A two-person zero-sum game is one in which the pay-offs add up to zero. They are strictly competitive in that what one player gains, the other loses. The game obeys a law of conservation of utility value, where utility value is never created or destroyed, only transferred from one player to another. The interests of the two players are always strictly opposed and competitive, with no possibility of, or benefit in, cooperation. One player must win and at the expense of the other; a feature known as pareto-efficiency. More precisely, a pareto-efficiency is a situation in which the lot of one player cannot be improved without worsening the lot of at least one other player.

Game theory is particularly well-suited to the analysis of zero-sum games and applications to everyday life (especially sporting contests) abound. Actually, ‘constant-sum games’ would be a better title since, in some circumstances, the pay-offs do not add up to zero because the game is unfair. However, they do sum to a constant, which is the prevalent feature of these strictly competitive games, so we will continue to use the term ‘zero-sum’ even in these instances, for the sake of simplicity.

The section that follows establishes a link between the tree representation described in the previous chapter and the normal form of the game, represented by pay-off matrices.