When Random Play is Optimal Against an Adversary

We analyze a sequential game between a Gambler and a Casino. The Gambler allocates bets from a limited budget over a fixed menu of gambling events that are offered at equal time intervals. The Casino chooses a binary loss outcome for each of the Gambler's bets. We derive the optimal minmax strategies for both participants and we find that the minimum cumulative loss of the Gambler given optimal play of the Casino leads to a wellknown combinatorial quantity: the expected number of draws needed to complete a multiple set of "cards" in the classical generalized Coupon Collector's Problem. Connections are also drawn with other random processes, such as the random positive walk on an n-dimensional finite hypercube, and the stages of an evolving random graph. We show that the optimal strategy of the Gambler is based on a random playout of the game from the current state and can be efficiently estimated.

• Type: Conference
• Year: 2008
• Where: COLT
• Authors: Jacob Abernethy, Manfred K. Warmuth, Joel Yellin
• Keywords: COLT 2008, Gambler, Gambler Allocates Bets, Gambler's Bets, Machine Learning