This would involve looking at new techniques to analyze the classic cops and robber problem. More specifically the use of graph searching to analyze the game with mutiple robbers instead on only one.
Reference: Richerby, David; Thilikos, Dimitrios M. Graph Searching In A Crime Wave. Siam J. Discrete Math. Vol 23, No.1 pp349-368.
Option 2 Stopping Time and Blackjack
This would involve analyzing the concept of exchangeable stopping time and a technique of dynamic programming based upon this concept for fast computation of the expected value of a payoff function upon stopping. Also applications to blackjack.
Reference: Li, Shuo-Yen Robert. Dynamic Progamming By Exchangeability. Siam J. Comput. Vol 18, No.3 pp463-472.
Option 3 Slot Bandit
In the multiarmed bandit problem, a gambler must decide which arm of K nonidentical slot machines to play in a sequence of trials so as to maximize his reward.
Reference: Auer, Peter. The Nonstochastic Multiarmed Bandit Problem. Siam J. Comput. Vol 32, No.1 pp48-77.