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All In: Give me your money!

Angel Y. He, Mark Holmes

TL;DR

To achieve this, the theoretical framework MeshItUp is introduced, and a two-stage reduction is performed to make MeshItUp computationally feasible, through the use of mixed-integer programming.

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.

All In: Give me your money!

TL;DR

To achieve this, the theoretical framework MeshItUp is introduced, and a two-stage reduction is performed to make MeshItUp computationally feasible, through the use of mixed-integer programming.

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 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 of player 1. They conjecture that if 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.

Paper Structure

This paper contains 21 sections, 10 theorems, 24 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminary Calculations
  3. The MeshItUp Algorithm
  4. Mixed-integer Linear Programming
  5. Algorithms for solving MILPs
  6. Branch-and-bound
  7. Cutting Planes
  8. Branch-and-cut
  9. Implementing MeshItUp
  10. The MILP Framework
  11. Towards Constrained Optimization
  12. Towards MILP
  13. Towards a solvable MILP
  14. Towards easier-to-solve MILPs
  15. Generalizing the proof
  16. ...and 6 more sections

Key Result

Theorem 1

If $0<x<y<z$ then $f(x,y,z)>f(y,x,z)>f(z,x,y)$. In particular, player 1 is less likely to lose if they swap their initial stack with that of another player who has a larger initial stack.

Figures (3)

  • Figure 1: Transition probabilities from a state $(x,y,z)$ with $0\le x\le y\le z$.
  • Figure 2: Visualization of ${\{ (x,y,z) : 0<x<y<z,\,\, x+y+z=2000,\,\, \Delta_n(\textbf{s})>\alpha_n \}}$
  • Figure 3: Parent states of $(x,y,z)$ in $h_n(x,y,z)$

Theorems & Definitions (20)

  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • Lemma 5
  • proof : Proof of Theorem \ref{['thm:want_to_swap']}
  • Definition 3.1
  • Lemma 6
  • ...and 10 more