Chris the Criminal: A mathematical puzzle

(Warning: the comments on this post contain spoilers!)

Chris the Criminal flips a fair coin. If it comes up heads he commits a crime. If it comes up tails, he doesn’t.

Now, Chris is being tried before a jury of n people (n is at least 2). The jury members all know the process by which Chris decided whether to commit the crime. The trial works as follows: all jury members simultaneously vote “guilty” or “not guilty” and Chris is convicted if and only if every jury member votes “guilty”.

Each jury member receives an independent signal of whether Chris is guilty; the signal is correct with probability 70%, both when Chris is guilty and when he’s innocent. That is, if Chris is guilty, each jury member independently receives the signal “guilty” with probability 70% and “innocent” with probability 30%; if Chris is innocent, each jury member independently receives “innocent” with probability 70% and “guilty” with probability 30%.

Before the trial starts, the jury gets together to come up with a strategy. However, after receiving their signals, the jury cannot deliberate and votes immediately.

The jury’s goal is to reach the correct verdict with as high a probability as possible. Find all strategies that achieve this goal. Note that jury members are allowed to use randomness to decide their vote: for example, one possible strategy is “everyone votes according to their signal with probability 2/3 and opposite their signal with probability 1/3.”

What inspired this problem (warning: includes a minor nudge about how to think about the problem): I once got into a heated argument about the board game Spyfall.

In Spyfall, several people who know a secret location (e.g. “bank” or “supermarket”) and one person — the spy — does not. The players exchange coded messages to try to figure out who the spy is and the spy tries to figure out the secret location.

For this problem, the precise mechanics of the game aren’t important. What’s important is the end mechanic. At the end of the game, someone accuses another player of being the spy. After that, everyone votes whether to convict the accused player or not. The goal of the non-spies is to convict the spy.

So let’s say you’re one of the non-spies and someone is accused. You aren’t sure — maybe they’re the spy, maybe not. Should you vote to convict?

I argued that it makes sense to preferentially vote to convict. To see why, observe that your vote only makes a difference if everyone else votes to convict. But if everyone else votes to convict, then the accused is probably a spy: after all, the fact that everyone else thinks they’re the spy is pretty substantial evidence that they’re in fact the spy.

But as my interlocutor pointed out, if everyone follows this strategy, then the accused player will pretty much be convicted no matter what, since everyone will preferentially vote to convict.

So maybe there’s some sort of equilibrium strategy here: perhaps you should vote to convict with some probability, and then with the remaining probability you should do what feels right, independently of what other people might be thinking.

The problem I posed doesn’t perfectly capture the dynamics of Spyfall, but I think it’s interesting nonetheless.