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From Citizendium, the Citizens' Compendium

A zero-sum game is an abstract game in which the sum of the payoffs for all the players is zero, whatever strategy they choose. The interests in a zero-sum game are diametrically opposed: a player can only gain at the expense of the other players. It is like dividing a cake, where one can only get more if another gets less. In games that are not zero-sum, there is the possibility to cooperate and thus increase the size of the cake.

For example, sports games are zero-sum, when considered on their own. The best result is to win and the worst is to lose, with a draw in between. When one side wins, the other side loses; this makes it a zero-sum game. However, when a series of games is played, then each individual game is not necessarily zero-sum. For instance, if in some competition both teams need only a draw to proceed to the next round and they do not get any advantages if they win, then they can cooperate and make sure that the game indeed ends in a draw.

Zero-sum games are easier to analyze than games that are not zero-sum. For instance, every zero-sum game has a Nash equilibrium if we allow mixed strategies. A Nash equilibrium is when all players have chosen a strategy so that none of the players can increase their payoff by changing their strategy unilaterally; it is natural to expect that "fair" outcomes of the games satisfy this condition. A mixed strategy is one in which a player does not commit to one strategy, but chooses randomly between two or more strategies. Furthermore, if a zero-sum game has more than one Nash equilibrium, then these equilibria have the same payoffs and so they are basically the same. Hence, a zero-sum game has a well-defined value to each of the players, namely their payoff in the equilibrium.