囚徒困境

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。

{ | class“ wikitable floatright”

Prisoner's dilemma payoff matrix
模板:Diagonal split header			B stays silent	B stays silent	B 保持安静	B betrays	B betrays	背叛
A stays silent	A stays silent	A 保持安静	模板:Diagonal split header			模板:Diagonal split header
A betrays	A betrays	一个叛徒	模板:Diagonal split header			模板:Diagonal split header

|}

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",^[1] presenting 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:

囚徒困境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：

/* Styling for Template:Quote */ .templatequote { overflow: hidden; margin: 1em 0; padding: 0 40px; } .templatequote .templatequotecite {

   line-height: 1.5em;
   /* @noflip */
   text-align: left;
   /* @noflip */
   padding-left: 1.6em;
   margin-top: 0;

}

! scope="col" style="width:60px;" | 模板:Color

! scope="col" style="width:60px;" |

!范围“ col”样式“ width: 60px; ” |

! scope="col" style="width:60px;" | 模板:Color

! scope="col" style="width:60px;" |

!范围“ col”样式“ width: 60px; ” |

|-

! scope="row" style="width:60px;" | 模板:Color

! scope="row" style="width:60px;" |

!作用域“ row”样式“ width: 60px; ” |

| 模板:Diagonal split header

| ||transparent}}

会透明的

| 模板:Diagonal split header

| ||transparent}}

会透明的

|-

! scope="row" | 模板:Color

! scope="row" |

!瞄准镜

| 模板:Diagonal split header

| ||transparent}}

会透明的

| 模板:Diagonal split header

| ||transparent}}

会透明的

|}

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:

模板:Tmath

The payoff relationship 模板:Tmath implies that mutual cooperation is superior to mutual defection, while the payoff relationships 模板:Tmath and 模板:Tmath 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.

回报关系意味着相互合作优于相互背叛，然而回报关系也意味着相互背叛背叛是双方的主导策略。

Special case: donation game

The "donation game"^[2] 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" 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

“捐赠博弈”是囚徒困境的一种形式，在这种博弈中，合作相当于以个人成本C为另一方提供一个收益B，而叛变意味着什么也不提供。回报矩阵是这样的 C

{ | 类“ wikitable”样式“ text-align: center”

模板:Diagonal split header	}}	}}	模板:Color			模板:Color
模板:Color			模板:Diagonal split header		transparent}} 会透明的	模板:Diagonal split header		transparent}} 会透明的
模板:Color		瞄准镜	模板:Diagonal split header		transparent}} 会透明的	模板:Diagonal split header		transparent}} 会透明的

|}

Note that 模板:Tmath (i.e. 模板:Tmath) 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游戏(见下一节)。博弈

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。

The iterated prisoner's dilemma

模板:More citations needed section

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.

如果两个参与者连续进行多次囚徒困境博弈，他们记住对手先前的行动并相应地改变策略，这种博弈被称为迭代囚徒困境。

In addition to the general form above, the iterative version also requires that 模板:Tmath, to prevent alternating cooperation and defection giving a greater reward than mutual cooperation.

In addition to the general form above, the iterative version also requires that , to prevent alternating cooperation and defection giving a greater reward than mutual cooperation.

除了上面的一般形式之外，迭代版本还要求，防止交替的合作和背叛能比相互合作带来更大的回报。

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".^[3]

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”。

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.

与标准的囚徒困境不同，在迭代的囚徒困境中，叛变策略是严重违反直觉的，以至于不能预测人类玩家的行为。然而，在标准的经济理论中，这是唯一正确的答案。具有固定次数 n 的迭代囚徒困境中的超理性策略是与超理性对手进行合作，在 n 很大的限制下，实验结果的策略与超理性结果的策略一致，而不是博弈论的理性结果。

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。

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.^[4]

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。

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.

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。

Axelrod discovered that when these encounters were repeated over a long period of time with many players, each with different strategies, greedy strategies tended to do very poorly in the long run while more altruistic strategies did better, as judged purely by self-interest. He used this to show a possible mechanism for the evolution of altruistic behaviour from mechanisms that are initially purely selfish, by natural selection.

阿克塞尔罗德发现，当这些遭遇在很长一段时间内在许多玩家身上重复发生时，每个玩家都有不同的策略，从长远来看，贪婪策略往往表现得非常糟糕，而更加利他的策略表现得更好，这完全是根据自身利益来判断的。他利用这一结果，揭示了通过自然选择，从最初纯粹自私行为向利他行为进化的可能机制。

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。

By analysing the top-scoring strategies, Axelrod stated several conditions necessary for a strategy to be successful.

通过分析得分最高的战略，阿克塞尔罗德阐述了战略成功的几个必要条件。

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。

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.

报复: 然而，阿克塞尔罗德认为，成功的战略决不能是盲目的乐观主义。它有时必须进行报复。非报复策略的一个例子就是永远合作。这是一个非常糟糕的选择，因为“肮脏”的策略会无情地剥削这些玩家。

Forgiving: Successful strategies must also be forgiving. Though players will retaliate, they will once again fall back to cooperating if the opponent does not continue to defect. This stops long runs of revenge and counter-revenge, maximizing points.

Forgiving: Successful strategies must also be forgiving. Though players will retaliate, they will once again fall back to cooperating if the opponent does not continue to defect. This stops long runs of revenge and counter-revenge, maximizing points.

宽容: 成功的策略也必须是宽容的。虽然玩家会报复，但如果对手不继续叛变，他们将再次回到合作的状态。这阻止了长时间的报复和反报复，最大限度地提高积分。

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。

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。

In the strategy called Pavlov, win-stay, lose-switch, faced with a failure to cooperate, the player switches strategy the next turn.^[5] In certain circumstances,模板:Specify Pavlov beats all other strategies by giving preferential treatment to co-players using a similar strategy.

In the strategy called Pavlov, win-stay, lose-switch, faced with a failure to cooperate, the player switches strategy the next turn. In certain circumstances, Pavlov beats all other strategies by giving preferential treatment to co-players using a similar strategy.

在所谓的巴甫洛夫策略中，赢-保持，输-变换，面对一次合作失败，玩家将在下一次变换策略。在某些情况下，巴甫洛夫通过给予使用类似策略的合作者优惠待遇打败了所有其他策略。

Deriving the optimal strategy is generally done in two ways:

得出最佳策略通常有两种方法:

Bayesian Nash equilibrium: If the statistical distribution of opposing strategies can be determined (e.g. 50% tit for tat, 50% always cooperate) an optimal counter-strategy can be derived analytically.模板:Efn

Monte Carlo simulations of populations have been made, where individuals with low scores die off, and those with high scores reproduce (a genetic algorithm for finding an optimal strategy). The mix of algorithms in the final population generally depends on the mix in the initial population. The introduction of mutation (random variation during reproduction) lessens the dependency on the initial population; empirical experiments with such systems tend to produce tit for tat players (see for instance Chess 1988),模板:Clarify but no analytic proof exists that this will always occur.^[6]

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.^[7] 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 – 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 – 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。

This strategy takes advantage of the fact that multiple entries were allowed in this particular competition and that the performance of a team was measured by that of the highest-scoring player (meaning that the use of self-sacrificing players was a form of minmaxing). In a competition where one has control of only a single player, tit for tat is certainly a better strategy. Because of this new rule, this competition also has little theoretical significance when analyzing single agent strategies as compared to Axelrod's seminal tournament. However, it provided a basis for analysing how to achieve cooperative strategies in multi-agent frameworks, especially in the presence of noise. In fact, long before this new-rules tournament was played, Dawkins, in his book The Selfish Gene, pointed out the possibility of such strategies winning if multiple entries were allowed, but he remarked that most probably Axelrod would not have allowed them if they had been submitted. It also relies on circumventing rules about the prisoner's dilemma in that there is no communication allowed between the two players, which the Southampton programs arguably did with their opening "ten move dance" to recognize one another; this only reinforces just how valuable communication can be in shifting the balance of the game.

This strategy takes advantage of the fact that multiple entries were allowed in this particular competition and that the performance of a team was measured by that of the highest-scoring player (meaning that the use of self-sacrificing players was a form of minmaxing). In a competition where one has control of only a single player, tit for tat is certainly a better strategy. Because of this new rule, this competition also has little theoretical significance when analyzing single agent strategies as compared to Axelrod's seminal tournament. However, it provided a basis for analysing how to achieve cooperative strategies in multi-agent frameworks, especially in the presence of noise. In fact, long before this new-rules tournament was played, Dawkins, in his book The Selfish Gene, pointed out the possibility of such strategies winning if multiple entries were allowed, but he remarked that most probably Axelrod would not have allowed them if they had been submitted. It also relies on circumventing rules about the prisoner's dilemma in that there is no communication allowed between the two players, which the Southampton programs arguably did with their opening "ten move dance" to recognize one another; this only reinforces just how valuable communication can be in shifting the balance of the game.

这种策略利用了这样一个事实，即在这场特殊的比赛中允许多个参赛项目，并且一支队伍的表现是由得分最高的项目来衡量的(这意味着使用自我牺牲的项目是一种分数最大化的形式)。在一个只能控制一个玩家的比赛中，以牙还牙当然是一个更好的策略。由于这一新规则的存在，与阿克塞尔罗德的具有深远影响的竞赛相比，这种竞赛在分析单个智能体策略时也就没有什么理论意义。然而，它为在分析多智能体框架下，特别是在存在干扰的情况下，如何实现协作策略提供了基础。事实上，早在这场新规则锦标赛开始之前，道金斯就在他的《自私的基因》一书中指出，如果允许多次参赛，这种策略就有可能获胜，但他说，如果提交这种策略的话，阿克塞尔罗德很可能不会允许。因为它依赖于规避囚徒困境的规则，即两个球员之间不允许交流，南安普顿的项目可以说在开场的“十步舞”中就是这样做以认识对方的; 这只是强调了交流对于改变游戏的平衡是多么有影响。

Stochastic iterated prisoner's dilemma

In a stochastic iterated prisoner's dilemma game, strategies are specified by in terms of "cooperation probabilities".^[8] 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]\displaystyle{ P=\{P_{cc},P_{cd},P_{dc},P_{dd}\} }[/math], where [math]\displaystyle{ 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]\displaystyle{ 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.^[8]

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]\displaystyle{ P=\{P_{cc},P_{cd},P_{dc},P_{dd}\} }[/math], where [math]\displaystyle{ 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]\displaystyle{ 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。

If we define P as the above 4-element strategy vector of X and [math]\displaystyle{ 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]\displaystyle{ 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.^[8]

If we define P as the above 4-element strategy vector of X and [math]\displaystyle{ 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]\displaystyle{ 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。

One result of stochastic theory is that there exists a stationary vector v for the matrix M such that [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ M^n }[/math] and particularly [math]\displaystyle{ M^\infty }[/math], so that each row of [math]\displaystyle{ M^\infty }[/math] will be equal to v. Thus the stationary vector specifies the equilibrium outcome probabilities for X. Defining [math]\displaystyle{ S_x=\{R,S,T,P\} }[/math] and [math]\displaystyle{ 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]\displaystyle{ s_x=v\cdot S_x }[/math] and [math]\displaystyle{ 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。

Zero-determinant strategies

文件:IPD Venn.svg

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。

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.^[8] 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]\displaystyle{ s_x=D(P,Q,S_x) }[/math] and [math]\displaystyle{ s_y=D(P,Q,S_y) }[/math], which do not involve the stationary vector v. Since the determinant function [math]\displaystyle{ s_y=D(P,Q,f) }[/math] is linear in f, it follows that [math]\displaystyle{ \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]\displaystyle{ 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]\displaystyle{ \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]\displaystyle{ s_x=D(P,Q,S_x) }[/math] and [math]\displaystyle{ s_y=D(P,Q,S_y) }[/math], which do not involve the stationary vector v. Since the determinant function [math]\displaystyle{ s_y=D(P,Q,f) }[/math] is linear in f, it follows that [math]\displaystyle{ \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]\displaystyle{ 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]\displaystyle{ \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。

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]\displaystyle{ D(P,Q,\beta S_y+\gamma U)=0 }[/math], unilaterally setting [math]\displaystyle{ 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]\displaystyle{ s_x }[/math] to a particular value, the range of possibilities is much smaller, only consisting of complete cooperation or complete defection.^[8])

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]\displaystyle{ D(P,Q,\beta S_y+\gamma U)=0 }[/math], unilaterally setting [math]\displaystyle{ 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]\displaystyle{ 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。)

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).^[9]

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)。

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.^[2]

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。

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.^[10] 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)^[11] 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.^[10]

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。

Continuous iterated prisoner's dilemma

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^[12] 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引用错误：没有找到与</ref>对应的<ref>标签) even though tit for tat seems robust in theoretical models.

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

尽管在理论模型中，以牙还牙似乎是有力的。

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.^[13] 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 – that is, if you can get away with cheating, repeat that behavior, however if you get caught, switch.^[14]

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. 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 – that is, if you can get away with cheating, repeat that behavior, however if you get caught, switch.

玩家似乎不能协调相互合作，因此常常陷入劣势但稳定的叛变策略。通过这种方式，迭代回合可以促进稳定策略的进化。多轮循环往往产生新颖的策略，这对复杂的社会互动有影响。其中一个策略就是“赢-保持-输”的转变。这个策略比一个简单的以牙还牙策略要好——也就是说，如果你能逃脱作弊的惩罚，重复这个行为，但是如果你被抓住了，就改变策略。

The only problem of this tit-for-tat strategy is that they are vulnerable to signal error. The problem arises when one individual cheats in retaliation but the other interprets it as cheating. As a result of this, the second individual now cheats and then it starts a see-saw pattern of cheating in a chain reaction.

这种以牙还牙策略的唯一问题是，它们很容易出现信号错误。当一个人因报复而作弊，而另一个人将其单纯解释为欺骗时，问题就出现了。结果，第二个人现在作弊，然后在接下来的连锁反应中开始了反复交替的作弊模式。

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 sciences 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.

囚犯的设置看起来似乎是人为的，但实际上在人类交往以及自然界的交互中有许多具有相同收益矩阵的例子。囚徒困境是经济学、政治学、社会学等社会科学以及动物行为学、进化生物学等生物学研究的热点问题。许多自然过程都被抽象为生物进行无休止的囚徒困境博弈的模型。这种广泛的适用性让博弈非常重要。

Environmental studies

In environmental studies, the PD is evident in crises such as global climate-change. It is argued all countries will benefit from a stable climate, but any single country is often hesitant to curb [[Carbon dioxide|模板: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.^[15]

In environmental studies, the PD is evident in crises such as global climate-change. It is argued all countries will benefit from a stable climate, but any single country is often hesitant to curb Carbon dioxide| 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.

在环境研究中，在诸如全球气候变化等危机中，这种差异显而易见。有人认为，所有国家都将从稳定的气候中受益，但是任何一个国家往往都在遏制二氧化碳排放方面犹豫不决。人们错误地认为，如果所有国家的行为都改变，任何一个国家保持目前的行为所带来的直接好处都会大于所谓的最终好处，这就解释了2007年气候变化方面的僵局。

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.^[16]

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。

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.^[17]

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。

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, guppies inspect predators cooperatively in groups, and they are thought to punish non-cooperative inspectors.

许多动物的合作行为可以被理解为囚徒困境的一个例子。通常动物会有长期的伙伴关系，这种关系可以更具体地模拟为重复的囚徒困境。例如，孔雀鱼成群结队地合作监察捕食者，它们被认为是在惩罚不合作的监察者。

Vampire bats are social animals that engage in reciprocal food exchange. Applying the payoffs from the prisoner's dilemma can help explain this behavior:^[18]

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:

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."

合作/合作：回报：我在不幸运的晚上得到了能让我果腹的血，那在幸运的晚上我也应该分出点血，那不会花费多少。

D/C: "Temptation: You save my life on my poor night. But then I get the added benefit of not having to pay the slight cost of feeding you on my good night."

叛变/合作：诱惑：你在我的不幸的夜里救了我，但在我的幸运夜我不会给你血以让我活的更好。

C/D: "Sucker's Payoff: I pay the cost of saving your life on my good night. But on my bad night you don't feed me and I run a real risk of starving to death."

合作/叛变：可怜者的回报：在我的幸运夜我救了你的命，但在我的不幸夜里你没有救我，我有饿死的风险。

D/D: "Punishment: I don't have to pay the slight costs of feeding you on my good nights. But I run a real risk of starving on my poor nights."

叛变/叛变：惩罚：我在我的幸运夜里不必付出代价来救你，但我在我的不幸夜里有挨饿的风险。

Psychology

In addiction research / behavioral economics, George Ainslie points out^[19] 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 – 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 – 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，但明天一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。

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(d，d)细胞或者至少不要陷入一个循环中。

Economics

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.^[20]

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.

囚徒困境被称为社会心理学中的大肠杆菌，它被广泛用于研究寡头垄断竞争和集体行动等问题，以产生集体利益。

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.^{[citation needed]}模板:Efn This analysis is likely to be pertinent in many other business situations involving advertising.^{[citation needed]}

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，那么两家公司的广告就会抵消对方的广告，收入保持不变，费用因广告成本而增加。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 公司可以通过广告获得巨大的利益。然而，一家公司的最佳广告数量取决于另一家公司承担了多少广告。由于最佳策略取决于其他公司的选择，因此没有占主导地位的策略，这使得它与囚徒困境略有不同。结果是相似的，尽管，在这两个公司会更好，如果他们的广告少于在均衡。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.^[21] '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。“叛逃”意味着在这个最低水平下销售，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背叛，确保消费者获得尽可能低的价格。

Sport

Doping in sport has been cited as an example of a prisoner's dilemma.^[22]

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

体育运动中的兴奋剂被认为是囚徒困境的一个例子。

Two competing athletes have the option to use an illegal and/or dangerous drug to boost their performance. If neither athlete takes the drug, then neither gains an advantage. If only one does, then that athlete gains a significant advantage over their competitor, reduced by the legal and/or medical dangers of having taken the drug. If both athletes take the drug, however, the benefits cancel out and only the dangers remain, putting them both in a worse position than if neither had used doping.^[22]

Two competing athletes have the option to use an illegal and/or dangerous drug to boost their performance. If neither athlete takes the drug, then neither gains an advantage. If only one does, then that athlete gains a significant advantage over their competitor, reduced by the legal and/or medical dangers of having taken the drug. If both athletes take the drug, however, the benefits cancel out and only the dangers remain, putting them both in a worse position than if neither had used doping.

两名参赛运动员可以选择使用非法和 / 或危险药物来提高成绩。如果两个运动员都没有服用这种药物，那么他们都不会获得优势。如果只有一个人这样做，那么这个运动员就比他们的竞争对手获得了明显的优势，减少了服用药物的法律和 / 或医疗危险。然而，如果两名运动员都服用了这种药物，那么好处就被抵消了，只剩下危险，这使得他们的处境比没有服用兴奋剂的情况更加糟糕。

International politics

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.^[23] 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。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，北约和华沙条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对方，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。

Although the 'best' overall outcome is for both sides to disarm, the rational course for both sides is to arm, and this is indeed what happened. Both sides poured enormous resources into military research and armament in a war of attrition for the next thirty years until the Soviet Union could not withstand the economic cost.^[24] The same logic could be applied in any similar scenario, be it economic or technological competition between sovereign states.

Although the 'best' overall outcome is for both sides to disarm, the rational course for both sides is to arm, and this is indeed what happened. Both sides poured enormous resources into military research and armament in a war of attrition for the next thirty years until the Soviet Union could not withstand the economic cost. The same logic could be applied in any similar scenario, be it economic or technological competition between sovereign states.

虽然最好的结果是双方解除武装，但是双方的理性选择是武装起来，事实也的确如此。在接下来的三十年里，双方都在埃以消耗战争的军事研究和武器装备上投入了大量的资源，直到苏联无法承受经济损失。同样的逻辑也适用于任何类似的情况，无论是主权国家之间的经济竞争还是技术竞争。

Multiplayer dilemmas

Many real-life dilemmas involve multiple players.^[25] 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。虽然比喻性的，Hardin 的公地悲剧可以被看作是 PD 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)叛逃的集体奖励是非常低的回报(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，集体成本是没有干净的盘子为任何人。

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.^[26]

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(叛逃) ，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 Elinor Ostrom 随后的研究假设，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，这是影响方案发展最佳结果的一个例子。

Related games

Closed-bag exchange

文件:Prisoner's Dilemma briefcase exchange (colorized).svg

The prisoner's dilemma as a briefcase exchange

囚徒困境是一个公文包的交换

Douglas Hofstadter^[27] 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 PD 问题，当它以一个简单游戏的形式表现出来时，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” :

/* Styling for Template:Quote */ .templatequote { overflow: hidden; margin: 1em 0; padding: 0 40px; } .templatequote .templatequotecite {

   line-height: 1.5em;
   /* @noflip */
   text-align: left;
   /* @noflip */
   padding-left: 1.6em;
   margin-top: 0;

}

Defection always gives a game-theoretically preferable outcome.^[28]

Defection always gives a game-theoretically preferable outcome.

叛逃总是会带来一个理论上更可取的结果。

Friend or Foe?

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？是一个游戏节目，从2002年至2005年在美国的游戏节目网络播出。这是囚徒困境游戏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有缺陷(敌人) ，那么叛逃者将得到所有的奖金，而合作者将一无所获。如果双方都有缺陷，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“双方都有缺陷”和“合作而对方有缺陷”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“两个缺陷”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，《敌友》在囚徒困境和“胆小鬼”博弈之间有一个奖励模型。

The rewards matrix is

奖励矩阵是

{ | class“ wikitable”

模板:Diagonal split header	}}	}}	模板:Color			模板:Color
模板:Color			模板:Diagonal split header		transparent}} 会透明的	模板:Diagonal split header		transparent}} 会透明的
模板:Color		瞄准镜	模板:Diagonal split header		transparent}} 会透明的	模板:Diagonal split header		transparent}} 会透明的

|}

This payoff matrix has also been used on the British television programmes Trust Me, Shafted, The Bank Job and Golden Balls, and on the American shows Bachelor Pad and Take It All. Game data from the Golden Balls series has been analyzed by a team of economists, who found that cooperation was "surprisingly high" for amounts of money that would seem consequential in the real world, but were comparatively low in the context of the game.^[29]

This payoff matrix has also been used on the British television programmes Trust Me, Shafted, The Bank Job and Golden Balls, and on the American shows Bachelor Pad and Take It All. Game data from the Golden Balls series has been analyzed by a team of economists, who found that cooperation was "surprisingly high" for amounts of money that would seem consequential in the real world, but were comparatively low in the context of the game.

英国电视节目《相信我》、《阴影》、《银行工作》和《黄金球》以及美国电视节目《单身公寓》和《全部拿走》也采用了这种盈利模式。一个经济学家团队分析了金球系列的游戏数据，他们发现，对于在现实世界中看似重要的金钱数量，合作程度“惊人地高” ，但在游戏的背景下，合作程度相对较低。

Iterated snowdrift

Researchers from the University of Lausanne and the University of Edinburgh have suggested that the "Iterated Snowdrift Game" may more closely reflect real-world social situations. Although this model is actually a chicken game, it will be described here. In this model, the risk of being exploited through defection is lower, and individuals always gain from taking the cooperative choice. The snowdrift game imagines two drivers who are stuck on opposite sides of a snowdrift, each of whom is given the option of shoveling snow to clear a path, or remaining in their car. A player's highest payoff comes from leaving the opponent to clear all the snow by themselves, but the opponent is still nominally rewarded for their work.

来自洛桑大学和爱丁堡大学的研究人员认为，“迭代雪堆游戏”可能更能反映现实世界的社会状况。虽然这个模型实际上是一个小鸡博弈，它将在这里描述。在这个模型中，由于背叛而被剥削的风险较低，个体总是从合作选择中获益。这个雪堆游戏设想两个司机被困在雪堆的两侧，每个司机都可以选择铲雪清理道路，或者留在自己的车里。一个玩家的最高回报来自于让对手自己清除所有的积雪，但是对手名义上仍然因为他们的工作而得到回报。

This may better reflect real world scenarios, the researchers giving the example of two scientists collaborating on a report, both of whom would benefit if the other worked harder. "But when your collaborator doesn’t do any work, it’s probably better for you to do all the work yourself. You’ll still end up with a completed project."^[30]

This may better reflect real world scenarios, the researchers giving the example of two scientists collaborating on a report, both of whom would benefit if the other worked harder. "But when your collaborator doesn’t do any work, it’s probably better for you to do all the work yourself. You’ll still end up with a completed project."

这可能更好地反映了现实世界的情景---- 研究人员举了两位科学家合作完成一份报告的例子，如果另一位科学家更加努力地工作，这两位科学家都会受益。“但当你的合作者不做任何工作时，你自己做所有的工作可能会更好。你最终还是会完成一个项目。”

{ | 类“ wikitable”样式“ text-align: center; ”

Example snowdrift payouts (A, B)
+ 例子雪堆补偿(a，b)	模板:Diagonal split header	Cooperates	Defects		Cooperates	Defects		合作！缺陷
Cooperates	Cooperates	合作	200, 200	100, 300	200, 200	100, 300	200, 200	100, 300
Defects	Defects	缺陷	300, 100	0, 0	300, 100	0, 0	300, 100	0, 0

{ | class“ wikitable”样式“ text-align: center; margin-left: 2em; ”

Example PD payouts (A, B)
+ PD 支出示例(a，b)	模板:Diagonal split header	Cooperates	Defects		Cooperates	Defects		合作！缺陷
Cooperates	Cooperates	合作	200, 200	-100, 300	200, 200	-100, 300	200, 200	-100, 300
Defects	Defects	缺陷	300, -100	0, 0	300, -100	0, 0	300, -100	0, 0

|}

Coordination games

In coordination games, players must coordinate their strategies for a good outcome. An example is two cars that abruptly meet in a blizzard; each must choose whether to swerve left or right. If both swerve left, or both right, the cars do not collide. The local left- and right-hand traffic convention helps to co-ordinate their actions.

在协调博弈中，参与者必须协调他们的策略以获得一个好的结果。一个例子是两辆车在暴风雪中突然相遇，每辆车必须选择是左转还是右转。如果两辆车都向左转弯，或者都向右转弯，那么两辆车就不会相撞。当地的左右手交通大会有助于协调他们的行动。

Symmetrical co-ordination games include Stag hunt and Bach or Stravinsky.

对称的协调游戏包括猎鹿和巴赫或斯特拉文斯基。

Asymmetric prisoner's dilemmas

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".^[31]

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".

一个更一般的游戏集是不对称的。就像在囚徒困境中一样，最好的结果是合作，而且背叛是有动机的。不像对称的囚徒困境，一方比另一方有更多的损失和 / 或获得。有些这样的游戏被描述为囚徒困境，其中一个囚徒有不在场证明，这就是术语“不在场证明游戏”的由来。

In experiments, players getting unequal payoffs in repeated games may seek to maximize profits, but only under the condition that both players receive equal payoffs; this may lead to a stable equilibrium strategy in which the disadvantaged player defects every X games, while the other always co-operates. Such behaviour may depend on the experiment's social norms around fairness.^[32]

In experiments, players getting unequal payoffs in repeated games may seek to maximize profits, but only under the condition that both players receive equal payoffs; this may lead to a stable equilibrium strategy in which the disadvantaged player defects every X games, while the other always co-operates. Such behaviour may depend on the experiment's social norms around fairness.

在实验中，重复博弈中获得不平等收益的参与者可能会寻求利润最大化，但是只有在两个参与者获得相同收益的条件下，这可能导致一个稳定的均衡策略，即弱势参与者在每个 x 博弈中都会缺陷，而另一个参与者总是合作。这种行为可能取决于实验围绕公平的社会规范。

Software

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

一些软件包已经被创建来运行囚徒困境模拟和比赛，其中一些有可用的源代码。

The source code for the second tournament run by Robert Axelrod (written by Axelrod and many contributors in Fortran) is available online

Prison, a library written in Java, last updated in 1998

Axelrod-Python, written in Python

play the Iterative Prisoner's Dilemma in the browser, play against strategies or let strategies play against other strategies

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 – 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 – 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.

汉努 · 拉贾尼埃米将他的《量子窃贼》三部曲的开场场景设置在一个“进退两难的监狱”中。该系列的主题被描述为“双重宇宙的不足” ，最终的对手是一个叫做全面叛逃者的角色。拉贾尼埃米作为一个处理这个问题的艺术家尤其有趣，因为他是剑桥大学培养的数学家，拥有数学物理学博士学位——物质和信息的可互换性是这本书的一个主要特征，它发生在“后奇点”的未来。该系列的第一本书于2010年出版，其续集《分形王子》和《因果天使》分别于2012年和2014年出版。

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.

一个模仿囚徒困境的游戏是2012年电子游戏《零度逃脱: 美德的最后奖励》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.

在特伦顿 · 李 · 斯图尔特(Trenton Lee Stewart)的《神秘的本尼迪克特社会和囚徒困境》(The Mysterious Benedict Society and The Prisoner’s Dilemma)中，主要角色从玩一个版本的游戏开始，然后一起逃离“监狱”。后来他们变成了真正的囚犯，再次越狱。

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: 苦难游戏中的平衡中，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，一次是叛逃。

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. 模板:Cn

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.

在作者詹姆斯 · s · a · 科里 · 提亚玛特的《愤怒》的第八部小说中。温斯顿•杜阿尔特(Winston Duarte)解释了他14岁的女儿特蕾莎(Teresa)面临的囚徒困境，以训练她的战略思维。

References

模板:Notelist

↑ Poundstone, 1992
↑ ^2.0 ^2.1 Hilbe, Christian; Martin A. Nowak; Karl Sigmund (April 2013). "Evolution of extortion in Iterated Prisoner's Dilemma games". PNAS. 110 (17): 6913–18. arXiv:1212.1067. Bibcode:2013PNAS..110.6913H. doi:10.1073/pnas.1214834110. PMC 3637695. PMID 23572576.
↑ Shy, Oz (1995). Industrial Organization: Theory and Applications. Massachusetts Institute of Technology Press. ISBN 978-0262193665. https://books.google.com/?id=tr4CjJ5LlRcC&pg=PR13&dq=industrial+organization+theory+and+applications. Retrieved February 27, 2013.
↑ Dal Bó, Pedro; Fréchette, Guillaume R. (2019). "Strategy Choice in the Infinitely Repeated Prisoner's Dilemma". American Economic Review (in English). 109 (11): 3929–3952. doi:10.1257/aer.20181480. ISSN 0002-8282.
↑ http://www.pnas.org/content/pnas/93/7/2686.full.pdf
↑ Wu, Jiadong; Zhao, Chengye (2019), Sun, Xiaoming; He, Kun; Chen, Xiaoyun (eds.), "Cooperation on the Monte Carlo Rule: Prisoner's Dilemma Game on the Grid", Theoretical Computer Science (in English), Springer Singapore, vol. 1069, pp. 3–15, doi:10.1007/978-981-15-0105-0_1, ISBN 978-981-15-0104-3
↑ 模板:Cite press release
↑ ^8.0 ^8.1 ^8.2 ^8.3 ^8.4 Press, WH; Dyson, FJ (26 June 2012). "Iterated Prisoner's Dilemma contains strategies that dominate any evolutionary opponent". Proceedings of the National Academy of Sciences of the United States of America. 109 (26): 10409–13. Bibcode:2012PNAS..10910409P. doi:10.1073/pnas.1206569109. PMC 3387070. PMID 22615375.
↑ Adami, Christoph; Arend Hintze (2013). "Evolutionary instability of Zero Determinant strategies demonstrates that winning isn't everything". Nature Communications. 4: 3. arXiv:1208.2666. Bibcode:2013NatCo...4.2193A. doi:10.1038/ncomms3193. PMC 3741637. PMID 23903782.
↑ ^10.0 ^10.1 Stewart, Alexander J.; Joshua B. Plotkin (2013). "From extortion to generosity, evolution in the Iterated Prisoner's Dilemma". Proceedings of the National Academy of Sciences of the United States of America. 110 (38): 15348–53. Bibcode:2013PNAS..11015348S. doi:10.1073/pnas.1306246110. PMC 3780848. PMID 24003115.
↑ Akin, Ethan (2013). "Stable Cooperative Solutions for the Iterated Prisoner's Dilemma". p. 9. arXiv:1211.0969 [math.DS]. 模板:Bibcode
↑ Le, S.; Boyd, R. (2007). "Evolutionary Dynamics of the Continuous Iterated Prisoner's Dilemma". Journal of Theoretical Biology. 245 (2): 258–67. doi:10.1016/j.jtbi.2006.09.016. PMID 17125798. {{cite journal}}: Unknown parameter |name-list-format= ignored (help)
↑ Spaniel, William (2011). Game Theory 101: The Complete Textbook.
↑ Nowak, Martin; Karl Sigmund (1993). "A strategy of win-stay, lose-shift that outperforms tit-for-tat in the Prisoner's Dilemma game". Nature. 364 (6432): 56–58. Bibcode:1993Natur.364...56N. doi:10.1038/364056a0. PMID 8316296.
↑ "Markets & Data". The Economist. 2007-09-27.
↑ Rehmeyer, Julie (2012-10-29). "Game theory suggests current climate negotiations won't avert catastrophe". Science News. Society for Science & the Public.
↑ Osang, Thomas; Nandyyz, Arundhati (August 2003). Environmental Regulation of Polluting Firms: Porter's Hypothesis Revisited (PDF) (paper).
↑ Dawkins, Richard (1976). The Selfish Gene. Oxford University Press.
↑ Ainslie, George (2001). Breakdown of Will. ISBN 978-0-521-59694-7.
↑ Axelrod, Robert (1980). "Effective Choice in the Prisoner's Dilemma". The Journal of Conflict Resolution. 24 (1): 3–25. doi:10.1177/002200278002400101. ISSN 0022-0027. JSTOR 173932.
↑ Nicholson, Walter (2000). Intermediate microeconomics and its application (8th ed.). Fort Worth, TX: Dryden Press : Harcourt College Publishers. ISBN 978-0-030-25916-6.
↑ ^22.0 ^22.1 Schneier, Bruce (2012-10-26). "Lance Armstrong and the Prisoners' Dilemma of Doping in Professional Sports | Wired Opinion". Wired. Wired.com. Retrieved 2012-10-29.
↑ Stephen J. Majeski (1984). "Arms races as iterated prisoner's dilemma games". Mathematical and Social Sciences. 7 (3): 253–66. doi:10.1016/0165-4896(84)90022-2.
↑ Kuhn, Steven (2019), "Prisoner's Dilemma", in Zalta, Edward N. (ed.), The Stanford Encyclopedia of Philosophy (Winter 2019 ed.), Metaphysics Research Lab, Stanford University, retrieved 2020-04-12
↑ Gokhale CS, Traulsen A. Evolutionary games in the multiverse. Proceedings of the National Academy of Sciences. 2010 Mar 23. 107(12):5500–04.
↑ "The Volokh Conspiracy " Elinor Ostrom and the Tragedy of the Commons". Volokh.com. 2009-10-12. Retrieved 2011-12-17.
↑ Hofstadter, Douglas R. (1985). "Ch.29 The Prisoner's Dilemma Computer Tournaments and the Evolution of Cooperation.". Metamagical Themas: questing for the essence of mind and pattern. Bantam Dell Pub Group. ISBN 978-0-465-04566-2.
↑ "Prisoner's dilemma - Wikipedia, the free encyclopedia". users.auth.gr. Retrieved 2020-04-12.
↑ Van den Assem, Martijn J. (January 2012). "Split or Steal? Cooperative Behavior When the Stakes Are Large" (PDF). Management Science. 58 (1): 2–20. doi:10.1287/mnsc.1110.1413. SSRN 1592456.
↑ Kümmerli, Rolf. "'Snowdrift' game tops 'Prisoner's Dilemma' in explaining cooperation". Retrieved 11 April 2012.
↑ Robinson, D.R.; Goforth, D.J. (May 5, 2004). Alibi games: the Asymmetric Prisoner' s Dilemmas (PDF). Meetings of the Canadian Economics Association, Toronto, June 4-6, 2004.
↑ 模板:Cite chapter

External links

模板:Commonscat-inline

Prisoner's Dilemma (Stanford Encyclopedia of Philosophy)

The Bowerbird's Dilemma The Prisoner's Dilemma in ornithology – mathematical cartoon by Larry Gonick.

The Prisoner's Dilemma The Prisoner's Dilemma with Lego minifigures.

Dixit, Avinash; Nalebuff, Barry (2008). "Prisoner's Dilemma". In David R. Henderson (ed.). Concise Encyclopedia of Economics (2nd ed.). Indianapolis: Library of Economics and Liberty. ISBN 978-0865976658. OCLC 237794267.

Game Theory 101: Prisoner's Dilemma

Dawkins: Nice Guys Finish First

Axelrod Iterated Prisoner's Dilemma Python library

Play the Iterated Prisoner's Dilemma on gametheorygames.nl

Play Prisoner's Dilemma on oTree (N/A 11-5-17)

Nicky Case's Evolution of Trust, an example of the donation game

Iterated Prisoner's Dilemma online game by Wayne Davis

模板:Decision theory paradoxes

模板:Game theory

Category:Non-cooperative games

类别: 非合作性游戏

Category:Thought experiments

类别: 思维实验

Category:Dilemmas

类别: 困境

Category:Environmental studies

类别: 环境研究

Category:Social psychology

类别: 社会心理学

Category:Moral psychology

范畴: 道德心理学

This page was moved from wikipedia:en:Prisoner's dilemma. Its edit history can be viewed at 囚徒困境/edithistory

[1] Poundstone, 1992

[Hilbe2013-2] 2.0 ^2.1 Hilbe, Christian; Martin A. Nowak; Karl Sigmund (April 2013). "Evolution of extortion in Iterated Prisoner's Dilemma games". PNAS. 110 (17): 6913–18. arXiv:1212.1067. Bibcode:2013PNAS..110.6913H. doi:10.1073/pnas.1214834110. PMC 3637695. PMID 23572576.

[Shy-3] Shy, Oz (1995). Industrial Organization: Theory and Applications. Massachusetts Institute of Technology Press. ISBN 978-0262193665. https://books.google.com/?id=tr4CjJ5LlRcC&pg=PR13&dq=industrial+organization+theory+and+applications. Retrieved February 27, 2013.

[4] Dal Bó, Pedro; Fréchette, Guillaume R. (2019). "Strategy Choice in the Infinitely Repeated Prisoner's Dilemma". American Economic Review (in English). 109 (11): 3929–3952. doi:10.1257/aer.20181480. ISSN 0002-8282.

[5] ttp://www.pnas.org/content/pnas/93/7/2686.full.pdf

[6] Wu, Jiadong; Zhao, Chengye (2019), Sun, Xiaoming; He, Kun; Chen, Xiaoyun (eds.), "Cooperation on the Monte Carlo Rule: Prisoner's Dilemma Game on the Grid", Theoretical Computer Science (in English), Springer Singapore, vol. 1069, pp. 3–15, doi:10.1007/978-981-15-0105-0_1, ISBN 978-981-15-0104-3

[7] 模板:Cite press release

[Press2012-8] 8.0 ^8.1 ^8.2 ^8.3 ^8.4 Press, WH; Dyson, FJ (26 June 2012). "Iterated Prisoner's Dilemma contains strategies that dominate any evolutionary opponent". Proceedings of the National Academy of Sciences of the United States of America. 109 (26): 10409–13. Bibcode:2012PNAS..10910409P. doi:10.1073/pnas.1206569109. PMC 3387070. PMID 22615375.

[9] Adami, Christoph; Arend Hintze (2013). "Evolutionary instability of Zero Determinant strategies demonstrates that winning isn't everything". Nature Communications. 4: 3. arXiv:1208.2666. Bibcode:2013NatCo...4.2193A. doi:10.1038/ncomms3193. PMC 3741637. PMID 23903782.

[Stewart2013-10] 10.0 ^10.1 Stewart, Alexander J.; Joshua B. Plotkin (2013). "From extortion to generosity, evolution in the Iterated Prisoner's Dilemma". Proceedings of the National Academy of Sciences of the United States of America. 110 (38): 15348–53. Bibcode:2013PNAS..11015348S. doi:10.1073/pnas.1306246110. PMC 3780848. PMID 24003115.

[Akin2013-11] Akin, Ethan (2013). "Stable Cooperative Solutions for the Iterated Prisoner's Dilemma". p. 9. arXiv:1211.0969 [math.DS]. 模板:Bibcode

[12] Le, S.; Boyd, R. (2007). "Evolutionary Dynamics of the Continuous Iterated Prisoner's Dilemma". Journal of Theoretical Biology. 245 (2): 258–67. doi:10.1016/j.jtbi.2006.09.016. PMID 17125798. {{cite journal}}: Unknown parameter |name-list-format= ignored (help)

[13] Spaniel, William (2011). Game Theory 101: The Complete Textbook.

[14] Nowak, Martin; Karl Sigmund (1993). "A strategy of win-stay, lose-shift that outperforms tit-for-tat in the Prisoner's Dilemma game". Nature. 364 (6432): 56–58. Bibcode:1993Natur.364...56N. doi:10.1038/364056a0. PMID 8316296.

[15] "Markets & Data". The Economist. 2007-09-27.

[16] Rehmeyer, Julie (2012-10-29). "Game theory suggests current climate negotiations won't avert catastrophe". Science News. Society for Science & the Public.

[17] Osang, Thomas; Nandyyz, Arundhati (August 2003). Environmental Regulation of Polluting Firms: Porter's Hypothesis Revisited (PDF) (paper).

[18] Dawkins, Richard (1976). The Selfish Gene. Oxford University Press.

[19] Ainslie, George (2001). Breakdown of Will. ISBN 978-0-521-59694-7.

[20] Axelrod, Robert (1980). "Effective Choice in the Prisoner's Dilemma". The Journal of Conflict Resolution. 24 (1): 3–25. doi:10.1177/002200278002400101. ISSN 0022-0027. JSTOR 173932.

[21] Nicholson, Walter (2000). Intermediate microeconomics and its application (8th ed.). Fort Worth, TX: Dryden Press : Harcourt College Publishers. ISBN 978-0-030-25916-6.

[wired-22] 22.0 ^22.1 Schneier, Bruce (2012-10-26). "Lance Armstrong and the Prisoners' Dilemma of Doping in Professional Sports | Wired Opinion". Wired. Wired.com. Retrieved 2012-10-29.

[23] Stephen J. Majeski (1984). "Arms races as iterated prisoner's dilemma games". Mathematical and Social Sciences. 7 (3): 253–66. doi:10.1016/0165-4896(84)90022-2.

[24] Kuhn, Steven (2019), "Prisoner's Dilemma", in Zalta, Edward N. (ed.), The Stanford Encyclopedia of Philosophy (Winter 2019 ed.), Metaphysics Research Lab, Stanford University, retrieved 2020-04-12

[25] Gokhale CS, Traulsen A. Evolutionary games in the multiverse. Proceedings of the National Academy of Sciences. 2010 Mar 23. 107(12):5500–04.

[26] "The Volokh Conspiracy " Elinor Ostrom and the Tragedy of the Commons". Volokh.com. 2009-10-12. Retrieved 2011-12-17.

[dh-27] Hofstadter, Douglas R. (1985). "Ch.29 The Prisoner's Dilemma Computer Tournaments and the Evolution of Cooperation.". Metamagical Themas: questing for the essence of mind and pattern. Bantam Dell Pub Group. ISBN 978-0-465-04566-2.

[28] "Prisoner's dilemma - Wikipedia, the free encyclopedia". users.auth.gr. Retrieved 2020-04-12.

[29] Van den Assem, Martijn J. (January 2012). "Split or Steal? Cooperative Behavior When the Stakes Are Large" (PDF). Management Science. 58 (1): 2–20. doi:10.1287/mnsc.1110.1413. SSRN 1592456.

[30] Kümmerli, Rolf. "'Snowdrift' game tops 'Prisoner's Dilemma' in explaining cooperation". Retrieved 11 April 2012.

[31] Robinson, D.R.; Goforth, D.J. (May 5, 2004). Alibi games: the Asymmetric Prisoner' s Dilemmas (PDF). Meetings of the Canadian Economics Association, Toronto, June 4-6, 2004.

[32] 模板:Cite chapter

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

[25]

[26]

[27]

[28]

[29]

[30]

[31]

[32]

囚徒困境

目录

Special case: donation game

The iterated prisoner's dilemma

Strategy for the iterated prisoner's dilemma

Stochastic iterated prisoner's dilemma

Zero-determinant strategies

Continuous iterated prisoner's dilemma

Emergence of stable strategies

Real-life examples

Environmental studies

Animals

Psychology

Economics

Sport

International politics

Multiplayer dilemmas

Related games

Closed-bag exchange

Friend or Foe?

Iterated snowdrift

Coordination games

Asymmetric prisoner's dilemmas

Software

In fiction

See also

References

Further reading

External links

导航菜单

搜索