Optimizing over iid distributions and the Beat the Average game

Abstract: A casino offers the following game. There are three cups each containing a die. You are being told that the dice in the cups are all the same, but possibly nonstandard. For a bet of \$1, the game master shakes all three cups and lets you choose one of them. You win \$2 if the die in your cup displays at least the average of the other two, and you lose otherwise. Is this game fair? If not, how should the casino design the dice to maximize their profit?

In this talk, I will answer this question, explain what it is an example of, and outline our partial results on a more difficult question of the same type: how likely can we make the event $X_1 + X_2 + X_3 < 2 X_4$, given the constraint that the random variables $X_1, ..., X_4$ must be iid? Surprisingly, obtaining good bounds involves solving challenging combinatorial optimization problems.

Based on joint work with Pierre C Bellec (arXiv:2412.15179), which has recently been featured as a test case for the AI tool AlphaEvolve (arXiv:2511.02864).

mathematical physicsalgebraic topologydifferential geometryrepresentation theorystatistics theory

Audience: researchers in the topic

( slides | video )

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Prague-Hradec Kralove seminar Cohomology in algebra, geometry, physics and statistics

Series comments: Virtual coffee starts on Zoom already 15 minutes before the seminar.

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