The old Michigan football coach had only two strategies, Run the Ball to the Left Side of the line, and Run the Ball to the Right Side.The defense can concentrate either on the left side or the right side of Michigan's line.If the opponent concentrates on the wrong side, Michigan is sure to gain at least 5 yards.If the defense defended the left side and Michigan ran left, Michigan would be stopped for no gain.But if the opponent defended the right side when Michigan ran right, Michigan would still gain at least 5 yards with probability .50.It is the last play of the game and Michigan needs to gain 5 yards to win.Both sides choose Nash equilibrium strategies.In Nash equilibrium, Michigan would
A)run to the two sides with equal probability.
B)be sure to run to the right side.
C)run to the right side with probability .67.
D)run to the right side with probability .80.
E)run to the right side with probability .50.

Question

Quizplus · Accepted Answer

In a Nash equilibrium, each player's strategy is optimal given the other player's strategy. Here, Michigan needs to mix its strategies to keep the defense guessing. If Michigan runs to the right, it has a 50% chance of gaining at least 5 yards even if the defense also focuses on the right. To make the defense indifferent between defending left or right, Michigan should choose a probability that makes the expected gain from either defensive choice equal. If Michigan runs right with probability p, it should run left with probability 1-p. The defense stops the run for no gain if it guesses correctly. For the defense to be indifferent, the expected gain from defending either side should be the same: 0.5p = (1-p)0, solving for p gives p = 2/3 or approximately 0.67. Thus, Michigan should run to the right side with a probability of 0.67 to maintain a Nash equilibrium.

The Old Michigan Football Coach Had Only Two Strategies, Run