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• Two perfectly rational gingerbread men, Crispy and Chewy,

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

• 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.

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

• They can discuss,

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

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

• 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.

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

• In game theory, this scenario is called thePrisoner's Dilemma.”

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

• we can map the outcomes of each decision.

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

• Meanwhile, the numbers in each cell

• represent the outcomes of their decisions,

• as measured in the number of limbs each would keep:

• So do we expect their friendship to last the game?

• First, let's consider Chewy's options.

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

• But if Crispy sacrifices him,

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

• No matter what Crispy decides,

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

• The same is true for Crispy.

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

• the two characters will betray one another.

• Their strategy to unconditionally sacrifice their companion

• is what game theorists call theNash Equilibrium,"

• meaning that neither can gain by deviating from it.

• 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.

• Normally, this is where the story would end,

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

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

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

• starting with all four limbs at each sunrise.

• Now what happens?

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

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

• as bargaining chips for the present ones.

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

• If one ever chooses to sacrifice,

• the other will retaliate by choosingsacrificefor the rest of eternity.

• So is that enough to get these poor sentient baked goods

• to agree to cooperate?

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

• less than they care about the present.

• In other words, they might discount

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

• which we'll call delta.

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

• 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.

• 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.

• 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.

• At some point in between they could go either way.

• 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.

• 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.

• This analysis isn't unique to cookies and wizards;

• we see it play out in real-life situations

• like trade negotiations and international politics.

• Rational leaders must assume that the decisions they make today

• will impact those of their adversaries tomorrow.

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

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

• 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.

Two perfectly rational gingerbread men, Crispy and Chewy,

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B2 crispy chewy gingerbread dilemma sacrifice spare

# How to outsmart the Prisoner’s Dilemma - Lucas Husted

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林宜悉 posted on 2020/10/26
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