Matching Pennies

John von Neumann had an astonishingly creative mind, even compared with other Hungarian mathematicians. By the age of 30 in 1933, he had developed the modern definition of ordinal numbers, specified an axiomatic foundation of set theory, and written a standard textbook on quantum physics. When he worked on the Manhattan Project, he had a key insight about how to make the atomic bomb work, and he also originated a fundamental concept of computer science, the "Von Neumann architecture." But these were just warm-up exercises for his work on the theory of games, which became the foundation of both modern economics and modern evolutionary biology

Von Neumann realized that many games are best played by randomizing what you do at each step. Consider a game called "Matching Pennies." In this game, there are two players, and they each have a penny. At each turn, each player secretly picks heads or tails: they turn their pennies heads-up or tails-up under their hands. Then the coins are revealed. If the first player, in the role of "matcher," has turned up the same side as the opponent (e.g. if both coins are heads), then the matcher wins the opponent's penny. If the coins don't match (e.g. if one is heads, the other tails), then the matcher must give a penny to the opponent. The first play is not so interesting, but as the game is repeated, one can form predictions about the opponent's behavior. The possibility of prediction makes Matching Pennies a strategically intricate game.

The roles of "matcher" and "non-matcher" seem different, but their goals are fundamentally the same: predict what the opponent will do, and then do whatever is appropriate (matching or not matching) to win the turn. All that matters is to find out the opponent's intentions. The ideal offensive strategy is to be the perfect predictor: figure out what the opponent is doing based on his or her past behavior, extrapolate that strategy to the next move, make the prediction, and win the money. But there is an easy way to defeat this prediction strategy: play unpredictably. Von Neumann remarked, "In playing Matching Pennies against an at least moderately intelligent opponent, the player will not attempt to find out the opponent's intentions, but will concentrate on avoiding having his own intentions found out, by playing irregularly heads and tails in successive games."

In particular, if a player picks heads half the time and tails half the time, then no opponent, no matter how good a predictor he or she is, can do better than break even in this game. This half-heads, half-tails strategy is an example of what game theorists call a "mixed strategy," because it mixes moves unpredictably. In their seminal 1944 book The Theory of Games and Economic Behavior, John von Neumann and Oskar Morgenstern proved an important theorem. Roughly speaking, they showed that in every competitive game between two players that has more than one equilibrium, the best strategy is mixed. We have already seen in the chapters on morality and language that many important games have more than one equilibrium. We know from evolution how important competition is. The theorem implies that when any two animals are interacting and they have a conflict of interest, they would often do well to randomize their behaviors at some level. When being predictable can make you lose a penny, unpredictability is recommended. When being predictable can make you lose your life to a predator, unpredictability is highly recommended.

The importance of randomness has long been appreciated in military strategy, competitive sports, and poker. In World War II, submarine captains sometimes threw dice to determine their patrol routes, generating a zigzagging course that would not be predictable to enemy ships. Some modern fighter aircraft are equipped with "electronic jinking" systems that can automatically randomize their evasion maneuvers when guided missiles try to intercept them ("jinking" means zigzagging very abruptly and randomly). Professional tennis players are coached to "mix it up" when they serve and return shots. Plays in American football are carefully randomized to be unpredictable. Random drug tests make it harder for Olympic athletes to predict when they can abuse steroids. These are all "mixed strategies" that work by being unpredictable. Game theory showed the common rationale for randomness in many situations like these, where players have conflicts of interest and benefit from predicting each other's behavior.

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