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 the “Prisoner'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 the “Nash 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 choosing “sacrifice” for 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.