A player confronts the following situation. A coin will be tossed at every time t, t = 1, 2, 3, . . . , T and the player will get a total reward Wt. He or she can either decide to stop or to continue to play. If he or she continues, a new coin will be tossed at time t + 1, and so on. The question is, what is the best time to stop? We consider several cases. We begin with the double-or-nothing game. The total reward received at time t = T is given by:

where the zt is a binomial random variable:

Thus, according to this, the reward either doubles or becomes zero at every stage.
(a) Can you calculate the expected reward at time T,E[WT], given this information?
(b) What is the best time to stop this game?
(c) Suppose now we sweeten the reward at every stage and-we-multiply the-WT by a number that increases and is greater than one. In fact, suppose the reward is now given by:

with T = 1, 2, 3, . . . Show that the expected reward if we stop at some time Tk is given by:

(Here, Tk is a stopping time such that one stops after the kth toss.)
(d) What is the maximum value this reward can reach?
(e) Is there an optimal stopping rule?

! with probability1 Zt-1-1 with probability 2n WT +1) 2k 221 k+1