Optimal Strategies from Random Walks

We analyze a sequential game between a Gam- bler and a Casino. The Gambler allocates bets from a limited budget over a fixed menu of gam- bling events that are offered at equal time intervals, and the Casino chooses a binary loss outcome for each of the events. We derive the optimal min-max strategies for both participants. We then prove that the minimum cumulative loss of the Gambler, as- suming optimal play by the Casino, is exactly a well-known combinatorial quantity: the expected number of draws needed to complete a multiple set of "cards" in the generalized Coupon Collec- tor's Problem. We show that this quantity and the optimal strategy of the Gambler can be efficiently estimated from a simple random walk.