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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)。

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)。

                 | 2L = Choice 1          | 2R = Choice 2          |

                 | 2L = Choice 1          | 2R = Choice 2          |

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。

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。

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>

|+ align=bottom |A zero-sum game

|+ align=bottom |A zero-sum game

| + align = bottom | 零和游戏

| + align = bottom | 零和博弈

 

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.

| {{diagonal split header|{{red|30}}|{{blue|−30}}|white}}

| {{diagonal split header|{{red|30}}|{{blue|−30}}|white}}

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，选择公布，每个玩家的总分受到这些选择的收益的影响。

进行的顺序如下: 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。

| {{diagonal split header|{{red|20}}|{{blue|−20}}|white}}

| {{diagonal split header|{{red|20}}|{{blue|−20}}|white}}

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.

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  where element }} is the payoff obtained when the minimizing player chooses pure strategy  and the maximizing player chooses pure strategy  (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  is positive.  The game will have at least one Nash equilibrium.  The Nash equilibrium can be found (Raghavan 1994, p.&nbsp;740) by solving the following linear program to find a vector :

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  where element }} is the payoff obtained when the minimizing player chooses pure strategy  and the maximizing player chooses pure strategy  (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  is positive.  The game will have at least one Nash equilibrium.  The Nash equilibrium can be found (Raghavan 1994, p.&nbsp;740) by solving the following linear program to find a vector :

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<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博弈<ref>Wenliang Wang (2015). Pooling Game Theory and Public Pension Plan. {{ISBN|978-1507658246}}. Chapter 4.</ref>。

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。

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博弈，一个人的收益就是另一个人的损失。

== See also又及 ==

== See also又及 ==

<small>This page was moved from [[wikipedia:en:Zero-sum game]]. Its edit history can be viewed at [[零和博弈/edithistory]]</small></noinclude>

<small>This page was moved from [[wikipedia:en:Zero-sum game]]. Its edit history can be viewed at [[零和博弈/edithistory]]</small></noinclude>

<small>此页摘自[[维基百科：英语：零和游戏]]。其编辑历史可在[[零和博弈编辑历史记录]]查看</small></noinclude>

<small>此页摘自[[维基百科：英语：零和博弈]]。其编辑历史可在[[零和博弈编辑历史记录]]查看</small></noinclude>

[[Category:待整理页面]]

[[Category:待整理页面]]