Two-person zero-sum games
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Two-person zero-sum games book
A zero-sum game, as its name suggests, is one in which, whatever the outcome, the payoffs to the players add up to zero, which means that what one player gains, the other(s) must necessarily lose. A constant-sum game is strategically equivalent to a zero-sum game, and the two terms are used almost interchangeably by some writers. The strategic equivalence becomes obvious as soon as one realizes that any constant-sum game can be interpreted as a situation in which each player is first fined or rewarded by a fixed amount for playing the game, depending on whether the constant sum is negative or positive, and the players are then allowed to play the resulting zero-sum game. A zero-sum or constant-sum game, in other words, is a closed system within which nothing of value to the players is created or destroyed: utilities merely change hands when the game is played. If there are just two players, this means that their interests are diametrically opposed, because an outcome that is favourable for one is bound to be correspondingly unfavourable for the other. Because each player can gain only at the expense of the other, there are no prospects of mutually profitable collaboration, and two-person zero-sum conflicts are therefore called strictly competitive games. They have proved especially amenable to formal analysis, and the most significant contributions to mathematical game theory relate to them. Zero-sum games with more than two players are not strictly competitive or straightforwardly soluble, as will be explained in chapter 8.