Gene says:
As Neller and Presser say, "There is no known general method for solving equations of the form x = max(Ax+b, A'x+b')."
Assuming A, A', b, b' are fixed given numbers, the graph of y = max(Ax+b, A'x+b') is continuous, piecewise linear, and consists of two segments. Its intersection with y = x consists of 0, 1, 2 or points or a continuum, and these are the solutions.
Sorry, I should have beeb clearer. Here x is a vector -- the vector of winning probabilities from every possible game state -- and Ax+b is a linear transformation. And max() is the coordinate-wise maximum of its two vector arguments. Of course it is still piecewise linear, but of 2^dim pieces now... --Michael Kleber -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.