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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]
In the mathematical theory of games, in particular the study of zero-sum continuous games, not every game has a minimax value. This is the expected value to one of the players when both play a perfect strategy (which is to choose from a particular PDF). This article gives an example of a zero-sum game that has no value. It is due to Sion and ...
Suppose player A plays x and player B plays y. Without loss of generality, assume player A chooses the larger number, so x ≥ y. Then the payoff to A is 0 if x = y, 1 if 1 < x/y < T and −ν if x/y ≥ T. Thus each player seeks to choose the larger number, but there is a penalty of ν for choosing too large a number.
An intransitive or non-transitive game is a zero-sum game in which pairwise competitions between the strategies contain a cycle. If strategy A beats strategy B, B beats C, and C beats A, then the binary relation "to beat" is intransitive, since transitivity would require that A beat C.
In normal play, the winning strategy is to finish every move with a nim-sum of 0. This is always possible if the nim-sum is not zero before the move. If the nim-sum is zero, then the next player will lose if the other player does not make a mistake. To find out which move to make, let X be the nim-sum of all the heap sizes.
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In game theory, a strictly determined game is a two-player zero-sum game that has at least one Nash equilibrium with both players using pure strategies.The value of a strictly determined game is equal to the value of the equilibrium outcome.
In the context of two-player zero-sum games, the sets and correspond to the strategy sets of the first and second player, respectively, which consist of lotteries over their actions (so-called mixed strategies), and their payoffs are defined by the payoff matrix.