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# Mathletics: How Gamblers, Managers, and Sports Enthusiasts Use Mathematics in Baseball, Basketball, and Football

By: Wayne Winston | Book details

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Page 158

23
IF PASSING IS BETTER THAN RUNNING,
WHY DON'T TEAMS ALWAYS PASS?
In football the offense selects a play and the defense lines up in a defensive formation. Let's consider a very simple model of play selection in which the offense and defense simultaneously select their play:
 • The offense may choose to run or pass. • The defense may choose a run or pass defense.

The number of yards gained is given in table 23.1, which we call a payoff matrix for the game. We see that if the defense makes the right call on a run, the opposing team loses 5 yards, and if the defense makes the wrong call, the team gains 5 yards. On a pass the right defensive call results in an incomplete pass for the opposing team while the wrong defensive call results in a 10- yard gain for the team. Games in which two players are in total conflict are called two- person zero sum games (TPZSGs). In our game every yard gained by the offense makes the defense one yard worse- off, so we have a TPZSG. The great mathematician James von Neumann and the brilliant economist Oskar Morgenstern discovered the solution concepts for TPZSG.1 We assume the row player wishes to maximize the payoff from the payoff matrix and the column player wants to minimize the payoff from the payoff matrix. We define the value of the game (v) to the row player as the maximum expected payoff the row player can assure himself. Suppose we choose a running play. Then the defense can choose run defense and we lose 5 yards. Suppose we choose a pass offense. Then the defense can choose pass defense and we gain 0 yards. Thus by throwing a pass the offense can ensure themselves of gaining 0 yards. Is there a way the offense can assure that on average they will gain more than 0 yards? A

Von Neumann and Morgenstern, Theory of Games and Economic Behavior.

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