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A zero-sum game is also called a strictly competitive game, while non-zero-sum games can be either competitive or non-competitive. Zero-sum games are most often solved with the minimax theorem which is closely related to linear programming duality, [5] or with Nash equilibrium. Prisoner's Dilemma is a classic non-zero-sum game. [6]
The first theorem in this sense is von Neumann's minimax theorem about two-player zero-sum games published in 1928, [2] which is considered the starting point of game theory. Von Neumann is quoted as saying "As far as I can see, there could be no theory of games
In zero-sum games, the total benefit goes to all players in a game, for every combination of strategies, and always adds to zero (more informally, a player benefits only at the equal expense of others). [20] Poker exemplifies a zero-sum game (ignoring the possibility of the house's cut), because one wins exactly the amount one's opponents lose.
Matching Pennies is a zero-sum game because each participant's gain or loss of utility is exactly balanced by the losses or gains of the utility of the other participants. If the participants' total gains are added up and their total losses subtracted, the sum will be zero.
Constant sum: A game is a constant sum game if the sum of the payoffs to every player are the same for every single set of strategies. In these games, one player gains if and only if another player loses. A constant sum game can be converted into a zero sum game by subtracting a fixed value from all payoffs, leaving their relative order unchanged.
The concept of a mixed-strategy equilibrium was introduced by John von Neumann and Oskar Morgenstern in their 1944 book The Theory of Games and Economic Behavior, but their analysis was restricted to the special case of zero-sum games. They showed that a mixed-strategy Nash equilibrium will exist for any zero-sum game with a finite set of ...