mind could be solved without communication or collaboration. Von Neumann had long believed that the RAND researchers ought to focus on cooperative games, conflicts in which players have the opportunity to communicate and collaborate and are able “to discuss the situation and agree on a rational joint plan of action, an agreement that is assumed to be enforceable.”16 In cooperative games, players form coalitions and reach agreements. The key assumption is that there’s an umpire around to enforce the agreement. The mathematics of cooperative games, like the mathematics of zero-sum games, is rich and elegant. But most economists, like Arrow, were cool to the idea.17 It was like saying, they thought, that the only hope for preventing a dangerous and wasteful nuclear arms race lay in appointing a world government with the power to enforce simultaneous disarmament. World government, as it happens, was a popular idea among mathematicians and scientists at the time. Albert Einstein, Bertrand Russell, and indeed much of the world’s intellectual elite subscribed to some version of “one worldism.”18 Even von Neumann tipped his hat to the notion, conservative hawk that he was. But most social scientists were dubious that any nation, much less the Soviets, would cede sovereignty to such an extent. Cooperative game theory also seemed to have little relevance to most economic, political, and military problems. As Arrow jokingly put it, “You did have cooperative game theory. But I couldn’t force the other side to cooperate.”19
By demonstrating that noncooperative games, games that did not involve joint actions, had stable solutions, said Arrow, “Nash suddenly provided a framework to ask the right questions.” At RAND, he added, it immediately led “a lot of people to calculate equilibrium points.”
• • •
News of Nash’s equilibrium result also inspired the most famous game of strategy in all of social science: the Prisoner’s Dilemma. The Prisoner’s Dilemma was partly invented at RAND, some months before Nash arrived, by two RAND mathematicians who responded to Nash’s idea with more skepticism than appreciation of the revolution that Nash’s concept of a game would inspire.20 The actual tale of prisoners used to illustrate the game’s significance was invented by Nash’s Princeton mentor, Al Tucker, who used it to explain what game theory was all about to an audience of psychologists at Stanford.21
As Tucker told the story, the police arrest two suspects and question them in separate rooms.22 Each one is given the choice of confessing, implicating the other, or keeping silent. The central feature of the game is that no matter what the other suspect does, each (considered alone) would be better off if he confessed. If the other confesses, the suspect in question ought to do the same and thereby avoid an especially harsh penalty for holding out. If the other remains silent, he can get especially lenient treatment for turning state’s witness. Confession is the dominant strategy. The irony is that both prisoners (considered together) would be better off if neither confessed — that is, if they cooperated — but since each is aware of the other’s incentive to confess, it is “rational” for both to confess.
Since 1950, the Prisoner’s Dilemma has spawned an enormous psychology literature on determinants of cooperation and defection.23 On a conceptual level; the game highlights the fact that Nash equilibria — defined as each player’s following his best strategy assuming that the other players will follow their best strategy — aren’t necessarily the best solution from the vantage point of the group of players.24 Thus, the Prisoner’s Dilemma contradicts Adam Smith’s metaphor of the Invisible Hand in economics. When each person in the game pursues his private interest, he does not necessarily promote the best interest of the collective.
The arms race between the Soviet Union and the United States could be thought of as a Prisoner’s Dilemma. Both nations might be better off if they cooperated and avoided the race. Yet the dominant strategy is for each to arm itself to the teeth. However, it doesn’t appear that Dresher and Flood, Tucker, or, for that matter, von Neumann, thought of the Prisoner’s Dilemma in the context of superpower rivalry.25 For them, the game was simply an interesting challenge to Nash’s idea.
The very afternoon that Dresher and Flood learned of Nash’s equilibrium idea, they ran an experiment using Williams and a UCLA economist, Armen Alchian, as guinea pigs.26 Poundstone says that Flood and Dresher “wondered if real people playing the game — especially people who had never heard of Nash or equilibrium points — would be drawn mysteriously to the equilibrium strategy. Flood and Dresher doubted it. The mathematicians ran their experiment one hundred times.”
Nash’s theory predicted that both players would play their dominant strategies, even though playing their dominated strategies would have left both better off. Though Williams and Alchian didn’t always cooperate, the results hardly resembled a Nash equilibrium. Dresher and Flood argued, and von Neumann apparently agreed, that their experiment showed that players tended not to choose Nash equilibrium strategies and instead were likely to “split the difference.”
As it turns out, Williams and Alchian chose to cooperate more often than they chose to cheat. Comments recorded after each player decided on strategy but before he learned the other player’s strategy show that Williams realized that players ought to cooperate to maximize their winnings. When Alchian didn’t cooperate, Williams punished him, then went back to cooperating next round.
Nash, who learned of the experiment from Tucker, sent Dresher and Flood a note — later published as a footnote in their report — disagreeing with their interpretation:27
The flaw in the experiment as a test of equilibrium point theory is that the experiment really amounts to having the players play one large multi-move game. One cannot just as well think of the thing as a sequence of independent games as one can in zero-sum cases. There is too much interaction. . . . It is really striking however how inefficient [Player One] and [Player Two] were in obtaining the rewards. One would have thought them more rational.
Nash managed to solve a problem at RAND that he and Shapley had both been working on the previous year. The problem was to devise a model of negotiation between two parties — whose interests neither coincided nor were diametrically opposed — that the players could use to determine what threats they should use in the process of negotiating. Nash beat Shapley to the punch. “We all worked on this problem,” Martin Shubik later wrote in a memoir of his Princeton experiences, “but Nash managed to formulate a good model of the two-person bargain utilizing threat moves to start with.”28
Instead of deriving the solution axiomatically — that is, listing desirable properties of a “reasonable” solution and then proving that these properties actually point to a unique outcome — as he had in formulating his original model of bargaining, Nash laid out a four-step negotiation.29 Stage One: Each player chooses a threat. This is what I’ll be forced to do if we can’t make a deal, that is, if our demands are incompatible. Stage Two: The players inform each other of the threats. Stage Three: Each player chooses a demand, that is, an outcome worth a certain amount to him. If the bargain doesn’t guarantee him that amount, he won’t agree to a deal. Stage Four: If it turns out that a deal exists that satisfies both players’ demands, the players get what they ask for. Otherwise, the threats have to be executed. It turns out that the game has an infinite number of Nash equilibria, but Nash gave an ingenious argument for selecting a unique stable equilibrium that coincides with the bargaining solution he previously derived axiomatically. He showed that each player had an “optimal” threat, that is, a threat that ensures that a deal is struck no matter what strategy the other player chooses.
Nash initially wrote up his results in a RAND memorandum dated August 31, 1950, suggesting that he managed to finish the paper just before leaving RAND for Bluefield.30 A longer and more descriptive version of the paper was eventually accepted by Econometrica, which had published “The Bargaining Problem” that April. Accepted for publication sometime during the following academic year, “Two Person Cooperative Games” did not in fact appear until January 1953.31 It was Nash’s last significant contribution to the theory of games.
• • •
Nobody at RAND solved any big new problems in the theory of noncooperative games. For all intents and purposes, Nash stopped working in the field in 1950. The dominant thrust of game theory at RAND came from the mathematicians, particularly Shapley, and they were guided less by applications than by the mathematics themselves. During the 1950s Shapley focused on cooperative games, which were necessarily of limited interest not only to economists but also to military strategists.
The justification of all mathematical models is that, oversimplified, unrealistic, and even false as they may be in some respect, they force analysts to confront possibilities that would not have occurred to them otherwise. The history of physics and medicine abounds with wrong or incomplete theories that throw just enough light to allow some other big breakthroughs. The atom bomb, for example, was built before physicists understood the structure of particles.
The most significant application of game theory to a military problem grew straight out of the theory of duels and helped shape what was probably RAND’s single most influential strategic study. The study was the brainchild of Al Wohlstetter, a mathematician who joined RAND’s economics group in early 1951, about six months after Nash joined the mathematics group.
According to Kaplan, the SAC operational plan in the early 1950s was to fly bombers from the United States to overseas bases and then to mobilize and launch an attack against the Soviet Union from there.32 The Air Force’s whole deterrence strategy was based on the idea of the power of the H-bomb and America’s ability to respond in kind to any attack. Apparently, no one before Wohlstetter had focused on vulnerability to a first strike aimed, not at American cities, but at wiping out the SAC force, then concentrated in a small number of foreign bases within striking distance of the Soviet Union. Kaplan writes:
Up to that point, most military applications of game theory had focused on tactics — the best way to plan a fighter-bomber duel, how to design bomber formations or execute anti-submarine warfare campaigns. But Wohlstetter would carry it further. It was this insistence on figuring out one’s own best moves in light of the enemy’s best moves that provoked Wohlstetter to look at a map and to conclude that the closer we are to them, the closer they are to us — the easier it is for us to hit them, the easier it is for them to hit us. Wohlstetter and his team estimated that a mere 120 bombs . . . could destroy 75 to 85 percent of the B-47 bombers while they casually sat on overseas bases. The SAC, seemingly the most powerful strike force in the world, was appearing to be so vulnerable in so many ways that merely putting the plan into action . . . created a target so concentrated that it invited a pre-emptive attack from the Soviet Union.33
Wohlstetter’s study had an electrifying effect on the Air Force establishment. With its focus on American vulnerability and the temptation of a Soviet surprise attack, the study also rationalized a paranoia in the military establishment that seeped into the body politic and wound up as national hysteria over the supposed “missile gap” in the second half of the 1950s. The RAND report, Fred Kaplan writes, “legitimized a basic fear of the enemy and the unknown through mathematical calculation and rational analysis, providing the techniques and the general perspective through which the new and rather scary situation — the Soviet Union’s acquisition of long range nuclear weapons — could be discussed and acted upon.”34
• • •
The golden age at RAND, from the point of view of the mathematicians, strategic thinkers, and economists, was already coming to a close.35 After a time, RAND’s sponsors grew less enthusiastic about pure research, less tolerant of idiosyncrasies, and more demanding. Mathematicians got bored and frustrated with game theory. Consultants stopped coming and permanent staffers drifted to universities. Nash never returned after the summer of 1954. Flood left for Columbia University in 1953. Von Neumann, who in any case had played a very small role in the group after inspiring it, dropped his RAND consultancy in 1954 when he accepted an appointment as a member of the Atomic Energy Commission.
Game theory, in any case, was going out of vogue at RAND. R. Duncan Luce and Howard Raiffa concluded in their 1957 book, Games and Decisions: “We have the historical fact that many social scientists have become disillusioned with game theory. Initially there was a naive band-wagon feeling that game theory solved innumerable problems of sociology and economics, or that, at least it made their solution a practical matter of a few years’ work. This has not turned out to be the case.”36 The military strategists were of the same mind. “Whenever we speak of deterrence, atomic blackmail, the balance of terror . . . we are evidently deep in game theory,” Thomas Schelling wrote in 1960, “yet formal game theory has contributed little to the clarification of these ideas.”37