Shoot your mate – game theory on TV shows

You are undercover and about to make a breakthrough with a mob boss.

But your partner, not knowing your mission, tries to save you and gets captured.

The boss suspects you might be working with the authorities.

He asks you to prove your allegiance. He gives you a gun and requests you shoot your partner. If you don’t fire, you will surely be caught.

Do you do it? Why or why not? Use game theory reasoning to figure it out.

Shoot your mate!

This pattern happens in many TV shows, and it is called the “Shoot your mate” trope.

As explained on the site tvtropes.org:

The hero is in the role of a Fake Defector. One of his friends has been captured. The Big Bad hands him or her a gun and tells them to kill them. The gun is almost always empty.

One of two things usually happens. They pull the trigger and nothing happens, or they point the gun at the Big Bad and nothing happens. In the latter case, their cover is pretty much blown.

One of the important details in this trope is that the gun is almost always empty.

Why is that?

It’s game theory time

Let’s think about the problem strategically. The boss either trusts you, or he does not, and he has either handed you a loaded gun or not.

Imagine for a second the boss has in fact handed you a loaded gun. That would only be reasonable if he truly trusted you. Right? After all, if he handed you a loaded gun but thought you were a spy, then he would have to be worried you could fire the gun at him.

It only makes sense to give a loaded gun to someone you deeply trust. But in that case, there is no reason to test the person’s loyalty!

The very fact you are being tested means the boss does not trust you. And in that case, the only sensible thing for the boss is to hand you an unloaded gun.

We can write out a matrix that shows handing you a loaded gun is a weakly dominant strategy. It is simply safer to do, whether he trusts you or not.

Therefore, if you are asked to shoot your mate, you can be reasonably sure the gun is not loaded. You should shoot at your partner to keep your cover and pray the boss was not crazy enough to hand you a loaded gun (of course, a villain as sadistic as the Joker might do this).

Examples in the show 24 (mild spoilers)

Jack Bauer is a game theorist. There are a couple of memorable instances of this trope that I want to mention here. (There are plenty of other examples in tv, movies, and literature at tvtrope.org)

Example from Season 4

In Season 4, a Muslim terrorist Dina defects to the American side to protect her son. She helps Jack Bauer to find the terrorist leader Marwan, who then questions her trust.

Marwan offers Dina a gun and tells her to shoot Jack to prove her loyalty. Dina gets nervous, because if she kills Jack then she would risk the federal protection on her son. Dina shoots at Marwan only to find the gun is not loaded. Her deception is revealed and Marwan orders her to be shot.

Example from Season 3

Another instance happens in Season 3 when Jack was in deep cover with the Salazar brothers. Jack’s partner, Chase, does not realize this and he makes a heroic effort to rescue Jack.

Unfortunately Chase is apprehended and it raises doubts whether Jack is secretly working with authorities. Ramon Salazar hands Jack a gun and tells him to shoot Chase to prove he is trustworthy. Jack takes fire, and it turns out the gun was not loaded so Chase survives. Jack keeps his cover and eventually saves the day as usual.

Chase later finds out Jack was undercover, and he is deeply angry that Jack took aim.

Jack reveals his game theoretic thinking all along, in Season 3, episode 13.

Chase: You put a gun to my head, and you pulled the trigger.

Jack: I made a judgment call that Ramon Salazar would not give me a loaded weapon–that he was testing me.

Chase: And it is was loaded? Then what?

Jack: Then I’d have finished my mission.

It is not an easy thing to shoot at your mate, but it is a judgment call that fits in line with strategic thinking.