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

在外面蹓躂時被狐狸抓到。

Seeing how happy they are, he decides that,

狐狸看它們這麼開心，

instead of simply eating them,

決定與其就這樣把它們吃掉，

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

不如用殘酷的困境 來考驗它們的友誼。

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

牠會讓每個薑餅人選擇 解救對方或是犧牲掉對方。

They can discuss,

它們可以討論，

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

但直到最後一刻， 它們都不會知道對方的決定。

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

如果兩位都選解救對方， 那麼狐狸就會吃掉它們各一肢 ;

if one chooses to spare while the other sacrifices,

如果一方選擇要解救對方， 但另一方選擇要犧牲對方，

the sparer will be fully eaten,

那麼選擇解救的薑餅人 會被整個吃掉，

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

背叛對方的薑餅人 就可以毫髮無傷地逃離。

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

最後，如果兩方都選擇要犧牲對方， 狐狸會吃掉它們雙方各三肢。

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

在博弈理論中，這種情況 被稱為「囚徒困境」。

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

為了理解這些完美理性的 薑餅人會如何行動，

we can map the outcomes of each decision.

我們可以用圖表 展示每個選擇的結果。

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

橫列代表脆脆的選擇， 直列則代表嚼嚼的選擇。

Meanwhile, the numbers in each cell

同時，格子裡面的數字

represent the outcomes of their decisions,

代表它們每個決定的結果，

as measured in the number of limbs each would keep:

也就是它們四肢的剩餘數：

So do we expect their friendship to last the game?

所以它們的友誼能撐到最後一刻嗎？

First, let's consider Chewy's options.

首先，我們來看嚼嚼的選擇。

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

如果脆脆解救它，嚼嚼就可以 藉犧牲脆脆而毫髮無傷地逃走。

But if Crispy sacrifices him,

但是如果脆脆犧牲它，

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

那麼嚼嚼也可以藉由 犧牲脆脆來保留自己的一肢。

No matter what Crispy decides,

不管脆脆如何選擇，

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

嚼嚼都能以選擇背叛同伴 來得到最利己的結果。

The same is true for Crispy.

同樣的情況也適用在脆脆身上。

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

這就是囚徒困境得出的標準結論：

the two characters will betray one another.

兩個人會互相背叛。

Their strategy to unconditionally sacrifice their companion

它們會選擇無條件背叛對方的策略

is what game theorists call the “Nash Equilibrium,"

在博弈理論中稱為「納許均衡」，

meaning that neither can gain by deviating from it.

即雙方除選擇背叛外均無更佳選擇。

Crispy and Chewy act accordingly

脆脆和嚼嚼各自做出選擇，

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

狐狸沾沾自喜帶著一肚子薑餅離開，

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

留下的兩位昔日好友 都只剩下一隻腿撐著身體。

Normally, this is where the story would end,

通常，故事就會這樣結束了，

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

但一位巫師剛好目擊整件事情的經過。

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

他告訴脆脆和嚼嚼， 因為互相背叛所受的懲罰是

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

它們餘生注定要重複經歷這場困境，

starting with all four limbs at each sunrise.

每當早上太陽升起， 四肢都會重新長出。

Now what happens?

現在事情會有所改變嗎？

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

這就是所謂的無限囚徒困境， 顛覆了之前的理論。

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

那是因為薑餅人可以用未來的決策

as bargaining chips for the present ones.

來作為當前的籌碼。

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

仔細想想這個策略： 每天雙方都同意解救對方。

If one ever chooses to sacrifice,

如果有一方選擇犧牲對方，

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

那麼另一方就會為了報復， 同樣一直選擇犧牲對方。

So is that enough to get these poor sentient baked goods

這足以讓兩位可憐有感情的烘培食品

to agree to cooperate?

願意互相合作嗎？

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

為了找出答案，我們必須 把另一個未知因素考慮進去：

the gingerbread men probably care about the future

與其重視未來，

less than they care about the present.

這些薑餅人更重視現在。

In other words, they might discount

換句話說，它們或許會

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

把自己未來肢體數量的 重視程度打折，

which we'll call delta.

我們稱這個打折的數為 ẟ。

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

這有點類似通貨膨脹 侵蝕貨幣價值的概念。

If delta is one half,

如果 ẟ 是 1/2，

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

它們對第二天的肢體的關心程度 是第一天的肢體的一半，

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

第三天的肢體是 第一天的 1/4，以此類推。

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

如果 ẟ 是 0，那代表 它們完全不在乎未來，

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

所以它們會重複選擇 無止盡地互相犧牲對方。

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

但如果 ẟ 接近 1，它們會盡可能

to avoid the pain of infinite triple limb consumption,

避免自己一直失去三肢的痛苦，

which means they'll choose to spare each other.

換言之，它們會選擇解救對方。

At some point in between they could go either way.

在某個時間點， 它們能選擇任何一種方式。

We can find out where that point is

為了找出那個時間點，

by writing the infinite series that represents each strategy,

我們可以列出每種策略的無窮級數，

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

把它們設為等式，進而求出 ẟ。

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

最後得出 1/3，意思是 只要脆脆和嚼嚼對明天的關心程度

at least 1/3 as much as today,

至少是今天的 1/3，

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

永遠合作解救對方是最理想的方法。

This analysis isn't unique to cookies and wizards;

這種分析並不只適用於 薑餅人及巫師；

we see it play out in real-life situations

我們也常在現實中看過，

like trade negotiations and international politics.

像是貿易談判和國際政治。

Rational leaders must assume that the decisions they make today

理智的領袖必須假設他今天做的決定

will impact those of their adversaries tomorrow.

會在明天影響到他的對手。

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

自私自利也許能帶來短期的利益， 但藉由適當的獎勵，

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

和平合作不僅是可行的， 也是數學證明最理想的方法。

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

對於兩位薑餅人， 它們無止盡的將來也許很慘，

but so long as they go out on a limb,

但只要它們願意捨己為人，

their friendship will never again be half-baked.

就能夠友誼長存。