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