How to outsmart the Prisoner’s Dilemma - Lucas Husted

翻译人员: Joyce He 校对人员: Yolanda Zhang

Two perfectly rational gingerbread men, Crispy and Chewy,

两个绝对理性的姜饼人， 嚼嚼和脆脆，

are out strolling when they’re caught by a fox.

在外面闲逛的时候 被一只狐狸抓住了。

Seeing how happy they are, he decides that,

狐狸看到它们这么开心，

就决定比起简单地把它们吃掉，

instead of simply eating them,

he’ll put their friendship to the test with a cruel dilemma.

它要通过一个残酷的窘境 来测试它们之间的友情。

He’ll ask each gingerbread man whether he’d opt to Spare or Sacrifice the other.

它分别问每只姜饼人是 选择解救还是牺牲对方。

They can discuss,

它们可以进行讨论，

but neither will know what the other chose until their decisions are locked in.

但直到它们的决定被确认后 才会知晓对方的选择。

If both choose to spare the other, the fox will eat just one of each of their limbs;

如果它们都选择解救对方， 那么狐狸只会吃掉每人一肢；

if one chooses to spare while the other sacrifices,

如果只有一个人选择解救对方，

the sparer will be fully eaten,

那么选择解救对方的人会被吃掉，

while the traitor will run away with all his limbs intact.

而叛徒却可以完好无损地跑掉。

Finally, if both choose to sacrifice, the fox will eat 3 limbs from each.

最后，如果它们都选择牺牲对方， 那么狐狸会吃掉每人三肢。

In game theory, this scenario is called the “Prisoner's Dilemma.”

在博弈论里，这个情况 被称之为“囚徒困境”。

To figure out how these gingerbread men will act in their perfect rationality,

为了搞清楚这些姜饼人 在绝对理性的情况下会怎么选择，

we can map the outcomes of each decision.

我们可以把每种情况的结果写出来。

The rows represent Crispy’s choices, and the columns are Chewy’s.

每一行代表的是脆脆的选择， 每一列代表的是嚼嚼的选择。

Meanwhile, the numbers in each cell

同时，每个单元格中的数字

represent the outcomes of their decisions,

代表的是每种选择所对应的结果，

as measured in the number of limbs each would keep:

通过每人残留的肢体数量来表示：

So do we expect their friendship to last the game?

你觉得它们的友情在游戏 结束后还能完好无损吗？

First, let’s consider Chewy’s options.

首先， 让我们来考虑嚼嚼的选项。

If Crispy spares him, Chewy can run away scot-free by sacrificing Crispy.

如果脆脆选择解救他，那么嚼嚼 就可以通过牺牲脆脆来逃脱惩罚。

But if Crispy sacrifices him,

但如果脆脆选择牺牲他，

Chewy can keep one of his limbs if he also sacrifices Crispy.

那么嚼嚼可以通过同时 牺牲脆脆来保留自己的一肢。

No matter what Crispy decides,

不管脆脆如何选择，

Chewy always experiences the best outcome by choosing to sacrifice his companion.

嚼嚼选择牺牲它的同伴 总能达到最优的结果。

The same is true for Crispy.

这一结论对脆脆来说也成立。

This is the standard conclusion of the Prisoner's Dilemma:

这就是囚徒困境的标准结论：

the two characters will betray one another.

两人都会选择出卖对方。

Their strategy to unconditionally sacrifice their companion

它们选择无条件牺牲对方的策略

is what game theorists call the “Nash Equilibrium,"

被博弈理论家称为 “纳什平衡”，

meaning that neither can gain by deviating from it.

意思是任何一方只要 背离这一策略都会有所损失。

Crispy and Chewy act accordingly

脆脆和嚼嚼遵照 这一理论做出决定

and the smug fox runs off with a belly full of gingerbread,

让沾沾自喜的狐狸 得以吃了一肚子的姜饼，

leaving the two former friends with just one leg to stand on.

而两位昔日好友都只剩下 一肢在支撑着它们的身体。

Normally, this is where the story would end,

通常情况下， 故事到这里就结束了。

but a wizard happened to be watching the whole mess unfold.

但有一个巫师 恰巧见证了这一切。

He tells Crispy and Chewy that, as punishment for betraying each other,

他告诉脆脆和嚼嚼， 作为背弃彼此的惩罚，

they’re doomed to repeat this dilemma for the rest of their lives,

它们余生都将注定 要一直重复这一窘境，

starting with all four limbs at each sunrise.

每天日出的时候 都将重新获得四肢。

Now what happens?

现在该如何是好？

This is called an Infinite Prisoner’s Dilemma, and it’s a literal game changer.

这被称为“无限囚徒困境”， 它颠覆了之前的结论。

That’s because the gingerbread men can now use their future decisions

这是因为姜饼人 可以用未来的决定

as bargaining chips for the present ones.

作为现在讨价还价的筹码。

Consider this strategy: both agree to spare each other every day.

让我们考虑下这个策略： 每人每天都同意互相解救对方。

If one ever chooses to sacrifice,

但凡有任何一个人选择牺牲对方，

the other will retaliate by choosing “sacrifice” for the rest of eternity.

那么另一人就可以通过余生 一直选择牺牲它来进行报复。

So is that enough to get these poor sentient baked goods

这样就足够让这些 可怜的，有意识的焙烤食品

to agree to cooperate?

同意合作了吗？

To figure that out, we have to factor in another consideration:

为了弄清楚，我们得 将另一因素考虑进来：

the gingerbread men probably care about the future

比起将来，这些姜饼人

less than they care about the present.

应该会更重视现在。

In other words, they might discount

换言之，它们可能会将

how much they care about their future limbs by some number,

自己所在乎的未来的 肢体数量换算成一个数字，

which we’ll call delta.

我们将其称为 δ 。

This is similar to the idea of inflation eroding the value of money.

这个点子类似于通货膨胀 会降低金钱的价值。

If delta is one half,

如果 δ 是 1/2 ，

on day one they care about day 2 limbs half as much as day 1 limbs,

那么第二天的每两个肢体对它们来说 都相当于是第一天的一个肢体，

day 3 limbs 1 quarter as much as day 1 limbs, and so on.

第三天的肢体是第一天肢体 价值的四分之一，以此类推。

A delta of 0 means that they don’t care about their future limbs at all,

δ 等于 0 则意味着它们根本 不在乎未来的肢体数量，

so they’ll repeat their initial choice of mutual sacrifice endlessly.

所以它们将会无止境地 重复最初的选择：互相牺牲。

But as delta approaches 1, they’ll do anything possible

但当 δ 趋近 1， 它们将会尽己所能地

to avoid the pain of infinite triple limb consumption,

避免自己每天无止境地 失去三肢的痛苦，

which means they’ll choose to spare each other.

于是他们会选择互相解救。

At some point in between they could go either way.

当 δ 取这两个值之间的某个点时， 任何一种选择都有可能发生。

We can find out where that point is

我们可以通过写出 代表每种策略的无穷级数，

by writing the infinite series that represents each strategy,

来找到那个点的位置，

setting them equal to each other, and solving for delta.

设它们的数值相等，来求解 δ 。

That yields 1/3, meaning that as long as Crispy and Chewy care about tomorrow

结果是 1/3 ，说明只要脆脆和嚼嚼 认为明天的重要性

at least 1/3 as much as today,

至少占今天的 1/3 ，

it’s optimal for them to spare and cooperate forever.

那么合作：互相解救 是对他们最有利的。

This analysis isn’t unique to cookies and wizards;

这个分析并不只 适用于饼干和巫师这则故事，

we see it play out in real-life situations

在现实生活中也经常出现于

like trade negotiations and international politics.

像是贸易谈判和国际政治的形势下。

Rational leaders must assume that the decisions they make today

理性的领导者必须假定 它们每天所做的决定

will impact those of their adversaries tomorrow.

会影响他们竞争对手明天的决定。

Selfishness may win out in the short-term, but with the proper incentives,

自私自利也许在短期内能带来利润， 但只要有恰当的激励措施，

peaceful cooperation is not only possible, but demonstrably and mathematically ideal.

和平的合作不只是可能的， 而且也被数学推导证实是更理想的。

As for the gingerbread men, their eternity may be pretty crumby,

对于姜饼人来说，它们 无穷无尽的故事看起来很糟糕，

but so long as they go out on a limb,

但只要它们肯为对方担风险，

their friendship will never again be half-baked.

那么它们的友谊就能地久天长。