2024/12/19 by Pierre Bellec, Bellec, Pierre C, T. A. Fritz +1
Computer Science · Decision Sciences · #Advanced Statistical Process Monitoring #Bayesian Modeling and Causal Inference #FOS: Mathematics #Multi-Criteria Decision Making #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2412.15179
openalex publication_date 2024/12/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 : Xn → \R, how large can the expectation of f under probability measures of the form μ⊗ 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 X1 + X2 + X3 < 2 X4 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.