In football the offense can either run the ball or pass the ball, whereas the de
ID: 1091821 • Letter: I
Question
In football the offense can either run the ball or pass the ball, whereas the defense can either anticipate (and prepare for) a run or anticipate (and prepare for) a pass. The defense wants to guess correctly to reduce the yards gained by the offense, whereas the offense wants its opponents to guess incorrectly so that it can gain more yards. Assume that the following table presented the expected payoffs (in yards) for the two teams (Denver Broncos and Seattle Seahawks) on any given down when Denvers offense and Seattles defense were on the field in Super Bowl XLVIII on Sunday, February 2, 2014.Explanation / Answer
a. Suppose the Denver play Run in the pure strategy NE, then it is optimal for Seattle to play Anticipate Run as this will minimize their losses. However, if Seattle were to play Anticipate Run it is not optimal for Denver to play Run. Hence, Run cannot be played by Denver in a pure strategy NE.
On the other hand, if Denver play Pass in the pure strategy NE, then it is optimal for Seattle to play Anticipate Pass as they will gain from this as compared to loosing from playing Anticipate Run. However, if Seattle were to play Anticipate Pass it is not optimal for Denver to play Pass. Hence, Pass cannot be played by Denver in a pure strategy NE.
Hence, there does not exist a Pure Strategy NE of this game.
b. Let the mixed strategy of Denver be:
play Run with probability p and play Pass with probability (1 - p), and
the mixed strategy of the Seattle be:
play Anticipate Run with probability q and play Anticipate Pass with probability (1 - q).
Then, Denver's pay-off from playing Run = q + 5(1 - q) = 5 - 4q
Also, Denver's pay-off from playing Pass = 9q - 3(1 - q) = 12q - 3
As this is a mixed-strategy equilibrium, we must have:
Denver's pay-off from playing Run = Denver's pay-off from playing Pass
=> 5 - 4q = 12q - 3
=> 16q = 8
=> q = 8/16 = 1/2
Likewise, Seattle's pay-off from playing Anticipate Run = Seattle's pay-off from playing Anticipate Pass
=> -p - 9(1 - p) = -5p + 3(1 - p)
=> 8p - 9 = -8p + 3
=> 16p = 12
=> p = 12/16 = 3/4
Mixed Strategy NE
Denver play Run with prob 3/4 and play Pass with probability 1/4
Seattle play Anticipate Run with prob 1/2 and play Anticipate Pass with probability 1/2
c. If Denver played the same strategy as employed by Seattle, then Seattle's pay-off from playing Anticipate Pass would have been greater than when playing Anticipate Run and they would have switched to play Anticipate Pass with probability 1.
d. Expected yards per down = 1q + 5(1 - q) = 1/2 + 5(1/2) = (1/2)(1 + 5) = 6/2 = 3