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Consider a game consisting of independent turns with even money payoffs inwhich the player wins with a fixed probability $p \geq 1/3$ and loses withprobability $1 - p$. The Labouchere system is a betting strategy which entailskeeping a list of positive real numbers and betting the sum of the first andthe last number on the list at every turn. In case of a victory, those twonumbers are erased from the list, and, in case of a loss, the bet amount isappended to the end of the list. The player finishes the game when the listbecomes empty. It is known that, in a game played with the Labouchere systemwith $p \leq 1/2$, both the sum of the bets and the maximal deficit haveinfinite expectation. Grimmett and Stirzaker raised the question of whether thesame is true for the maximal bet. In this paper we show the expectation of themaximal 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.

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