Office Hours: Game Theory

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.