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.
Categories: math.PR cs.DM cs.IT math.CO math.IT