79 research outputs found
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.Comment: 47 page
- …