**2.1** In a simplified analysis of a football game, suppose that the offense can only choose a pass or run, and the defense can choose only to defend a pass or run. Here is the matrix in which the payoffs are the average yards gained:

The offense’s goal is to maximize the average yards gained per play. Find *v*(*A*) and the optimal strategies using the explicit formulas. Check your answers by solving graphically as well.

**2.1 Answer:** If we use the 2 × 2 formulas, we get det(*A*) = –80, , so

If we use the graphical method, the lines cross where *x* + 10(1 − *x*) = 8*x* ⇒ *x* = . The rest also checks.

**2.2** Suppose that an offensive pass against a run defense now gives 12 yards per play on average to the offense (so the 10 in the previous matrix changes to a 12). Believe it or not, the offense should pass less, not more. Verify that and give a game theory (not math) explanation of why this is so.

**2.2 Answer:** Using the formulas ...

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