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第1行： − 此词条暂由水流心不竞初译，翻译字数共1421，未经审校，带来阅读不便，请见谅。+ + + + 第11行：
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第22行： − 相比之下，非零和描述了（另）一种情况，在这种情况下，相互作用的各方的总体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,+ 第29行：
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第48行： − 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游戏（例如战争游戏，国际象棋)和非零和游戏(例如:集合选择游戏)。+ 第54行：
第57行： + − 对于双人有限零和对策，'''<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，博弈总是存在均衡。+ − + + + + − 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. − 零和博弈是常和博弈的一个具体例子，其中每个结果的总和总是零。这种游戏是分配的，而不是综合的；并不能通过良好的谈判扩大（收益）馅饼。 第85行：
第90行： − 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）。+ + 第177行：
第183行： − 博弈的支付矩阵是一种方便的表示形式。例如，考虑图中右上方的两人零和游戏。+ 第185行：
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第207行： − 在这个例子中，两个玩家都知道支付矩阵，并试图最大化他们的分数。红队的理由如下: “在第二场比赛中，我可能输掉20分，只能赢20分，而在第一场比赛中，我只能输掉10分，但可以赢得30分，所以第一场比赛看起来要好得多。”根据类似的推理，蓝方会选择动作 c。如果两个玩家都采取这些动作，红方会赢得20分。如果蓝色预料到红色的推理和行动1的选择，蓝色可能会选择行动 b，从而赢得10点。如果红色，反过来，预测这个把戏，并去行动2，这赢得红色20点。+ 第207行：
第213行： − + − 和约翰·冯·诺伊曼的基本观点是概率提供了一条解决这个难题的途径。两个参与者不是决定采取什么明确的行动，而是分配各自行动的可能性，然后使用一个随机设备，根据这些可能性，为他们选择一个行动。每个参与人计算概率，以最小化最大期望点损失独立于对手的策略。这就导致了每个玩家的最优策略的线性规划问题。这种'''<font color="#ff8000"> 极大极小法Minimax method</font>'''可以计算出所有两人零和游戏的最优策略。+ + − |} + − For the example given above, it turns out that Red should choose action 1 with probability and action 2 with probability , and Blue should assign the probabilities 0, , and to the three actions A, B, and C. Red will then win points on average per game.+ − 对于上面给出的例子，结果表明，红色应该选择行动1的概率和行动2的概率，和蓝色应该指定的概率为0，和三个行动的 a，b，和 c 红色将赢得平均每场比赛的分数。+ − A game's [[payoff matrix]] is a convenient representation. Consider for example the two-player zero-sum game pictured at right or above. 第233行：
第239行： − “例如：红色选择动作2，蓝色选择动作B。当分配到回报时，红色获得20点，蓝色失去20点。”+ − − Minimize: − − 最小化: − − In this example game, both players know the payoff matrix and attempt to maximize the number of their points. Red could reason as follows: "With action 2, I could lose up to 20 points and can win only 20, and with action 1 I can lose only 10 but can win up to 30, so action 1 looks a lot better." With similar reasoning, Blue would choose action C. If both players take these actions, Red will win 20 points. If Blue anticipates Red's reasoning and choice of action 1, Blue may choose action B, so as to win 10 points. If Red, in turn, anticipates this trick and goes for action 2, this wins Red 20 points. − − 在这个示例游戏中，两个玩家都知道收益矩阵，并试图最大化他们的点数。瑞德可以这样解释：“在第二个动作中，我最多可以输20分，但只能赢20分，而在第一个动作中，我只能输10分，但最多可以赢30分，所以第一个动作看起来好多了。”根据类似的推理，蓝队会选择动作C。如果两个选手都采取这些动作，红色将赢得20分。如果蓝色预测到红色的推理和行动1的选择，蓝色可以选择行动B，从而赢得10分。如果雷德，反过来，预期这一技巧和行动2，这赢得红20分。 − − <math>\sum_{i} u_i</math> − − [ math ] sum { i } u i − − − − Subject to the constraints: − − 视乎限制而定: − − [[Émile Borel]] and [[John von Neumann]] had the fundamental insight that [[probability]] provides a way out of this conundrum. Instead of deciding on a definite action to take, the two players assign probabilities to their respective actions, and then use a random device which, according to these probabilities, chooses an action for them. Each player computes the probabilities so as to minimize the maximum [[expected value|expected]] point-loss independent of the opponent's strategy. This leads to a [[linear programming]] problem with the optimal strategies for each player. This [[minimax]] method can compute probably optimal strategies for all two-player zero-sum games. − − [[Émile Borel]]和[[John von Neumann]]的基本见解是[[概率]]提供了一种解决这个难题的方法。这两个玩家没有决定要采取的明确行动，而是给他们各自的行动分配概率，然后使用一个随机装置，根据这些概率，为他们选择一个行动。每个玩家计算概率，以使最大[[期望值|预期]]点损失最小化，与对手的策略无关。这就导致了一个[[线性规划]]问题，每个参与者的最优策略。这种[[极大极小]]方法可以计算所有两人零和博弈的可能最优策略。 − − . − − . − For the example given above, it turns out that Red should choose action 1 with probability {{sfrac|4|7}} and action 2 with probability {{sfrac|3|7}}, and Blue should assign the probabilities 0, {{sfrac|4|7}}, and {{sfrac|3|7}} to the three actions A, B, and C. Red will then win {{sfrac|20|7}} points on average per game. − 上面给出的例子中，红色应该选择行动1，概率 {{sfrac|4|7}} 和行动2，概率{{sfrac|3|7}}, 而蓝色应将概率0，{{sfrac|4|7}}, 和 {{sfrac|3|7}} 分配给A、B、C三个行动。红色平均每场比赛将赢得{sfrac | 20 | 7}点。 第275行：
第253行： − 如果博弈矩阵没有所有的正元素，只要在每个元素上加一个常数，这个常数足够大，使得它们都是正的。这个常数会增加博弈的价值，对均衡的混合策略没有影响。+ 第281行：
第259行： − + − 通过求解给定线性规划的对偶问题，可以找到最小化问题的均衡混合策略。或者，可以用上述方法求解一个修正的支付矩阵，它是(加一个常数使其为正)的转置和否定，然后求解结果博弈。+ − ：最小化：+ 第304行：
第282行： − 如果回避零和博弈是一个具有一定概率的行动选择，那么在零和博弈中，至少一个参与者的回避总是一个均衡策略。对于任何一个零和游戏的玩家来说，在游戏开始后零和游戏是不可能的或者不可信的，比如说扑克，除了回避游戏之外没有其他的纳什均衡点策略。即使在零和博弈开始后出现了可信的零比零平局，这也不比回避策略好。从这个意义上说，在最优选择计算中找到随走随奖是一件很有趣的事情，在开始与否的问题上，所有两个玩家的零和游戏都优先。+ 第318行：
第296行： − The most common or simple example from the subfield of social psychology is the concept of "social traps". In some cases pursuing individual personal interest can enhance the collective well-being of the group, but in other situations all parties pursuing personal interest results in mutually destructive behavior. − − 社会心理学分支领域最常见或最简单的例子就是“社会陷阱”的概念。在某些情况下，追求个人利益可以增强集体的幸福感，但在其他情况下，追求个人利益的各方都会导致相互破坏的行为。 − + 第332行：
第307行： − 罗伯特 · 赖特在他的《非零: 人类命运的逻辑》一书中提出了这样的理论: 当社会变得越来越复杂、专门化和相互依存时，它就会变得越来越非零和。+ − + 第345行：
第320行： − 1944年，约翰·冯·诺伊曼和奥斯卡·摩根斯腾证明了 n 个玩家的任何非零和博弈等价于 n + 1个玩家的零和博弈，第(n + 1)个玩家代表全球的盈亏。+ 第351行：
第326行： − + − − Zero-sum games and particularly their solutions are commonly misunderstood by critics of game theory, usually with respect to the independence and rationality of the players, as well as to the interpretation of utility functions. Furthermore, the word "game" does not imply the model is valid only for recreational games. − − 零和博弈，尤其是零和博弈的解决方案，常常被博弈论的批评家们所误解，通常涉及到玩家的独立性和理性，以及效用函数的解释。此外，“游戏”一词并不意味着该模型仅适用于娱乐游戏。 − − + − Politics is sometimes called zero sum. − 政治有时被称为零和游戏。 第387行：
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无编辑摘要
此词条暂由水流心不竞初译，翻译字数共1421，带来阅读不便，请见谅。
此词条暂由Miyasaki审校
--[[用户:Miyasaki|Miyasaki]]（[[用户讨论:Miyasaki|讨论]]）此词条存在重复部分，有些重复段落不能确切定位，暂搁置。
{{short description|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}}
{{short description|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}}
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).
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>'''是一种数学描述，其中每个参与者的效用收益与其他参与者的效用收益的损失完全平衡。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，如果所有的参与者都平等地评价每一块蛋糕，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数量，同时也会增加给那个接受者的边际效用。
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 linear programming duality,
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。
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.
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.
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; 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。
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.
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，如果一个国家有过量的香蕉与另一个国家进行交易以换取其过剩的苹果，而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他们开始时的水平。
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。
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).
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).
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游戏（例如战争游戏，国际象棋)和非零和游戏(例如:集合选择游戏)。
| 2L = Choice 1 | 2R = Choice 2 |
| 2L = Choice 1 | 2R = Choice 2 |
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.
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.
对于双人有限零和游戏来说，'''<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，游戏中总是存在平衡。
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>
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。
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>。
{| class="wikitable" style="float:right; margin-left:1em;"
{| class="wikitable" style="float:right; margin-left:1em;"
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>
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>
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>。
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A game's payoff matrix is a convenient representation. Consider for example the two-player zero-sum game pictured at right or above.
A game's payoff matrix is a convenient representation. Consider for example the two-player zero-sum game pictured at right or above.
一场游戏的收益矩阵是一种方便的表示形式。让我们以图中右上方的两人零和游戏为例来考虑一下。
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The order of play proceeds as follows: The first player (red) chooses in secret one of the two actions 1 or 2; the second player (blue), unaware of the first player's choice, chooses in secret one of the three actions A, B or C. Then, the choices are revealed and each player's points total is affected according to the payoff for those choices.
The order of play proceeds as follows: The first player (red) chooses in secret one of the two actions 1 or 2; the second player (blue), unaware of the first player's choice, chooses in secret one of the three actions A, B or C. Then, the choices are revealed and each player's points total is affected according to the payoff for those choices.
游戏的顺序如下: 第一个玩家(红色)秘密地在两个动作1或2中选择一个; 第二个玩家(蓝色)不知道第一个玩家的选择，秘密地在三个动作 a、 b 或 c 中选择一个，然后，选择被显示出来，每个玩家的总分受到这些选择的回报的影响。
游戏的顺序如下: 第一个玩家(红色)秘密地在两个动作1或2中选择一个; 第二个玩家(蓝色)不知道第一个玩家的选择，秘密地在三个动作 a、 b 或 c 中选择一个，然后，选择公布，每个玩家的总分受到这些选择的收益的影响。
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Example: Red chooses action 2 and Blue chooses action B. When the payoff is allocated, Red gains 20 points and Blue loses 20 points.
Example: Red chooses action 2 and Blue chooses action B. When the payoff is allocated, Red gains 20 points and Blue loses 20 points.
例如: 红选择操作2，蓝选择操作 b。当回报被分配时，红色获得20点，蓝色失去20点。
例如: 玩家红选择操作2，玩家蓝选择操作B。当回报被分配时，红色获得20点，蓝色失去20点。
! {{red|''2''}}
! {{red|''2''}}
In this example game, both players know the payoff matrix and attempt to maximize the number of their points. Red could reason as follows: "With action 2, I could lose up to 20 points and can win only 20, and with action 1 I can lose only 10 but can win up to 30, so action 1 looks a lot better." With similar reasoning, Blue would choose action C. If both players take these actions, Red will win 20 points. If Blue anticipates Red's reasoning and choice of action 1, Blue may choose action B, so as to win 10 points. If Red, in turn, anticipates this trick and goes for action 2, this wins Red 20 points.
In this example game, both players know the payoff matrix and attempt to maximize the number of their points. Red could reason as follows: "With action 2, I could lose up to 20 points and can win only 20, and with action 1 I can lose only 10 but can win up to 30, so action 1 looks a lot better." With similar reasoning, Blue would choose action C. If both players take these actions, Red will win 20 points. If Blue anticipates Red's reasoning and choice of action 1, Blue may choose action B, so as to win 10 points. If Red, in turn, anticipates this trick and goes for action 2, this wins Red 20 points.
在这个例子中，两个玩家都知道收益矩阵，并试图最大化他们的分数。红队的理由如下: “在第二场比赛中，我可能输掉20分，只能赢20分，而在第一场比赛中，我只能输掉10分，但可以赢得30分，所以第一场比赛看起来要好得多。”根据类似的推理，蓝方会选择动作 c。如果两个玩家都采取这些动作，红方会赢得20分。如果蓝色预料到红色的推理和行动1的选择，蓝色可能会选择行动 b，从而赢得10点。如果红色，反过来，预测到这个计策，并选择行动2，这将为红色赢得20点。
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Émile Borel and John von Neumann had the fundamental insight that probability provides a way out of this conundrum. Instead of deciding on a definite action to take, the two players assign probabilities to their respective actions, and then use a random device which, according to these probabilities, chooses an action for them. Each player computes the probabilities so as to minimize the maximum expected point-loss independent of the opponent's strategy. This leads to a linear programming problem with the optimal strategies for each player. This minimax method can compute probably optimal strategies for all two-player zero-sum games.
[[Émile Borel]] and [[John von Neumann]] had the fundamental insight that [[probability]] provides a way out of this conundrum. Instead of deciding on a definite action to take, the two players assign probabilities to their respective actions, and then use a random device which, according to these probabilities, chooses an action for them. Each player computes the probabilities so as to minimize the maximum [[expected value|expected]] point-loss independent of the opponent's strategy. This leads to a [[linear programming]] problem with the optimal strategies for each player. This [[minimax]] method can compute probably optimal strategies for all two-player zero-sum games.
[[Émile Borel]]和[[John von Neumann]]的基本见解是[[概率]]提供了一种解决这个难题的方法。这两个玩家没有决定要采取的明确行动，而是给他们各自的行动分配概率，然后使用一个随机装置，根据这些概率，为他们选择一个行动。每个玩家计算概率，以使最大[[期望值|预期]]点损失最小化，与对手的策略无关。这就导致了一个[[线性规划]]问题，每个参与者的最优策略。这种[[极大极小]]方法可以计算所有两人零和博弈的可能最优策略。
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For the example given above, it turns out that Red should choose action 1 with probability and action 2 with probability , and Blue should assign the probabilities 0, and to the three actions A, B, and C. Red will then win points on average per game.
对于上面给出的例子，结果表明，红色应该选择行动1的概率和行动2的概率，和蓝色应该将概率0，分配给三个行动A，B，C。红色将赢得平均每场比赛的分数。
--[[用户:Miyasaki|Miyasaki]]（[[用户讨论:Miyasaki|讨论]]）这里原文应该有缺漏。
''Example: Red chooses action 2 and Blue chooses action B. When the payoff is allocated, Red gains 20 points and Blue loses 20 points.''
''Example: Red chooses action 2 and Blue chooses action B. When the payoff is allocated, Red gains 20 points and Blue loses 20 points.''
例如：红色选择动作2，蓝色选择动作B。分配收益时，红色获得20点，蓝色失去20点。
The first constraint says each element of the vector must be nonnegative, and the second constraint says each element of the vector must be at least 1. For the resulting vector, the inverse of the sum of its elements is the value of the game. Multiplying by that value gives a probability vector, giving the probability that the maximizing player will choose each of the possible pure strategies.
The first constraint says each element of the vector must be nonnegative, and the second constraint says each element of the vector must be at least 1. For the resulting vector, the inverse of the sum of its elements is the value of the game. Multiplying by that value gives a probability vector, giving the probability that the maximizing player will choose each of the possible pure strategies.
If the game matrix does not have all positive elements, simply add a constant to every element that is large enough to make them all positive. That will increase the value of the game by that constant, and will have no effect on the equilibrium mixed strategies for the equilibrium.
If the game matrix does not have all positive elements, simply add a constant to every element that is large enough to make them all positive. That will increase the value of the game by that constant, and will have no effect on the equilibrium mixed strategies for the equilibrium.
如果游戏矩阵不具备所有的正元素，只要在每个元素上加一个足够大的常数，使得它们都是正的。这个常数会增加游戏的价值，对均衡的混合策略没有影响。
The [[Nash equilibrium]] for a two-player, zero-sum game can be found by solving a [[linear programming]] problem. Suppose a zero-sum game has a payoff matrix {{mvar|M}} where element {{math|''M''{{sub|''i'',''j''}}}} is the payoff obtained when the minimizing player chooses pure strategy {{mvar|i}} and the maximizing player chooses pure strategy {{mvar|j}} (i.e. the player trying to minimize the payoff chooses the row and the player trying to maximize the payoff chooses the column). Assume every element of {{mvar|M}} is positive. The game will have at least one Nash equilibrium. The Nash equilibrium can be found (Raghavan 1994, p. 740) by solving the following linear program to find a vector {{mvar|u}}:
The [[Nash equilibrium]] for a two-player, zero-sum game can be found by solving a [[linear programming]] problem. Suppose a zero-sum game has a payoff matrix {{mvar|M}} where element {{math|''M''{{sub|''i'',''j''}}}} is the payoff obtained when the minimizing player chooses pure strategy {{mvar|i}} and the maximizing player chooses pure strategy {{mvar|j}} (i.e. the player trying to minimize the payoff chooses the row and the player trying to maximize the payoff chooses the column). Assume every element of {{mvar|M}} is positive. The game will have at least one Nash equilibrium. The Nash equilibrium can be found (Raghavan 1994, p. 740) by solving the following linear program to find a vector {{mvar|u}}:
一个两人零和博弈的[[纳什均衡]]可以通过求解一个[[线性规划]]问题得到。假设一个零和博弈有一个支付矩阵{{mvar|M}}，其中元素{{math|''M''{{sub|''i'',''j''}}}}是当最小化的玩家选择纯策略 {{mvar|i}}而最大化的玩家选择纯策略 {{mvar|j}} 时获得的收益（即，试图最小化收益的玩家选择行，而试图最大化收益的玩家选择列）。假设 {{mvar|M}}的每个元素都是正的。博弈至少有一个纳什均衡。纳什均衡可以通过求解以下线性规划找到向量 {{mvar|u}}来找到（Raghavan 1994，p.740）：
一个两人零和博弈的[[纳什均衡]]可以通过求解一个[[线性规划]]问题得到。假设一个零和博弈有一个支付矩阵{{mvar|M}}，其中元素{{math|''M''{{sub|''i'',''j''}}}}是当最小化的玩家选择纯策略 {{mvar|i}}而最大化的玩家选择纯策略 {{mvar|j}} 时获得的收益（即，试图最小化收益的玩家选择行，而试图最大化收益的玩家选择列）。假设 {{mvar|M}}的每个元素都是正的。博弈至少有一个纳什均衡。纳什均衡可以通过求解以下线性规划找到向量 {{mvar|u}}来找到（Raghavan 1994，p.740）：
The equilibrium mixed strategy for the minimizing player can be found by solving the dual of the given linear program. Or, it can be found by using the above procedure to solve a modified payoff matrix which is the transpose and negation of (adding a constant so it's positive), then solving the resulting game.
The equilibrium mixed strategy for the minimizing player can be found by solving the dual of the given linear program. Or, it can be found by using the above procedure to solve a modified payoff matrix which is the transpose and negation of (adding a constant so it's positive), then solving the resulting game.
通过求解给定线性规划的对偶问题，可以找到最小化问题的均衡混合策略。或者，可以用上述方法求解一个修正后的收益矩阵，它是(加一个常数使其为正)的转置和否定，然后求解结果博弈。
: Minimize:
: Minimize:
: 最小化：
::: <math>\sum_{i} u_i</math>
::: <math>\sum_{i} u_i</math>
If avoiding a zero-sum game is an action choice with some probability for players, avoiding is always an equilibrium strategy for at least one player at a zero-sum game. For any two players zero-sum game where a zero-zero draw is impossible or non-credible after the play is started, such as poker, there is no Nash equilibrium strategy other than avoiding the play. Even if there is a credible zero-zero draw after a zero-sum game is started, it is not better than the avoiding strategy. In this sense, it's interesting to find reward-as-you-go in optimal choice computation shall prevail over all two players zero-sum games with regard to starting the game or not.
If avoiding a zero-sum game is an action choice with some probability for players, avoiding is always an equilibrium strategy for at least one player at a zero-sum game. For any two players zero-sum game where a zero-zero draw is impossible or non-credible after the play is started, such as poker, there is no Nash equilibrium strategy other than avoiding the play. Even if there is a credible zero-zero draw after a zero-sum game is started, it is not better than the avoiding strategy. In this sense, it's interesting to find reward-as-you-go in optimal choice computation shall prevail over all two players zero-sum games with regard to starting the game or not.
如果回避零和博弈是一个具有一定概率的行动选择，那么在零和博弈中，至少一个参与者的回避总是一个均衡策略。对于任何一个零和游戏的玩家来说，在游戏开始后零和游戏是不可能的或者不可信的，比如说扑克，除了回避游戏之外没有其他的纳什均衡点策略。即使在零和博弈开始后出现了可信的零比零平局，这也不比回避策略好。从这个意义上说，有趣的是，在最优选择计算中找到随走随奖在开始与否的问题上将比所有所有双人零和游戏。
The first constraint says each element of the {{mvar|u}} vector must be nonnegative, and the second constraint says each element of the {{mvar|M u}} vector must be at least 1. For the resulting {{mvar|u}} vector, the inverse of the sum of its elements is the value of the game. Multiplying {{mvar|u}} by that value gives a probability vector, giving the probability that the maximizing player will choose each of the possible pure strategies.
The first constraint says each element of the {{mvar|u}} vector must be nonnegative, and the second constraint says each element of the {{mvar|M u}} vector must be at least 1. For the resulting {{mvar|u}} vector, the inverse of the sum of its elements is the value of the game. Multiplying {{mvar|u}} by that value gives a probability vector, giving the probability that the maximizing player will choose each of the possible pure strategies.
通过求解给定线性规划的对偶问题，可以找到最小化博弈者的均衡混合策略。或者，也可以通过使用上述过程来求解修正的支付矩阵，即 {{mvar|M}}的转置和否定（添加一个常数使其为正），然后求解结果博弈。
通过求解给定线性规划的对偶问题，可以找到最小化博弈者的均衡混合策略。或者，也可以通过使用上述过程来求解修正的支付矩阵，即 {{mvar|M}}的转置和否定（添加一个常数使其为正），然后求解结果博弈。
If all the solutions to the linear program are found, they will constitute all the Nash equilibria for the game. Conversely, any linear program can be converted into a two-player, zero-sum game by using a change of variables that puts it in the form of the above equations. So such games are equivalent to linear programs, in general. {{citation needed|date=October 2010}}
If all the solutions to the linear program are found, they will constitute all the Nash equilibria for the game. Conversely, any linear program can be converted into a two-player, zero-sum game by using a change of variables that puts it in the form of the above equations. So such games are equivalent to linear programs, in general. {{citation needed|date=October 2010}}
如果找到线性规划的所有解，它们将构成博弈的所有'''<font color="#ff8000"> 纳什均衡</font>'''。相反，任何线性程序都可以通过使用变量上述方程形式的变化，将其转换为两人零和博弈。所以，一般来说，这种游戏相当于线性程序。{{citation needed|date=October 2010}}
如果找到线性规划的所有解，它们将构成博弈的所有'''<font color="#ff8000"> 纳什均衡</font>'''。相反，任何线性程序都可以通过使用变量上述方程形式的变化，将其转换为两人零和博弈。所以，一般来说，这种游戏相当于线性程序。{{citation needed|date=October 2010}}
It has been theorized by Robert Wright in his book Nonzero: The Logic of Human Destiny, that society becomes increasingly non-zero-sum as it becomes more complex, specialized, and interdependent.
It has been theorized by Robert Wright in his book Nonzero: The Logic of Human Destiny, that society becomes increasingly non-zero-sum as it becomes more complex, specialized, and interdependent.
Robert Wright在他的《非零: 人类命运的逻辑》一书中提出了这样的理论: 当社会变得越来越复杂、专门化和相互依存时，它就会变得越来越非零和。
If avoiding a zero-sum game is an action choice with some probability for players, avoiding is always an equilibrium strategy for at least one player at a zero-sum game. For any two players zero-sum game where a zero-zero draw is impossible or non-credible after the play is started, such as poker, there is no Nash equilibrium strategy other than avoiding the play. Even if there is a credible zero-zero draw after a zero-sum game is started, it is not better than the avoiding strategy. In this sense, it's interesting to find reward-as-you-go in optimal choice computation shall prevail over all two players zero-sum games with regard to starting the game or not.<ref>Wenliang Wang (2015). Pooling Game Theory and Public Pension Plan. {{ISBN|978-1507658246}}. Chapter 4.</ref>
If avoiding a zero-sum game is an action choice with some probability for players, avoiding is always an equilibrium strategy for at least one player at a zero-sum game. For any two players zero-sum game where a zero-zero draw is impossible or non-credible after the play is started, such as poker, there is no Nash equilibrium strategy other than avoiding the play. Even if there is a credible zero-zero draw after a zero-sum game is started, it is not better than the avoiding strategy. In this sense, it's interesting to find reward-as-you-go in optimal choice computation shall prevail over all two players zero-sum games with regard to starting the game or not.<ref>Wenliang Wang (2015). Pooling Game Theory and Public Pension Plan. {{ISBN|978-1507658246}}. Chapter 4.</ref>
如果避免零和博弈对玩家来说是一种有一定概率的行为选择，那么在零和博弈中，回避总是至少一个参与者的均衡策略。对于任何两个玩家的零和游戏，在游戏开始后零-零平局是不可能或不可信的，例如扑克，没有纳什均衡策略，除非不做游戏。即使在零和博弈开始后出现了可信的零-零平局，也不比回避策略好。从这个意义上说，有趣的是，在最优选择计算中，在开始游戏或不开始游戏时，最佳选择计算应优先于所有两个玩家的零和游戏。
如果避免零和博弈对玩家来说是一种有一定概率的行为选择，那么在零和博弈中，回避总是至少一个参与者的均衡策略。对于任何两个玩家的零和游戏，在游戏开始后零-零平局是不可能或不可信的，例如扑克，没有纳什均衡策略，除非不做游戏。即使在零和博弈开始后出现了可信的零-零平局，也不比回避策略好。从这个意义上说，有趣的是，在最优选择计算中，在开始游戏或不开始游戏时，最佳选择计算应优先于所有两个玩家的零和游戏<ref>Wenliang Wang (2015). Pooling Game Theory and Public Pension Plan. {{ISBN|978-1507658246}}. Chapter 4.</ref>。
In 1944, John von Neumann and Oskar Morgenstern proved that any non-zero-sum game for n players is equivalent to a zero-sum game with n + 1 players; the (n + 1)th player representing the global profit or loss.
In 1944, John von Neumann and Oskar Morgenstern proved that any non-zero-sum game for n players is equivalent to a zero-sum game with n + 1 players; the (n + 1)th player representing the global profit or loss.
1944年，John von Neumann和Oskar Morgenstern证明了 n 个玩家的任何非零和博弈等价于 n + 1个玩家的零和博弈，第(n + 1)个玩家代表全球的盈亏。
The most common or simple example from the subfield of [[social psychology]] is the concept of "[[social trap]]s". In some cases pursuing individual personal interest can enhance the collective well-being of the group, but in other situations all parties pursuing personal interest results in mutually destructive behavior.
The most common or simple example from the subfield of [[social psychology]] is the concept of "[[social trap]]s". In some cases pursuing individual personal interest can enhance the collective well-being of the group, but in other situations all parties pursuing personal interest results in mutually destructive behavior.
[[社会心理学]]子领域中最常见或最简单的例子是“[[社会陷阱]]s”的概念。在某些情况下，追求个人利益可以增进群体的集体福祉，但在其他情况下，追求个人利益的各方都会导致相互破坏的行为。
[[社会心理学]]子领域中最常见或最简单的例子是“[[社会陷阱]]”的概念。在某些情况下，追求个人利益可以增进群体的集体福祉，但在其他情况下，追求个人利益的各方都会导致相互破坏的行为。
=== Complexity 复杂性===
=== Complexity 复杂性===
It has been theorized by [[Robert Wright (journalist)|Robert Wright]] in his book ''[[Nonzero: The Logic of Human Destiny]]'', that society becomes increasingly non-zero-sum as it becomes more complex, specialized, and interdependent.
It has been theorized by [[Robert Wright (journalist)|Robert Wright]] in his book ''[[Nonzero: The Logic of Human Destiny]]'', that society becomes increasingly non-zero-sum as it becomes more complex, specialized, and interdependent.
[[罗伯特·赖特（记者）|罗伯特·赖特]]在他的著作“[[非零：人类命运的逻辑]]”中提出，随着社会变得更加复杂、专业化和相互依存，社会变得越来越非零和。
[[Robert Wright（记者）|Robert Wright]]在他的著作“[[非零：人类命运的逻辑]]”中提出，随着社会变得更加复杂、专业化和相互依存，社会变得越来越非零和。
== Extensions 扩展==
== Extensions 扩展==
Zero-sum games and particularly their solutions are commonly misunderstood by critics of [[game theory]], usually with respect to the independence and [[Rational choice theory|rationality]] of the players, as well as to the interpretation of utility functions. Furthermore, the word "game" does not imply the model is valid only for recreational [[game]]s.<ref name="Binmore2007">{{cite book|author=Ken Binmore|title=Playing for real: a text on game theory|url=https://books.google.com/?id=eY0YhSk9ujsC|year=2007|publisher=Oxford University Press US|isbn=978-0-19-530057-4|authorlink=Ken Binmore}}, chapters 1 & 7</ref>
Zero-sum games and particularly their solutions are commonly misunderstood by critics of [[game theory]], usually with respect to the independence and [[Rational choice theory|rationality]] of the players, as well as to the interpretation of utility functions. Furthermore, the word "game" does not imply the model is valid only for recreational [[game]]s.<ref name="Binmore2007">{{cite book|author=Ken Binmore|title=Playing for real: a text on game theory|url=https://books.google.com/?id=eY0YhSk9ujsC|year=2007|publisher=Oxford University Press US|isbn=978-0-19-530057-4|authorlink=Ken Binmore}}, chapters 1 & 7</ref>
零和博弈，尤其是它们的解决方案经常被[[博弈论]]的批评者误解，通常是关于参与者的独立性和[[理性选择理论|理性]]，以及对效用函数的解释。此外，“游戏”一词并不意味着该模型仅对娱乐[[游戏]]有效。
零和博弈，尤其是它们的解决方案经常被[[博弈论]]的批评者误解，通常是关于参与者的独立性和[[理性选择理论|理性]]，以及对效用函数的解释。此外，“游戏”一词并不意味着该模型仅对娱乐[[游戏]]有效<ref name="Binmore2007">{{cite book|author=Ken Binmore|title=Playing for real: a text on game theory|url=https://books.google.com/?id=eY0YhSk9ujsC|year=2007|publisher=Oxford University Press US|isbn=978-0-19-530057-4|authorlink=Ken Binmore}}, chapters 1 & 7</ref>。
Politics is sometimes called zero sum.<ref>{{cite news|last=Rubin |first=Jennifer |url=https://www.washingtonpost.com/blogs/right-turn/wp/2013/10/04/the-flaw-in-zero-sum-politics/ |title=The flaw in zero sum politics |newspaper=The Washington Post |date=2013-10-04 |accessdate=2017-03-08}}</ref><ref>{{cite news|url=https://www.economist.com/news/united-states/21595973-voters-think-both-parties-are-telling-truth-about-how-awful-other-lot-are-zero-sum |title=Lexington: Zero-sum politics |newspaper=The Economist |date=2014-02-08 |accessdate=2017-03-08}}</ref><ref>{{cite dictionary|url=http://www.dictionary.com/browse/zero-sum-game |title=Zero-sum game | Define Zero-sum game at |dictionary=Dictionary.com |accessdate=2017-03-08}}</ref>
Politics is sometimes called zero sum.<ref>{{cite news|last=Rubin |first=Jennifer |url=https://www.washingtonpost.com/blogs/right-turn/wp/2013/10/04/the-flaw-in-zero-sum-politics/ |title=The flaw in zero sum politics |newspaper=The Washington Post |date=2013-10-04 |accessdate=2017-03-08}}</ref><ref>{{cite news|url=https://www.economist.com/news/united-states/21595973-voters-think-both-parties-are-telling-truth-about-how-awful-other-lot-are-zero-sum |title=Lexington: Zero-sum politics |newspaper=The Economist |date=2014-02-08 |accessdate=2017-03-08}}</ref><ref>{{cite dictionary|url=http://www.dictionary.com/browse/zero-sum-game |title=Zero-sum game | Define Zero-sum game at |dictionary=Dictionary.com |accessdate=2017-03-08}}</ref>
政治有时被称为零和。
政治有时被称为零和。<ref>{{cite news|last=Rubin |first=Jennifer |url=https://www.washingtonpost.com/blogs/right-turn/wp/2013/10/04/the-flaw-in-zero-sum-politics/ |title=The flaw in zero sum politics |newspaper=The Washington Post |date=2013-10-04 |accessdate=2017-03-08}}</ref><ref>{{cite news|url=https://www.economist.com/news/united-states/21595973-voters-think-both-parties-are-telling-truth-about-how-awful-other-lot-are-zero-sum |title=Lexington: Zero-sum politics |newspaper=The Economist |date=2014-02-08 |accessdate=2017-03-08}}</ref><ref>{{cite dictionary|url=http://www.dictionary.com/browse/zero-sum-game |title=Zero-sum game | Define Zero-sum game at |dictionary=Dictionary.com |accessdate=2017-03-08}}</ref>
== Zero-sum thinking 零和思维==
== Zero-sum thinking 零和思维==
In psychology, [[zero-sum thinking]] refers to the perception that a situation is like a zero-sum game, where one person's gain is another's loss.
In psychology, [[zero-sum thinking]] refers to the perception that a situation is like a zero-sum game, where one person's gain is another's loss.
在心理学中，[[零和思维]]指的是一种感觉，即一种情况就像一场零和游戏，一个人的收益就是另一个人的损失。
在心理学中，[[零和思维]]指的是一种感觉，即感觉某情形就像一场零和游戏，一个人的收益就是另一个人的损失。
== See also又及 ==
== See also又及 ==