Two perfectly rational gingerbread men, Crispy and Chewy,
兩個絕對理性的薑餅人， 嚼嚼和脆脆，
are out strolling when they’re caught by a fox.
在外面閒逛的時候 被一隻狐狸抓住了。
Seeing how happy they are, he decides that,
狐狸看到它們這麼開心，
instead of simply eating them,
就決定比起簡單地把它們吃掉，
he』ll put their friendship to the test with a cruel dilemma.
它要通過一個殘酷的窘境 來測試它們之間的友情。
He』ll ask each gingerbread man whether he』d opt to Spare or Sacrifice the other.
它分別問每隻薑餅人是 選擇解救還是犧牲對方。
They can discuss,
它們可以進行討論，
but neither will know what the other chose until their decisions are locked in.
但直到它們的決定被確認後 才會知曉對方的選擇。
If both choose to spare the other, the fox will eat just one of each of their limbs;
如果它們都選擇解救對方， 那麼狐狸只會吃掉每人一肢；
if one chooses to spare while the other sacrifices,
如果只有一個人選擇解救對方，
the sparer will be fully eaten,
那麼選擇解救對方的人會被吃掉，
while the traitor will run away with all his limbs intact.
而叛徒卻可以完好無損地跑掉。
Finally, if both choose to sacrifice, the fox will eat 3 limbs from each.
最後，如果它們都選擇犧牲對方， 那麼狐狸會吃掉每人三肢。
In game theory, this scenario is called the 「Prisoner's Dilemma.」
在博弈論裡，這個情況 被稱之為「囚徒困境」。
To figure out how these gingerbread men will act in their perfect rationality,
為了搞清楚這些薑餅人 在絕對理性的情況下會怎麼選擇，
we can map the outcomes of each decision.
我們可以把每種情況的結果寫出來。
The rows represent Crispy’s choices, and the columns are Chewy’s.
每一行代表的是脆脆的選擇， 每一列代表的是嚼嚼的選擇。
Meanwhile, the numbers in each cell
同時，每個單元格中的數字
represent the outcomes of their decisions,
代表的是每種選擇所對應的結果，
as measured in the number of limbs each would keep:
通過每人殘留的肢體數量來表示：
So do we expect their friendship to last the game?
你覺得它們的友情在遊戲 結束後還能完好無損嗎？
First, let’s consider Chewy’s options.
首先， 讓我們來考慮嚼嚼的選項。
If Crispy spares him, Chewy can run away scot-free by sacrificing Crispy.
如果脆脆選擇解救他，那麼嚼嚼 就可以通過犧牲脆脆來逃脫懲罰。
But if Crispy sacrifices him,
但如果脆脆選擇犧牲他，
Chewy can keep one of his limbs if he also sacrifices Crispy.
那麼嚼嚼可以通過同時 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.
嚼嚼選擇犧牲它的同伴 總能達到最優的結果。
The same is true for Crispy.
這一結論對脆脆來說也成立。
This is the standard conclusion of the Prisoner's Dilemma:
這就是囚徒困境的標準結論：
the two characters will betray one another.
兩人都會選擇出賣對方。
Their strategy to unconditionally sacrifice their companion
它們選擇無條件犧牲對方的策略
is what game theorists call the 「Nash Equilibrium,"
被博弈理論家稱為 「納什平衡」，
meaning that neither can gain by deviating from it.
意思是任何一方只要 背離這一策略都會有所損失。
Crispy and Chewy act accordingly
脆脆和嚼嚼遵照 這一理論做出決定
and the smug fox runs off with a belly full of gingerbread,
讓沾沾自喜的狐狸 得以吃了一肚子的薑餅，
leaving the two former friends with just one leg to stand on.
而兩位昔日好友都只剩下 一肢在支撐著它們的身體。
Normally, this is where the story would end,
通常情況下， 故事到這裡就結束了。
but a wizard happened to be watching the whole mess unfold.
但有一個巫師 恰巧見證了這一切。
He tells Crispy and Chewy that, as punishment for betraying each other,
他告訴脆脆和嚼嚼， 作為背棄彼此的懲罰，
they’re doomed to repeat this dilemma for the rest of their lives,
它們餘生都將註定 要一直重複這一窘境，
starting with all four limbs at each sunrise.
每天日出的時候 都將重新獲得四肢。
Now what happens?
現在該如何是好？
This is called an Infinite Prisoner’s Dilemma, and it’s a literal game changer.
這被稱為「無限囚徒困境」， 它顛覆了之前的結論。
That’s because the gingerbread men can now use their future decisions
這是因為薑餅人 可以用未來的決定
as bargaining chips for the present ones.
作為現在討價還價的籌碼。
Consider this strategy: both agree to spare each other every day.
讓我們考慮下這個策略：每人每天都同意互相解救對方。
If one ever chooses to sacrifice,
但凡有任何一個人選擇犧牲對方，
the other will retaliate by choosing 「sacrifice」 for the rest of eternity.
那麼另一人就可以通過餘生 一直選擇犧牲它來進行報復。
So is that enough to get these poor sentient baked goods
這樣就足夠讓這些 可憐的，有意識的焙烤食品
to agree to cooperate?
同意合作了嗎？
To figure that out, we have to factor in another consideration:
為了弄清楚，我們得 將另一因素考慮進來：
the gingerbread men probably care about the future
比起將來，這些薑餅人
less than they care about the present.
應該會更重視現在。
In other words, they might discount
換言之，它們可能會將
how much they care about their future limbs by some number,
自己所在乎的未來的 肢體數量換算成一個數字，
which we』ll call delta.
我們將其稱為 δ 。
This is similar to the idea of inflation eroding the value of money.
這個點子類似於通貨膨脹 會降低金錢的價值。
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,
那麼第二天的每兩個肢體對它們來說 都相當於是第一天的一個肢體，
day 3 limbs 1 quarter as much as day 1 limbs, and so on.
第三天的肢體是第一天肢體 價值的四分之一，以此類推。
A delta of 0 means that they don’t care about their future limbs at all,
δ 等於 0 則意味著它們根本 不在乎未來的肢體數量，
so they』ll repeat their initial choice of mutual sacrifice endlessly.
所以它們將會無止境地 重複最初的選擇：互相犧牲。
But as delta approaches 1, they』ll do anything possible
但當 δ 趨近 1， 它們將會盡己所能地
to avoid the pain of infinite triple limb consumption,
避免自己每天無止境地 失去三肢的痛苦，
which means they』ll choose to spare each other.
於是他們會選擇互相解救。
At some point in between they could go either way.
當 δ 取這兩個值之間的某個點時， 任何一種選擇都有可能發生。
We can find out where that point is
我們可以通過寫出 代表每種策略的無窮級數，
by writing the infinite series that represents each strategy,
來找到那個點的位置，
setting them equal to each other, and solving for delta.
設它們的數值相等，來求解 δ 。
That yields 1/3, meaning that as long as Crispy and Chewy care about tomorrow
結果是 1/3 ，說明只要脆脆和嚼嚼 認為明天的重要性
at least 1/3 as much as today,
至少佔今天的 1/3 ，
it’s optimal for them to spare and cooperate forever.
那麼合作：互相解救 是對他們最有利的。
This analysis isn’t unique to cookies and wizards;
這個分析並不只 適用於餅乾和巫師這則故事，
we see it play out in real-life situations
在現實生活中也經常出現於
like trade negotiations and international politics.
像是貿易談判和國際政治的形勢下。
Rational leaders must assume that the decisions they make today
理性的領導者必須假定 它們每天所做的決定
will impact those of their adversaries tomorrow.
會影響他們競爭對手明天的決定。
Selfishness may win out in the short-term, but with the proper incentives,
自私自利也許在短期內能帶來利潤， 但只要有恰當的激勵措施，
peaceful cooperation is not only possible, but demonstrably and mathematically ideal.
和平的合作不只是可能的， 而且也被數學推導證實是更理想的。
As for the gingerbread men, their eternity may be pretty crumby,
對於薑餅人來說，它們 無窮無盡的故事看起來很糟糕，
but so long as they go out on a limb,
但只要它們肯為對方擔風險，
their friendship will never again be half-baked.
那麼它們的友誼就能地久天長。