Max-Average Games with Random Payoffs

Consider the following simple 2-person sequential game with i.i.d. payoffs. The 2 players, Max and Average, each have exactly 2 options for each
move. Max plays optimally, i.e., to maximize her payoff, and Average plays randomly. How does the expected payoff for Max depend on the distribution on payoffs?

I will describe the complexity-theoretic motivation for this question, and describe some preliminary results when the distribution on payoffs is Bernoulli.

Joint work with Andy Drucker.