Game theory -- the abstract study of games, or the mathematics of competition and cooperation -- analyzes situations in terms of gains and losses of opposing players. Two major theories about modern life have come out of games. The first is probability theory, which was first developed out of games of chance in the 17th century by Blaise Pascal. The strategies used to achieve success on the game board can also be applied in many real-life situations. The branch of mathematics that analyzes a range of problems involving decision-making; also called games theory. Although often illustrated by games of chance, there are important applications to military strategy, economics, ecology, and other applied sciences.
Game theory is a system for predicting how people should optimally behave in situations of conflict. In a typical game, decision-making "players," who each have their own goals, try to outsmart one another by anticipating each other's decisions; the game is resolved as a consequence of the players' decisions. Games involving one, two, or more players are distinguished, as in patience, chess, and roulette respectively. Game theory analyses the strategies each player uses to maximize the chance of winning, and attempts to predict outcomes. A solution to a game prescribes the decisions the players should make and describes the game's appropriate outcome.
It is applied widely in economics, operations research, military and political science, organization theory, the study of negotiation, warfare, economic competition, to determine the formation of political coalitions or business conglomerates, the optimum price at which to sell products or services, the power of a voter or a bloc of voters, the selection of a jury, the best site for a manufacturing plant, and even the behaviour of certain species in the struggle for survival.
Game theory was developed in the last century, principally by French mathematician Emile Borel (1871-1956) and US mathematician Johann von Neumann. Game theory was developed out of games of strategy in the 1920s by mathematician John von Neumann and Oskar Morgenstern. It did't become well known until the publication in 1944 of their Theory of Games and Economic Behavior.
Its importance in economic theory, for example, was shown by the awarding of the 1993/4 Nobel Prize for economics to three prominent game-theoreticians: American mathematician John F. Nash, Hungarian-American economist John C. Harsanyi, and German economist Reinhard Selten. Nash's contribution was the development of an idea now known as the Nash Equilibrium, a key component in the study of game theory.
The Minimax Theorem discovered by von Neumann in 1928 asserts that every finite, zero-sum, two-player game has a minimax value if mixed strategies are allowed. This means that every such game has a solution (an optimal strategy) -- but it may be hard to find the solution. Zero-sum means that any gain for one player represents an equal loss for the other. Many parlor games are zero-sum, but the "games" found in economics or in operations research usually are not, since wealth may be created or destroyed.
The Minimax Theorem does't apply to nonzero-sum games or games with more than two players. John Nash showed in 1950 that such games do have a weaker solution, a noncooperative equilibrium in which no player, acting on the assumption that the other players' strategies are fixed, can gain anything by changing his or her own strategy. These solutions are often called Nash equilibria.
Examples
Game theory can be seen most clearly by considering a simple contest like Sumo wrestling. To somebody seeing it for the first time, it would seem to be obvious that a light-weight could win every time, just by waiting for the heavier wrestler to charge, and then stepping to one side.
While this sort of approach may work for bull fighting, the heavier wrestler is also a thinking individual, who is likely to notice the lighter wrestler's methods, and stop charging, but rather to move in slowly and deliberately, at which point the lighter wrestler will need to come up with some other strategies.
An even simpler game is the children's game of "paper, rock and scissors", where two players each reveal a hand with a flat palm ("paper"), a clenched fist ("rock") or two separated fingers ("scissors"). Scissors cut paper, so "scissors" beats "paper"; paper can wrap rock, so "paper" beats "rock"; rock can blunt scissors, so "rock" beats "scissors".
Game theory will normally dictate a mixed strategy, and often requires a conscious or unconscious use of randomisation.
Given an opponent who is playing logically, what strategy should you seek to adopt, in order to win a majority of encounters? If your opponent always offers "paper", it will be tempting to offer "scissors", but this may be just what your opponent wants you to do. This, of course, is a simple example, a finite two-person zero-sum game, where the gains of one player equal the losses of another. Real life is often more complex, and may involve a situation where the result is not necessarily zero-sum:
The Prisoner's Dilemma
This is the name given to an interesting paradox in game theory. The paradox was originally formulated by Melvin Dresher and Merrill Flood of the RAND Corporation, and was give its name by Albert W. Tucker.
In its simplest form, two prisoners are each given exactly the same information: there is enough evidence against each of them to get them sentenced for a gaol term of two years. If one is prepared to give evidence against the other, then that prisoner will get off free, while the other prisoner serves five years. On the other hand, if each provides evidence against the other, each will be convicted and serve four years. The dilemma is that each prisoner knows that the other prisoner has the same information, and will act in some way: so what is the best choice to make?
The problem has a number of practical applications: a child immunised against a certain disease may run a small risk (let us say one chance in a million) of dying as a result of an infection caused by the immunisation. On the other hand, if nobody is immunised against the disease, one child in ten thousand will certainly die in an epidemic. If 95% of the population are immunised, an epidemic will not take place, as the disease will be eliminated.
To many people, the selfish choice is best: avoid immunisation, and allow others to take the risks for you, but if nobody is immunised, then those who are without immunity run a much greater chance of dying. Other applications are to be found in economics, evolutionary biology, and international relations.
