Optimizing over iid distributions and the Beat the Average game

TL;DR

This paper investigates optimal strategies and bounds for a game involving iid distributions and extends the analysis to a general measurable space, providing methods to maximize expectations of functions under product measures.

Contribution

It introduces a general method for optimizing expectations over iid distributions and applies it to a complex probability problem, deriving bounds and conjecturing exact values.

Findings

Maximal probability of a specific event is between 0.400695 and 0.417.

Upper bound obtained via mixed integer linear programming.

Conjecture that the lower bound is the exact probability.

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 in your favor? If not, how should the casino design the dice to maximize their profit? This problem is a special case of the following more general question: given a measurable space $X$ and a bounded measurable function $f : X^n \to \R$, how large can the expectation of $f$ under probability measures of the form $\mu^{\otimes n}$ be? We develop a general method to answer this kind of question. As an example application that is harder than the casino problem, we show that the maximal probability of the event $X_1 + X_2 + X_3 < 2 X_4$ for nonnegative iid random variables lies between $0.400695$ and $0.417$, where the upper bound is obtained by mixed integer linear programming. We conjecture the lower bound to be the exact value.