Game Theory is based largely on the idea of John von Neumann. von Neumann wrote a book explaining his ideas in a publication entitled "Theory of Games and Economic Behaviour" which was co-authored with Oskar Morgernsten in 1934, shortly after he re-located to the United States. For more information on him I recommend reading Teiresias's excellent node, linked above.

Game Theory is an interesting branch of mathematics that attempts to study the decision making process of two or more players in a game. In this case a game is a conflict situation where the players are either attemping to fulfil different, mutually exclusive, objectives on the same system, or they are playing for control of a finite resource. Game Theory attempts to provide a mathematical structure for selecting an Optimum Strategy in the face of an opponent who will also be trying to play to an Optimum Strategy.

In Game Theory there are certain assumptions that must be made:

  1. Each player has two or more choices at each stage of the game.
  2. Every possible combination of plays leads to a defined end-state, either win, lose or draw.
  3. Every end-state results in a pre-defined payoff for each player (such as gaining control over a certain amount of a resource). A Zero-Sum game is when, at a given end-state, the sum total of the payoffs to all the players is zero (i.e. the winnings are equal to the loses).
  4. Each player has full knowledge of the game and all its possible end-states. This means that they know the payoffs both to themselves and their opponents at any given end-state.
  5. All decisions are rational. A player will always choose the option that would result in a greater payoff to himself.

The last two points limit (to an extent) the usefullness of Game Theory in terms of its application to real-life situations. They have, however, been used to do research in economics and psycology.

There are several different types of game, but the most commonly seen type of game is an NxM game. This is any game where the outcomes can be plotted onto a Matrix of dimentions N by M, where N and M represtent the number of options that each player has available to him at that point.

Two common examples of NxM games are 'The Prisoners Dilemma' and 'Matching Pennies'

The "Prisoners Dilemma" is a NxM game of dimentions 2x2. In its simplest form it works thus: There are two prisoners who have been arrested by the police. If neither of them admit to the crime then they will both walk free. If they both admit to the crime then they will each get a sentence of five years. If one of them admits to the crime and the other denys it then the one who admitted the crime gets three years and the one who denied it gets nine years. What is the optimal strategy that they should follow?

This is best represented as a table:

   | D | A |
  ----------     Where D = Deny, A = Admit
  D|0/0|9/5|     Outcomes are given as (A/B)
A ----------

From this you can easily see that while it is temping to deny the crime as this will give the greatest payoff, the strategy with the best payoff-to-risk ratio is for both sides to admit the crime. (Personal story - I played a game like this for a group of 10yr old kids. Even after playing five times niether side was willing to take the optimal strategy. Both sides kept playing the outcome where they both lost, rather than the less risky one where both sides would have won a little. This is why it doesn't work so well in real life)

The "Matching Pennies" game is apparently played by children. It is also a 2x2 games. Each player secretly turns a coin heads up or tails up. They then show the coins to each other. If they match then A has to pay B a penny. If they differ then B pays A a penny. This games continues untill they get bored.

As a table:

   | H  |  T |
  ------------     Where H = Heads, A = Tails
  H|-1/1|1/-1|     Outcomes are given as (A/B)
A ------------

From this it is easy to see that there is no optimal strategy. In the long run it is best to play completely randomly as this means that both sides should come out even. Given that long term strategy this becomes a zero sum game.

For more information about Game Theory is a good place to start.