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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:

  • the gingerbread men probably care about the future

  • 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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