GAMES MATHEMATICIANS PLAY

Now to another aspect of Mathias Rust's flight to Red Square, which will take us into game theory. Mathematically speaking, a game is a conflict in which two parties pursue a goal that no more than one of them can achieve. There are certain moves defined by the rules of the game. Some games have only "personal" moves, re-quiring a decision (chess); others have only random moves (roulette); still others have both (poker). A particular sequence of moves is called a strategy. Such games are of two broad classes: those with perfect information, where each player always knows of the moves his opponent has made; and all others. Chess and soccer are games with perfect information; bridge and war are not.

One of the basic theorems of game theory says that every game with perfect information and non-random moves has a "simple" solution, meaning a single optimum strategy of fixed personal moves rather than a "mixed" strategy, in which a player uses a number of selected simple strategies at random, choosing each with a certain probability.

And this is where it begins to be fascinating, for that theorem says that if two players play a game with non-random moves and perfect information then there exists an optimum strategy always guaranteeing the same outcome (usually a win for the player with the first move). For example, two players agree to take turns in placing coins on a round or rectangular table until there is no more room left. The one who placed the last coin wins.

This game offers perfect information and has no random moves.

Therefore it must have a "simple" optimum strategy, and here it is: the first player puts his coin in the exact middle of the table. He then apes every move of his opponent by placing a coin of the same size symmetrically to this middle coin until. . . see?

But now that you know the solution, the game has lost its in-terest: unless you are a politician and therefore enjoy victimizing the ignorant, you will be bored to death. In fact, simple games without random moves and perfect information (tic-tac-toe) are played only by children, and games with random moves only, i.e. without per-sonal decision making (roulette), are played only by idiots.

The real challenge are games with hidden information and with both personal and random moves, such as blackjack or war. In such games the optimum strategy must be a probabilistic mix of several "simple" strategies. [If you kept repeating the same strategy in every game, you would, in effect, turn it into a game of perfect information for your opponent, who would use your predictability against you.] Blackjack is so simple that one can now buy computer programs (or dedicated toys) against which you cannot win in a very large number of games. The fact that you can win some games is what lures people to Las Vegas; and the law of large numbers is what its palaces are built on.

The basic theorems of game theory essentially reveal the types of games that have a"simple" solution, and which are of little interest. (Mathematically, chess belongs to this class and must have a "sim-ple" solution; but the vast number of possible strategies keeps that solution unknown even in the computer age, so that chess remains the King of Games in spite of its category.) The theory is mainly devoted to the games that are solved by a mixed strategy. It is a very fascinating subject, but requires advanced probability theory.

Here we will note only that the people who use terms like "zero-sum game" and "calculated risk" in the mass media are most often politicians, journalists and other brainwashers. They have picked up a few words of jargon and use them for what they know best: impressing the impressionable by talking about matters of which they have not the slightest inkling.