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The prisoner's dilemma is a standard example of a game analyzed in game theory that shows why two completely rational individuals might not cooperate, even if it appears that it is in their best interests to do so. It was originally framed by Merrill Flood and Melvin Dresher while working at RAND in 1950. Albert W. Tucker formalized the game with prison sentence rewards and named it "prisoner's dilemma", prensenting it as follows:

The prisoner's dilemma is a standard example of a game analyzed in game theory that shows why two completely rational individuals might not cooperate, even if it appears that it is in their best interests to do so. It was originally framed by Merrill Flood and Melvin Dresher while working at RAND in 1950. Albert W. Tucker formalized the game with prison sentence rewards and named it "prisoner's dilemma", prensenting it as follows:

<font color="#ff8000"> 囚徒困境</font>是博弈论分析的一个代表性例子，它揭示了为什么两个完全理性的个体可能不会合作，即使这样做似乎对他们最有利。它最初是由 Merrill Flood 和 Melvin Dresher 于1950年在兰德公司工作时构建的。Albert W. Tucker将这种博弈以囚徒的方式加以阐述，并将其命名为“囚徒困境” ，具体阐述如下：

<font color="#ff8000"> 囚徒困境</font>是<font color="#ff8000"> 博弈论</font>分析的一个代表性例子，它揭示了为什么两个完全理性的个体可能不会合作，即使这样做似乎对他们最有利。它最初是由梅里尔·弗勒德和 梅文·加舍尔于1950年在兰德公司工作时构建的。阿尔伯特.W.塔克将这种博弈以囚徒的方式加以阐述，并将其命名为“<font color="#ff8000">囚徒困境</font>” ，具体阐述如下：

 

{{quote|Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of communicating with the other. The prosecutors lack sufficient evidence to convict the pair on the principal charge, but they have enough to convict both on a lesser charge. Simultaneously, the prosecutors offer each prisoner a bargain. Each prisoner is given the opportunity either to betray the other by testifying that the other committed the crime, or to cooperate with the other by remaining silent. The possible outcomes are:

{{quote|Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary confinement with no means of communicating with the other. The prosecutors lack sufficient evidence to convict the pair on the principal charge, but they have enough to convict both on a lesser charge. Simultaneously, the prosecutors offer each prisoner a bargain. Each prisoner is given the opportunity either to betray the other by testifying that the other committed the crime, or to cooperate with the other by remaining silent. The possible outcomes are:

{{引号 | 一个犯罪团伙的两名成员被捕入狱。每个囚犯都被关在各自的禁闭室里，没有任何与其他囚犯沟通的方式。检察官缺乏足够的证据来定罪这两人的主要指控，但有足够的证据来定罪两人较轻的指控。同时，检察官向每个犯人提供了一个交易。每个囚犯都有机会出卖对方，证明对方犯下罪行，或者他们可以进行合作，保持沉默。可能的结果是:

{{引号 | 一个犯罪团伙的两名成员被捕入狱。每个囚犯都被关在各自的禁闭室里，没有任何与其他囚犯沟通的方式。检察官缺乏足够的证据来定罪这两人的主要指控，但有足够的证据来定罪两人的次要指控。同时，检察官向每个犯人提供了一个交易。每个囚犯都有机会出卖对方，证明对方犯下罪行，或者他们可以进行合作，保持沉默。可能的结果是:

 

* If A and B each betray the other, each of them serves two years in prison

* If A and B each betray the other, each of them serves two years in prison

如果A和B都背叛了对方，他们都会在监狱服刑两年。

如果A和B都背叛了对方，他们都会在监狱服刑两年。

An extended "iterated" version of the game also exists. In this version, the classic game is played repeatedly between the same prisoners, who continuously have the opportunity to penalize the other for previous decisions. If the number of times the game will be played is known to the players, then (by backward induction) two classically rational players will betray each other repeatedly, for the same reasons as the single-shot variant. In an infinite or unknown length game there is no fixed optimum strategy, and prisoner's dilemma tournaments have been held to compete and test algorithms for such cases.

An extended "iterated" version of the game also exists. In this version, the classic game is played repeatedly between the same prisoners, who continuously have the opportunity to penalize the other for previous decisions. If the number of times the game will be played is known to the players, then (by backward induction) two classically rational players will betray each other repeatedly, for the same reasons as the single-shot variant. In an infinite or unknown length game there is no fixed optimum strategy, and prisoner's dilemma tournaments have been held to compete and test algorithms for such cases.

一个扩展的“迭代”版本的博弈由此衍生出来。在这个版本中，经典博弈会在在同一组囚犯之间重复进行，他们不断有机会因为以前的决定而惩罚另一个。如果参与者知道博弈的次数，那么(通过逆向归纳法)两个经典理性的玩家就会因为和在单次博弈相同的原因反复背叛对方。在无限次或未知次数的博弈中，没有固定的最优策略，囚徒困境竞赛因而能被用来竞赛并且检验这种情况下的算法。

一个扩展的“迭代”版本的博弈由此衍生出来。在这个版本中，经典博弈会在在同一组囚犯之间重复进行，他们不断有机会因为以前的决定而惩罚另一个囚犯。如果参与者知道博弈的次数，那么(通过逆向归纳法)两个经典理性的玩家就会因为和在单次博弈相同的原因反复背叛对方。在无限次或未知次数的博弈中，没有固定的最优策略，囚徒困境锦标赛因而能被用来竞赛并且检验这种情况下的算法。

 

The prisoner's dilemma game can be used as a model for many [[#Real-life examples|real world situations]] involving cooperative behavior. In casual usage, the label "prisoner's dilemma" may be applied to situations not strictly matching the formal criteria of the classic or iterative games: for instance, those in which two entities could gain important benefits from cooperating or suffer from the failure to do so, but find it difficult or expensive—not necessarily impossible—to coordinate their activities.

The prisoner's dilemma game can be used as a model for many [[#Real-life examples|real world situations]] involving cooperative behavior. In casual usage, the label "prisoner's dilemma" may be applied to situations not strictly matching the formal criteria of the classic or iterative games: for instance, those in which two entities could gain important benefits from cooperating or suffer from the failure to do so, but find it difficult or expensive—not necessarily impossible—to coordinate their activities.

The prisoner's dilemma game can be used as a model for many real world situations involving cooperative behavior. In casual usage, the label "prisoner's dilemma" may be applied to situations not strictly matching the formal criteria of the classic or iterative games: for instance, those in which two entities could gain important benefits from cooperating or suffer from the failure to do so, but find it difficult or expensive—not necessarily impossible—to coordinate their activities.

The prisoner's dilemma game can be used as a model for many real world situations involving cooperative behavior. In casual usage, the label "prisoner's dilemma" may be applied to situations not strictly matching the formal criteria of the classic or iterative games: for instance, those in which two entities could gain important benefits from cooperating or suffer from the failure to do so, but find it difficult or expensive—not necessarily impossible—to coordinate their activitie

 

囚徒困境博弈可以作为许多现实世界中涉及合作行为的模型。在非正式用法中，”囚徒困境”一词可适用于不严格符合传统或迭代博弈的正式标准的情况: 例如，两个实体可以从合作中获得巨大利益或者会因为不能合作而遭受损失，但却发现协调其活动很困难或代价昂贵（并非不可能存在这种情况）。

<font color="#ff8000"> 囚徒困境</font>博弈可以作为许多现实世界中涉及合作行为的模型。在非正式用法中，”囚徒困境”一词可适用于不严格符合传统或迭代博弈的正式标准的情况: 例如，两个实体可以从合作中获得巨大利益或者会因为不能合作而遭受损失，但却发现协调其活动很困难或代价昂贵（并非不可能存在这种情况）。

==Strategy for the prisoner's dilemma==

==Strategy for the prisoner's dilemma==

==Generalized form==

==Generalized form==

 

The structure of the traditional prisoner's dilemma can be generalized from its original prisoner setting. Suppose that the two players are represented by the colors red and blue, and that each player chooses to either "cooperate" or "defect".

The structure of the traditional prisoner's dilemma can be generalized from its original prisoner setting. Suppose that the two players are represented by the colors red and blue, and that each player chooses to either "cooperate" or "defect".

and to be a prisoner's dilemma game in the strong sense, the following condition must hold for the payoffs:

and to be a prisoner's dilemma game in the strong sense, the following condition must hold for the payoffs:

要成为强意义下的囚徒困境博弈，收益必须满足以下条件:

要成为强意义下的<font color="#ff8000"> 囚徒困境</font>博弈，收益必须满足以下条件:

The payoff relationship  implies that mutual cooperation is superior to mutual defection, while the payoff relationships  and  imply that defection is the dominant strategy for both agents.

The payoff relationship  implies that mutual cooperation is superior to mutual defection, while the payoff relationships  and  imply that defection is the dominant strategy for both agents.

===Special case: donation game===

===Special case: donation game===

 

The "donation game"<ref name=Hilbe2013>{{cite journal|last=Hilbe|first=Christian |author2=Martin A. Nowak |author3=Karl Sigmund|title=Evolution of extortion in Iterated Prisoner's Dilemma games|journal=PNAS|date=April 2013|volume=110|issue=17|pages=6913–18|doi=10.1073/pnas.1214834110|pmid=23572576 |pmc=3637695 |bibcode=2013PNAS..110.6913H |arxiv=1212.1067}}</ref> is a form of prisoner's dilemma in which cooperation corresponds to offering the other player a benefit ''b'' at a personal cost ''c'' with ''b'' > ''c''. Defection means offering nothing. The payoff matrix is thus

The "donation game"<ref name=Hilbe2013>{{cite journal|last=Hilbe|first=Christian |author2=Martin A. Nowak |author3=Karl Sigmund|title=Evolution of extortion in Iterated Prisoner's Dilemma games|journal=PNAS|date=April 2013|volume=110|issue=17|pages=6913–18|doi=10.1073/pnas.1214834110|pmid=23572576 |pmc=3637695 |bibcode=2013PNAS..110.6913H |arxiv=1212.1067}}</ref> is a form of prisoner's dilemma in which cooperation corresponds to offering the other player a benefit ''b'' at a personal cost ''c'' with ''b'' > ''c''. Defection means offering nothing. The payoff matrix is thus

Note that  (i.e. ) which qualifies the donation game to be an iterated game (see next section).

Note that  (i.e. ) which qualifies the donation game to be an iterated game (see next section).

请注意(即)这使得捐赠博弈成为一个迭代博弈(见下一节)。

请注意(即)这使得<font color="#ff8000"> 捐赠博弈</font>成为一个迭代博弈(见下一节)。

The donation game may be applied to markets. Suppose X grows oranges, Y grows apples. The marginal utility of an apple to the orange-grower X is b, which is higher than the marginal utility (c) of an orange, since X has a surplus of oranges and no apples. Similarly, for apple-grower Y, the marginal utility of an orange is b while the marginal utility of an apple is c. If X and Y contract to exchange an apple and an orange, and each fulfills their end of the deal, then each receive a payoff of b-c. If one "defects" and does not deliver as promised, the defector will receive a payoff of b, while the cooperator will lose c. If both defect, then neither one gains or loses anything.

The donation game may be applied to markets. Suppose X grows oranges, Y grows apples. The marginal utility of an apple to the orange-grower X is b, which is higher than the marginal utility (c) of an orange, since X has a surplus of oranges and no apples. Similarly, for apple-grower Y, the marginal utility of an orange is b while the marginal utility of an apple is c. If X and Y contract to exchange an apple and an orange, and each fulfills their end of the deal, then each receive a payoff of b-c. If one "defects" and does not deliver as promised, the defector will receive a payoff of b, while the cooperator will lose c. If both defect, then neither one gains or loses anything.

捐赠博弈可能适用于市场。假设 X种橘子，Y 种苹果。苹果对橙子种植者 X 的边际效用是 b，这比橙子的边际效用c高，因为 x 有橙子剩余而没有苹果。同样，对于苹果种植者 y 来说，橙子的边际效用是 b，而苹果的边际效用是 c。 如果 X 和Y签约交换一个苹果和一个橙子，并且每个人都完成了交易，那么每个人都会得到 从c到b的效用收益。如果一方违约没有按照承诺交货，那么这个违约者将得到 b 效用的收益，而合作者将失去 c的效用收益。 如果两者都违约，那么谁也不会得到或失去任何东西。

<font color="#ff8000"> 捐赠博弈</font>可能适用于市场。假设 X种橘子，Y 种苹果。苹果对橙子种植者 X 的边际效用是 b，这比橙子的边际效用c高，因为 x 有橙子剩余而没有苹果。同样，对于苹果种植者 y 来说，橙子的边际效用是 b，而苹果的边际效用是 c。 如果 X 和Y签约交换一个苹果和一个橙子，并且每个人都完成了交易，那么每个人都会得到 从c到b的效用收益。如果一方违约没有按照承诺交货，那么这个违约者将得到 b 效用的收益，而合作者将失去 c的效用收益。 如果两者都违约，那么谁也不会得到或失去任何东西。

==The iterated prisoner's dilemma==

==The iterated prisoner's dilemma==

 

迭代<font color="#ff8000"> 囚徒困境</font>{{more citations needed section|date=November 2012}}

{{more citations needed section|date=November 2012}}

If two players play prisoner's dilemma more than once in succession and they remember previous actions of their opponent and change their strategy accordingly, the game is called iterated prisoner's dilemma.

If two players play prisoner's dilemma more than once in succession and they remember previous actions of their opponent and change their strategy accordingly, the game is called iterated prisoner's dilemma.

If two players play prisoner's dilemma more than once in succession and they remember previous actions of their opponent and change their strategy accordingly, the game is called iterated prisoner's dilemma.

If two players play prisoner's dilemma more than once in succession and they remember previous actions of their opponent and change their strategy accordingly, the game is called iterated prisoner's dilemma.

The iterated prisoner's dilemma game is fundamental to some theories of human cooperation and trust. On the assumption that the game can model transactions between two people requiring trust, cooperative behaviour in populations may be modeled by a multi-player, iterated, version of the game. It has, consequently, fascinated many scholars over the years. In 1975, Grofman and Pool estimated the count of scholarly articles devoted to it at over 2,000. The iterated prisoner's dilemma has also been referred to as the "peace-war game".

The iterated prisoner's dilemma game is fundamental to some theories of human cooperation and trust. On the assumption that the game can model transactions between two people requiring trust, cooperative behaviour in populations may be modeled by a multi-player, iterated, version of the game. It has, consequently, fascinated many scholars over the years. In 1975, Grofman and Pool estimated the count of scholarly articles devoted to it at over 2,000. The iterated prisoner's dilemma has also been referred to as the "peace-war game".

迭代囚徒困境博弈是一些人类合作与信任理论的基础。假设博弈可以为两个需要信任的人之间的交易建模，那么群体中的合作行为也可以由多个参与者迭代的博弈模型来建模。因此，这些年来，它吸引了许多学者。1975年，葛夫曼和普尔估计关于它的学术文章超过2000篇。迭代的的囚徒困境也被称为“和平-战争博弈”。

迭代<font color="#ff8000"> 囚徒困境</font>博弈是一些人类合作与信任理论的基础。假设博弈可以为两个需要信任的人之间的交易建模，那么群体中的合作行为也可以由多个参与者迭代的博弈模型来建模。因此，这些年来，它吸引了许多学者。1975年，葛夫曼和普尔估计关于它的学术文章超过2000篇。迭代的的<font color="#ff8000"> 囚徒困境</font>也被称为“和平-战争博弈”。

If the game is played exactly N times and both players know this, then it is optimal to defect in all rounds. The only possible Nash equilibrium is to always defect. The proof is inductive: one might as well defect on the last turn, since the opponent will not have a chance to later retaliate. Therefore, both will defect on the last turn. Thus, the player might as well defect on the second-to-last turn, since the opponent will defect on the last no matter what is done, and so on.  The same applies if the game length is unknown but has a known upper limit.

If the game is played exactly N times and both players know this, then it is optimal to defect in all rounds. The only possible Nash equilibrium is to always defect. The proof is inductive: one might as well defect on the last turn, since the opponent will not have a chance to later retaliate. Therefore, both will defect on the last turn. Thus, the player might as well defect on the second-to-last turn, since the opponent will defect on the last no matter what is done, and so on.  The same applies if the game length is unknown but has a known upper limit.

如果这个游戏正好玩了 n 次，并且两个玩家都知道这一点，那么在所有回合中最佳的策略就是叛变。唯一可能的纳什均衡点就是永远叛变。证据是归纳的: 一个人不妨在最后一回合叛变，因为对手以后没有机会反击。因此，双方都会在最后一个回合叛变。所以玩家同样也会在倒数第二回合时变节，因为无论做什么，对手都会在倒数第三回合变节，依此类推。同样方法适用于博弈次

如果这个游戏正好玩了 n 次，并且两个玩家都知道这一点，那么在所有回合中最佳的策略就是叛变。唯一可能的纳什均衡点就是永远叛变。证据是归纳出来的: 一个人不妨在最后一回合叛变，因为对手以后没有机会反击。因此，双方都会在最后一个回合叛变。所以玩家同样也会在倒数第二回合时叛变，因为无论做什么，对手都会在倒数第三回合变节，依此类推。同样方法适用于博弈次数未知但次数有限的情况。

Unlike the standard prisoner's dilemma, in the iterated prisoner's dilemma the defection strategy is counter-intuitive and fails badly to predict the behavior of human players. Within standard economic theory, though, this is the only correct answer.  The [[superrational]] strategy in the iterated prisoner's dilemma with fixed ''N'' is to cooperate against a superrational opponent, and in the limit of large ''N'', experimental results on strategies agree with the superrational version, not the game-theoretic rational one.

Unlike the standard prisoner's dilemma, in the iterated prisoner's dilemma the defection strategy is counter-intuitive and fails badly to predict the behavior of human players. Within standard economic theory, though, this is the only correct answer.  The [[superrational]] strategy in the iterated prisoner's dilemma with fixed ''N'' is to cooperate against a superrational opponent, and in the limit of large ''N'', experimental results on strategies agree with the superrational version, not the game-theoretic rational one.

For cooperation to emerge between game theoretic rational players, the total number of rounds N must be unknown to the players. In this case "always defect" may no longer be a strictly dominant strategy, only a Nash equilibrium. Amongst results shown by Robert Aumann in a 1959 paper, rational players repeatedly interacting for indefinitely long games can sustain the cooperative outcome.

For cooperation to emerge between game theoretic rational players, the total number of rounds N must be unknown to the players. In this case "always defect" may no longer be a strictly dominant strategy, only a Nash equilibrium. Amongst results shown by Robert Aumann in a 1959 paper, rational players repeatedly interacting for indefinitely long games can sustain the cooperative outcome.

为了使博弈论理性参与者之间出现合作，参与者必须不知道 n 回合的总数。在这种情况下，“总是叛变”可能不再是一个严格占主导地位的策略，而只是一个纳什均衡点。罗伯特 · 奥曼在1959年的一篇论文中表明，理性参与者在无限多次的博弈中通过反复互动可以维持合作的结果。

为了使<font color="#ff8000"> 博弈论</font>理性参与者之间出现合作，参与者必须不知道 n 回合的总数。在这种情况下，“总是叛变”可能不再是一个严格占主导地位的策略，而只是一个纳什均衡点。罗伯特·奥曼在1959年的一篇论文中表明，理性参与者在无限多次的博弈中通过反复互动可以维持合作的结果。

According to a 2019 experimental study in the American Economic Review which tested what strategies real-life subjects used in iterated prisoners' dilemma situations with perfect monitoring, the majority of chosen strategies were always defect, tit-for-tat, and Grim trigger. Which strategy the subjects chose depended on the parameters of the game.

According to a 2019 experimental study in the American Economic Review which tested what strategies real-life subjects used in iterated prisoners' dilemma situations with perfect monitoring, the majority of chosen strategies were always defect, tit-for-tat, and Grim trigger. Which strategy the subjects chose depended on the parameters of the game.

《美国经济评论》(American Economic Review)2019年的一项实验研究测试了现实生活中的实验对象在完全监控的情况下在迭代的囚徒困境中使用的策略，结果显示，大多数选择的策略都是叛变、针锋相对抑或是触发策略。受试者选择的策略取决于博弈的参数。

《美国经济评论》(American Econo以牙还牙锋相对抑或是<font color="#ff8000"> 冷酷触发策略</font>。受试者选择的策略取决于博弈的参数。

===Strategy for the iterated prisoner's dilemma===

===Strategy for the iterated prisoner's dilemma===

 

Interest in the iterated prisoner's dilemma (IPD) was kindled by [[Robert Axelrod]] in his book ''[[The Evolution of Cooperation]]'' (1984). In it he reports on a tournament he organized of the ''N'' step prisoner's dilemma (with ''N'' fixed) in which participants have to choose their mutual strategy again and again, and have memory of their previous encounters. Axelrod invited academic colleagues all over the world to devise computer strategies to compete in an IPD tournament. The programs that were entered varied widely in algorithmic complexity, initial hostility, capacity for forgiveness, and so forth.

Interest in the iterated prisoner's dilemma (IPD) was kindled by [[Robert Axelrod]] in his book ''[[The Evolution of Cooperation]]'' (1984). In it he reports on a tournament he organized of the ''N'' step prisoner's dilemma (with ''N'' fixed) in which participants have to choose their mutual strategy again and again, and have memory of their previous encounters. Axelrod invited academic colleagues all over the world to devise computer strategies to compete in an IPD tournament. The programs that were entered varied widely in algorithmic complexity, initial hostility, capacity for forgiveness, and so forth.

Interest in the iterated prisoner's dilemma (IPD) was kindled by Robert Axelrod in his book The Evolution of Cooperation (1984). In it he reports on a tournament he organized of the N step prisoner's dilemma (with N fixed) in which participants have to choose their mutual strategy again and again, and have memory of their previous encounters. Axelrod invited academic colleagues all over the world to devise computer strategies to compete in an IPD tournament. The programs that were entered varied widely in algorithmic complexity, initial hostility, capacity for forgiveness, and so forth.

Interest in the iterated prisoner's dilemma (IPD) was kindled by Robert Axelrod in his book The Evolution of Cooperation (1984). In it he reports on a tournament he organized of the N step prisoner's dilemma (with N fixed) in which participants have to choose their mutual strategy again and again, and have memory of their previous encounters. Axelrod invited academic colleagues all over the world to devise computer strategies to compete in an IPD tournament. The programs that were entered varied widely in algorithmic complexity, initial hostility, capacity for forgiveness, and so forth.

罗伯特 · 阿克塞尔罗德在他的著作《合作的进化》(1984)中激起了了人们对迭代的囚徒困境(IPD)的兴趣。在这篇文章中，他报道了一个关于 n 次囚徒困境的比赛，参赛者必须一次又一次地选择他们共同的策略，并且要记住他们之前的遭遇。阿克塞尔罗德邀请世界各地的学术界同仁设计计算机策略来参加此次比赛。输入的程序在算法复杂性、最初敌意、宽恕能力等方面差异很大。

罗伯特·阿克塞尔罗德在他的著作《合作的进化》(1984)中激起了了人们对迭代的<font color="#ff8000"> 囚徒困境</font>(IPD)的兴趣。在这篇文章中，他报道了一个关于 n 次<font color="#ff8000"> 囚徒困境</font>的比赛，参赛者必须一次又一次地选择他们共同的策略，并且要记住他们之前的遭遇。阿克塞尔罗德邀请世界各地的学术界同仁设计计算机策略来参加此次比赛。输入的程序在算法复杂性、最初敌意、宽恕能力等方面差异很大。

The winning deterministic strategy was tit for tat, which Anatol Rapoport developed and entered into the tournament. It was the simplest of any program entered, containing only four lines of BASIC, and won the contest. The strategy is simply to cooperate on the first iteration of the game; after that, the player does what his or her opponent did on the previous move. Depending on the situation, a slightly better strategy can be "tit for tat with forgiveness". When the opponent defects, on the next move, the player sometimes cooperates anyway, with a small probability (around 1–5%). This allows for occasional recovery from getting trapped in a cycle of defections. The exact probability depends on the line-up of opponents.

The winning deterministic strategy was tit for tat, which Anatol Rapoport developed and entered into the tournament. It was the simplest of any program entered, containing only four lines of BASIC, and won the contest. The strategy is simply to cooperate on the first iteration of the game; after that, the player does what his or her opponent did on the previous move. Depending on the situation, a slightly better strategy can be "tit for tat with forgiveness". When the opponent defects, on the next move, the player sometimes cooperates anyway, with a small probability (around 1–5%). This allows for occasional recovery from getting trapped in a cycle of defections. The exact probability depends on the line-up of opponents.

最终获胜的决定性策略是以牙还牙，这是阿纳托尔 · 拉波波特开发并参加比赛的策略。这是所有参赛程序中最简单的一个，只有四行 BASIC 语言，并且赢得了比赛。策略很简单，就是在游戏的第一次迭代中进行合作; 在此之后，玩家做他或她的对手在前一步中所做的事情。根据具体情况，一个稍微好一点的策略可以是“带着宽恕的心以牙还牙”。当对手叛变时，在下一次博弈中，玩家有时还是会合作，但概率很小(大约1-5%)。这允许博弈能偶尔从陷入叛变循环中恢复过来。确切的概率取决于对手的安排。

最终获胜的决定性策略是以牙还牙，这是阿纳托尔·拉波波特开发并参加比赛的策略。这是所有参赛程序中最简单的一个，只有四行 BASIC 语言，并且赢得了比赛。策略很简单，就是在游戏的第一次迭代中进行合作; 在此之后，玩家做他或她的对手在前一步中所做的事情。根据具体情况，一个稍微好一点的策略可以是“<font color="#32CD32">带着宽恕的心以牙还牙</font>”。当对手叛变时，在下一次博弈中，玩家有时还是会合作，但概率很小(大约1-5%)。这允许博弈能偶尔从陷入叛变循环中恢复过来。确切的概率取决于对手的安排。

  Nice: The most important condition is that the strategy must be "nice", that is, it will not defect before its opponent does (this is sometimes referred to as an "optimistic" algorithm). Almost all of the top-scoring strategies were nice; therefore, a purely selfish strategy will not "cheat" on its opponent, for purely self-interested reasons first.

  Nice: The most important condition is that the strategy must be "nice", that is, it will not defect before its opponent does (this is sometimes referred to as an "optimistic" algorithm). Almost all of the top-scoring strategies were nice; therefore, a purely selfish strategy will not "cheat" on its opponent, for purely self-interested reasons first.

友好： 最重要的条件是策略必须是好的 ，也就是说，它不会在对手之前叛变(这有时被称为“乐观”算法)。几乎所有得分最高的策略都是友好的; 因此，一个纯粹自私的策略不会出于纯粹自身利益的原因而“欺骗”对手。

友好： 最重要的条件是策略必须是好的 ，也就是说，它不会在对手之前叛变(这有时被称为“乐观”算法)。几乎所有得分最高的策略都是友好的; 因此，<font color="#32CD32">一个纯粹自私的策略不会出于纯粹自身利益的原因而“欺骗”对手</font>。

; Retaliating: However, Axelrod contended, the successful strategy must not be a blind optimist. It must sometimes retaliate. An example of a non-retaliating strategy is Always Cooperate. This is a very bad choice, as "nasty" strategies will ruthlessly exploit such players.

; Retaliating: However, Axelrod contended, the successful strategy must not be a blind optimist. It must sometimes retaliate. An example of a non-retaliating strategy is Always Cooperate. This is a very bad choice, as "nasty" strategies will ruthlessly exploit such players.

  Non-envious: The last quality is being non-envious, that is not striving to score more than the opponent.

  Non-envious: The last quality is being non-envious, that is not striving to score more than the opponent.

不嫉妒: 最后一个品质是不嫉妒，不比对手得分更多。

不嫉妒: 最后一个品质是不嫉妒，不强求比对手得分更多。

The optimal (points-maximizing) strategy for the one-time PD game is simply defection; as explained above, this is true whatever the composition of opponents may be. However, in the iterated-PD game the optimal strategy depends upon the strategies of likely opponents, and how they will react to defections and cooperations. For example, consider a population where everyone defects every time, except for a single individual following the tit for tat strategy. That individual is at a slight disadvantage because of the loss on the first turn. In such a population, the optimal strategy for that individual is to defect every time. In a population with a certain percentage of always-defectors and the rest being tit for tat players, the optimal strategy for an individual depends on the percentage, and on the length of the game.

The optimal (points-maximizing) strategy for the one-time PD game is simply defection; as explained above, this is true whatever the composition of opponents may be. However, in the iterated-PD game the optimal strategy depends upon the strategies of likely opponents, and how they will react to defections and cooperations. For example, consider a population where everyone defects every time, except for a single individual following the tit for tat strategy. That individual is at a slight disadvantage because of the loss on the first turn. In such a population, the optimal strategy for that individual is to defect every time. In a population with a certain percentage of always-defectors and the rest being tit for tat players, the optimal strategy for an individual depends on the percentage, and on the length of the game.

对于一次性的囚徒困境博弈，最优(点数最大化)策略就是简单的叛变; 正如上面所说，无论对手的构成如何，这都是正确的。然而，在迭代囚徒困境博弈中，最优策略取决于可能的对手的策略，以及他们对叛变和合作的反应。例如，考虑一个群体，其中每个人每次都会叛变，除了一个人遵循以牙还牙的策略。那个人就会由于第一回合的失利而处于轻微的不利地位。在这样一个群体中，个体的最佳策略是每次都叛变。在一定比例的总是选择背叛的玩家和其余组成为以牙还牙的玩家的人群中，个人的最佳策略取决于这一比例和博弈的次数。

对于一次性的<font color="#ff8000"> 囚徒困境</font>博弈，最优(点数最大化)策略就是简单的叛变; 正如上面所说，无论对手的构成如何，这都是正确的。然而，在迭代<font color="#ff8000"> 囚徒困境</font>博弈中，最优策略取决于可能的对手的策略，以及他们对叛变和合作的反应。例如，考虑一个群体，其中每个人每次都会叛变，除了一个人遵循以牙还牙的策略。那个人就会由于第一回合的失利而处于轻微的不利地位。在这样一个群体中，个体的最佳策略是每次都叛变。在一定比例的总是选择背叛的玩家和其余组成为以牙还牙的玩家的人群中，个人的最佳策略取决于这一比例和博弈的次数。

Although tit for tat is considered to be the most robust basic strategy, a team from Southampton University in England introduced a new strategy at the 20th-anniversary iterated prisoner's dilemma competition, which proved to be more successful than tit for tat. This strategy relied on collusion between programs to achieve the highest number of points for a single program. The university submitted 60 programs to the competition, which were designed to recognize each other through a series of five to ten moves at the start. Once this recognition was made, one program would always cooperate and the other would always defect, assuring the maximum number of points for the defector. If the program realized that it was playing a non-Southampton player, it would continuously defect in an attempt to minimize the score of the competing program. As a result, the 2004 Prisoners' Dilemma Tournament results show University of Southampton's strategies in the first three places, despite having fewer wins and many more losses than the GRIM strategy. (In a PD tournament, the aim of the game is not to "win" matches&nbsp;– that can easily be achieved by frequent defection). Also, even without implicit collusion between software strategies (exploited by the Southampton team) tit for tat is not always the absolute winner of any given tournament; it would be more precise to say that its long run results over a series of tournaments outperform its rivals. (In any one event a given strategy can be slightly better adjusted to the competition than tit for tat, but tit for tat is more robust). The same applies for the tit for tat with forgiveness variant, and other optimal strategies: on any given day they might not "win" against a specific mix of counter-strategies. An alternative way of putting it is using the Darwinian ESS simulation. In such a simulation, tit for tat will almost always come to dominate, though nasty strategies will drift in and out of the population because a tit for tat population is penetrable by non-retaliating nice strategies, which in turn are easy prey for the nasty strategies. Richard Dawkins showed that here, no static mix of strategies form a stable equilibrium and the system will always oscillate between bounds.}} this strategy ended up taking the top three positions in the competition, as well as a number of positions towards the bottom.

Although tit for tat is considered to be the most robust basic strategy, a team from Southampton University in England introduced a new strategy at the 20th-anniversary iterated prisoner's dilemma competition, which proved to be more successful than tit for tat. This strategy relied on collusion between programs to achieve the highest number of points for a single program. The university submitted 60 programs to the competition, which were designed to recognize each other through a series of five to ten moves at the start. Once this recognition was made, one program would always cooperate and the other would always defect, assuring the maximum number of points for the defector. If the program realized that it was playing a non-Southampton player, it would continuously defect in an attempt to minimize the score of the competing program. As a result, the 2004 Prisoners' Dilemma Tournament results show University of Southampton's strategies in the first three places, despite having fewer wins and many more losses than the GRIM strategy. (In a PD tournament, the aim of the game is not to "win" matches&nbsp;– that can easily be achieved by frequent defection). Also, even without implicit collusion between software strategies (exploited by the Southampton team) tit for tat is not always the absolute winner of any given tournament; it would be more precise to say that its long run results over a series of tournaments outperform its rivals. (In any one event a given strategy can be slightly better adjusted to the competition than tit for tat, but tit for tat is more robust). The same applies for the tit for tat with forgiveness variant, and other optimal strategies: on any given day they might not "win" against a specific mix of counter-strategies. An alternative way of putting it is using the Darwinian ESS simulation. In such a simulation, tit for tat will almost always come to dominate, though nasty strategies will drift in and out of the population because a tit for tat population is penetrable by non-retaliating nice strategies, which in turn are easy prey for the nasty strategies. Richard Dawkins showed that here, no static mix of strategies form a stable equilibrium and the system will always oscillate between bounds.}} this strategy ended up taking the top three positions in the competition, as well as a number of positions towards the bottom.

尽管以牙还牙认为是最有力的基本策略，来自英格兰南安普敦大学的一个团队在20周年的迭代囚徒困境竞赛中提出了一个新策略，这个策略被证明比以牙还牙更为成功。这种策略依赖于程序之间的串通，以获得单个程序的最高分数。这所大学向比赛提交了60个程序，这些程序的设计目的是在比赛开始时通过一系列的5到10个动作来互相认识。一旦认识建立，一个程序总是合作，另一个程序总是叛变，保证叛变者得到最多的分数。如果这个程序意识到它正在和一个非南安普顿的球员比赛，它会不断地叛变，试图最小化与之竞争程序的得分。因此，2004年囚徒困境锦标赛的结果显示了南安普敦大学战略位居前三名，尽管它比冷酷战略赢得更少，输的更多。(在囚徒困境锦标赛中，比赛的目的不是“赢”比赛——这一点频繁叛变很容易实现)。此外，即使没有软件策略之间的暗中串通(南安普顿队利用了这一点) ，以牙还牙并不总是任何特定锦标赛的绝对赢家; 更准确地说，它是在一系列锦标赛中的长期结果超过了它的竞争对手。(在任何一个事件中，一个给定的策略可以比以牙还牙稍微更好地适应竞争，但是以牙还牙更有力)。这同样适用于带有宽恕变量的以牙还牙，和其他最佳策略: 在任何特定的一天，他们可能不会“赢”一个特定的混合反战略。另一种方法是使用达尔文的 ESS 模拟。在这样的模拟中，以牙还牙几乎总是占主导地位，尽管讨厌的策略会在人群中进进出出，因为使用以牙还牙策略的人群可以通过非报复性的好策略进行渗透，这反过来使他们容易成为讨厌策略的猎物。理查德·道金斯指出，在这里，没有静态的混合策略会形成一个稳定的平衡，系统将始终在界限之间振荡。这种策略最终在比赛中获得了前三名的位置，以及一些接近垫底的位置。

尽管以牙还牙认为是最有力的基本策略，来自英格兰南安普敦大学的一个团队在20周年的迭代<font color="#ff8000"> 囚徒困境</font>竞赛中提出了一个新策略，这个策略被证明比以牙还牙更为成功。这种策略依赖于程序之间的串通，以获得单个程序的最高分数。这所大学向比赛提交了60个程序，这些程序的设计目的是在比赛开始时通过一系列的5到10个动作来互相认识。一旦认识建立，一个程序总是合作，另一个程序总是叛变，保证叛变者得到最多的分数。如果这个程序意识到它正在和一个非南安普顿的球员比赛，它会不断地叛变，试图最小化与之竞争程序的得分。因此，2004年囚徒困境锦标赛的结果显示了南安普敦大学战略位居前三名，尽管它比冷酷战略赢得更少，输的更多。(在囚徒困境锦标赛中，比赛的目的不是“赢”比赛——这一点频繁叛变很容易实现)。此外，即使没有软件策略之间的暗中串通(南安普顿队利用了这一点) ，以牙还牙并不总是任何特定锦标赛的绝对赢家; 更准确地说，它是在一系列锦标赛中的长期结果超过了它的竞争对手。(在任何一个事件中，一个给定的策略可以比以牙还牙稍微更好地适应竞争，但是以牙还牙更有力)。这同样适用于带有宽恕变量的以牙还牙，和其他最佳策略: 在任何特定的一天，他们可能不会“赢”一个特定的混合反战略。另一种方法是使用达尔文的<font color="#ff8000"> ESS模拟</font>。在这样的模拟中，以牙还牙几乎总是占主导地位，尽管讨厌的策略会在人群中进进出出，因为使用以牙还牙策略的人群可以通过非报复性的好策略进行渗透，这反过来使他们容易成为讨厌策略的猎物。理查德·道金斯指出，在这里，没有静态的混合策略会形成一个稳定的平衡，系统将始终在界限之间振荡。这种策略最终在比赛中获得了前三名的位置，以及一些接近垫底的位置。

In a stochastic iterated prisoner's dilemma game, strategies are specified by in terms of "cooperation probabilities". In an encounter between player X and player Y, X 's strategy is specified by a set of probabilities P of cooperating with Y. P is a function of the outcomes of their previous encounters or some subset thereof. If P is a function of only their most recent n encounters, it is called a "memory-n" strategy. A memory-1 strategy is then specified by four cooperation probabilities:  <math>P=\{P_{cc},P_{cd},P_{dc},P_{dd}\}</math>, where <math>P_{ab}</math> is the probability that X will cooperate in the present encounter given that the previous encounter was characterized by (ab). For example, if the previous encounter was one in which X cooperated and Y defected, then <math>P_{cd}</math> is the probability that X will cooperate in the present encounter. If each of the probabilities are either 1 or 0, the strategy is called deterministic. An example of a deterministic strategy is the tit for tat strategy written as P={1,0,1,0}, in which X responds as Y did in the previous encounter. Another is the win–stay, lose–switch strategy written as P={1,0,0,1}, in which X responds as in the previous encounter, if it was a "win" (i.e. cc or dc) but changes strategy if it was a loss (i.e. cd or dd). It has been shown that for any memory-n strategy there is a corresponding memory-1 strategy which gives the same statistical results, so that only memory-1 strategies need be considered.

In a stochastic iterated prisoner's dilemma game, strategies are specified by in terms of "cooperation probabilities". In an encounter between player X and player Y, X 's strategy is specified by a set of probabilities P of cooperating with Y. P is a function of the outcomes of their previous encounters or some subset thereof. If P is a function of only their most recent n encounters, it is called a "memory-n" strategy. A memory-1 strategy is then specified by four cooperation probabilities:  <math>P=\{P_{cc},P_{cd},P_{dc},P_{dd}\}</math>, where <math>P_{ab}</math> is the probability that X will cooperate in the present encounter given that the previous encounter was characterized by (ab). For example, if the previous encounter was one in which X cooperated and Y defected, then <math>P_{cd}</math> is the probability that X will cooperate in the present encounter. If each of the probabilities are either 1 or 0, the strategy is called deterministic. An example of a deterministic strategy is the tit for tat strategy written as P={1,0,1,0}, in which X responds as Y did in the previous encounter. Another is the win–stay, lose–switch strategy written as P={1,0,0,1}, in which X responds as in the previous encounter, if it was a "win" (i.e. cc or dc) but changes strategy if it was a loss (i.e. cd or dd). It has been shown that for any memory-n strategy there is a corresponding memory-1 strategy which gives the same statistical results, so that only memory-1 strategies need be considered.

在随机迭代囚徒困境博弈中，策略由“合作概率”来确定。在玩家 x 和玩家 y 之间的遭遇中，x 的策略由一组与 y 合作的概率 p 确定，p 是他们之前遭遇的结果的函数，或者是其中的一些子集。如果 p 只是它们最近遇到次数 n 的函数，那么它被称为“记忆-n”策略。我们可以用四个合作概率确定一个记忆-1策略:p { cc }、 p { cd }、 p { dc }、 p { dd } ，其中 math遭遇中合作的概率。如果每个概率都是1或0，这种策略称为确定性策略。确定性策略的一个例子是以牙还牙策略，写成 p {1,0,1,0} ，其中 x 的反应和 y 在前一次遭遇中的反应一样。另一种是胜-保持-败-转换策略，它被写成 p {1,0,0,1} ，在这种策略中，如果 x 获得胜利(即:cc 或 dc)，x会做出与上一次遭遇一样的反应 ，但如果失败，x会改变策略(即cd 或 dd)。研究表明，对于任何一种记忆-n 策略，存在一个相应的记忆-1策略，这个策略给出相同的统计结果，因此只需要考虑记忆-1策略。

在随机迭代<font color="#ff8000"> 囚徒困境</font>博弈中，策略由“合作概率”来确定。在玩家 x 和玩家 y 之间的遭遇中，x 的策略由一组与 y 合作的概率 p 确定，p 是他们之前遭遇的结果的函数，或者是其中的一些子集。如果 Pab 只是它们最近遇到次数 n 的函数，那么它被称为“记忆-n”策略。我们可以用四个合作概率确定一个记忆-1策略:p { cc }、 p { cd }、 p { dc }、 p { dd } ，其中 math遭遇中合作的概率。如果每个概率都是1或0，这种策略称为确定性策略。确定性策略的一个例子是以牙还牙策略，写成 p {1,0,1,0} ，其中 x 的反应和 y 在前一次遭遇中的反应一样。另一种是胜-保持-败-转换策略，它被写成 p {1,0,0,1} ，在这种策略中，如果 x 获得胜利(即:cc 或 dc)，x会做出与上一次遭遇一样的反应 ，但如果失败，x会改变策略(即cd 或 dd)。研究表明，对于任何一种记忆-n 策略，存在一个相应的记忆-1策略，这个策略给出相同的统计结果，因此只需要考虑记忆-1策略。

If we define P as the above 4-element strategy vector of X and <math>Q=\{Q_{cc},Q_{cd},Q_{dc},Q_{dd}\}</math> as the 4-element strategy vector of Y, a transition matrix M may be defined for X whose ij th entry is the probability that the outcome of a particular encounter between X and Y will be j given that the previous encounter was i, where i and j are one of the four outcome indices: cc, cd, dc, or dd. For example, from X 's point of view, the probability that the outcome of the present encounter is cd given that the previous encounter was cd is equal to <math>M_{cd,cd}=P_{cd}(1-Q_{dc})</math>. (The indices for Q are from Y 's point of view: a cd outcome for X is a dc outcome for Y.)  Under these definitions, the iterated prisoner's dilemma qualifies as a stochastic process and M is a stochastic matrix, allowing all of the theory of stochastic processes to be applied.

If we define P as the above 4-element strategy vector of X and <math>Q=\{Q_{cc},Q_{cd},Q_{dc},Q_{dd}\}</math> as the 4-element strategy vector of Y, a transition matrix M may be defined for X whose ij th entry is the probability that the outcome of a particular encounter between X and Y will be j given that the previous encounter was i, where i and j are one of the four outcome indices: cc, cd, dc, or dd. For example, from X 's point of view, the probability that the outcome of the present encounter is cd given that the previous encounter was cd is equal to <math>M_{cd,cd}=P_{cd}(1-Q_{dc})</math>. (The indices for Q are from Y 's point of view: a cd outcome for X is a dc outcome for Y.)  Under these definitions, the iterated prisoner's dilemma qualifies as a stochastic process and M is a stochastic matrix, allowing all of the theory of stochastic processes to be applied.

如果我们将 p 定义为 x 的上述4元策略向量，并将  q { cc }、 q { cd }、 q { dc }、 q { dd }  定义为 y 的4元策略向量，则对于 x 可以定义一个转移矩阵 m，该 x 的第 j 项是 x 和 y 之间特定遭遇的结果为 j 的概率，前一次遭遇为 i，其中 i 和 j 是 cc、 cd、 dc 或 dd 四个结果索引中的一个。例如，从 x 的角度来看，如果前一次遭遇的结果是 cd，那么这次遭遇的结果是 cd 的概率等于 m { cd，cd } p { cd }(1-Q { dc }) 。(q 的指数是 y 的观点: x 的 cd 结果是 y 的 dc 结果)在这些定义下，重复的囚徒困境被定义为一个随机过程，m 是一个转移矩阵，允许所有的随机过程理论被应用。

如果我们将 p 定义为 x 的上述4元策略向量，并将  q { cc }、 q { cd }、 q { dc }、 q { dd }  定义为 y 的4元策略向量，则对于 x 可以定义一个转移矩阵 m，该 x 的第 j 项是 x 和 y 之间特定遭遇的结果为 j 的概率，前一次遭遇为 i，其中 i 和 j 是 cc、 cd、 dc 或 dd 四个结果索引中的一个。例如，从 x 的角度来看，如果前一次遭遇的结果是 cd，那么这次遭遇的结果是 cd 的概率等于Mcd，cd=Pcd(1−Qdc 。(q 的指数是 y 的观点: x 的 cd 结果是 y 的 dc 结果)在这些定义下，重复的囚徒困境被定义为一个随机过程，m 是一个转移矩阵，允许所有的随机过程理论被应用。

One result of stochastic theory is that there exists a stationary vector v for the matrix M such that <math>v\cdot M=v</math>. Without loss of generality, it may be specified that v is normalized so that the sum of its four components is unity. The ij th entry in <math>M^n</math> will give the probability that the outcome of an encounter between X and Y will be j given that the encounter n steps previous is i. In the limit as n approaches infinity, M will converge to a matrix with fixed values, giving the long-term probabilities of an encounter producing j which will be independent of i. In other words, the rows of <math>M^\infty</math> will be identical, giving the long-term equilibrium result probabilities of the iterated prisoners dilemma without the need to explicitly evaluate a large number of interactions. It can be seen that v is a stationary vector for <math>M^n</math> and particularly <math>M^\infty</math>, so that each row of <math>M^\infty</math> will be equal to v. Thus the stationary vector specifies the equilibrium outcome probabilities for X. Defining <math>S_x=\{R,S,T,P\}</math> and <math>S_y=\{R,T,S,P\}</math> as the short-term payoff vectors for the {cc,cd,dc,dd} outcomes (From X 's point of view), the equilibrium payoffs for X and Y can now be specified as <math>s_x=v\cdot S_x</math> and <math>s_y=v\cdot S_y</math>, allowing the two strategies P and Q to be compared for their long term payoffs.

One result of stochastic theory is that there exists a stationary vector v for the matrix M such that <math>v\cdot M=v</math>. Without loss of generality, it may be specified that v is normalized so that the sum of its four components is unity. The ij th entry in <math>M^n</math> will give the probability that the outcome of an encounter between X and Y will be j given that the encounter n steps previous is i. In the limit as n approaches infinity, M will converge to a matrix with fixed values, giving the long-term probabilities of an encounter producing j which will be independent of i. In other words, the rows of <math>M^\infty</math> will be identical, giving the long-term equilibrium result probabilities of the iterated prisoners dilemma without the need to explicitly evaluate a large number of interactions. It can be seen that v is a stationary vector for <math>M^n</math> and particularly <math>M^\infty</math>, so that each row of <math>M^\infty</math> will be equal to v. Thus the stationary vector specifies the equilibrium outcome probabilities for X. Defining <math>S_x=\{R,S,T,P\}</math> and <math>S_y=\{R,T,S,P\}</math> as the short-term payoff vectors for the {cc,cd,dc,dd} outcomes (From X 's point of view), the equilibrium payoffs for X and Y can now be specified as <math>s_x=v\cdot S_x</math> and <math>s_y=v\cdot S_y</math>, allowing the two strategies P and Q to be compared for their long term payoffs.

随机理论的一个结果是，矩阵 m 存在一个平稳向量 v，使得矩阵 m 是一个平稳向量，并且不失一般性，我们可以指定 v 是标准化的，因此它的4个组成部分之和是单位。数学 m ^ n 中的 ij 项给出了 x 和 y 相遇的结果是 j 的概率，前面相遇 n 步的概率是 i。当 n 趋于无穷时，m 收敛于一个具有固定值的矩阵，并给出了产生 j 的长期概率，j 与 i 无关。换句话说， m 的无限次方的行将是相同的，给出了重复囚徒困境的长期平衡结果概率，而不需要明确地计算大量的相互作用。可以看出，v 是数学 m ^ n 特别是 m ^ infty  的平稳向量，因此数学 m的无限次方的每一行都等于 v，因此平稳向量指定 x 的平衡结果概率。将 s，r，s，t，p 和 s y，r，t，s，p  定义为{ cc，cd，dc，dd }结果的短期收益向量(从 x 的角度来看) ，x 和 y 的均衡收益现在可以指定为s_x=v\cdot S_x和s_y=v\cdot S_y ，使得两种P、Q策略能比较他们的长期回报。

随机理论的一个结果是，矩阵 m 存在一个平稳向量 v，使得矩阵 m 是一个平稳向量，并且不失一般性，我们可以指定 v 是标准化的，因此它的4个组成部分之和是单位。数学 m ^ n 中的 ij 项给出了 x 和 y 相遇的结果是 j 的概率，前面相遇 n 步的概率是 i。当 n 趋于无穷时，m 收敛于一个具有固定值的矩阵，并给出了产生 j 的长期概率，j 与 i 无关。换句话说， m^∞的行将是相同的，给出了重复囚徒困境的长期平衡结果概率，而不需要明确地计算大量的相互作用。可以看出，v 是数学 m ^ n 特别是 m^∞, 的平稳向量，因此数学 m的无限次方的每一行都等于 v，因此平稳向量指定 x 的平衡结果概率。将Sx={R,S,T,P}和Sy={R,T,S,P}定义为{ cc，cd，dc，dd }结果的短期收益向量(从 x 的角度来看) ，x 和 y 的均衡收益现在可以指定为sx=v⋅Sx 和sy=v⋅Sy,使得两种P、Q策略能比较他们的长期回报。

====Zero-determinant strategies====

====Zero-determinant strategies====

 

The relationship between zero-determinant (ZD), cooperating and defecting strategies in the iterated  prisoner's dilemma (IPD) illustrated in a [[Venn diagram. Cooperating strategies always cooperate with other cooperating strategies, and defecting strategies always defect against other defecting strategies. Both contain subsets of strategies that are robust under strong selection, meaning no other memory-1 strategy is selected to invade such strategies when they are resident in a population. Only cooperating strategies contain a subset that are always robust, meaning that no other memory-1 strategy is selected to invade and replace such strategies, under both strong and weak selection. The intersection between ZD and good cooperating strategies is the set of generous ZD strategies. Extortion strategies are the intersection between ZD and non-robust defecting strategies. Tit-for-tat lies at the intersection of cooperating, defecting and ZD strategies.]]

The relationship between zero-determinant (ZD), cooperating and defecting strategies in the iterated  prisoner's dilemma (IPD) illustrated in a [[Venn diagram. Cooperating strategies always cooperate with other cooperating strategies, and defecting strategies always defect against other defecting strategies. Both contain subsets of strategies that are robust under strong selection, meaning no other memory-1 strategy is selected to invade such strategies when they are resident in a population. Only cooperating strategies contain a subset that are always robust, meaning that no other memory-1 strategy is selected to invade and replace such strategies, under both strong and weak selection. The intersection between ZD and good cooperating strategies is the set of generous ZD strategies. Extortion strategies are the intersection between ZD and non-robust defecting strategies. Tit-for-tat lies at the intersection of cooperating, defecting and ZD strategies.]]

利用文献[1]中的维恩图，讨论了迭代囚徒困境(IPD)中零行列式(ZD)、合作策略和变节策略之间的关系。合作策略总是与其他合作策略相互配合，而变通策略总是与其他变通策略相抵触。这两种策略都包含在强选择下具有鲁棒性的策略子集，这意味着当它们驻留在一个种群中时，没有其他记忆1策略被选择来入侵这样的策略。只有协作策略包含一个总是鲁棒的子集，这意味着在强选择和弱选择情况下，没有选择其他的记忆-1策略来入侵和替换这些策略。ZD和好的合作策略之间的交集是一套慷慨的 ZD 策略。敲诈策略是 ZD 策略和非鲁棒性叛逃策略的交集。以牙还牙是合作、背叛和 ZD 策略的交集。

利用文献[1]中的<font color="#ff8000"> 维恩图</font>，讨论了迭代<font color="#ff8000"> 囚徒困境</font>(IPD)中零行列式(ZD)、合作策略和变节策略之间的关系。合作策略总是与其他合作策略相互配合，而变通策略总是与其他变通策略相抵触。这两种策略都包含在强选择下具有鲁棒性的策略子集，这意味着当它们驻留在一个种群中时，没有其他记忆-1策略被选择来入侵这样的策略。只有协作策略包含一个总是鲁棒的子集，这意味着在强选择和弱选择情况下，没有选择其他的记忆-1策略来入侵和替换这些策略。ZD和好的合作策略之间的交集是一套慷慨的 ZD 策略。敲诈策略是 ZD 策略和非鲁棒性叛逃策略的交集。以牙还牙是合作、背叛和 ZD 策略的交集。

In 2012, William H. Press and Freeman Dyson published a new class of strategies for the stochastic iterated prisoner's dilemma called "zero-determinant" (ZD) strategies. The long term payoffs for encounters between X and Y can be expressed as the determinant of a matrix which is a function of the two strategies and the short term payoff vectors: <math>s_x=D(P,Q,S_x)</math> and <math>s_y=D(P,Q,S_y)</math>, which do not involve the stationary vector v. Since the determinant function <math>s_y=D(P,Q,f)</math> is linear in f, it follows that <math>\alpha s_x+\beta s_y+\gamma=D(P,Q,\alpha S_x+\beta S_y+\gamma U)</math> (where U={1,1,1,1}). Any strategies for which <math>D(P,Q,\alpha S_x+\beta S_y+\gamma U)=0</math> is by definition a ZD strategy, and the long term payoffs obey the relation  <math>\alpha s_x+\beta s_y+\gamma=0</math>.

In 2012, William H. Press and Freeman Dyson published a new class of strategies for the stochastic iterated prisoner's dilemma called "zero-determinant" (ZD) strategies. The long term payoffs for encounters between X and Y can be expressed as the determinant of a matrix which is a function of the two strategies and the short term payoff vectors: <math>s_x=D(P,Q,S_x)</math> and <math>s_y=D(P,Q,S_y)</math>, which do not involve the stationary vector v. Since the determinant function <math>s_y=D(P,Q,f)</math> is linear in f, it follows that <math>\alpha s_x+\beta s_y+\gamma=D(P,Q,\alpha S_x+\beta S_y+\gamma U)</math> (where U={1,1,1,1}). Any strategies for which <math>D(P,Q,\alpha S_x+\beta S_y+\gamma U)=0</math> is by definition a ZD strategy, and the long term payoffs obey the relation  <math>\alpha s_x+\beta s_y+\gamma=0</math>.

2012年，威廉· h·普莱斯和弗里曼 · 戴森针对随机迭代囚徒困境提出了一类新的策略，称为“零决定因素”策略。x 和 y 之间的长期收益可以表示为一个矩阵的决定因素，它是两个策略和短期收益向量的函数: 不涉及平稳向量 v 的 s s x d (p，q，sx) 和 s y d (p，q，sy)。 由于行列式函数 s y d (p，q，f)  在 f 中是线性的，因此可以推出alpha s x + βs y + γd (p，q，αs x + βs y + γu)(其中 u {1,1,1})。任何策略的数学 d (p，q， αsx + βsy + gamma u)0 被定义为 ZD 策略，长期收益服从关系式。

2012年，威廉·H·普莱斯和弗里曼·戴森针对随机迭代囚徒困境提出了一类新的策略，称为“零决定因素”策略。x 和 y 之间的长期收益可以表示为一个矩阵的决定因素，它是两个策略和短期收益向量的函数: 不涉及平稳向量 v 的 sx=D(P,Q,Sx) 和sy=D(P,Q,Sy）。 由于行列式函数 sy=D(P,Q,Sy）在 f 中是线性的，因此可以推出αsx+βsy+γ=D(P,Q,αSx+βSy+γU) (当 U={1,1,1,1}))。任何策略的D(P,Q,αSx+βSy+γU)=0 被定义为 ZD 策略，长期收益服从关系式αsx+βsy+γ=0。

Tit-for-tat is a ZD strategy which is "fair" in the sense of not gaining advantage over the other player. However, the ZD space also contains strategies that, in the case of two players, can allow one player to unilaterally set the other player's score or alternatively, force an evolutionary player to achieve a payoff some percentage lower than his own. The extorted player could defect but would thereby hurt himself by getting a lower payoff. Thus, extortion solutions turn the iterated prisoner's dilemma into a sort of ultimatum game. Specifically, X is able to choose a strategy for which <math>D(P,Q,\beta S_y+\gamma U)=0</math>, unilaterally setting <math>s_y</math>  to a specific value within a particular range of values, independent of Y 's strategy, offering an opportunity for X to "extort" player Y (and vice versa). (It turns out that if X tries to set <math>s_x</math> to a particular value, the range of possibilities is much smaller, only consisting of complete cooperation or complete defection.)

Tit-for-tat is a ZD strategy which is "fair" in the sense of not gaining advantage over the other player. However, the ZD space also contains strategies that, in the case of two players, can allow one player to unilaterally set the other player's score or alternatively, force an evolutionary player to achieve a payoff some percentage lower than his own. The extorted player could defect but would thereby hurt himself by getting a lower payoff. Thus, extortion solutions turn the iterated prisoner's dilemma into a sort of ultimatum game. Specifically, X is able to choose a strategy for which <math>D(P,Q,\beta S_y+\gamma U)=0</math>, unilaterally setting <math>s_y</math>  to a specific value within a particular range of values, independent of Y 's strategy, offering an opportunity for X to "extort" player Y (and vice versa). (It turns out that if X tries to set <math>s_x</math> to a particular value, the range of possibilities is much smaller, only consisting of complete cooperation or complete defection.)

以牙还牙是 ZD 战略，这是“公平”的意义上说，没有占其他玩家的便宜。然而，ZD 空间也包含一些策略，在两个玩家的情况下，允许一个玩家单方面设置另一个玩家的分数，或者强迫一个进化的玩家获得比他自己的分数低一定百分比的回报。被敲诈的玩家可能会叛变，但因此获得较低的回报而受到伤害。因此，敲诈的解决方案将迭代的囚徒困境转化为一种最后通牒博弈。具体来说，x 能够选择一种策略，对于这种策略，数学 d (p，q， beta sy + gamma u)0 单方面地将 s y  设置为一个特定值范围内的特定值，与 y 的策略无关，为 x 提供了一个“勒索”玩家 y 的机会(反之亦然)。(事实证明，如果 x 试图将 s x 设置为一个特定的值，那么可能性的范围要小得多，只包括完全合作或完全叛变。)

以牙还牙是零决定战略，这是“公平”的意义上说，没有占其他玩家的便宜。然而，ZD 空间也包含一些策略，在两个玩家的情况下，允许一个玩家单方面设置另一个玩家的分数，或者强迫一个进化的玩家获得比他自己的分数低一定百分比的回报。被敲诈的玩家可能会叛变，但因此获得较低的回报而受到伤害。因此，敲诈的解决方案将迭代的囚徒困境转化为一种最后通牒博弈。具体来说，x 能够选择一种策略，对于这种策略，数学 D(P,Q,βSy+γU)=0 单方面地将 sy设置为一个特定值范围内的特定值，与 y 的策略无关，为 x 提供了一个“勒索”玩家 y 的机会(反之亦然)。(事实证明，如果 x 试图 sx设置为一个特定的值，那么可能性的范围要小得多，只包括完全合作或完全叛变。)

An extension of the IPD is an evolutionary stochastic IPD, in which the relative abundance of particular strategies is allowed to change, with more successful strategies relatively increasing. This process may be accomplished by having less successful players imitate the more successful strategies, or by eliminating less successful players from the game, while multiplying the more successful ones. It has been shown that unfair ZD strategies are not evolutionarily stable. The key intuition is that an evolutionarily stable strategy must not only be able to invade another population (which extortionary ZD strategies can do) but must also perform well against other players of the same type (which extortionary ZD players do poorly, because they reduce each other's surplus).

An extension of the IPD is an evolutionary stochastic IPD, in which the relative abundance of particular strategies is allowed to change, with more successful strategies relatively increasing. This process may be accomplished by having less successful players imitate the more successful strategies, or by eliminating less successful players from the game, while multiplying the more successful ones. It has been shown that unfair ZD strategies are not evolutionarily stable. The key intuition is that an evolutionarily stable strategy must not only be able to invade another population (which extortionary ZD strategies can do) but must also perform well against other players of the same type (which extortionary ZD players do poorly, because they reduce each other's surplus).

IPD的一个扩展是进化随机 IPD，其中允许特定策略的相对丰度发生变化，更成功的策略相对增加。这个过程可以通过让不那么成功的玩家模仿更成功的策略来完成，或者通过从游戏中淘汰不那么成功的玩家，同时让更成功的玩家成倍增加。研究表明，不公平的 ZD 策略不是进化稳定策略。关键的直觉告诉我们，简化稳定策略不仅要能够入侵另一个群体(这是敲诈 ZD 策略可以做到的) ，而且还要在同类型的其他玩家面前表现良好(敲诈 ZD 的玩家表现不佳，因为他们减少了彼此的盈余)。

<font color="#ff8000"> 迭代囚徒困境</font>的一个扩展是进化的随机<font color="#ff8000">迭代囚徒困境</font>，其中允许特定策略的相对丰度发生变化，更成功的策略相对增加。这个过程可以通过让不那么成功的玩家模仿更成功的策略来完成，或者通过从游戏中淘汰不那么成功的玩家，同时让更成功的玩家成倍增加。研究表明，不公平的零决定策略不是进化稳定策略。关键的直觉告诉我们，简化稳定策略不仅要能够入侵另一个群体(这是敲诈零决定策略可以做到的) ，而且还要在同类型的其他玩家面前表现良好(敲诈 ZD 的玩家表现不佳，因为他们减少了彼此的盈余)。

Theory and simulations confirm that beyond a critical population size, ZD extortion loses out in evolutionary competition against more cooperative strategies, and as a result, the average payoff in the population increases when the population is larger. In addition, there are some cases in which extortioners may even catalyze cooperation by helping to break out of a face-off between uniform defectors and win–stay, lose–switch agents.

Theory and simulations confirm that beyond a critical population size, ZD extortion loses out in evolutionary competition against more cooperative strategies, and as a result, the average payoff in the population increases when the population is larger. In addition, there are some cases in which extortioners may even catalyze cooperation by helping to break out of a face-off between uniform defectors and win–stay, lose–switch agents.

理论和模拟证实，超过一个临界种群规模，ZD 勒索失去了在进化竞争对更多的合作策略，结果，平均回报在种群增加。此外，在有些情况下，勒索者甚至可能通过帮助打破一致叛变者与赢-保持-输-转换代理人之间的对峙而促进合作。

理论和模拟证实，超过一个临界种群规模，零决定勒索失去了在进化竞争对更多的合作策略，结果，平均回报在种群增加。此外，在有些情况下，<font color="#32CD32">勒索者甚至可能通过帮助打破一致叛变者与赢-保持-输-转换代理人之间的对峙而促进合作</font>。

While extortionary ZD strategies are not stable in large populations, another ZD class called "generous" strategies is both stable and robust.  In fact, when the population is not too small, these strategies can supplant any other ZD strategy and even perform well against a broad array of generic strategies for iterated prisoner's dilemma, including win–stay, lose–switch. This was proven specifically for the donation game by Alexander Stewart and Joshua Plotkin in 2013. Generous strategies will cooperate with other cooperative players, and in the face of defection, the generous player loses more utility than its rival. Generous strategies are the intersection of ZD strategies and so-called "good" strategies, which were defined by Akin (2013) to be those for which the player responds to past mutual cooperation with future cooperation and splits expected payoffs equally if he receives at least the cooperative expected payoff. Among good strategies, the generous (ZD) subset performs well when the population is not too small. If the population is very small, defection strategies tend to dominate.

While extortionary ZD strategies are not stable in large populations, another ZD class called "generous" strategies is both stable and robust.  In fact, when the population is not too small, these strategies can supplant any other ZD strategy and even perform well against a broad array of generic strategies for iterated prisoner's dilemma, including win–stay, lose–switch. This was proven specifically for the donation game by Alexander Stewart and Joshua Plotkin in 2013. Generous strategies will cooperate with other cooperative players, and in the face of defection, the generous player loses more utility than its rival. Generous strategies are the intersection of ZD strategies and so-called "good" strategies, which were defined by Akin (2013) to be those for which the player responds to past mutual cooperation with future cooperation and splits expected payoffs equally if he receives at least the cooperative expected payoff. Among good strategies, the generous (ZD) subset performs well when the population is not too small. If the population is very small, defection strategies tend to dominate.

虽然被敲诈的 ZD 策略在人口众多的情况下并不稳定，但另一种被称为“慷慨”的 ZD 策略既稳定又强健。事实上，当人口不是太少的时候，这些策略可以取代任何其他 ZD 策略，甚至在一系列针对迭代囚徒困境的通用策略中表现良好，包括赢-保持，输-转换策略。亚历山大·斯图尔特和约书亚·普洛特金在2013年的捐赠博弈中证明了这一点。慷慨的策略会与其他合作的玩家合作，面对叛变，慷慨的玩家比他的对手失去更多的效用。慷慨策略是 ZD 策略和所谓的“好”策略的交集，Akin (2013)将这两种策略定义为玩家对过去的相互合作作出回应，并在至少获得合作预期收益的情况下平均分配预期收益的策略。在好的策略中，当总体不太小时，慷慨(ZD)子集表现良好。如果总体很少，叛变策略往往占主导地位。

虽然被敲诈的零决定策略在人口众多的情况下并不稳定，但另一种被称为“慷慨”的零决定策略既稳定又强健。事实上，当人口不是太少的时候，这些策略可以取代任何其他零决定策略，甚至在一系列针对迭代囚徒困境的通用策略中表现良好，包括赢-保持，输-转换策略。亚历山大·斯图尔特和约书亚·普洛特金在2013年的捐赠博弈中证明了这一点。慷慨的策略会与其他合作的玩家合作，面对叛变，慷慨的玩家比他的对手失去更多的效用。慷慨策略是零决定策略和所谓的“好”策略的交集，阿金(2013)将这两种策略定义为玩家对过去的相互合作作出回应，并在至少获得合作预期收益的情况下平均分配预期收益的策略。在好的策略中，当总体不太小时，慷慨(零决定)子集表现良好。如果总体很少，叛变策略往往占主导地位。

Most work on the iterated prisoner's dilemma has focused on the discrete case, in which players either cooperate or defect, because this model is relatively simple to analyze. However, some researchers have looked at models of the continuous iterated prisoner's dilemma, in which players are able to make a variable contribution to the other player. Le and Boyd found that in such situations, cooperation is much harder to evolve than in the discrete iterated prisoner's dilemma. The basic intuition for this result is straightforward: in a continuous prisoner's dilemma, if a population starts off in a non-cooperative equilibrium, players who are only marginally more cooperative than non-cooperators get little benefit from assorting with one another. By contrast, in a discrete prisoner's dilemma, tit for tat cooperators get a big payoff boost from assorting with one another in a non-cooperative equilibrium, relative to non-cooperators. Since nature arguably offers more opportunities for variable cooperation rather than a strict dichotomy of cooperation or defection, the continuous prisoner's dilemma may help explain why real-life examples of tit for tat-like cooperation are extremely rare in nature (ex. Hammerstein<ref>Hammerstein, P. (2003). Why is reciprocity so rare in social animals? A protestant appeal. In: P. Hammerstein, Editor, Genetic and Cultural Evolution of Cooperation, MIT Press. pp. 83–94.

Most work on the iterated prisoner's dilemma has focused on the discrete case, in which players either cooperate or defect, because this model is relatively simple to analyze. However, some researchers have looked at models of the continuous iterated prisoner's dilemma, in which players are able to make a variable contribution to the other player. Le and Boyd found that in such situations, cooperation is much harder to evolve than in the discrete iterated prisoner's dilemma. The basic intuition for this result is straightforward: in a continuous prisoner's dilemma, if a population starts off in a non-cooperative equilibrium, players who are only marginally more cooperative than non-cooperators get little benefit from assorting with one another. By contrast, in a discrete prisoner's dilemma, tit for tat cooperators get a big payoff boost from assorting with one another in a non-cooperative equilibrium, relative to non-cooperators. Since nature arguably offers more opportunities for variable cooperation rather than a strict dichotomy of cooperation or defection, the continuous prisoner's dilemma may help explain why real-life examples of tit for tat-like cooperation are extremely rare in nature (ex. Hammerstein<ref>Hammerstein, P. (2003). Why is reciprocity so rare in social animals? A protestant appeal. In: P. Hammerstein, Editor, Genetic and Cultural Evolution of Cooperation, MIT Press. pp. 83–94.

关于迭代囚徒困境的研究大多集中在离散情况下，在这种情况下，参与者要么合作，要么叛变，因为这个模型分析起来比较简单。然而，一些研究人员已经研究了连续迭代囚徒困境的模型，在这个模型中，玩家能够对另一个玩家做出可变的贡献。le和Boyd发现，在这种情况下，合作比离散迭代的囚徒困境更难发展。这个结果的基本直觉很简单: 在一个持续的囚徒困境中，如果一个人群开始处于非合作均衡状态，那么只比非合作者稍微多一点合作性的玩家从相互配合中获益不大。相比之下，在离散的囚徒困境中，以牙还牙的合作者相对于非合作者，在非合作均衡中相互配合会获得较大的回报提升。由于自然可以提供更多的机会来进行各种各样的合作，而不是严格地将合作或叛变分为两类，持续的囚徒困境可以帮助解释为什么现实生活中以牙还牙的合作的例子在自然界中极其罕见。（参考汉默斯坦，p.(2003)。《为什么互惠在群居动物中如此罕见？新教徒的呼吁：合作的基因和文化进化》 ，麻省理工学院出版社。83–94页）。

关于迭代囚徒困境的研究大多集中在离散情况下，在这种情况下，参与者要么合作，要么叛变，因为这个模型分析起来比较简单。然而，一些研究人员已经研究了连续迭代囚徒困境的模型，在这个模型中，玩家能够对另一个玩家做出可变的贡献。乐和博伊德发现，在这种情况下，合作比离散迭代的囚徒困境更难发展。这个结果的基本直觉很简单: 在一个持续的囚徒困境中，如果一个人群开始处于非合作均衡状态，那么只比非合作者稍微多一点合作性的玩家从相互配合中获益不大。相比之下，在离散的囚徒困境中，以牙还牙的合作者相对于非合作者，在非合作均衡中相互配合会获得较大的回报提升。由于自然可以提供更多的机会来进行各种各样的合作，而不是严格地将合作或叛变分为两类，持续的囚徒困境可以帮助解释为什么现实生活中以牙还牙的合作的例子在自然界中极其罕见。（参考汉默斯坦·P，(2003)。《为什么互惠在群居动物中如此罕见？新教徒的呼吁：合作的基因和文化进化》 ，麻省理工学院出版社。83–94页）。

</ref>) even though tit for tat seems robust in theoretical models.

</ref>) even though tit for tat seems robust in theoretical models.

===Emergence of stable strategies===

===Emergence of stable strategies===

 

Players cannot seem to coordinate mutual cooperation, thus often get locked into the inferior yet stable strategy of defection.  In this way, iterated rounds facilitate the evolution of stable strategies.<ref>{{cite book|last=Spaniel|first=William|title=Game Theory 101: The Complete Textbook|year=2011}}</ref> Iterated rounds often produce novel strategies, which have implications to complex social interaction. One such strategy is win-stay lose-shift. This strategy outperforms a simple Tit-For-Tat strategy&nbsp;– that is, if you can get away with cheating, repeat that behavior, however if you get caught, switch.<ref>{{cite journal|last=Nowak|first=Martin|author2=Karl Sigmund|title=A strategy of win-stay, lose-shift that outperforms tit-for-tat in the Prisoner's Dilemma game|journal=Nature|year=1993|volume=364|issue=6432|doi=10.1038/364056a0|pages=56–58|pmid=8316296|bibcode=1993Natur.364...56N}}</ref>

Players cannot seem to coordinate mutual cooperation, thus often get locked into the inferior yet stable strategy of defection.  In this way, iterated rounds facilitate the evolution of stable strategies.<ref>{{cite book|last=Spaniel|first=William|title=Game Theory 101: The Complete Textbook|year=2011}}</ref> Iterated rounds often produce novel strategies, which have implications to complex social interaction. One such strategy is win-stay lose-shift. This strategy outperforms a simple Tit-For-Tat strategy&nbsp;– that is, if you can get away with cheating, repeat that behavior, however if you get caught, switch.<ref>{{cite journal|last=Nowak|first=Martin|author2=Karl Sigmund|title=A strategy of win-stay, lose-shift that outperforms tit-for-tat in the Prisoner's Dilemma game|journal=Nature|year=1993|volume=364|issue=6432|doi=10.1038/364056a0|pages=56–58|pmid=8316296|bibcode=1993Natur.364...56N}}</ref>

==Real-life examples==

==Real-life examples==

 

The prisoner setting may seem contrived, but there are in fact many examples in human interaction as well as interactions in nature that have the same payoff matrix. The prisoner's dilemma is therefore of interest to the [[social science]]s such as [[economics]], [[politics]], and [[sociology]], as well as to the biological sciences such as [[ethology]] and [[evolutionary biology]]. Many natural processes have been abstracted into models in which living beings are engaged in endless games of prisoner's dilemma. This wide applicability of the PD gives the game its substantial importance.

The prisoner setting may seem contrived, but there are in fact many examples in human interaction as well as interactions in nature that have the same payoff matrix. The prisoner's dilemma is therefore of interest to the [[social science]]s such as [[economics]], [[politics]], and [[sociology]], as well as to the biological sciences such as [[ethology]] and [[evolutionary biology]]. Many natural processes have been abstracted into models in which living beings are engaged in endless games of prisoner's dilemma. This wide applicability of the PD gives the game its substantial importance.

===Environmental studies===

===Environmental studies===

 

In [[environmental studies]], the PD is evident in crises such as global [[climate change|climate-change]]. It is argued all countries will benefit from a stable climate, but any single country is often hesitant to curb [[Carbon dioxide|{{Co2}}]] emissions. The immediate benefit to any one country from maintaining current behavior is wrongly perceived to be greater than the purported eventual benefit to that country if all countries' behavior was changed, therefore explaining the impasse concerning climate-change in 2007.<ref>{{cite news|newspaper=[[The Economist]]|url=http://www.economist.com/finance/displaystory.cfm?story_id=9867020|title=Markets & Data|date=2007-09-27}}</ref>

In [[environmental studies]], the PD is evident in crises such as global [[climate change|climate-change]]. It is argued all countries will benefit from a stable climate, but any single country is often hesitant to curb [[Carbon dioxide|{{Co2}}]] emissions. The immediate benefit to any one country from maintaining current behavior is wrongly perceived to be greater than the purported eventual benefit to that country if all countries' behavior was changed, therefore explaining the impasse concerning climate-change in 2007.<ref>{{cite news|newspaper=[[The Economist]]|url=http://www.economist.com/finance/displaystory.cfm?story_id=9867020|title=Markets & Data|date=2007-09-27}}</ref>

An important difference between climate-change politics and the prisoner's dilemma is uncertainty; the extent and pace at which pollution can change climate is not known. The dilemma faced by government is therefore different from the prisoner's dilemma in that the payoffs of cooperation are unknown. This difference suggests that states will cooperate much less than in a real iterated prisoner's dilemma, so that the probability of avoiding a possible climate catastrophe is much smaller than that suggested by a game-theoretical analysis of the situation using a real iterated prisoner's dilemma.

An important difference between climate-change politics and the prisoner's dilemma is uncertainty; the extent and pace at which pollution can change climate is not known. The dilemma faced by government is therefore different from the prisoner's dilemma in that the payoffs of cooperation are unknown. This difference suggests that states will cooperate much less than in a real iterated prisoner's dilemma, so that the probability of avoiding a possible climate catastrophe is much smaller than that suggested by a game-theoretical analysis of the situation using a real iterated prisoner's dilemma.

气候变化政治与囚徒困境之间的一个重要区别是不确定性; 污染对气候变化的影响程度和速度尚不清楚。因此，政府面临的困境不同于囚徒困境，因为合作的回报是未知的。这种差异表明，各国之间的合作远远少于真正的迭代囚徒困境中的合作，因此避免可能发生的气候灾难的可能性远远小于使用真正的迭代囚徒困境进行的博弈论分析所提出的可能性。

气候变化政治与<font color="#ff8000"> 囚徒困境</font>之间的一个重要区别是不确定性; 污染对气候变化的影响程度和速度尚不清楚。因此，政府面临的困境不同于囚徒困境，因为合作的回报是未知的。这种差异表明，各国之间的合作远远少于真正的迭代<font color="#ff8000"> 囚徒困境/font>中的合作，因此避免可能发生的气候灾难的可能性远远小于使用真正的迭代<font color="#ff8000"> 囚徒困境</font><font color="#ff8000"> 博弈论</font>情景分析

 

Osang and Nandy (2003) provide a theoretical explanation with proofs for a regulation-driven win-win situation along the lines of Michael Porter's hypothesis, in which government regulation of competing firms is substantial.

Osang and Nandy (2003) provide a theoretical explanation with proofs for a regulation-driven win-win situation along the lines of Michael Porter's hypothesis, in which government regulation of competing firms is substantial.

Osang 和 Nandy (2003)提供了一个理论解释，并根据 Michael Porter 的假设，即政府对竞争企业的监管是实质性的，提供了监管驱动的双赢局面的证据。

欧桑和南迪(2003)提供了一个理论解释，并根据迈克尔·波特的假设，即政府对竞争企业的监管是实质性的，提供了监管驱动的双赢局面的证据。

===Animals===

===Animals===

 

Cooperative behavior of many animals can be understood as an example of the prisoner's dilemma. Often animals engage in long term partnerships, which can be more specifically modeled as iterated prisoner's dilemma. For example, [[guppy|guppies]] inspect predators cooperatively in groups, and they are thought to punish non-cooperative inspectors.

Cooperative behavior of many animals can be understood as an example of the prisoner's dilemma. Often animals engage in long term partnerships, which can be more specifically modeled as iterated prisoner's dilemma. For example, [[guppy|guppies]] inspect predators cooperatively in groups, and they are thought to punish non-cooperative inspectors.

Cooperative behavior of many animals can be understood as an example of the prisoner's dilemma. Often animals engage in long term partnerships, which can be more specifically modeled as iterated prisoner's dilemma. For example, guppies inspect predators cooperatively in groups, and they are thought to punish non-cooperative inspectors.

Cooperative behavior of many animals can be understood as an example of the prisoner's dilemma. Often animals engage in long term partnerships, which can be more specifically modeled as iterated prisoner's dilemma. For example, guppies inspect predators cooperatively in groups, and they are thought to punish non-cooperative inspectors.

Vampire bats are social animals that engage in reciprocal food exchange. Applying the payoffs from the prisoner's dilemma can help explain this behavior:

Vampire bats are social animals that engage in reciprocal food exchange. Applying the payoffs from the prisoner's dilemma can help explain this behavior:

吸血蝙蝠是群居动物，从事相互的食物交换。应用囚徒困境的收益可以帮助解释这种行为:

吸血蝙蝠是群居动物，从事相互的食物交换。应用的<font color="#ff8000"> 囚徒困境</font>收益可以帮助解释这种行为:

* C/C: "Reward: I get blood on my unlucky nights, which saves me from starving. I have to give blood on my lucky nights, which doesn't cost me too much."

* C/C: "Reward: I get blood on my unlucky nights, which saves me from starving. I have to give blood on my lucky nights, which doesn't cost me too much."

===Psychology===

===Psychology===

 

In [[addiction]] research / [[behavioral economics]], [[George Ainslie (psychologist)|George Ainslie]] points out<ref>{{cite book |first=George|last=Ainslie |title=Breakdown of Will |year=2001 |isbn=978-0-521-59694-7}}</ref> that addiction can be cast as an intertemporal PD problem between the present and future selves of the addict.  In this case, ''defecting'' means ''relapsing'', and it is easy to see that not defecting both today and in the future is by far the best outcome. The case where one abstains today but relapses in the future is the worst outcome&nbsp;– in some sense the discipline and self-sacrifice involved in abstaining today have been "wasted" because the future relapse means that the addict is right back where he started and will have to start over (which is quite demoralizing, and makes starting over more difficult).  Relapsing today and tomorrow is a slightly "better" outcome, because while the addict is still addicted, they haven't put the effort in to trying to stop. The final case, where one engages in the addictive behavior today while abstaining "tomorrow" will be familiar to anyone who has struggled with an addiction.  The problem here is that (as in other PDs) there is an obvious benefit to defecting "today", but tomorrow one will face the same PD, and the same obvious benefit will be present then, ultimately leading to an endless string of defections.

In [[addiction]] research / [[behavioral economics]], [[George Ainslie (psychologist)|George Ainslie]] points out<ref>{{cite book |first=George|last=Ainslie |title=Breakdown of Will |year=2001 |isbn=978-0-521-59694-7}}</ref> that addiction can be cast as an intertemporal PD problem between the present and future selves of the addict.  In this case, ''defecting'' means ''relapsing'', and it is easy to see that not defecting both today and in the future is by far the best outcome. The case where one abstains today but relapses in the future is the worst outcome&nbsp;– in some sense the discipline and self-sacrifice involved in abstaining today have been "wasted" because the future relapse means that the addict is right back where he started and will have to start over (which is quite demoralizing, and makes starting over more difficult).  Relapsing today and tomorrow is a slightly "better" outcome, because while the addict is still addicted, they haven't put the effort in to trying to stop. The final case, where one engages in the addictive behavior today while abstaining "tomorrow" will be familiar to anyone who has struggled with an addiction.  The problem here is that (as in other PDs) there is an obvious benefit to defecting "today", but tomorrow one will face the same PD, and the same obvious benefit will be present then, ultimately leading to an endless string of defections.

In addiction research / behavioral economics, George Ainslie points out that addiction can be cast as an intertemporal PD problem between the present and future selves of the addict.  In this case, defecting means relapsing, and it is easy to see that not defecting both today and in the future is by far the best outcome. The case where one abstains today but relapses in the future is the worst outcome&nbsp;– in some sense the discipline and self-sacrifice involved in abstaining today have been "wasted" because the future relapse means that the addict is right back where he started and will have to start over (which is quite demoralizing, and makes starting over more difficult).  Relapsing today and tomorrow is a slightly "better" outcome, because while the addict is still addicted, they haven't put the effort in to trying to stop. The final case, where one engages in the addictive behavior today while abstaining "tomorrow" will be familiar to anyone who has struggled with an addiction.  The problem here is that (as in other PDs) there is an obvious benefit to defecting "today", but tomorrow one will face the same PD, and the same obvious benefit will be present then, ultimately leading to an endless string of defections.

In addiction research / behavioral economics, George Ainslie points out that addiction can be cast as an intertemporal PD problem between the present and future selves of the addict.  In this case, defecting means relapsing, and it is easy to see that not defecting both today and in the future is by far the best outcome. The case where one abstains today but relapses in the future is the worst outcome&nbsp;– in some sense the discipline and self-sacrifice involved in abstaining today have been "wasted" because the future relapse means that the addict is right back where he started and will have to start over (which is quite demoralizing, and makes starting over more difficult).  Relapsing today and tomorrow is a slightly "better" outcome, because while the addict is still addicted, they haven't put the effort in to trying to stop. The final case, where one engages in the addictive behavior today while abstaining "tomorrow" will be familiar to anyone who has struggled with an addiction.  The problem here is that (as in other PDs) there is an obvious benefit to defecting "today", but tomorrow one will face the same PD, and the same obvious benefit will be present then, ultimately leading to an endless string of defections.

在成瘾研究/行为经济学中，乔治·安斯利指出，成瘾可以被描述为成瘾者现在和未来自我之间的跨期囚徒困境问题。在这种情况下，叛变意味着反复，很容易看出，不在今天和未来叛变是迄今为止最好的结果。如果一个人今天戒了，但在将来又复吸，这是最糟糕的结果——从某种意义上来说，今天戒瘾所包含的纪律和自我牺牲已经被“浪费”了，因为未来的复吸意味着瘾君子又回到了他开始的地方，将不得不重新开始(这相当令人沮丧，也使得重新开始更加困难)。今天和明天复发是一个稍微“更好”的结果，因为当瘾君子仍然上瘾时，他们没有努力去尝试停止。最后一种情况，一个人在今天进行成瘾行为，而在明天弃权，这对于任何一个与成瘾作斗争的人来说都是熟悉的。这里的问题是(和其他囚徒困境一样) ，背叛“今天”有一个明显的好处，但明天这个个人将面临同样的囚徒困境，同样明显的好处将出现，最终导致一连串无休止的叛变。

在成瘾研究/行为经济学中，乔治·安斯利指出，成瘾可以被描述为成瘾者现在和未来自我之间的跨期<font color="#ff8000"> 专业名词＋对应英文</font>问题。在这种情况下，叛变意味着反复，很容易看出，不在今天和未来叛变是迄今为止最好的结果。如果一个人今天戒了，但在将来又复吸，这是最糟糕的结果——从某种意义上来说，今天戒瘾所包含的纪律和自我牺牲已经被“浪费”了，因为未来的复吸意味着瘾君子又回到了他开始的地方，将不得不重新开始(这相当令人沮丧，也使得重新开始更加困难)。今天和明天复发是一个稍微“更好”的结果，因为当瘾君子仍然上瘾时，他们没有努力去尝试停止。最后一种情况，一个人在今天进行成瘾行为，而在明天放弃，这对于任何一个与成瘾作斗争的人来说都是熟悉的。这里的问题是(和其他<font color="#ff8000">囚徒困境</font>一样) ，背叛“今天”有一个明显的好处，但明天这个个人将面临同样的<font color="#ff8000">囚徒困境</font>，同样明显的好处将出现，最终导致一连串无休止的叛变。

John Gottman in his research described in "the science of trust" defines good relationships as those where partners know not to enter the (D,D) cell or at least not to get dynamically stuck there in a loop.

John Gottman in his research described in "the science of trust" defines good relationships as those where partners know not to enter the (D,D) cell or at least not to get dynamically stuck there in a loop.

John Gottman 在他的研究《信任的科学》中将良好的关系定义为伴侣知道不要进入(叛变，叛变）牢房中或者至少不要陷入这样一个循环中。

越好·高特曼在他的研究《信任的科学》中将良好的关系定义为伴侣知道<font color="#32CD32">不要进入(叛变，叛变）牢房中或者至少不要陷入这样一个循环中</font>。

===Economics===

===Economics===

 

The prisoner's dilemma has been called the ''[[Escherichia coli|E. coli]]'' of social psychology, and it has been used widely to research various topics such as [[Oligopoly|oligopolistic]] competition and collective action to produce a collective good.<ref>{{Cite journal|last=Axelrod|first=Robert|date=1980|title=Effective Choice in the Prisoner's Dilemma|journal=The Journal of Conflict Resolution|volume=24|issue=1|pages=3–25|issn=0022-0027|jstor=173932|doi=10.1177/002200278002400101|url=https://semanticscholar.org/paper/fd1ab82470446bfb12c39f0c577644291027cf76}}</ref>  

The prisoner's dilemma has been called the ''[[Escherichia coli|E. coli]]'' of social psychology, and it has been used widely to research various topics such as [[Oligopoly|oligopolistic]] competition and collective action to produce a collective good.<ref>{{Cite journal|last=Axelrod|first=Robert|date=1980|title=Effective Choice in the Prisoner's Dilemma|journal=The Journal of Conflict Resolution|volume=24|issue=1|pages=3–25|issn=0022-0027|jstor=173932|doi=10.1177/002200278002400101|url=https://semanticscholar.org/paper/fd1ab82470446bfb12c39f0c577644291027cf76}}</ref>  

The prisoner's dilemma has been called the E. coli of social psychology, and it has been used widely to research various topics such as oligopolistic competition and collective action to produce a collective good.  

The prisoner's dilemma has been called the E. coli of social psychology, and it has been used widely to research various topics such as oligopolistic competition and collective action to produce a collective good.  

Advertising is sometimes cited as a real-example of the prisoner's dilemma.  When cigarette advertising was legal in the United States, competing cigarette manufacturers had to decide how much money to spend on advertising.  The effectiveness of Firm A's advertising was partially determined by the advertising conducted by Firm B.  Likewise, the profit derived from advertising for Firm B is affected by the advertising conducted by Firm A.  If both Firm A and Firm B chose to advertise during a given period, then the advertisement from each firm negates the other's, receipts remain constant, and expenses increase due to the cost of advertising.  Both firms would benefit from a reduction in advertising.  However, should Firm B choose not to advertise, Firm A could benefit greatly by advertising. Nevertheless, the optimal amount of advertising by one firm depends on how much advertising the other undertakes. As the best strategy is dependent on what the other firm chooses there is no dominant strategy, which makes it slightly different from a prisoner's dilemma. The outcome is similar, though, in that both firms would be better off were they to advertise less than in the equilibrium. Sometimes cooperative behaviors do emerge in business situations.  For instance, cigarette manufacturers endorsed the making of laws banning cigarette advertising, understanding that this would reduce costs and increase profits across the industry. This analysis is likely to be pertinent in many other business situations involving advertising.

Advertising is sometimes cited as a real-example of the prisoner's dilemma.  When cigarette advertising was legal in the United States, competing cigarette manufacturers had to decide how much money to spend on advertising.  The effectiveness of Firm A's advertising was partially determined by the advertising conducted by Firm B.  Likewise, the profit derived from advertising for Firm B is affected by the advertising conducted by Firm A.  If both Firm A and Firm B chose to advertise during a given period, then the advertisement from each firm negates the other's, receipts remain constant, and expenses increase due to the cost of advertising.  Both firms would benefit from a reduction in advertising.  However, should Firm B choose not to advertise, Firm A could benefit greatly by advertising. Nevertheless, the optimal amount of advertising by one firm depends on how much advertising the other undertakes. As the best strategy is dependent on what the other firm chooses there is no dominant strategy, which makes it slightly different from a prisoner's dilemma. The outcome is similar, though, in that both firms would be better off were they to advertise less than in the equilibrium. Sometimes cooperative behaviors do emerge in business situations.  For instance, cigarette manufacturers endorsed the making of laws banning cigarette advertising, understanding that this would reduce costs and increase profits across the industry. This analysis is likely to be pertinent in many other business situations involving advertising.

广告有时被引用为囚徒困境的一个真实例子。当香烟广告在美国是合法的时候，相互竞争的香烟制造商必须决定在广告上花多少钱。公司 a 的广告效果部分取决于公司 b 的广告效果。同样，公司 b 的广告带来的利润也受到公司 a 的广告影响。如果公司 a 和公司 b 都选择在给定的时间段内做广告，那么一家公司的广告就会抵消另一方的广告，倘若收入保持不变，费用就会因广告成本而增加。两家公司都将从广告减少中获益。然而，如果 b 公司选择不做广告，a 公司就可以通过广告获得巨大的利益。尽管如此，一家公司的最佳广告数量仍取决于另一家公司从事了多少广告。由于最佳策略取决于其他公司的选择，因此这里没有占主导地位的策略，这使得它与囚徒困境略有不同。但结果是相似的，如果两家公司的广告都少于均衡他们的处境会更好。有时合作行为确实会在商业环境中出现。例如，香烟制造商支持立法禁止香烟广告，理解这将降低成本和增加整个行业的利润。这种分析可能适用于许多其他涉及广告的商业情况。

广告有时被引用为<font color="#ff8000"> 囚徒困境</font>的一个真实例子。当香烟广告在美国是合法的时候，相互竞争的香烟制造商必须决定在广告上花多少钱。公司 a 的广告效果部分取决于公司 b 的广告效果。同样，公司 b 的广告带来的利润也受到公司 a 的广告影响。如果公司 a 和公司 b 都选择在给定的时间段内做广告，那么一家公司的广告就会抵消另一方的广告，倘若收入保持不变，费用就会因广告成本而增加。两家公司都将从广告减少中获益。然而，如果 b 公司选择不做广告，a 公司就可以通过广告获得巨大的利益。尽管如此，一家公司的最佳广告数量仍取决于另一家公司从事了多少广告。由于最佳策略取决于其他公司的选择，因此这里没有占主导地位的策略，这使得它与<font color="#ff8000"> 囚徒困境</font>略有不同。但结果是相似的，如果两家公司的广告都少于均衡他们的处境会更好。有时合作行为确实会在商业环境中出现。例如，香烟制造商支持立法禁止香烟广告，理解这将降低成本和增加整个行业的利润。这种分析可能适用于许多其他涉及广告的商业情况。

Without enforceable agreements, members of a cartel are also involved in a (multi-player) prisoner's dilemma. 'Cooperating' typically means keeping prices at a pre-agreed minimum level. 'Defecting' means selling under this minimum level, instantly taking business (and profits) from other cartel members. Anti-trust authorities want potential cartel members to mutually defect, ensuring the lowest possible prices for consumers.

Without enforceable agreements, members of a cartel are also involved in a (multi-player) prisoner's dilemma. 'Cooperating' typically means keeping prices at a pre-agreed minimum level. 'Defecting' means selling under this minimum level, instantly taking business (and profits) from other cartel members. Anti-trust authorities want potential cartel members to mutually defect, ensuring the lowest possible prices for consumers.

没有可强制执行的协议，卡特尔的成员国也会陷入(多玩家)囚徒困境。“合作”通常意味着将价格保持在预先商定的最低水平。“叛变”意味着在这个最低价格水平下方销售，立即从其他卡特尔成员那里获得业务(和利润)。反垄断机构希望潜在的卡特尔成员相互叛变，确保消费者获得尽可能低的价格。

没有可强制执行的协议，卡特尔的成员国也会陷入(多玩家)囚徒困境。“合作”通常意味着将价格保持在预先商定的最低水平。“叛变”意味着在这个最低价格水平下方销售，并立即从其他卡特尔成员那里获得业务(和利润)。反垄断机构希望潜在的卡特尔成员相互叛变，确保消费者获得尽可能低的价格。

===Sport===

===Sport===

 

[[Doping in sport]] has been cited as an example of a prisoner's dilemma.<ref name="wired">{{cite journal|last=Schneier |first=Bruce |url=https://www.wired.com/opinion/2012/10/lance-armstrong-and-the-prisoners-dilemma-of-doping-in-professional-sports/ |title=Lance Armstrong and the Prisoners' Dilemma of Doping in Professional Sports &#124; Wired Opinion |journal=Wired |publisher=Wired.com |date=2012-10-26 |accessdate=2012-10-29}}</ref>

[[Doping in sport]] has been cited as an example of a prisoner's dilemma.<ref name="wired">{{cite journal|last=Schneier |first=Bruce |url=https://www.wired.com/opinion/2012/10/lance-armstrong-and-the-prisoners-dilemma-of-doping-in-professional-sports/ |title=Lance Armstrong and the Prisoners' Dilemma of Doping in Professional Sports &#124; Wired Opinion |journal=Wired |publisher=Wired.com |date=2012-10-26 |accessdate=2012-10-29}}</ref>

Doping in sport has been cited as an example of a prisoner's dilemma.

Doping in sport has been cited as an example of a prisoner's dilemma.

===International politics===

===International politics===

 

In [[international politics|international political theory]], the Prisoner's Dilemma is often used to demonstrate the coherence of [[strategic realism]], which holds that in international relations, all states (regardless of their internal policies or professed ideology), will act in their rational self-interest given [[anarchy (international relations)|international anarchy]]. A classic example is an arms race like the [[Cold War]] and similar conflicts.<ref>{{cite journal| title = Arms races as iterated prisoner's dilemma games | author = Stephen J. Majeski | journal = Mathematical and Social Sciences | volume = 7 | issue = 3 | pages = 253–66 | year = 1984 | doi=10.1016/0165-4896(84)90022-2}}</ref> During the Cold War the opposing alliances of [[NATO]] and the [[Warsaw Pact]] both had the choice to arm or disarm. From each side's point of view, disarming whilst their opponent continued to arm would have led to military inferiority and possible annihilation. Conversely, arming whilst their opponent disarmed would have led to superiority. If both sides chose to arm, neither could afford to attack the other, but both incurred the high cost of developing and maintaining a nuclear arsenal. If both sides chose to disarm, war would be avoided and there would be no costs.

In [[international politics|international political theory]], the Prisoner's Dilemma is often used to demonstrate the coherence of [[strategic realism]], which holds that in international relations, all states (regardless of their internal policies or professed ideology), will act in their rational self-interest given [[anarchy (international relations)|international anarchy]]. A classic example is an arms race like the [[Cold War]] and similar conflicts.<ref>{{cite journal| title = Arms races as iterated prisoner's dilemma games | author = Stephen J. Majeski | journal = Mathematical and Social Sciences | volume = 7 | issue = 3 | pages = 253–66 | year = 1984 | doi=10.1016/0165-4896(84)90022-2}}</ref> During the Cold War the opposing alliances of [[NATO]] and the [[Warsaw Pact]] both had the choice to arm or disarm. From each side's point of view, disarming whilst their opponent continued to arm would have led to military inferiority and possible annihilation. Conversely, arming whilst their opponent disarmed would have led to superiority. If both sides chose to arm, neither could afford to attack the other, but both incurred the high cost of developing and maintaining a nuclear arsenal. If both sides chose to disarm, war would be avoided and there would be no costs.

In international political theory, the Prisoner's Dilemma is often used to demonstrate the coherence of strategic realism, which holds that in international relations, all states (regardless of their internal policies or professed ideology), will act in their rational self-interest given international anarchy. A classic example is an arms race like the Cold War and similar conflicts. During the Cold War the opposing alliances of NATO and the Warsaw Pact both had the choice to arm or disarm. From each side's point of view, disarming whilst their opponent continued to arm would have led to military inferiority and possible annihilation. Conversely, arming whilst their opponent disarmed would have led to superiority. If both sides chose to arm, neither could afford to attack the other, but both incurred the high cost of developing and maintaining a nuclear arsenal. If both sides chose to disarm, war would be avoided and there would be no costs.

In international political theory, the Prisoner's Dilemma is often used to demonstrate the coherence of strategic realism, which holds that in international relations, all states (regardless of their internal policies or professed ideology), will act in their rational self-interest given international anarchy. A classic example is an arms race like the Cold War and similar conflicts. During the Cold War the opposing alliances of NATO and the Warsaw Pact both had the choice to arm or disarm. From each side's point of view, disarming whilst their opponent continued to arm would have led to military inferiority and possible annihilation. Conversely, arming whilst their opponent disarmed would have led to superiority. If both sides chose to arm, neither could afford to attack the other, but both incurred the high cost of developing and maintaining a nuclear arsenal. If both sides chose to disarm, war would be avoided and there would be no costs.

在国际政治理论中，囚徒困境经常被用来证明战略现实主义的一致性，这种战略现实主义认为，在国际关系中，由于国际无政府状态，所有国家(无论其国内政策或公开宣称的意识形态如何)都会为了自身的理性利益来行动。一个典型的例子是类似冷战和类似冲突的军备竞赛。在冷战期间，北约和华约组织的对立联盟都可以选择武装或解除武装。从双方的观点来看，解除武装而对手继续武装将导致军事劣势和可能的被歼灭。相反，如果武装的时候对手已经解除了武装，那么就会获得优势。如果双方都选择武装自己，那么任何一方都承担不起攻击对方的代价，但是双方都为发展和维持核武库付出了高昂的代价。如果双方都选择裁军，战争就可以避免，也不会有任何代价。

在国际政治理论中，<font color="#ff8000"> 囚徒困境</font>经常被用来证明战略现实主义的一致性，这种战略现实主义认为，在国际关系中，由于国际无政府状态，所有国家(无论其国内政策或公开宣称的意识形态如何)都会为了自身的理性利益来行动。一个典型的例子是类似冷战和类似冲突的军备竞赛。在冷战期间，北约和华约组织的对立联盟都可以选择武装或解除武装。从双方的观点来看，解除武装而对手继续武装将导致军事劣势和可能的被歼灭。相反，如果武装的时候对手已经解除了武装，那么就会获得优势。如果双方都选择武装自己，那么任何一方都承担不起攻击对方的代价，但是双方都为发展和维持核武库付出了高昂的代价。如果双方都选择裁军，战争就可以避免，也不会有任何代价。

===Multiplayer dilemmas===

===Multiplayer dilemmas===

 

Many real-life dilemmas involve multiple players.<ref>Gokhale CS, Traulsen A. Evolutionary games in the multiverse. Proceedings of the National Academy of Sciences. 2010 Mar 23. 107(12):5500–04.</ref> Although metaphorical, [[Garrett Hardin|Hardin's]] [[tragedy of the commons]] may be viewed as an example of a multi-player generalization of the PD: Each villager makes a choice for personal gain or restraint. The collective reward for unanimous (or even frequent) defection is very low payoffs (representing the destruction of the "commons"). A commons dilemma most people can relate to is washing the dishes in a shared house.  By not washing dishes an individual can gain by saving his time, but if that behavior is adopted by every resident the collective cost is no clean plates for anyone.

Many real-life dilemmas involve multiple players.<ref>Gokhale CS, Traulsen A. Evolutionary games in the multiverse. Proceedings of the National Academy of Sciences. 2010 Mar 23. 107(12):5500–04.</ref> Although metaphorical, [[Garrett Hardin|Hardin's]] [[tragedy of the commons]] may be viewed as an example of a multi-player generalization of the PD: Each villager makes a choice for personal gain or restraint. The collective reward for unanimous (or even frequent) defection is very low payoffs (representing the destruction of the "commons"). A commons dilemma most people can relate to is washing the dishes in a shared house.  By not washing dishes an individual can gain by saving his time, but if that behavior is adopted by every resident the collective cost is no clean plates for anyone.

Many real-life dilemmas involve multiple players. Although metaphorical, Hardin's tragedy of the commons may be viewed as an example of a multi-player generalization of the PD: Each villager makes a choice for personal gain or restraint. The collective reward for unanimous (or even frequent) defection is very low payoffs (representing the destruction of the "commons"). A commons dilemma most people can relate to is washing the dishes in a shared house.  By not washing dishes an individual can gain by saving his time, but if that behavior is adopted by every resident the collective cost is no clean plates for anyone.

Many real-life dilemmas involve multiple players. Although metaphorical, Hardin's tragedy of the commons may be viewed as an example of a multi-player generalization of the PD: Each villager makes a choice for personal gain or restraint. The collective reward for unanimous (or even frequent) defection is very low payoffs (representing the destruction of the "commons"). A commons dilemma most people can relate to is washing the dishes in a shared house.  By not washing dishes an individual can gain by saving his time, but if that behavior is adopted by every resident the collective cost is no clean plates for anyone.

许多现实生活中的困境牵涉到多个参与者。虽然是比喻性的，哈丁的公地悲剧可以被看作是囚徒困境的多人泛化的一个例子: 每个村民做出选择是为了个人利益还是为了克制。对于一致(甚至频繁)叛变的集体回报是非常低的回报(代表了对“公共资源”的破坏)。一个大多数人都能理解的共同困境就是在一个共享的房子里洗碗。通过不洗碗，个人可以节省时间，但如果这种行为被每个居民采纳，集体成本是任何人都没有干净的盘子。

许多现实生活中的困境牵涉到多个参与者。虽然是比喻性的，哈丁的<font color="#ff8000"> 公地悲剧</font>可以被看作是<font color="#ff8000"> 囚徒困境</font>的多人泛化的一个例子: 每个村民做出选择是为了个人利益还是为了克制。对于一致(甚至频繁)叛变的集体回报是非常低的回报(代表了对“公共资源”的破坏)。一个大多数人都能理解的共同困境就是在一个共享的房子里洗碗。通过不洗碗，个人可以节省时间，但如果这种行为被每个居民采纳，集体成本是任何人都没有干净的盘子。

The commons are not always exploited: William Poundstone, in a book about the prisoner's dilemma (see References below), describes a situation in New Zealand where newspaper boxes are left unlocked. It is possible for people to take a paper without paying (defecting) but very few do, feeling that if they do not pay then neither will others, destroying the system. Subsequent research by Elinor Ostrom, winner of the 2009 Nobel Memorial Prize in Economic Sciences, hypothesized that the tragedy of the commons is oversimplified, with the negative outcome influenced by outside influences. Without complicating pressures, groups communicate and manage the commons among themselves for their mutual benefit, enforcing social norms to preserve the resource and achieve the maximum good for the group, an example of effecting the best case outcome for PD.

The commons are not always exploited: William Poundstone, in a book about the prisoner's dilemma (see References below), describes a situation in New Zealand where newspaper boxes are left unlocked. It is possible for people to take a paper without paying (defecting) but very few do, feeling that if they do not pay then neither will others, destroying the system. Subsequent research by Elinor Ostrom, winner of the 2009 Nobel Memorial Prize in Economic Sciences, hypothesized that the tragedy of the commons is oversimplified, with the negative outcome influenced by outside influences. Without complicating pressures, groups communicate and manage the commons among themselves for their mutual benefit, enforcing social norms to preserve the resource and achieve the maximum good for the group, an example of effecting the best case outcome for PD.

公共资源并不总是被利用: 威廉·庞德斯通在一本关于囚徒困境的书(见下文参考文献)中描述了新西兰的一种情况，报纸盒子没有上锁。人们可以不付钱就拿报纸(叛变) ，但很少有人这样做，他们觉得如果他们不付钱，那么其他人也不会付钱，这会摧毁整个体系。2009年诺贝尔经济学奖获得者埃莉诺·奥斯特罗姆随后的研究认为公地悲剧经济学过于简单化，其负面结果受到外部影响。在没有复杂压力的情况下，团体之间为了共同利益进行沟通和管理，执行社会规范以保护资源并为团体实现最大利益，这是影响囚徒困境发展最佳结果的一个例子。

公共资源并不总是被利用: 威廉·庞德斯通在一本关于<font color="#ff8000"> 囚徒困境</font>的书(见下文参考文献)中描述了新西兰的一种情况，报纸盒子没有上锁。人们可以不付钱就拿报纸(叛变) ，但很少有人这样做，他们觉得如果他们不付钱，那么其他人也不会付钱，这会摧毁整个体系。2009年诺贝尔经济学奖获得者埃莉诺·奥斯特罗姆随后的研究认为公地悲剧经济学过于简单化，其负面结果受到外部影响。在没有复杂压力的情况下，团体之间为了共同利益进行沟通和管理，执行社会规范以保护资源并为团体实现最大利益，这是影响囚徒困境发展最佳结果的一个例子。

==Related games==

==Related games==

 

===Closed-bag exchange===

===Closed-bag exchange===

 

[[File:Prisoner's Dilemma briefcase exchange (colorized).svg|thumb|The prisoner's dilemma as a briefcase exchange]]

[[File:Prisoner's Dilemma briefcase exchange (colorized).svg|thumb|The prisoner's dilemma as a briefcase exchange]]

The prisoner's dilemma as a briefcase exchange

The prisoner's dilemma as a briefcase exchange

[[Douglas Hofstadter]]<ref name="dh">{{cite book | first=Douglas R. | last=Hofstadter| authorlink=Douglas Hofstadter | title= Metamagical Themas: questing for the essence of mind and pattern | publisher= Bantam Dell Pub Group| year=1985 | isbn=978-0-465-04566-2|chapter= Ch.29 ''The Prisoner's Dilemma Computer Tournaments and the Evolution of Cooperation''.| title-link=Metamagical Themas}}</ref> once suggested that people often find problems such as the PD problem easier to understand when it is illustrated in the form of a simple game, or trade-off. One of several examples he used was "closed bag exchange":

[[Douglas Hofstadter]]<ref name="dh">{{cite book | first=Douglas R. | last=Hofstadter| authorlink=Douglas Hofstadter | title= Metamagical Themas: questing for the essence of mind and pattern | publisher= Bantam Dell Pub Group| year=1985 | isbn=978-0-465-04566-2|chapter= Ch.29 ''The Prisoner's Dilemma Computer Tournaments and the Evolution of Cooperation''.| title-link=Metamagical Themas}}</ref> once suggested that people often find problems such as the PD problem easier to understand when it is illustrated in the form of a simple game, or trade-off. One of several examples he used was "closed bag exchange":

Douglas Hofstadter once suggested that people often find problems such as the PD problem easier to understand when it is illustrated in the form of a simple game, or trade-off. One of several examples he used was "closed bag exchange":

Douglas Hofstadter once suggested that people often find problems such as the PD problem easier to understand when it is illustrated in the form of a simple game, or trade-off. One of several examples he used was "closed bag exchange":

侯世达曾经指出，人们通常会发现问题，比如囚徒困境问题，当它以一个简单博弈的形式表现出来时，或者以权衡的方式表现出来时，会更容易理解。他使用的几个例子之一是“封闭式袋子交换” :

侯世达曾经指出，人们通常会发现问题，比如，当它以一个简单<font color="#ff8000"> 囚徒困境</font>博弈的形式表现出来时，或者以权衡的方式表现出来时，会更容易理解。他使用的几个例子之一是“封闭式袋子交换” :

{{quote|Two people meet and exchange closed bags, with the understanding that one of them contains money, and the other contains a purchase. Either player can choose to honor the deal by putting into his or her bag what he or she agreed, or he or she can defect by handing over an empty bag.}}

{{quote|Two people meet and exchange closed bags, with the understanding that one of them contains money, and the other contains a purchase. Either player can choose to honor the deal by putting into his or her bag what he or she agreed, or he or she can defect by handing over an empty bag.}}

 

Defection always gives a game-theoretically preferable outcome.<ref>{{Cite web|url=https://users.auth.gr/kehagiat/Research/GameTheory/06GamesToPlay/Prisoner%27s_dilemma.htm#Closed_Bag_Exchange|title=Prisoner's dilemma - Wikipedia, the free encyclopedia|website=users.auth.gr|access-date=2020-04-12}}</ref>

Defection always gives a game-theoretically preferable outcome.<ref>{{Cite web|url=https://users.auth.gr/kehagiat/Research/GameTheory/06GamesToPlay/Prisoner%27s_dilemma.htm#Closed_Bag_Exchange|title=Prisoner's dilemma - Wikipedia, the free encyclopedia|website=users.auth.gr|access-date=2020-04-12}}</ref>

===''Friend or Foe?''===

===''Friend or Foe?''===

 

''[[Friend or Foe? (TV series)|Friend or Foe?]]'' is a game show that aired from 2002 to 2005 on the [[Game Show Network]] in the US.  It is an example of the prisoner's dilemma game tested on real people, but in an artificial setting. On the game show, three pairs of people compete. When a pair is eliminated, they play a game similar to the prisoner's dilemma to determine how the winnings are split. If they both cooperate (Friend), they share the winnings 50–50. If one cooperates and the other defects (Foe), the defector gets all the winnings and the cooperator gets nothing. If both defect, both leave with nothing. Notice that the reward matrix is slightly different from the standard one given above, as the rewards for the "both defect" and the "cooperate while the opponent defects" cases are identical. This makes the "both defect" case a weak equilibrium, compared with being a strict equilibrium in the standard prisoner's dilemma. If a contestant knows that their opponent is going to vote "Foe", then their own choice does not affect their own winnings. In a specific sense, ''Friend or Foe'' has a rewards model between prisoner's dilemma and the [[Chicken (game)|game of Chicken]].

''[[Friend or Foe? (TV series)|Friend or Foe?]]'' is a game show that aired from 2002 to 2005 on the [[Game Show Network]] in the US.  It is an example of the prisoner's dilemma game tested on real people, but in an artificial setting. On the game show, three pairs of people compete. When a pair is eliminated, they play a game similar to the prisoner's dilemma to determine how the winnings are split. If they both cooperate (Friend), they share the winnings 50–50. If one cooperates and the other defects (Foe), the defector gets all the winnings and the cooperator gets nothing. If both defect, both leave with nothing. Notice that the reward matrix is slightly different from the standard one given above, as the rewards for the "both defect" and the "cooperate while the opponent defects" cases are identical. This makes the "both defect" case a weak equilibrium, compared with being a strict equilibrium in the standard prisoner's dilemma. If a contestant knows that their opponent is going to vote "Foe", then their own choice does not affect their own winnings. In a specific sense, ''Friend or Foe'' has a rewards model between prisoner's dilemma and the [[Chicken (game)|game of Chicken]].

Friend or Foe? is a game show that aired from 2002 to 2005 on the Game Show Network in the US.  It is an example of the prisoner's dilemma game tested on real people, but in an artificial setting. On the game show, three pairs of people compete. When a pair is eliminated, they play a game similar to the prisoner's dilemma to determine how the winnings are split. If they both cooperate (Friend), they share the winnings 50–50. If one cooperates and the other defects (Foe), the defector gets all the winnings and the cooperator gets nothing. If both defect, both leave with nothing. Notice that the reward matrix is slightly different from the standard one given above, as the rewards for the "both defect" and the "cooperate while the opponent defects" cases are identical. This makes the "both defect" case a weak equilibrium, compared with being a strict equilibrium in the standard prisoner's dilemma. If a contestant knows that their opponent is going to vote "Foe", then their own choice does not affect their own winnings. In a specific sense, Friend or Foe has a rewards model between prisoner's dilemma and the game of Chicken.

Friend or Foe? is a game show that aired from 2002 to 2005 on the Game Show Network in the US.  It is an example of the prisoner's dilemma game tested on real people, but in an artificial setting. On the game show, three pairs of people compete. When a pair is eliminated, they play a game similar to the prisoner's dilemma to determine how the winnings are split. If they both cooperate (Friend), they share the winnings 50–50. If one cooperates and the other defects (Foe), the defector gets all the winnings and the cooperator gets nothing. If both defect, both leave with nothing. Notice that the reward matrix is slightly different from the standard one given above, as the rewards for the "both defect" and the "cooperate while the opponent defects" cases are identical. This makes the "both defect" case a weak equilibrium, compared with being a strict equilibrium in the standard prisoner's dilemma. If a contestant knows that their opponent is going to vote "Foe", then their own choice does not affect their own winnings. In a specific sense, Friend or Foe has a rewards model between prisoner's dilemma and the game of Chicken.

朋友还是敌人？是一个竞赛节目，从2002年至2005年在美国的竞赛节目网络播出。这是囚徒困境博弈在真人身上测试的一个例子，但是是在人为的环境中。在游戏节目中，有三对选手参加比赛。当一对被淘汰时，他们会玩一个类似囚徒困境的游戏来决定奖金如何分配。如果他们都合作(朋友) ，他们分享奖金50-50。如果一方合作而另一方叛变(敌人) ，那么叛变者将得到所有的奖金，而合作者将一无所获。如果双方都叛变，那么双方都将一无所有。请注意，奖励矩阵与上面给出的标准矩阵略有不同，因为“双方都叛变”和“合作而对方叛变”情况下的奖励是相同的。与标准囚徒困境中的严格均衡相比，这使得“两个都叛变”情况成为一个弱均衡。如果一个参赛者知道他们的对手将投票给“敌人” ，那么他们自己的选择不会影响他们自己的奖金。从特定意义上讲，“朋友还是敌人”节目在囚徒困境和“胆小鬼”博弈之间有一个奖励模型。

朋友还是敌人？是一个竞赛节目，从2002年至2005年在美国的竞赛节目网络播出。这是<font color="#ff8000"> 囚徒困境</font>博弈在真人身上测试的一个例子，但是是在人为的环境中。在游戏节目中，有三对选手参加比赛。当一对被淘汰时，他们会玩一个类似囚徒困境的游戏来决定奖金如何分配。如果他们都合作(朋友) ，他们分享奖金50-50。如果一方合作而另一方叛变(敌人) ，那么叛变者将得到所有的奖金，而合作者将一无所获。如果双方都叛变，那么双方都将一无所有。请注意，奖励矩阵与上面给出的标准矩阵略有不同，因为“双方都叛变”和“合作而对方叛变”情况下的奖励是相同的。与标准囚徒困境中的严格均衡相比，这使得“两个都叛变”情况成为一个弱均衡。如果一个参赛者知道他们的对手将投票给“敌人” ，那么他们自己的选择不会影响他们自己的奖金。从特定意义上讲，“朋友还是敌人”节目在囚徒困境和“胆小鬼”博弈之间有一个奖励模型。

===Iterated snowdrift===

===Iterated snowdrift===

 

{{main|snowdrift game}}

{{main|snowdrift game}}

===Coordination games===

===Coordination games===

 

{{main|coordination games}}

{{main|coordination games}}

A more general set of games are asymmetric. As in the prisoner's dilemma, the best outcome is co-operation, and there are motives for defection. Unlike the symmetric prisoner's dilemma, though, one player has more to lose and/or more to gain than the other. Some such games have been described as a prisoner's dilemma in which one prisoner has an alibi, whence the term "alibi game".

A more general set of games are asymmetric. As in the prisoner's dilemma, the best outcome is co-operation, and there are motives for defection. Unlike the symmetric prisoner's dilemma, though, one player has more to lose and/or more to gain than the other. Some such games have been described as a prisoner's dilemma in which one prisoner has an alibi, whence the term "alibi game".

==Software==

==Software==

 

Several software packages have been created to run prisoner's dilemma simulations and tournaments, some of which have available source code.

Several software packages have been created to run prisoner's dilemma simulations and tournaments, some of which have available source code.

* The source code for the [[The Evolution of Cooperation|second tournament]] run by Robert Axelrod (written by Axelrod and many contributors in [[Fortran]]) is available [http://www-personal.umich.edu/~axe/research/Software/CC/CC2.html online]

* The source code for the [[The Evolution of Cooperation|second tournament]] run by Robert Axelrod (written by Axelrod and many contributors in [[Fortran]]) is available [http://www-personal.umich.edu/~axe/research/Software/CC/CC2.html online]

==In fiction==

==In fiction==

 

[[Hannu Rajaniemi]] set the opening scene of his ''[[The Quantum Thief]]'' trilogy in a "dilemma prison". The main theme of the series has been described as the "inadequacy of a binary universe" and the ultimate antagonist is a character called the All-Defector.  Rajaniemi is particularly interesting as an artist treating this subject in that he is a Cambridge-trained mathematician and holds a PhD in [[mathematical physics]]&nbsp;– the interchangeability of matter and information is a major feature of the books, which take place in a "post-singularity" future.  The first book in the series was published in 2010, with the two sequels, ''[[The Fractal Prince]]'' and ''[[The Causal Angel]]'', published in 2012 and 2014, respectively.

[[Hannu Rajaniemi]] set the opening scene of his ''[[The Quantum Thief]]'' trilogy in a "dilemma prison". The main theme of the series has been described as the "inadequacy of a binary universe" and the ultimate antagonist is a character called the All-Defector.  Rajaniemi is particularly interesting as an artist treating this subject in that he is a Cambridge-trained mathematician and holds a PhD in [[mathematical physics]]&nbsp;– the interchangeability of matter and information is a major feature of the books, which take place in a "post-singularity" future.  The first book in the series was published in 2010, with the two sequels, ''[[The Fractal Prince]]'' and ''[[The Causal Angel]]'', published in 2012 and 2014, respectively.

Hannu Rajaniemi set the opening scene of his The Quantum Thief trilogy in a "dilemma prison". The main theme of the series has been described as the "inadequacy of a binary universe" and the ultimate antagonist is a character called the All-Defector.  Rajaniemi is particularly interesting as an artist treating this subject in that he is a Cambridge-trained mathematician and holds a PhD in mathematical physics&nbsp;– the interchangeability of matter and information is a major feature of the books, which take place in a "post-singularity" future.  The first book in the series was published in 2010, with the two sequels, The Fractal Prince and The Causal Angel, published in 2012 and 2014, respectively.

Hannu Rajaniemi set the opening scene of his The Quantum Thief trilogy in a "dilemma prison". The main theme of the series has been described as the "inadequacy of a binary universe" and the ultimate antagonist is a character called the All-Defector.  Rajaniemi is particularly interesting as an artist treating this subject in that he is a Cambridge-trained mathematician and holds a PhD in mathematical physics&nbsp;– the interchangeability of matter and information is a major feature of the books, which take place in a "post-singularity" future.  The first book in the series was published in 2010, with the two sequels, The Fractal Prince and The Causal Angel, published in 2012 and 2014, respectively.

A game modeled after the (iterated) prisoner's dilemma is a central focus of the 2012 video game Zero Escape: Virtue's Last Reward and a minor part in its 2016 sequel Zero Escape: Zero Time Dilemma.

A game modeled after the (iterated) prisoner's dilemma is a central focus of the 2012 video game Zero Escape: Virtue's Last Reward and a minor part in its 2016 sequel Zero Escape: Zero Time Dilemma.

一个模仿迭代囚徒困境的游戏《零度逃脱: 美德的最后奖励》是2012年电子游戏的中心焦点，也是2016年续集《零度逃脱: 极限脱出刻之困境》的一个次要部分。

一个模仿迭代<font color="#ff8000"> 囚徒困境</font>的游戏《零度逃脱: 美德的最后奖励》是2012年电子游戏的中心焦点，也是2016年续集《零度逃脱: 极限脱出刻之困境》的一个次要部分。

In The Mysterious Benedict Society and the Prisoner's Dilemma by Trenton Lee Stewart, the main characters start by playing a version of the game and escaping from the "prison" altogether. Later they become actual prisoners and escape once again.

In The Mysterious Benedict Society and the Prisoner's Dilemma by Trenton Lee Stewart, the main characters start by playing a version of the game and escaping from the "prison" altogether. Later they become actual prisoners and escape once again.

In The Adventure Zone: Balance during The Suffering Game subarc, the player characters are twice presented with the prisoner's dilemma during their time in two liches' domain, once cooperating and once defecting.

In The Adventure Zone: Balance during The Suffering Game subarc, the player characters are twice presented with the prisoner's dilemma during their time in two liches' domain, once cooperating and once defecting.

在冒险区: 苦难游戏的平衡中，玩家角色在他们在两个领域的时间内两次被呈现囚徒困境，一次是合作，一次是叛变。

在冒险区: 苦难游戏的平衡中，玩家角色在他们在两个领域的时间内两次被呈现<font color="#ff8000"> 囚徒困境</font>，一次是合作，一次是叛变。

In the 8th novel from the author James S. A. Corey Tiamat's Wrath . Winston Duarte explains the prisoners dilemma in his 14-year-old daughter, Teresa, to train her in strategic thinking.  

In the 8th novel from the author James S. A. Corey Tiamat's Wrath . Winston Duarte explains the prisoners dilemma in his 14-year-old daughter, Teresa, to train her in strategic thinking.