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.
This is just one of 70 fascinating problems in a new book, Math puzzles: classic riddles in counting, geometry, probability, and game theory, by Presh Talwalkar. Presh writes the blog, Mind Your Decisions, which discusses a lot of decision making problems, including a lot of game theory. The book is a collection of some of the best problems from the blog.
There are three sections in the book. Section one looks at Counting and Geometry problems, section two covers probability problems and section three, most interestingly for readers of this blog, covers Strategy and Game Theory problems.
The problems are interesting challenges for those of you who want something challenging to get your brain around. What really makes the book stand out is Presh’s explanations of the answers which go beyond a simple text-book answer and often give more context and different methods of solving the problems.
Back to the gun problem, do you pull the trigger?
Here’s Presh’s answer:
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
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 what if 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