Friday, January 13, 2012

Game Theory

by Richard Crews
The serious study of games has invaded the daily news and, in fact, many aspects of our daily lives. The field of Game Theory is defined as the study of conflict and cooperation between decision makers. It involves not just Scrabble and Monopoly and not just how to win at poker and at the race track , but maneuvers in wartime (and in peacetime negotiations), and in high-stakes business deals and political campaign strategies as well.

Let us consider the simplest kind of game: two people flip a coin; if it comes up heads then one wins, if tails, the other. Is this entirely a game of chance, or is there a strategy that can give one player a winning edge in the long run? Surprisingly, there is a winning strategy, even if it is a fair flip each time (that is, even if there really is a random, equal chance the coin will come up heads or tails on each flip) because human beings cannot choose randomly. There are inevitable biases that creep, consciously or unconsciously, into a person's call for heads or tails on the next flip. For example, if a coin toss has come up heads five times in a row, there is a powerful psychological inclination to expect that it will come up tails next time. But in fact the odds are still 50:50. So the winning strategy is: Let the other person make the call.

Some games are far more complex than this. But anyone who has played a lot of "serious" Monopoly knows that a strategy of buying as many properties as one can early in a game, and loading them up with houses and hotels as quickly as possible, will usually produce a win. Similarly in Scrabble, thinking up high word values and blocking an opponent from using the double- and triple-score spaces is usually a winning strategy.

Some terms from Game Theory have crept into everyday discourse; we hear them regularly on the evening news. For example, the distinction is made between a "zero-sum game" in which if one side wins, then the other side loses, and a "win-win game" in which both sides can come out ahead. If two players each bet $5 and then cut a deck of cards with the agreement that the one who gets the higher card wins the $10 and the other gets nothing, that is a zero-sum game. On the other hand, if they agree that the one who gets the higher card gets the $10 and the other gets to keep the deck of cards (also worth more than $5, but provided by some outside source), that would be a win-win game.

The principle of the game of Russian Roulette is well known: players alternately take a chance at suffering a big loss, such as taking turns firing a pistol that has one bullet in a six-chambered gun at ones own head. Situations in which any participants risk serious loss is often referred to as a game of Russian roulette.

One also hears business or political strategies likened to a game of Rope-a-Dope, a term borrowed from boxing in which one boxer protects his face and head with his arms and leans against the rope, letting his opponent reign blows on him. His hope is that his opponent will tire himself out or get careless, and the fighter employing the Rope-a-Dope strategy can take advantage of that and win in the end.

Another more complicated game-theory situation is called a Prisoner's Dilemma. To understand this, imagine that two men are arrested for a crime, but the police do not have enough evidence for a conviction. The police, secretly, offer both the same deal—
(1) if one testifies against his partner and the other remains silent, the betrayer goes free and the silent one receives the full one-year sentence.
(2) If both remain silent, both are sentenced to only one month in jail for a minor charge.
(3) If each snitches on the other, each receives a three-month sentence.
Each prisoner must choose either to betray or remain silent; if he snitches on his partner, the most he can get is a three-month jail sentence and he might go free; if he remains silent, he is sure to spend at least one month in jail and, depending on what his partner does, he might have to serve a full year.

The term Prisoner's Dilemma is commonly used to refer to a simpler scenario in which two negotiators will both benefit if they cooperate, but either one will lose heavily if he expresses a willingness to negotiate and the other backs out.