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第10行： − In [[game theory]] and [[economic theory]], a '''zero-sum game''' is a [[Mathematical model|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 participants. If the total gains of the participants are added up and the total losses are subtracted, they will sum to zero. Thus, [[Fair cake-cutting|cutting a cake]], where taking a larger piece reduces the amount of cake available for others as much as it increases the amount available for that taker, is a zero-sum game if all participants value each unit of cake equally (see [[marginal utility]]).+ − 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 participants. 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 as much as it increases the amount available for that taker, is a zero-sum game if all participants value each unit of cake equally (see marginal utility).+ − − 在博弈论和经济理论中，'''<font color="#ff8000"> 零和博弈Zero-sum game</font>'''是对某种情形的一种数学描述，在这种情形中每个参与者的效用增减与其他参与者的效用的增减互相平衡。如果将参与者的总收益加起来，再减去总损失，则它们之和为零。因此，如果公认蛋糕每一部分都具有同等价值，那么切蛋糕就是一个零和游戏，切一块蛋糕会减少给其他人的蛋糕量，同时也会增加给那个接受者的蛋糕量。 − − − − 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 [[LP duality|linear programming duality]],<ref name="Binmore2007"/> or with [[Nash equilibrium]]. − − 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, − − 相比之下，非零和描述了另一种情形，在这种情形中，相互作用的各方的总计收益和损失可能小于或大于零。零和博弈也称为严格竞争博弈，而非零和博弈可以是竞争博弈，也可以是非竞争博弈。零和博弈通常是用极大极小定理来解决的，这个定理与线性规划二元性密切相关。 − − − − Many people have a [[cognitive bias]] towards seeing situations as zero-sum, known as [[zero-sum bias]]. − − 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. + + − == Definition 定义==+ − − 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. − − 参与者可以共同获益或共同受苦的情况称为'''<font color="#ff8000"> 非零和Non-zero-sum</font>'''。因此，如果一个香蕉过剩的国家与另一个国家进行交易以换取其过剩的苹果，这两个国家都从交易中受益，那么这个国家就处于一种非零和情况。其他'''<font color="#ff8000"> 非零和博弈</font>'''是这样一种博弈，在这种博弈中，参与者的得与失之和有时大于或小于开始时的水平。 − − − − {{Payoff matrix | Name = Generic zero-sum game − 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 (e.g. warfare game, chess) and non-zero-sum games (e.g. pooling selection games).+ + − − − For two-player finite zero-sum games, the different [[game theory|game theoretic]] [[solution concept]]s of [[Nash equilibrium]], [[minimax]], and [[maximin (decision theory)|maximin]] all give the same solution. If the players are allowed to play a [[mixed strategy]], the game always has an equilibrium.+ − − For two-player finite zero-sum games, the different game theoretic solution concepts of Nash equilibrium, minimax, and maximin all give the same solution. If the players are allowed to play a mixed strategy, the game always has an equilibrium. − − − − 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.<ref>{{cite book |first=Samuel |last=Bowles |title=Microeconomics: Behavior, Institutions, and Evolution |url=https://archive.org/details/microeconomicsbe00bowl |url-access=limited |location= |publisher=[[Princeton University Press]] |pages=[https://archive.org/details/microeconomicsbe00bowl/page/n47 33]–36 |year=2004 |isbn=0-691-09163-3 }}</ref> − − 零和属性（如果一个获得，另一个失败）意味着零和情况的任何结果都是[[帕累托最优]]。一般来说，所有策略都是帕累托最优的博弈称为冲突博弈.<ref>{{cite book |first=Samuel |last=Bowles |title=Microeconomics: Behavior, Institutions, and Evolution |url=https://archive.org/details/microeconomicsbe00bowl |url-access=limited |location= |publisher=[[Princeton University Press]] |pages=[https://archive.org/details/microeconomicsbe00bowl/page/n47 33]–36 |year=2004 |isbn=0-691-09163-3 }}</ref>。 − − − − 第87行：
第52行： − 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 (e.g. warfare game, chess) and non-zero-sum games (e.g. pooling selection games).<ref>Wenliang Wang (2015). Pooling Game Theory and Public Pension Plan. {{ISBN|978-1507658246}}. Chapter 1 and Chapter 4.</ref>
无编辑摘要
{{other uses|Zero sum (disambiguation)}}
{{other uses|Zero sum (disambiguation)}}
在[[博弈论]]和经济理论中，''' 零和博弈Zero-sum game'''是对某种情形的一种数学描述，在这种情形中每个参与者的效用增减与其他参与者的效用的增减互相平衡。如果将参与者的总收益加起来，再减去总损失，则它们之和为零。因此，如果公认蛋糕每一部分都具有同等价值，那么切蛋糕就是一个零和游戏，切一块蛋糕会减少给其他人的蛋糕量，同时也会增加给那个接受者的蛋糕量。
相比之下，'''非零和Non-zero-sum'''描述了另一种情形，在这种情形中，相互作用的各方的总计收益和损失可能小于或大于零。零和博弈也称为严格竞争博弈，而非零和博弈可以是竞争博弈，也可以是非竞争博弈。零和博弈通常是用极大极小定理来解决的，这个定理与线性规划二元性<ref name="Binmore2007"/>密切相关。
许多人对将情况视为零和有[[认知偏差]]，称为[[零和偏差]]。
许多人对将情况视为零和有[[认知偏差]]，称为[[零和偏差]]。
零和博弈是常数和博弈的一个具体例子，其中每个结果的和总是为零。这种游戏是分配性的，而不是综合性的; 良好的谈判无法扩大这块蛋糕。
零和博弈是常数和博弈的一个具体例子，其中每个结果的和总是为零。这种游戏是分配性的，而不是综合性的; 良好的谈判无法扩大这块蛋糕。
== 定义==
零和属性（如果一个获得，另一个失败）意味着零和情况的任何结果都是[[帕累托最优]]。一般来说，所有策略都是[[帕累托最优]]的博弈称为冲突博弈.<ref>{{cite book |first=Samuel |last=Bowles |title=Microeconomics: Behavior, Institutions, and Evolution |url=https://archive.org/details/microeconomicsbe00bowl |url-access=limited |location= |publisher=[[Princeton University Press]] |pages=[https://archive.org/details/microeconomicsbe00bowl/page/n47 33]–36 |year=2004 |isbn=0-691-09163-3 }}</ref>。
零和博弈是恒定和博弈的特定示例，其中每个结果的总和始终为零。这种游戏是分布式的，而不是集成的；不能通过良好的谈判来扩大派。
参与者可以共同获益或共同受苦的情况称为''' 非零和'''。因此，如果一个香蕉过剩的国家与另一个国家进行交易以换取其过剩的苹果，这两个国家都从交易中受益，那么这个国家就处于一种非零和情况。其他'''非零和博弈'''是这样一种博弈，在这种博弈中，参与者的得与失之和有时大于或小于开始时的水平。
在零和博弈中，帕累托最优收益的概念引出了一个广义的相对自私的理性标准，即惩罚对手的标准，在这个标准中，双方总是以对自己较有利的代价来寻求最小化对手的收益，而不是偏好多于少。惩罚对手标准可以同时用在零和博弈（例如战争游戏，国际象棋)和非零和博弈(例如:集合选择游戏)。
在零和博弈中，帕累托最优收益的概念引出了一个广义的相对自私的理性标准，即惩罚对手的标准，在这个标准中，双方总是以对自己较有利的代价来寻求最小化对手的收益，而不是偏好多于少。惩罚对手标准可以同时用在零和博弈（例如战争游戏，国际象棋)和非零和博弈(例如:集合选择游戏)。
{{Payoff matrix | Name = 基因零和博弈
| 2L = Choice 1 | 2R = Choice 2 |
| 2L = Choice 1 | 2R = Choice 2 |
1U =Choice 1 | UL = −A, A | UR = B, −B |
1U =Choice 1 | UL = −A, A | UR = B, −B |
1D = Choice 2 | DL = C, −C | DR = −D, D }}
1D = Choice 2 | DL = C, −C | DR = −D, D }}
对于双人有限零和博弈来说，''' 纳什均衡点Nash equilibrium'''、极大极小和极大的不同对策理论解概念都给出了相同的解。如果允许参与者采用混合策略，博弈中总是存在平衡。
对于双人有限零和博弈来说，'''<font color="#ff8000"> 纳什均衡点Nash equilibrium</font>'''、极大极小和极大的不同对策理论解概念都给出了相同的解。如果允许参与者采用混合策略，博弈中总是存在平衡。
!|}}
!|}}
零和博弈中帕累托最优收益的思想产生了一种广义的相对自私理性标准，即惩罚对手标准，在这种标准中，双方总是以对自己有利的成本来寻求最小化对手的回报，而不是偏好多而少。惩罚对手标准可用于零和博弈（如战争博弈、国际象棋）和非零和博弈（如池选博弈）<ref>Wenliang Wang (2015). Pooling Game Theory and Public Pension Plan. {{ISBN|978-1507658246}}. Chapter 1 and Chapter 4.</ref>。
零和博弈中帕累托最优收益的思想产生了一种广义的相对自私理性标准，即惩罚对手标准，在这种标准中，双方总是以对自己有利的成本来寻求最小化对手的回报，而不是偏好多而少。惩罚对手标准可用于零和博弈（如战争博弈、国际象棋）和非零和博弈（如池选博弈）<ref>Wenliang Wang (2015). Pooling Game Theory and Public Pension Plan. {{ISBN|978-1507658246}}. Chapter 1 and Chapter 4.</ref>。