Opera

Player 2 Left Right

Player 1

Up Down

Player 2 Left Right

x, —

y,~

z, —

strategies. We turn next to a graphical argument that any such game has a Nash equilibrium (possibly involving mixed strategies).

Consider the payoffs for player 1 given in Figure 1.3.8. There are two important comparisons: x versus z, and y versus w. Based on these comparisons, we can define four main cases: (i) x > z and y > w, (ii) x < z and y < w, (iii) x > z and y < w, and (iv) x < z and y > w. We first discuss these four main cases, and then turn to the remaining cases involving x — z or y = w.

(Left)

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