I explore a game where two players simultaneously choose between two actions in finitely many stages. Players have incomplete information about their payoffs. In each stage, they receive a noisy signal about a parameter that determines their payoffs; the signal's noise is a random variable that is uniformly distributed over the closed interval from 0 to 1 and independent of the payoff parameter. Players share a common prior belief that the payoff parameter is uniformly distributed over the closed interval from 0 to 1 and they commonly update this belief given the full history of actions. Players are restricted to "cutoff" strategies: They play action "1" if they see that the average of their signals up to that stage is strictly above a cutoff value, and they play action "0" otherwise. I describe how players obtain cutoffs that are optimal given the publicly known posterior beliefs about the payoff parameter, their privately observed average of signals in each stage, and the opposing player's cutoff strategy. Each player is further restricted to choosing the maximum, "pessimistic," cutoff if multiple exist. I give the exact pessimistic equilibrium cutoffs in Stages 1 and 2 for all possible average of signals. Using Monte Carlo integration methods, I simulate this game and approximate the pessimistic cutoffs for each player in Stages 3 and beyond. I give a lower bound on the possible values of any nonzero pessimistic cutoffs for each player. I show that there is a range of average signals where each player almost always plays action "1." With certain assumptions, I give a range of signal averages for any stage such that players will always choose action "0" when a nonzero cutoff exists and I provide numerical evidence for this property. Finally, I look at two alternative equilibria---one where each player always chooses a cutoff of 0 and one where each player chooses the smallest nonzero cutoff when multiple nonzero cutoffs exist.