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

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.