What is game theory and what are some of its applications?

 Game: A competitive activity involving skill, chance, or endurance on the part of two or more persons who play according to a set of rules, usually for their own amusement or for that of spectators (The Random House Dictionary of the English Language,1967).

Consider the following real-world competitive situations: missile defense, sales price wars for new cars, energy regulation, auditing tax payers, the TV show "Survivor," terrorism, NASCAR racing, labor- management negotiations, military conflicts, bidding at auction, arbitration, advertising, elections and voting, agricultural crop selection, conflict resolution, stock market, insurance, and telecommunications. What do they have in common?

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A basic example helps to illustrate the point. After learning how to play the game tick-tack-toe, you probably discovered a strategy of play that enables you to achieve at least a draw and even win if your opponent makes a mistake and you notice it. Sticking to that strategy ensures that you will not lose.

This simple game illustrates the essential aspects of what is now called game theory. In it, a game is the set of rules that describe it. An instance of the game from beginning to end is known as a play of the game. And a pure strategy--such as the one you found for tick-tack-toe--is an overall plan specifying moves to be taken in all eventualities that can arise in a play of the game. A game is said to have perfect information if, throughout its play, all the rules, possible choices, and past history of play by any player are known to all participants. Games like tick-tack-toe, backgammon and chess are games with perfect information and such games are solved by pure strategies. But whereas you may be able to describe all such pure strategies for tick-tack-toe, it is not possible to do so for chess, hence the latter's age-old intrigue.

Games without perfect information, such as matching pennies, stone-paper-scissors or poker offer the players a challenge because there is no pure strategy that ensures a win. For matching pennies you have two pure strategies: play heads or tails. For stone-paper-scissors you have three pure strategies: play stone or paper or scissors. In both instances you cannot just continually play a pure strategy like heads or stone because your opponent will soon catch on and play the associated winning strategy. What to do? We soon learn to try to confound our opponent by randomizing our choice of strategy for each play (for heads-tails, just toss the coin in the air and see what happens for a 50-50 split). There are also other ways to control how we randomize. For example, for stone-paper-scissors we can toss a six-sided die and decide to select stone half the time (the numbers 1, 2 or 3 are tossed), select paper one third of the time (the numbers 4 or 5 are tossed) or select scissors one sixth of the time (the number 6 is tossed). Doing so would tend to hide your choice from your opponent. But, by mixing strategies in this manner, should you expect to win or lose in the long run? What is the optimal mix of strategies you should play? How much would you expect to win? This is where the modern mathematical theory of games comes into play.