In this \(2 \times 2\) game, \(A, B, C, D\) are the payoffs to player I, which are real numbers, no two of which are equal. Similarly, \(a, b, c, d\) are the payoffs to player II, which are real numbers, also no two of which are equal.

(a) Under which conditions does this game have a mixed equilibrium which is not a pure-strategy equilibrium?

(b) Under which conditions in (a) is this the only equilibrium of the game?

(c) Consider one of the situations in (b) and compute the probabilities \(1-p\) and \(p\) for playing Top and Bottom, respectively, and \(1-q\) and \(q\) for playing left and right, respectively, that hold in equilibrium.

(d) For the solution in (c), give a simple formula for the quotients \((1-p) / p\) and \((1-q) / q\) in terms of the payoff parameters. Try to write this formula such that the denominator and numerator are both positive.

(e) Show that the mixed equilibrium strategies do not change if player I's payoffs \(A\) and \(C\) are replaced by \(A+E\) and \(C+E\), respectively, for some constant \(E\), and similarly if player II's payoffs \(a\) and \(b\) are replaced by \(a+e\) and \(b+e\), respectively, for some constant \(e\).