Proof of a conjecture about Parrondo's paradox for two-armed slot machines

The 1936 Mills Futurity slot machine had the feature that, if a player loses 10 times in a row, the 10 lost coins are returned. Ethier and Lee (2010) studied a generalized version of this machine, with 10 replaced by deterministic parameter J. They established the Parrondo effect for a hypothetical two-armed machine with the Futurity award. Specifically, arm A and arm B, played individually, are asymptotically fair, but when alternated randomly (the so-called random mixture strategy), the casino makes money in the long run. They also considered the nonrandom periodic pattern strategy for patterns with r As and s Bs (e.g., $ABABB$ if $r=2$ and $s=3$). They established the Parrondo effect if $r+s$ divides J, and conjectured it in four other situations, including the case $J=2$ with $r\ge 1$ and $s\ge 1$. We prove the conjecture in the latter case.