This exercise asks you to work through the characterization of all the Nash equilibria of general…

This exercise asks you to work through the characterization of all the Nash equilibria of general two-layer games in which each player has two actions (i.e., 2 × 2 matrix games). This process is time consuming but straightforward and is recommended to the student who is unfamiliar with the mechanics of determining Nash equilibria.

Let the game be as illustrated in figure 1.22.

The pure-strategy Nash equilibria are easily found by testing each cell of the matrix; e.g., (U,L) is a Nash equilibrium if and only if a ≥ e and b ≥d.

To determine the mixed-strategy equilibria requires more work. Let x be the probability player 1 plays U and let y be the probability player 2 plays L. We provide an outline, which the student should complete: (i) Compute each player's reaction correspondence as a function of his opponent's randomizing probability.

(ii) For which parameters is player i indifferent between his two strategies regardless of the play of his opponent?

(iii) For which parameters does player i have a strictly dominant strategy?

(iv) Show that if neither player has a strictly dominant strategy, and the game has a unique equilibrium, the equilibrium must be in mixed strategies.

(v) Consider the particular example illustrated in figure 1.23.

 (a) Derive the best-response correspondences graphically by plotting player i's payoff to his two pure strategies as a function of his opponent's mixed strategy.

(b) Plot the two reaction correspondences in the (x,y) space. What are the Nash equilibria?