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.

翻译：假设一位赌徒每次下注支付一枚金币，在一台双臂未来型老虎机（一种古董级赌博机）上进行游戏，且每当赌徒连续两次输局时，会退还两枚金币作为未来奖励。赌场庄家诚实宣称，其双臂机器的每只手臂都是公平的，即若赌徒仅使用任意一只手臂下注，则赌徒和庄家的渐近期望收益均为零。赌徒可轮流使用任意一只手臂，按确定性顺序或随机方式完成每次下注。自1936年未来型老虎机设计以来的近九十年间，一个长期未解决的开放问题在于：这种老虎机是否遵循赌场中常见的"长赌必输"现象。Ethier与Lee[Ann. Appl. Proba. 20(2010), pp.1098-1125]猜想：若赌徒采用任何非随机混合策略，长期而言也必然失败。本文将对Ethier与Lee的猜想给出证明。我们的结果与Ethier和Lee的结论直接表明：若赌徒在游戏开始前决定使用随机或非随机双臂策略，且中途不中断地重复执行，则即使考虑未来奖励，赌场庄家始终能够盈利。本工作的贡献在于有助于完整揭示赌场盈利的机理，同时为赌场庄家优化赌博机设计、以及为赌徒有效参与赌博活动奠定基础。