Chances Are (41 page)

This solution introduced two very powerful ideas: first, that despite not being able to specify an expectation from the game, a player could act to restrict the maximum loss; second, that in the absence of any single best strategy, a player should choose one course or another randomly—but weighting that choice according to the different payoffs from each alternative, in what is now called a
mixed strategy
.

Soon Daniel Bernoulli, in considering the Saint Petersburg paradox, would come up with the idea of utility as a component of probabilistic calculation—and utility works far better than all those ducats and pistoles as a means of calculating the value to each player of the various outcomes from strategic choices. Taken together, these ideas provided the foundation for modern game theory—though it took another two hundred years before it came formally into being.

 

One puzzle of war is how the individual soldier's utility is overtaken by that of the army as a whole. Socrates, in Plato's
Laches,
pointed out how odd it seems to individual rationality that one man, however brave, should choose to stay and help other men defend a dangerous position when his best interest is to save himself and let them fight. Tolstoy growled in his beard at the historian's lazy assumption that generals “won” or “lost” battles, as if the experience of the thousands of participants had no meaning. His source for much of
War and Peace,
the French exile Joseph de Maistre, denied that anyone could really tell what was going on in a battle:

 

And yet armies
do
seem to bind themselves into a single will, at least for limited military purposes. Generals may always find more constraint and muddle than they wish, but it is still possible for them to set massed forces in motion with a word and a gesture.

Tolstoy, echoing de Maistre, insisted that victory or defeat had to do not with the final dispositions of men and resources but with the beliefs of the opposing forces. There are many examples of the stronger army willing itself into defeat or the weaker stubbornly refusing to accept it. After losing the battle of Albuera in 1811, Marshal Soult complained: “The British were completely beaten and the day was mine, but they did not know it and would not run.”

The disciplines of war are anti-heroic; they aim to reduce the individual fighter to a known, repeatable quantity with predictable force. But discipline soon congeals into rigidity. Cromwell's red-coated Puritans and Gustavus Adolphus' ruthless Swedish Lutherans generated their own uniformity through a shared culture of self-sacrifice for a cause. Later armies aped the form but lost the motive power. An eighteenth-century European officer had all his accomplishments dictated to him—from siegecraft to the minuet—at the Academy of Pages or the École des Nobles. He would know his Vegetius as a contemporary peasant would know the Lord's Prayer: every word by heart, but with no great expectation of putting its spirit into effect. Armies, too, tend to entomb the live principle in the casket of form: the Austrians, seeing how their Croat irregulars won great success against the Turks, mustered them into regiments and drilled them into such soldierly perfection that they lost all their former effectiveness. As Socrates had noticed, discipline means forgoing private ends like safety in favor of shared ones like victory—but if
everyone
loses sight of the ends, the means take over. Effective discipline becomes petty rule-mongering.

It was at this point that Napoleon came along to unsettle the military mind. He did not dance. He favored unfashionable services like field artillery and supply. He
cheated,
innocently and openly at chess, but also in the more formal game of war. He divided his forces in the face of the enemy; through forced marches, he caught his opponents between battlefields; he broke properly formed lines of battle with cannon and columns of attack; and from the first, he negotiated while he campaigned, making politics a branch of strategy.

Most of all, Napoleon was an individual, waging war according to his own will and ideas. The great commanders of the previous century had worked within a system, but Napoleon's battles, even when fought by his marshals, were his personal creation; and the generals who succeeded against him were those who had learned to work within his terms. Napoleon opened with artillery—Wellington took positions behind ridges and crests. Napoleon attacked in column—Wellington perfected the technique of alternating volleys from two lines, concentrating his fire; Napoleon swept late with cavalry—Wellington formed into squares, as invulnerable as a porcupine. The cheating had become predictable through repetition: “He came at us in the same old way,” said Wellington after Waterloo, “and we saw him off in the same old way.”

Napoleon's wars reopened the nagging question: Is there a science of combat? He had claimed his method was “I engage and then I see”—but that was just another ruse to retain the glory and baffle the hidebound. He had discovered how to make luck work for him: to gamble on the probabilities, not just of force and terrain, but of his opponents' preconceptions. Once his luck gave out and the great individual was chained to his desert rock, the victorious powers sought to replicate his genius in new doctrines, rules, and exercises.

The various German courts and schools of pages had long amused themselves with a pastime called “war chess.” In 1811 the Prussian Baron von Reisswitz—ironically, a civilian—came up with a new game of war on a sand table, sculpted to allow its little porcelain armies to use terrain stealthily, as Napoleon had shown they could. Von Reisswitz's son made the final improvements. A lieutenant of artillery, he had that arm's willingness to substitute calculation for flair. He replaced the beautiful but distracting model landscape with the newly invented military contour map, the porcelain troops with small metal strips, the assumptions of the players with detailed descriptions of every type of engagement—and the rule of the umpire with the roll of a die.

Probability had arrived on the field of combat. After all, the race is not always to the swift, nor the battle to the strong; it just
tends
to be that way. Napoleon had shown the vital role of luck in war, so the new game, “Kriegspiel,” enshrined it in the random, although properly weighted, decision of each encounter. “The object is for the player who has good luck to seize and use it, and for the player who has the misfortune to meet with bad luck to take those correct measures which in reality would be required.”

The game was an immediate success—soon, all the Prussian officer corps (and, after Prussia's victories, all the world) was Kriegspiel-mad. This madness took two forms: one school thought Kriegspiel a meticulously measured blueprint for war; the other an inspired freehand sketch of its variability. This divergence in opinion also reflected the opposed traditions of German and French military education in the increasingly frantic decades as Europe spurred on toward the Great War.

Victoire, c'est la volonté
. France, said her generals, would prevail through self-belief; the spirit of all-out attack, distilled through the fiery aphorisms of inspired superiors, would fill the troops with
élan vital
and bias the dice in their favor. Even their trousers were an aggressive red: “
Le pantalon rouge, c'est la France!”
exclaimed M. Étienne, an ex-War Minister. These patriotic trousers would prove a great aid to the aim of the field-gray Germans in 1914.

In Germany, though, Kriegspiel exacerbated the sense of godlike control that is the occupational disease of all General Staffs. This had its highest expression in the Plan Cabinet, where rested, each in its sealed parcel, the detailed instructions for waging all conceivable wars; there was, apparently, a plan for an amphibious invasion of New York. Germany's army numbered one and a half million men, to be propelled toward France in 11,000 trains on schedules exact to the minute. There could be no question now of Napoleonic flair, of “engage and then see”; everything had to be calculated. The blueprint for the war to come, the Schlieffen Plan, deliberately violated Belgian neutrality in part because there was simply not enough room to maneuver such large forces on the Franco-German border. Kriegspiel had created war in its own image: numerical, directed, inevitable.

But let's return for a moment to von Reisswitz's true invention. He had allowed fate—randomness—to have its say at the most local level, when players rolled a die to decide the outcome of each engagement. Above that level, everything was a multiplication of these local results. The weights of the die, the implicit probabilities of victory or defeat, were determined by an informed assessment of the firepower, the killing potential, of each unit and its weapons; but this assessment was not a fact. It was a guess—and it could be wrong.

The Maxim gun can fire 450 rounds of .303 ammunition in one minute; before the war, the German army estimated this to be the equivalent of 80 rifles. This is true in simple terms of weight of fire, but it was a tragic misjudgment. Soldiers were soon to find there was a great difference between attacking an entrenched position held by eighty separate infantrymen with their varied skills, attention, stamina, and bravery—and one held by a single determined machine gunner, content to blaze away forever. If a blueprint errs in one critical dimension, it is no more useful than a freehand sketch: both versions of Kriegspiel had failed to capture reality.

 

Émile Borel was a man perfectly constituted to rediscover game theory: he had worked in cooperation with Lebesgue on measure, the mystic chain by which probability swung from the world of men to the abstract, multidimensional realm of set theory. Borel studied cards, wrote a probabilistic analysis of bridge, was Minister of Marine in the mid-twenties, and came out of the Second World War a seventy-year-old with the Resistance medal.

Borel wrote five papers during the 1920s on the theory of games. Knowing nothing of Waldegrave, he returned to the idea of entirely symmetrical games: games without a house advantage, where, if both players adopt the same strategy, their chance of winning is equal. He assumed that players would begin by eliminating those strategies that would provide a lower return no matter what the opponent did; and then noticed the interesting point that doing so might itself weaken other strategies. For example, every schoolchild knows that running away or running to the teacher at every affront is a weak strategy; yet the rule of the playground also states that exploding with rage at each shove or kick only invites more. Eliminating weaker choices may leave no single best strategy; a mix of submission, reporting, and resistance may be the only way to keep the bullies off balance, with the proportions in the mix determined by their probability of success, but chosen randomly each time to prevent predictability. As Borel said: “to follow [a mixed strategy] to the letter, a complete incoherence of mind would be needed, combined, of course, with the intelligence necessary to eliminate those methods we have called bad.” This Brer Rabbit method of calibrated incoherence turns out to govern many of childhood's most popular games; you should follow it, for instance, next time you are challenged to play rock/paper/scissors.

You cannot have your heart's desire and win every time—since that is not your opponent's desire, and you are locked together in this enterprise. The best you can get is the minimum of your maximum risk—and, as in Waldegrave's solution for Le Her, there is a mixed strategy that will achieve it for you.

 

John von Neumann was born in 1903 to a family of Jewish bankers in Budapest; in his boyhood, he played a homemade version of Kriegspiel with his brothers, marching model soldiers across the map to the dictates of a die. He was always a competitive man—like many game theorists, he seemed to think of genius as a kind of muscle, to be demonstrated in feats of mental arm wrestling. When he read Borel's work, he noticed the claim that a choice among more than three strategies guaranteeing the least bad outcome was impossible. He could not let this challenge pass; and in 1928, he published an article in which he proved—using mathematics few but he would find easy—that every finite, two-person, zero-sum game has a rule-determined, preordained solution under the use of mixed strategy types.

This is easiest to understand using the pictorial form of game theory: the matrix. Across the top are the various strategies open to the opponent (bluff, call, attack, maneuver, flirt, commit); down the side are the options for our side (fold, raise, defend, harass, misunderstand, surrender). The combination of choices yields payoffs for each side (ours first, theirs second) defined in terms of our utility—whatever it is we hope to have most of, whether it be money, territory, or self-respect.