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Two perfectly rational gingerbread men, Crispy and Chewy, are out strolling when they're caught by a fox.
两个绝对理性的姜饼人，嚼嚼和脆脆，在外面闲逛的时候被一只狐狸抓住了。
Seeing how happy they are, he decides that, instead of simply eating them, he'll put their friendship to the test with a cruel dilemma.
狐狸看到它们这么开心，就决定比起简单地把它们吃掉，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, but neither will know what the other chose until their decisions are locked in.
它们可以进行讨论，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, the sparer will be fully eaten, while the traitor will run away with all his limbs intact.
如果只有一个人选择解救对方，那么选择解救对方的人会被吃掉，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, we can map the outcomes of each decision.
为了搞清楚这些姜饼人在绝对理性的情况下会怎么选择，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 represent the outcomes of their decisions, as measured in the number of limbs each would keep:
同时，每个单元格中的数字代表的是每种选择所对应的结果，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. If Crispy spares him, Chewy can run away scot-free by sacrificing Crispy.
首先，让我们来考虑嚼嚼的选项。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, Chewy can keep one of his limbs if he also sacrifices Crispy.
但如果脆脆选择牺牲他，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, Chewy always experiences the best outcome by choosing to sacrifice his companion. The same is true for Crispy.
不管脆脆如何选择，嚼嚼选择牺牲它的同伴总能达到最优的结果。这一结论对脆脆来说也成立。
This is the standard conclusion of the Prisoner's Dilemma: the two characters will betray one another.
这就是囚徒困境的标准结论：两人都会选择出卖对方。
Their strategy to unconditionally sacrifice their companion is what game theorists call the "Nash Equilibrium," meaning that neither can gain by deviating from it.
它们选择无条件牺牲对方的策略被博弈理论家称为“纳什平衡”，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 and the smug fox runs off with a belly full of gingerbread, leaving the two former friends with just one leg to stand on.
脆脆和嚼嚼遵照这一理论做出决定让沾沾自喜的狐狸得以吃了一肚子的姜饼，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, but a wizard happened to be watching the whole mess unfold.
通常情况下，故事到这里就结束了。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, starting with all four limbs at each sunrise.
它们余生都将注定要一直重复这一窘境，每天日出的时候都将重新获得四肢。
Now what happens? This is called an Infinite Prisoner's Dilemma, and it's a literal game changer.
现在该如何是好？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 as bargaining chips for the present ones.
这是因为姜饼人可以用未来的决定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, the other will retaliate by choosing "sacrifice" for the rest of eternity.
但凡有任何一个人选择牺牲对方，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 to agree to cooperate?
这样就足够让这些可怜的，有意识的焙烤食品同意合作了吗？
To figure that out, we have to factor in another consideration: the gingerbread men probably care about the future less than they care about the present.
为了弄清楚，我们得将另一因素考虑进来：比起将来，这些姜饼人应该会更重视现在。
In other words, they might discount how much they care about their future limbs by some number, which we'll call delta.
换言之，它们可能会将自己所在乎的未来的肢体数量换算成一个数字，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, on day one they care about day 2 limbs half as much as day 1 limbs, day 3 limbs 1 quarter as much as day 1 limbs, and so on.
如果δ是1/2，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，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, so they'll repeat their initial choice of mutual sacrifice endlessly.
δ等于0则意味着它们根本不在乎未来的肢体数量，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 to avoid the pain of infinite triple limb consumption, which means they'll choose to spare each other.
但当δ趋近1，它们将会尽己所能地避免自己每天无止境地失去三肢的痛苦，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 by writing the infinite series that represents each strategy, setting them equal to each other, and solving for delta.
我们可以通过写出代表每种策略的无穷级数，来找到那个点的位置，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 at least 1/3 as much as today, it's optimal for them to spare and cooperate forever.
结果是1/3，说明只要脆脆和嚼嚼认为明天的重要性至少占今天的1/3，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; we see it play out in real-life situations like trade negotiations and international politics.
这个分析并不只适用于饼干和巫师这则故事，在现实生活中也经常出现于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 will impact those of their adversaries tomorrow.
理性的领导者必须假定它们每天所做的决定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, peaceful cooperation is not only possible, but demonstrably and mathematically ideal.
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，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, but so long as they go out on a limb, their friendship will never again be half-baked.
对于姜饼人来说，它们无穷无尽的故事看起来很糟糕，但只要它们肯为对方担风险，那么它们的友谊就能地久天长。
