Expectation of the Largest bet size in Labouchere System

TL;DR

This paper proves that the expected largest bet size in the Labouchere betting system is finite when the winning probability exceeds 50%, and infinite otherwise, resolving a long-standing conjecture.

Contribution

It establishes the finiteness or infiniteness of the expected largest bet size for the Labouchere system based on winning probability, extending to a broader class of betting systems.

Findings

Finite expectation for p > 1/2

Infinite expectation for p ≤ 1/2

Generalization to a family of betting systems

Abstract

For Labouchere system with winning probability $p$ at each coup, we prove that the expectation of the largest bet size under any initial list is finite if $p>\frac{1}{2}$, and is infinite if $p\le \frac{1}{2}$, solving the open conjecture in Grimmett and Stirzaker (2001). The same result holds for a general family of betting systems, and the proof builds upon a recursive representation of the optimal betting system in the larger family.