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第1行： − 此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。+ + + 第9行：
第11行： − 在博弈论和经济学理论中，零和博弈是一种数学描述，在这种情况下，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，切蛋糕是一个零和游戏，如果所有的参与者都平等地评价每一块蛋糕的价值，那么切大块的蛋糕会减少其他人可以得到的蛋糕数量，同时也会增加那个人可以得到的蛋糕边际效用。+ 第21行：
第23行： − Humans have a [[cognitive bias]] towards seeing situations as zero-sum, known as [[zero-sum bias]].+ − Humans have a cognitive bias towards seeing situations as zero-sum, known as zero-sum bias.+ − 人类有一种认知偏见，认为情况是零和的，也就是所谓的零和偏见。+ 第37行：
第39行： − + 第43行：
第45行： − + 第55行：
第57行： − + − + 第79行：
第81行： − 参与者可以共同获益或共同受苦的情况称为非零和。因此，一个香蕉过剩的国家与另一个国家交易过剩的苹果，双方都从交易中受益，这是一种非零和情况。其他非零和博弈是这样一种博弈，在这种博弈中，参与者的得失之和有时大于或小于他们开始时的得失之和。+ 第87行：
第89行： − 在零和博弈中，帕累托最优收益的概念产生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。惩罚对手标准可以用在零和游戏中。战争游戏，国际象棋)和非零和游戏(例如:。集合选择游戏)。+ 第99行：
第101行： − 对于双人有限零和对策，纳什均衡点、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。+ 第111行：
第113行： − + 第117行：
第119行： − + 第159行：
第161行： − 会发生什么+ 第165行：
第167行： − 会发生什么+ 第171行：
第173行： − 会发生什么+ 第189行：
第191行： − 会发生什么+ 第195行：
第197行： − 会发生什么+ 第239行：
第241行： − + 第247行：
第249行： − 和约翰·冯·诺伊曼的基本观点是概率提供了解决这个难题的方法。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。这就导致了每个参与者的最优策略的线性规划问题。这种极大极小方法可以计算出所有两人零和对策的最优策略。+ 第267行：
第269行： − + 第281行：
第283行： − 数学，数学+ 第339行：
第341行： − 如果回避一个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。即使在一场零和游戏开始后出现了可信的零比零平局，这也不比回避策略好。在这个意义上，有趣的是在最优选择计算中找到随行奖励将优先于所有两个玩家的零和游戏，关于是否开始游戏。+ 第370行：
第372行： − − − − === Movies === − − − − See the plot of [[Arrival_(film) | Arrival]] − − See the plot of Arrival − − 看看《抵达》的情节 第405行：
第395行： − 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。+
Moved page from wikipedia:en:Zero-sum game (history)
此词条暂由彩云小译翻译，翻译字数共1521，未经人工整理和审校，带来阅读不便，请见谅。
{{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}}
{{distinguish|Empty sum|Zero game}}
{{distinguish|Empty sum|Zero game}}
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，零和博弈是一种数学描述，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，那么它们之和为零。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，切得越大，其他人得到的蛋糕数量就越少，同时也增加了那个人得到的边际效用。
Many people have a [[cognitive bias]] towards seeing situations as zero-sum, known as [[zero-sum bias]].
Many people have a cognitive bias towards seeing situations as zero-sum, known as zero-sum bias.
许多人有一种认知偏见，认为情况是零和的，也就是所谓的零和偏见。
{{Payoff matrix | Name = Generic zero-sum game
{{Payoff matrix | Name = Generic zero-sum game
{支付矩阵 | 命名一般零和对策
{{支付矩阵 | 名称 = 一般零和博弈
| 2L = Choice 1 | 2R = Choice 2 |
| 2L = Choice 1 | 2R = Choice 2 |
| 2L = Choice 1 | 2R = Choice 2 |
| 2L = Choice 1 | 2R = Choice 2 |
2 l Choice 1 | 2 r Choice 2 |
2 l = Choice 1 | 2 r = 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 }}
1D Choice 2 | DL c,-c | DR-d，d }
1D = 选择2 | DL = c,-c | DR =-d，d }
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 |location= |publisher=[[Princeton University Press]] |pages=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>
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.
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.
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非零和。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非零和情况。其他非零和博弈是一种博弈，其中参与者的得与失之和有时大于或小于他们开始时的数值。
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).
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，在这个标准下，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。惩罚对手标准可以用在零和游戏中。战争游戏，国际象棋)和非零和游戏(例如:。集合选择游戏)。
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.
对于双人有限零和对策，纳什均衡点、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，则博弈总是存在均衡。
{| class="wikitable" style="float:right; margin-left:1em;"
{| class="wikitable" style="float:right; margin-left:1em;"
{ | class“ wikitable”样式“ float: right; margin-left: 1em; ”
{ | class = “ wikitable” style = “ float: right; margin-left: 1em; ”
|+ align=bottom |''A zero-sum game''
|+ align=bottom |''A zero-sum game''
|+ align=bottom |A zero-sum game
|+ align=bottom |A zero-sum game
| + 对齐底部 | 零和游戏
| + align = bottom | 零和游戏
! {{diagonal split header|{{red|Red}}|{{blue|Blue}}}}
! {{diagonal split header|{{red|Red}}|{{blue|Blue}}}}
| ||white}}
| ||white}}
我会开枪的
| {{diagonal split header|{{red|−10}}|{{blue|10}}|white}}
| {{diagonal split header|{{red|−10}}|{{blue|10}}|white}}
| ||white}}
| ||white}}
我会开枪的
| {{diagonal split header|{{red|20}}|{{blue|−20}}|white}}
| {{diagonal split header|{{red|20}}|{{blue|−20}}|white}}
| ||white}}
| ||white}}
我会开枪的
|-
|-
| ||white}}
| ||white}}
我会开枪的
| {{diagonal split header|{{red|20}}|{{blue|−20}}|white}}
| {{diagonal split header|{{red|20}}|{{blue|−20}}|white}}
| ||white}}
| ||white}}
我会开枪的
| {{diagonal split header|{{red|−20}}|{{blue|20}}|white}}
| {{diagonal split header|{{red|−20}}|{{blue|20}}|white}}
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。如果蓝色预测红色的推理和行动选择1，蓝色可能会选择行动 b，以赢得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。
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。
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 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.
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。这种极大极小方法可以计算出所有两人零和对策的最优策略。
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. 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. 740) by solving the following linear program to find a vector :
一个双人零和游戏的纳什均衡点可以通过解决一个线性规划问题来找到。假设一个零和对策有一个支付矩阵，其中元素}是当最小化对策者选择纯策略而最大化对策者选择纯策略时所获得的支付。试图最小化回报的参与人选择行，而试图最大化回报的参与人选择列)。假设每个元素都是正数。这个游戏至少有一个纳什均衡点。可以通过解决下面的线性程序找到一个向量来找到纳什均衡点:
一个双人零和游戏的纳什均衡点可以通过解决一个线性规划问题来找到。假设一个零和对策有一个支付矩阵，其中元素}是最小化对策者选择纯策略而最大化对策者选择纯策略(即最小化对策者选择纯策略)所获得的支付。试图最小化回报的参与人选择行，而试图最大化回报的参与人选择列)。假设元素的每个元素都是正的。这个游戏至少有一个纳什均衡点。可以通过解决下面的线性程序找到一个向量来找到纳什均衡点:
<math>\sum_{i} u_i</math>
<math>\sum_{i} u_i</math>
[ math ] sum { i } u i
: Subject to the constraints:
: Subject to the constraints:
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.
如果回避一个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。在这个意义上，有趣的是在最优选择计算中找到随走随奖，将优先于所有两个玩家的零和游戏，关于是否开始游戏。
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.
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，特别是零和博弈的解决方案，常常被博弈论的批评家们所误解，通常涉及到博弈主体的独立性和理性，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适用于娱乐游戏。