囚犯困境The Prisoner's Dilemma

注意：这篇文章上次更新于2044天前，文章内容可能已经过时。

This article was last updated2044 days ago, the content may be outdated.

坦白从宽，牢底坐穿。

抗拒从严，回家过年。

Confess for leniency, and wear out the prison floor.

Resist and face severity, and be home for the New Year.

什么是囚犯困境

这个关于囚犯的杜撰例子，经过几十年的传播和修改后，现在基本上是以下面的例子呈献的：

两个小偷（也称为“囚犯”）一起去盗窃，结果很不幸，被警察逮了个正着。然后，警察在两个独立的不能互通信息的审讯室里分别对这两个小偷进行审讯。在这种情形下，两个小偷都可以作出自己的选择：选择坦白，供述以往的犯罪事实（即与警察合作，从而背叛他的同伙），或者选择抗拒，保持沉默（也就是与他的同伙合作，而不是与警察合作）。这两个囚犯都知道，如果他俩都能保持抗拒的话，警察只能以本次的犯罪事实定罪，判少量的徒刑，两人都坐 2 年牢。警察为了激励他们招供，就对他们说：如果他们中的一个人坦白，即告发他的同伙，那么他就可以被当场释放。而他的同伙则被加重处罚，坐 10 年牢（坦白从宽、抗拒从严）。当然，如果这两个人都选择招供、互相背叛的话，就会按照以往的犯罪事实来定罪，即两人都坐 5 年牢，既不会被奖励，也不会被加重处罚。双方的损益矩阵表如下：

很明显，对于囚犯们整体而言，双方都选择沉默是全局最优策略。但是，对于每个个体来说，如果对方选择沉默，自己应该选择坦白认罪即可获得释放；如果对方选择坦白认罪，此时自己如果选择抗拒就会服刑10年，坦白认罪服刑5年，因此，选择坦白认罪是此时的最优策略。也就是无论对方怎么做，自己选择坦白认罪永远是最有利于自身的选择。由于驴也是这么想的，因此博弈双方就会自然的选择坦白认罪，最终获得了一个最坏的结局。

攻守同盟是否有用

试想一下，如果在审讯过程中给小偷们 5 分钟的时间商量一下对策，即小偷双方达成战时攻守同盟，约定好一起选择沉默不认罪。这样的攻守同盟是否有用？

然后，双方再单独被提审。我们不妨设想囚犯A的心理，他一定会认为，如果囚犯B遵守约定的话，则自己坦白就可获得自由；如果囚犯B不遵守约定的话，若选择抗拒就会被判坐更长时间的牢，因此也应该选择坦白。也就是说，无论对方是否遵守约定，自己不遵守约定都是最好的选择。特别是囚犯A认为囚犯B也会有跟他同样的想法，也会选择坦白时，更不会有单独背叛对方的道德负担。

事实上，双方并不能依靠简单的沟通或约定而摆脱困境。恋人们在恋爱中海誓山盟，最终还是会分手；企业之间相互沟通信誓旦旦，价格战仍然会爆发；美苏（俄）两国经常会晤，甚至签订核不扩散条约，但军费一年高过一年……这些现象都反映了上面所说明的问题。

原因分析

对于出现囚犯困境的原因，人们一般理解为是个体对于自身利益的追求。在上表中，每个囚犯选择坦白的目的是为了自己能够少坐牢。如果每个囚犯不是为了自己的利益考虑，而是能够为别人的利益考虑，哪怕自己多坐牢也要让对方先出去，那么，无论对方选择什么，自己都会选择抗拒，最后的结果就是（抗拒、抗拒），两个人都会得到最好的结果。或许，这就是所谓的困境：每个人都想少坐牢，选择了坦白，结果多坐牢；每个人都想多坐牢，选择了抗拒，反而可以少坐牢。真乃“事与愿违”！

如果人们把囚犯困境的原因归结为囚犯对自身利益的追求，那么，要想避免出现囚犯困境，就应该把囚犯们改造成“毫不利己、专门利人”的人，也就是说，要改变的是“人性”。从延安时期的白求恩、张思德，到解放战争时期的董存瑞和抗美援朝的黄继光、邱少云，再到新中国成立后不久的焦裕禄、雷锋，以及改革开放之后的孔繁森、牛玉儒、任长霞等，众多让几代中国人耳熟能详的模范人物最大的人格魅力，就是他们都具有无私奉献的精神。然而，要把13亿中国人中的大多数都改造成和这些模范人物一样，是不可能的。表面上看，囚犯对自身利益的追求是导致囚犯困境的原因，其实真正的原因是：囚犯们在追求自身利益的同时，以更多地损害对方的利益为代价。

看下表中的例子，如果囚犯们面对的是真正的“坦白从宽、抗拒从严”，那么每个人选择占优策略所形成的占优策略均衡就是一个对双方来说最好的结局。

	坦白	抗拒
坦白	（-0.5 ，-0.5）	（0， -5）
抗拒	（-5 ，0）	（-3， -3）

由上表可知，无论对方选择坦白还是抗拒，选择坦白都是对我们最有利的结果。于是，双方依然是都选择坦白，但此时双方都选择坦白就是整个博弈的最优策略。对比与原囚犯困境模型，可以发现不同之处在于，囚犯们的利己选择是否以更大程度上的损人为代价。

在原模型中，如果 A 选择沉默 ，B选择认罪所获得的利益要小于因此使对方遭受的损失，在这种情况下，由于双方的利己选择都是以更大程度上损失对方的利益为代价，最终会使得双方获得整体上的最差结果。

而在新的博弈模型中，假设 A 选择抗拒，此时B选择坦白会比选择抗拒少坐3年牢，而 A 会因为 B 的坦白多坐牢2年。也正是因为此时双方的利己选择所得到的利益大于对方遭受的损失，于是双方即使都做出利己选择，也会达到全局最优解。

以上分析给了我们一个非常重要的启示：损人利己和损己利人从本质上是一样的，是一个硬币的正反两面；从一方看来是损人利己的行为，在另一方看来就是损己利人的行为，问题的关键是所损（失）的部分和所利（得）的部分哪个更多。

如何摆脱囚犯困境

德国达豪集中营入口处，刻着17世纪一位诗人的警世名言：“当一个政权开始烧书的时候，若不加以阻止，它的下一步就要烧人；当一个政权开始禁言的时候，若不加以阻止，它的下一步就要灭口！”出口处还有一条：“当世人忘掉这些事的时候，就意味着，这些事还会发生。”

1. 道德教化

   初次接触囚犯困境的读者很容易去谴责小偷背信弃义、只顾自己利益的行为，并对最后的结局有一种幸灾乐祸的感觉。对囚犯困境的进一步思考，又让人们意识到囚犯困境不仅仅是囚犯的困境，而是整个人类面临的困境。如果人类不能把他人的利益纳入自己的损益函数中来，那么这个社会的前景是令人担忧的。金迪斯（H.Gintis）是桑塔菲学派的一位经济学家，在最近的一系列论文里论证了一个被叫作“社会学基本定理”的看法。他论证，如果一个社会都是由自利主义者构成的，那么，长期而言，这个社会将会消亡。而如果一个社会通过“说服教育”以及其他说教机构的努力，长期保持一定比例的利他主义者，它就能够稳定地繁衍下去。

2. 财产私有

   回顾中国人民公社的那段历史，社员们“出工不出力”，“一不做、二不休”又何尝不是一种囚犯困境。正如苏联的工人所说：“他们假装给我卢布，我假装努力工作！”中国比苏联走得更远，最后宣告了人民公社制度的彻底失败。

3. 温故知新

   如果人类不能从过去的历史中获得遥望未来的能力，那么每一代人都只能重复过去的故事。如何从过去总结教训、习得经验是人不同于其他动物的重要区别。今天的狮子与2000年前的狮子相比，在智力水平上并没有明显的变化，而今天的人与2000年前的人相比却有了智力上的巨大变化，因为我们通过符号记载了过去的历史，传承了过去的经验和教训，让我们不仅仅像狮子那样能从父辈那里习得生活的能力，还可以从2000多年前的孔子、孟子和孙子那里，学会做人做事的道理。

如何利用他人的囚犯困境

囚犯困境对囚犯们而言当然是一种困境，而对警察而言，就是乐境了。那么，对你而言，如何能够利用别人的囚犯困境让自己获得更大的利益呢？

最常见的一种是你作为消费者货比三家，让商家给你报价，选最便宜的买。这样，商家就陷入了囚犯困境，给了你一个相对低的价格。特别是许多耐用消费品，消费者往往通过持币待购的方式逼迫厂家降价。航空公司和旅游公司在消费淡季的时候，也会陷入囚犯困境中，被消费者任意宰割。当然，在旺季的时候，两者的情况又反了过来，消费者之间的竞价让航空公司和旅游公司得到了好处。房地产行业也是如此，房子热卖的时候，开发商采用捂盘的方式导致购房人纷纷抢购；而房子销售形势不好时，开发商又纷纷采取打折等各种优惠措施来吸引人们购买。

What Is the Prisoner’s Dilemma

This fabricated example about prisoners, after decades of spreading and modification, is now basically presented as follows:

Two thieves (also called “prisoners”) go burglary together and, unfortunately, get caught red-handed by the police. Then the police interrogate the two thieves separately in two independent interrogation rooms where no information can be exchanged. In this situation, each thief can make his own choice: confess, disclosing past crimes (that is, cooperating with the police and thereby betraying his accomplice), or resist and keep silent (that is, cooperating with his accomplice rather than with the police). Both prisoners know that if they both keep silent, the police can only convict them of the current crime, and both will serve 2 years. To encourage them to confess, the police tell them: if one of them confesses, i.e., informs on his accomplice, he will be released on the spot, while his accomplice gets a heavier sentence of 10 years (confess and get leniency, resist and get severity). Of course, if both choose to confess and betray each other, they will be convicted based on past crimes, i.e., both serve 5 years — neither rewarded nor punished more severely. The payoff matrix for both parties is as follows:

Obviously, for the prisoners as a whole, both staying silent is the globally optimal strategy. But for each individual, if the other stays silent, you should confess and plead guilty to be released; if the other confesses and pleads guilty, then resisting would mean 10 years in prison while confessing and pleading guilty means 5 years. Therefore, confessing and pleading guilty is the optimal strategy at that point. That is, no matter what the other does, confessing and pleading guilty is always the most favorable choice for yourself. Since the donkey thinks the same way too, both sides of the game naturally choose to confess and plead guilty, ultimately reaching a worst-case outcome.

Is a Pact of Mutual Defense Useful

Imagine that during the interrogation the thieves were given 5 minutes to discuss a strategy, i.e., the two thieves form a wartime pact of mutual defense, agreeing to both stay silent and plead not guilty. Would such a pact be useful?

Then the two are interrogated separately again. Let’s imagine Prisoner A’s psychology: he will surely think that if Prisoner B keeps the agreement, then confessing would set him free; if Prisoner B breaks the agreement, then resisting would mean an even longer prison sentence, so he should also confess. In other words, no matter whether the other party keeps the agreement, breaking it yourself is the best choice. Especially since Prisoner A believes Prisoner B would have the same thought and also choose to confess, he won’t even have the moral burden of betraying the other party alone.

In fact, the two parties cannot escape the dilemma through simple communication or agreements. Lovers who swear eternal vows in romance still end up breaking up; companies that communicate and pledge solemnly to each other still end up in price wars; the US and the USSR (Russia) held frequent summits and even signed nuclear non-proliferation treaties, yet military spending rose year after year… All these phenomena reflect the problem described above.

Cause Analysis

People generally understand the cause of the prisoner’s dilemma as individuals’ pursuit of their own interests. In the table above, each prisoner confesses in order to serve less time. If each prisoner considered not his own interests but the other’s — even willing to serve more time himself so that the other goes free first — then no matter what the other chooses, he would choose to resist, and the final outcome would be (resist, resist), the best outcome for both. Perhaps this is the so-called dilemma: everyone wants to serve less time, so they confess, and end up serving more; everyone is willing to serve more time, so they resist, and end up serving less. Truly “things turn out contrary to one’s wishes”!

If people attribute the cause of the prisoner’s dilemma to the prisoners’ pursuit of self-interest, then to avoid the dilemma, the prisoners should be remade into people who are “utterly devoted to others without any thought of self” — in other words, what needs to change is “human nature”. From Bethune and Zhang Side in the Yan’an period, to Dong Cunrui in the War of Liberation and Huang Jiguang and Qiu Shaoyun in the War to Resist US Aggression and Aid Korea, to Jiao Yulu and Lei Feng shortly after the founding of New China, and to Kong Fansen, Niu Yuru, Ren Changxia and others after the reform and opening up, the greatest personal charm of the many model figures familiar to generations of Chinese people is their spirit of selfless dedication. However, it is impossible to remake the majority of 1.3 billion Chinese people into such model figures. On the surface, the prisoners’ pursuit of self-interest causes the dilemma, but the real cause is: the prisoners, while pursuing their own interests, do so at the cost of damaging the other party’s interests to a greater extent.

Look at the example in the table below. If the prisoners face a genuine “confess and get leniency, resist and get severity”, then the dominant-strategy equilibrium formed by everyone choosing the dominant strategy is a best outcome for both parties.

	坦白	抗拒
坦白	（-0.5 ，-0.5）	（0， -5）
抗拒	（-5 ，0）	（-3， -3）

From the table above, whether the other party chooses to confess or resist, choosing to confess is the most favorable outcome for us. So both sides still choose to confess, but this time both confessing is the optimal strategy of the whole game. Compared with the original prisoner’s dilemma model, the difference is whether the prisoners’ self-interested choices come at the cost of harming the other party to a greater extent.

In the original model, if A stays silent, the benefit B gains from confessing is less than the loss inflicted on the other party. In this case, since both parties’ self-interested choices come at the cost of damaging the other party’s interests to a greater extent, both end up with the overall worst outcome.

In the new game model, suppose A chooses to resist; then B confessing would serve 3 fewer years than resisting, while A would serve 2 more years because of B’s confession. Precisely because each party’s self-interested choice now yields more benefit than the loss inflicted on the other, even if both make self-interested choices, they reach the global optimum.

The analysis above gives us a very important insight: harming others to benefit oneself and harming oneself to benefit others are essentially the same — two sides of the same coin. What looks like harming others to benefit oneself from one side looks like harming oneself to benefit others from the other side. The key is which part is greater: the lost part or the gained part.

How to Escape the Prisoner’s Dilemma

At the entrance of the Dachau concentration camp in Germany is engraved the warning quote of a 17th-century poet: “When a regime starts burning books, if not stopped, its next step is to burn people; when a regime starts silencing speech, if not stopped, its next step is to silence mouths!” At the exit there is another one: “When the world forgets these things, it means these things will happen again.”

1. Moral education

   Readers new to the prisoner’s dilemma tend to condemn the thieves for betraying their accomplices and caring only about their own interests, and feel a sense of schadenfreude at the final outcome. Further reflection on the prisoner’s dilemma makes people realize that it is not just the prisoners’ dilemma, but a dilemma facing all of humanity. If people cannot incorporate others’ interests into their own payoff functions, then the prospects of this society are worrying. H. Gintis is an economist of the Santa Fe school; in a series of recent papers he has argued for a view called the “fundamental theorem of sociology”. He argues that if a society is composed entirely of self-interested individuals, then, in the long run, this society will die out. And if a society, through the efforts of “persuasion and education” and other didactic institutions, maintains a certain proportion of altruists over the long term, it can reproduce stably.

2. Private property

   Looking back at the history of the people’s communes in China, the commune members’ “showing up for work without exerting effort” and “doing nothing at all” is in itself a kind of prisoner’s dilemma. As Soviet workers said: “They pretend to pay me rubles, and I pretend to work hard!” China went further than the Soviet Union and finally declared the complete failure of the people’s commune system.

3. Review the old and learn the new

   If people cannot gain the ability to look into the future from past history, then every generation can only repeat the stories of the past. Drawing lessons and gaining experience from the past is an important difference between humans and other animals. Today’s lion, compared with the lion of 2,000 years ago, shows no obvious change in intelligence, whereas today’s humans have changed enormously in intelligence compared with people 2,000 years ago, because we record past history through symbols and pass on past experience and lessons. We can not only learn the skills of living from our fathers like lions do, but also learn the principles of conducting ourselves in society from Confucius, Mencius, and Sun Tzu from over 2,000 years ago.

How to Exploit Others’ Prisoner’s Dilemma

The prisoner’s dilemma is of course a dilemma for the prisoners, but for the police it is a happy situation. So for you, how can you exploit others’ prisoner’s dilemmas to gain greater benefit for yourself?

The most common one is that as a consumer you shop around, asking merchants for quotes and buying the cheapest. This way, the merchants fall into a prisoner’s dilemma and give you a relatively low price. Especially for many durable consumer goods, consumers often force manufacturers to cut prices by holding onto their money and waiting. Airlines and travel companies also fall into a prisoner’s dilemma during the off-season and are “slaughtered” at will by consumers. Of course, in the peak season, the situation is reversed, and the price competition among consumers benefits the airlines and travel companies. The real estate industry is similar: when houses are selling hot, developers hoard properties, causing home buyers to rush to snap them up; when the housing market is sluggish, developers compete with discounts and various promotional measures to attract people to buy.