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# Solution Methods for Matrix Games

## 2.1   Solution of Some Special Games

#### Problems

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) = 8xx = . 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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