All In: Give me your money!

TL;DR

This paper proves a conjecture about a three-player betting game, showing how initial capital arrangements affect the probability of player elimination, using a novel computational framework and optimization techniques.

Contribution

It introduces MeshItUp, a new theoretical framework, and employs mixed-integer programming for a computer-assisted proof of a conjecture in game theory.

Findings

Probability of elimination is non-monotone with initial capital.

Swapping initial capitals can reduce a player's risk of early elimination.

Computational methods can rigorously prove complex probabilistic conjectures.

Abstract

We present a computer assisted proof for a result concerning a three player betting game, introduced by Angel and Holmes. The three players start with initial capital $x, y, z > 0$ respectively. At each step of this game two players are selected at random to bet on the outcome of a fair coin toss, with the size of the bet being the largest possible, namely the total capital held by the poorer of the two players at that time. The main quantity of interest is the probability of player 1 being eliminated (reaching 0 capital) first. Angel and Holmes have shown that this probability is not monotone decreasing as a function of the initial capital $x$ of player 1. They conjecture that if $x < y < z$ then player 1 would be better off (less likely to be eliminated first) by swapping their capital with another player. In this paper we present a computer-assisted proof of this conjecture. To achieve this, we introduce the theoretical framework MeshItUp, and then perform a two-stage reduction to make MeshItUp computationally feasible, through the use of mixed-integer programming.