On the Expected Value of the Maximal Bet in the Labouchere System

TL;DR

This paper analyzes the Labouchere betting system in a probabilistic game, showing that the expected maximum bet remains finite when the win probability exceeds approximately 0.6138, addressing a question about the system's risk.

Contribution

It provides a partial answer to whether the expected maximum bet in the Labouchere system is finite for certain win probabilities, identifying a specific threshold.

Findings

Expected maximum bet is finite for p > 0.613763

The threshold p is derived from a specific algebraic equation

Addresses a previously open question about the Labouchere system's risk

Abstract

Consider a game consisting of independent turns with even money payoffs in which the player wins with a fixed probability $p \geq 1/3$ and loses with probability $1 - p$. The Labouchere system is a betting strategy which entails keeping a list of positive real numbers and betting the sum of the first and the last number on the list at every turn. In case of a victory, those two numbers are erased from the list, and, in case of a loss, the bet amount is appended to the end of the list. The player finishes the game when the list becomes empty. It is known that, in a game played with the Labouchere system with $p \leq 1/2$, both the sum of the bets and the maximal deficit have infinite expectation. Grimmett and Stirzaker raised the question of whether the same is true for the maximal bet. In this paper we show the expectation of the maximal bet is finite for $p > c$, where $c \approx 0.613763$ solves $(c-1)^2 c = \frac{8}{27(1+\sqrt{5})}$, thus partially answering this question.