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- [Mary Clare] Today, we're going to learn more about Game Theory
by using it to solve a simple problem.
Bob and Al are two prestigious rival magicians
who have developed a new trick that is quite popular.
They've then agreed to limit performances
so they can charge more.
If both magicians perform only one show a week,
each will earn $10,000.
However, if one magician breaks the agreement
and performs five times a week
while the other continues to perform once a week --
that double-crosser will make $15,000
while the other magician will make only $1,000.
And if both magicians break the agreement
and perform five times a week, each will earn $6,000.
So, what is the Nash equilibrium of how many shows
they will each perform?
The Nash equilibrium means that no person has an incentive
to change their behavior or strategy
unless someone else changes their behavior or strategy.
In order to find the Nash equilibrium
of Bob and Al's performances,
we have to first analyze Bob's behavior
based on Al's behavior and vice versa.
It will be easier to track everything
if we fill out a 2-by-2 matrix.
There are two individuals with two options.
In each box of the matrix we'll list each person's path
given the state of the world.
So we'll list Bob's path first and Al's second.
So let's first look at Bob's best strategy
based on Al's behavior.
Al will either keep her promise to perform once a week,
or she'll break her promise and perform five shows.
If she cooperates and performs one show,
what is Bob's best strategy?
Again, if we just look at what he stands to gain,
then his best option would be to cheat
and perform five times a week
and make $15,000 versus performing once a week
and making $10,000.
Now, what if Al backstabs Bob and performs five shows?
Bob's best strategy here is also to perform five shows a week
and make $6,000 versus performing once a week
and making only $1,000.
Given that Bob's best strategy is to cheat and perform five shows
regardless of what Al does, cheating is his dominant strategy.
Now, let's look at it from Al's perspective.
I bet you can see where this is going.
If Bob keeps his promise and performs one show per week,
then Al's best option is to perform five shows.
She'll earn $15,000 instead of $10,000.
And, if Bob decides to break his promise
and perform five shows,
Al's best option is also to cheat and perform five shows
because she'll earn $6,000 instead of $1,000.
Given that Al's best strategy is to perform five times per week --
again, regardless of what Bob does --
this is also considered her dominant strategy.
So if Bob's dominant strategy is to cheat as well,
then the Nash equilibrium in this game
is for both of them to break their promises.
They'll each perform five shows and earn $6,000.
Notice that this isn't an optimal outcome.
It would be so much better for them to each perform
only one show per week.
They'd earn a lot more money,
and they'd also have a lot more free time on their hands.
But if we're just evaluating what to do
from the payoffs listed in our matrix,
it is in both Bob's best interest and Al's best interest to cheat.
Thus, it's the Nash equilibrium.
Of course, there's a real world outside of the matrix.
The world is much more complicated than this.
People care about keeping promises,
and they think about the long run,
rather than just week to week.
So think of this example as just a simple
but powerful starting point
to better understand human decision-making.
As always, let us know what you think.
And, if you'd like more practice,
check out our additional challenge questions
at the end of this video.
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