Office Hours: Game Theory

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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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