Chances Are (42 page)

In this game we would obviously prefer to follow strategy B, since it offers the highest payoff and the most we could lose is 1 unit. We would not be so blind as to choose strategy C, since the most we could win is 1 and we might lose 4. Similar considerations drive our opponents' choice, seeking to minimize their potential loss by never choosing strategy 3. In the absence of mutual expectation, the combination B2 would represent the minimized maximum loss (or “minimax,” to use the jargon) for both sides—a pointless but safe position.

If, however, our opponents
know
we will choose strategy B, they will choose strategy 1; they gain, we lose; but if
we
know they will choose strategy 1, we will choose strategy A and win three units; but if
they
know we will choose strategy A, they will choose strategy 2. We are in the same situation we found playing Le Her: a permanent cycle of mutual second-guessing. The answer here, as it was there, is a mixed strategy. We should choose strategy A one-sixth of the time and strategy B five-sixths of the time; they should choose strategy 1 one-third of the time and strategy 2 two-thirds of the time. On average, we will lose and they gain one third of a unit each game. This may not seem very satisfactory, but it does mark the game's center of gravity: if both sides play skillfully with the aim of minimizing maximum loss, the game will tend in this direction as assuredly as tic-tac-toe heads for a draw.

The first application of game theory outside the competitively charged atmosphere of a Budapest parlor was in economics, because here, unlike in war, there were no clear maxims to link an individual decision to an overall result. If a general neglects a cardinal rule (by, say, invading China or Russia), the blunder itself leads logically to defeat. But if a government or industry somehow collectively fails to respond to a change in the habits of the market—yes, unemployment or deflation results, but whose exactly was the decision that produced this outcome? Demand is composed of competing consumers, supply of competing producers. Their strategies include straightforward monetary decisions about price and investment, but also more complicated issues of reputation, emulation and rivalry. Money is only one measure of success; a better general yardstick is utility. So you can see the appeal of game theory as a model for economics: it explains, by a few axioms, how a given arrangement of payoffs can influence the millions of rational competitive decisions that add up to the workings of a market or business.

Classical economics, like classical physics, seemed to describe circumstances that never actually occur on earth: frictionless markets, with no entry costs and with every agent content to operate individually under the rules of supply and demand. By providing a way to study the mutual influence of strategic choices, game theory brought economics out of this clockwork universe. It did so with the publication, in 1944, of
The Theory of Games and Economic Behavior
by von Neumann and a Princeton colleague, the Viennese economist Oskar Morgenstern. The book sets out a body of axioms governing rational strategic choice in economic life and is one of the great unread masterpieces, saluted as the most influential work in its field while selling fewer than 4,000 copies.

In the meantime, there was a war to win. Borel had warned that military affairs were too complicated to be reduced to the equivalent of poker, but the atomic bomb, it seemed, simplified things greatly. To drop or not to drop; to strike first, or guarantee mutual annihilation; to make the strong hand tell or bluff out the weak one. This wasn't like poker; it
was
poker.

Game theory was already being applied by von Neumann's students to the selection of bombing targets in Japan. If you bomb only the important target, defense can be concentrated there; if you bomb an unimportant target, you scatter the defense but waste a raid. Now von Neumann was called in to help choose where the atomic bomb would fall. It was a task he apparently welcomed.

In retrospect, it is questionable whether the atomic bomb actually made strategic conflict as “scientific” as was assumed at the time. At the beginning of World War II, the bombing of cities held an equivalent significance in the military imagination: the dreadful trump to be played only in extremity. When it came at last, the results were indeed appalling in the loss of lives and beauty, but the earlier assumption that people would panic and society collapse proved wrong. Even in Japan, the two atomic explosions may have hastened the end of the war only through bluff—the false implication that there were many more bombs in reserve. In terms of sheer destruction, Japan had already suffered raids more horrible than those on Hiroshima and Nagasaki. Yet the immediate end of hostilities made atomic weapons seem a potent war winner; especially since the nature of the new enemy, communism, made a conventional military response impossible: above all, never invade Russia or China.

The world's conflict narrowed to a 2-by-2 matrix and military thinking was taken over by civilians: the staffers at RAND (which stands simply for “R and D”) and, later, Robert McNamara's “whiz kids” at the Pentagon. The
a
-or-
b
quality of strategic problems—decisions made in minutes bringing destruction to millions—handed the intellectual baton to those who were used to considering large, powerful things in the abstract: mathematicians and physicists. There would be no time to learn in battle, as the American army had always done. The next war had to be winnable in
theory
.

Von Neumann was a regular if inexpert poker player, and he was also a connoisseur of the uses of power. One passage he knew by heart and liked particularly to recite was the bald message of the Athenians to the dithering Melians in Thucydides'
History of the Peloponnesian War
: “The strong do what they have the power to do, and the weak accept what they have to accept.” After the war, von Neumann saw the world as a simple two-person, zero-sum game with perfect information. The United States had the atomic bomb. The Soviet Union did not, but, he suspected, soon would—thanks to Klaus Fuchs, its spy in Los Alamos. The carving up of eastern Europe made clear that Stalin's intentions were aggressive and expansionist: there could be no peaceful coexistence. What game could be simpler? If the Soviets' strategic choice was to build bombs or not, we knew they would; if our choice was to launch a preemptive strike or not, the contrast in outcomes made our preferred course clear. Millions of Russian dead were worth a secure world. “If you say why not bomb them tomorrow, I say why not today?” von Neumann argued; “If you say today at 5 o'clock, I say why not 1 o'clock?”

The matrix proved his point: as long as we had the bomb and Stalin did not, we had a strategy that maximized our utility no matter what he might do:

Once the Soviets perfected their weapon, the payoffs would change:

Von Neumann took his matrix to the White House—but neither of the two presidents, Truman and Eisenhower, to whom he made these arguments acted on them. Why? Perhaps because they had an innate sense of the things left uncovered by game theory: questions of repetition and of the nature of utility. You bomb the Russians at 1 o'clock—then what? In what way is the game over? Truman had been an artillery captain, proud to have kept his gun firing until the very last minute of the first World War, but he felt, with a politician's optimism, that: “When we understand the other fellow's viewpoint, and he understands ours, then we can sit down and work out our differences.” Eisenhower knew even more what he was talking about: “I hate war as only a soldier who has lived it can, only as one who has seen its brutality, its futility, its stupidity.” Both knew that the game does not end in the quiet after the last explosion.

Interestingly, the one conflict of the Cold War period that most clearly followed game-theory principles was entirely conventional: the Korean War. In the three years that conflict washed up and down the peninsula, both the UN and Chinese forces came to understand their minimax solution: a return to the
status quo ante
. The North Koreans and General MacArthur, each holding out for ultimate victory, both lost.

 

Von Neumann's minimax theorem had applied only to zero-sum games, in which each player's gain is the other's loss. His RAND colleague John Nash extended the idea to include games in which it is possible for
both
players to benefit by a combination of strategies. Nash showed that these, too, must have equilibria: situations where both players effectively say after the game, “Well, given what the opposition did, I got the best result possible.”

These equilibria need not be the outcomes that both players find most desirable or most efficient. The good, the simple, and the possible are three different things and rarely coincide; so, if you find yourself in a situation where they do not, your definition of the good—your utility function—may have to change to suit the equilibria.

Richmond—capital, rail center, and workshop of the Confederacy and less than a hundred miles from Washington—was the great problem for the Union high command in the American Civil War. The essential components of any campaign against it were straightforward and well known: the attacking Army of the Potomac would be more numerous and better equipped than the defending Army of Northern Virginia. The defenders would have the advantage of knowing the country intimately and being under the orders of a general, Robert E. Lee, who made no unforced errors. God being on the side of the big battalions, this was a game with an equilibrium that should favor the Union—but it took time to find a general willing to do what this equilibrium demanded.

The first player was General George McClellan, “the little Napoleon.” Second in his class at West Point when Robert E. Lee had been its Superintendent, he saw war as a clash of generals and brilliance as a general's supreme virtue. He knew what his opponents knew, and he knew them, too.

In April of 1862, McClellan first risked his hand in a thrust toward Richmond: the Peninsular campaign. His purpose, of course, was to win the war; but he had other purposes, other utilities: to show his old class-mates his powers, to conserve the lives of his soldiers, and to calm the fears of his political masters. He therefore advanced cautiously, securing his lines, probing toward the front, and trusting to his superiority in numbers and firepower.

Lee understood this complex of utilities perfectly and promptly set about breaking it up. He menaced Washington with a small but alarming force; he sent “Prince John” Magruder to flit about in front of the Union advance, convincing McClellan that he was facing an army twice its actual size; he stripped defenders away from the Richmond perimeter, joined them to his own force and then struck the Federals unexpectedly, hard, and repeatedly. McClellan got most of his troops out by a combination of good staff work and staunch defense—but this was not the kind of brilliance he had intended to show. By valuing lives and reputation over victory, he had failed to find the equilibrium.

Two years later, a very different man crossed the Rapidan and advanced into northern Virginia: Ulysses S. Grant was a man inured to failure and unburdened by a name for brilliance. Although he, too, had been a West Pointer, it had been near the bottom of his class. His army career had not prospered: he showed a talent for mathematics and horsemanship, but had no conversation and could not hold his liquor. War found him bankrupt in Ohio, bossed around the family leather store by his two younger brothers. Grant, though, knew a few things that McClellan did not: that politicians would forgive anything for victory; that Lee had fewer lives to spare than he did; that superior force, if it bears down unrelentingly, must prevail.

Grant had a simple strategy: to outflank Lee's right. Lee, always one notion ahead, had no intention of letting this happen and attacked the Federal army while it was still toiling through the dense scrub of the Wilderness. It was a bloody, ferocious struggle in undergrowth that was soon aflame. Grant lost 17,500 men; Lee 8,000. Outthought and out-fought, a conventional general would have pulled back across the river; Grant extended to the East and pushed South.