Proofs of the Ethier and Lee slot machine conjectures

Suppose a gambler pays one coin per coup to play a two-armed Futurity slot machine, an antique casinos, and two coins are refunded for every two consecutive gambler losses. This payoff is called the Futurity award. The casino owner honestly advertises that each arm on his/her two-armed machine is fair in the sense that the asymptotic expected profit of both gambler and dealer is 0 if the gambler only plays either arm. The gambler is allowed to play either arm on each coup alternatively in some deterministic order or at random. For almost 90 years, since Futurity slot machines is designed in 1936, an open problem that has not been solved for a long time is whether the slot machine will obey the so-called "long bet will lose" phenomenon so common to casino games. Ethier and Lee [Ann. Appl. Proba. 20(2010), pp.1098-1125] conjectured that a player will also definitely lose in the long run by applying any non-random-mixture strategy. In this paper, we shall prove Ethier and Lee's conjecture. Our result with Ethier and Lee's conclusion straightforwardly demonstrates that players decide to use either random or non-random two-arm strategies before playing and then repeated without interruption, the casino owners are always profitable even when the Futurity award is taken into account. The contribution of this work is that it helps complete the demystification of casino profitability. Moreover, it paves the way for casino owners to improve casino game design and for players to participate effectively in gambling.