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 min- max strategies for both participants and we find that the minimum cumulative loss of the Gambler given optimal play of the Casino leads to a well- known combinatorial quantity: the expected num- ber of draws needed to complete a multiple set of "cards" in the classical generalized Coupon Col- lector's Problem. Connections are also drawn with other random processes, such as the random pos- itive 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.