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In game theory and economic theory, a zero-sum game is a mathematical representation of a situation in which each participant's gain (or loss) of utility is exactly balanced by the losses (or gains) of the utility of the other participant(s). If the total gains of the participants are added up and the total losses are subtracted, they will sum to zero. Thus cutting a cake, where taking a larger piece reduces the amount of cake available for others, is a zero-sum game if all participants value each unit of cake equally (see marginal utility). In contrast, non-zero-sum describes a situation in which the interacting parties' aggregate gains and losses can be less than or more than zero. 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,〔 or with Nash equilibrium. == Definition == The zero-sum property (if one gains, another loses) means that any result of a zero-sum situation is Pareto optimal (generally, any game where all strategies are Pareto optimal is called a conflict game). Zero-sum games are a specific example of constant sum games where the sum of each outcome is always zero. Such games are distributive, not integrative; the pie cannot be enlarged by good negotiation. Situations where participants can all gain or suffer together are referred to as non-zero-sum. Thus, a country with an excess of bananas trading with another country for their excess of apples, where both benefit from the transaction, is in a non-zero-sum situation. Other non-zero-sum games are games in which the sum of gains and losses by the players are sometimes more or less than what they began with. The idea of Pareto optimal payoff in a zero-sum game gives rise to a generalized relative selfish rationality standard, the punishing-the-opponent standard, where both players always seek to minimize the opponent’s payoff at a favorable cost to himself rather to prefer more than less. The punishing-the-opponent standard can be used in both zero-sum games (i.e. warfare game, Chess) and non-zero-sum games (i.e. pooling selection games).〔Wenliang Wang (2015). Pooling Game Theory and Public Pension Plan. ISBN 978-1507658246. Chapter 1 and Chapter 4.〕 A complete definition of zero-sum game is, a situation, especially a competitive one, which there is no net gain among the participants. If one gains, it means others have to lose an equivalent amount. For example, if the only way for you to gain $1,000 is to deprive someone else of $1,000, you're in a zero-sum game. The term is also sometimes used to refer to situations in which one's own gains offset one's losses. These are ''zero-sum games popular meanings, anyway. Followers of game theory, where the phrase derives, might find these definitions to be oversimplifications. ''Zero-sum game'' is sometimes misspelled ''zero-sum gain''. This sort of makes sense, as a zero-sum game results in a gain of a sum of zero, but it's not the conventional form of the phrase. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Zero-sum game」の詳細全文を読む スポンサード リンク
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