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♪ [music] ♪

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- [Mary Clare] Today, we're going
to learn more about Game Theory

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by using it to solve
a simple problem.

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Bob and Al are two
prestigious rival magicians

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who have developed a new trick
that is quite popular.

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They've then agreed
to limit performances

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so they can charge more.

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If both magicians perform
only one show a week,

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each will earn $10,000.

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However, if one magician
breaks the agreement

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and performs five times a week

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while the other continues
to perform once a week --

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that double-crosser
will make $15,000

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while the other magician
will make only $1,000.

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And if both magicians
break the agreement

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and perform five times a week,
each will earn $6,000.

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So, what is the Nash equilibrium
of how many shows

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they will each perform?

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The Nash equilibrium means
that no person has an incentive

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to change their behavior
or strategy

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unless someone else changes
their behavior or strategy.

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In order to find
the Nash equilibrium

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of Bob and Al's performances,

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we have to first analyze
Bob's behavior

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based on Al's behavior
and vice versa.

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It will be easier
to track everything

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if we fill out a 2-by-2 matrix.

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There are two individuals
with two options.

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In each box of the matrix
we'll list each person's path

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given the state of the world.

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So we'll list Bob's path first
and Al's second.

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So let's first look
at Bob's best strategy

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based on Al's behavior.

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Al will either keep her promise
to perform once a week,

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or she'll break her promise
and perform five shows.

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If she cooperates
and performs one show,

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what is Bob's best strategy?

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Again, if we just look
at what he stands to gain,

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then his best option
would be to cheat

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and perform five times a week

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and make $15,000 versus
performing once a week

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and making $10,000.

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Now, what if Al backstabs Bob
and performs five shows?

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Bob's best strategy here is also
to perform five shows a week

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and make $6,000 versus
performing once a week

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and making only $1,000.

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Given that Bob's best strategy
is to cheat and perform five shows

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regardless of what Al does,
cheating is his dominant strategy.

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Now, let's look at it
from Al's perspective.

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I bet you can see
where this is going.

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If Bob keeps his promise
and performs one show per week,

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then Al's best option
is to perform five shows.

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She'll earn $15,000
instead of $10,000.

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And, if Bob decides
to break his promise

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and perform five shows,

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Al's best option is also to cheat
and perform five shows

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because she'll earn $6,000
instead of $1,000.

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Given that Al's best strategy
is to perform five times per week --

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again, regardless of what Bob does --

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this is also considered
her dominant strategy.

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So if Bob's dominant strategy
is to cheat as well,

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then the Nash equilibrium
in this game

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is for both of them
to break their promises.

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They'll each perform five shows
and earn $6,000.

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Notice that this isn't
an optimal outcome.

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It would be so much better
for them to each perform

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only one show per week.

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They'd earn a lot more money,

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and they'd also have a lot more
free time on their hands.

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But if we're just
evaluating what to do

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from the payoffs
listed in our matrix,

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it is in both Bob's best interest
and Al's best interest to cheat.

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Thus, it's the Nash equilibrium.

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Of course, there's a real world
outside of the matrix.

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The world is much more
complicated than this.

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People care about keeping promises,

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and they think about the long run,

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rather than just week to week.

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So think of this example
as just a simple

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but powerful starting point

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to better understand
human decision-making.

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As always, let us know
what you think.

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And, if you'd like more practice,

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check out our additional
challenge questions

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at the end of this video.

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♪ [music] ♪