Consider a repeated game with standard information, where each player maximizes his expected 8-discounted average payoff, for some 8 that is close to but less than 1. Payoffs at each round depend on the players' moves as shown in Table 1. Consider the following scenario.

At the first round, and at each round where the outcome of the preceding round was (x₁,x₂) or (y₁,y₂), the players independently randomize, each player i choosing y, with probability q or ; with probability 1 - q. At each round where the outcome of the preceding round was (x₁,y₂), the players choose y₁ and x₂. At each round where the outcome of the preceding round was (y₁,x₂), the players choose x1 and

Write a system of equations that determine the probability q that makes this scenario a subgame-perfect equilibrium, once 8 is specified, as long as 8 is not too small. Show that these equations have a solution where q = .664183 when 8 = .9.

What is the smallest discount factor 8 such that the q determined by these equations will be a subgame-perfect equilibrium?