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 .70.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) be sure to run to the right side.
B) run to the right side with probability .77.
C) run to the right side with probability .87.
D) run to the two sides with equal probability.
E) run to the right side with probability .70.

Question

serita steele · Accepted Answer

Final Answer : B, Explanation :   In a Nash equilibrium, each player's strategy is optimal given the other player's strategy. Let   p  p  p   be the probability that Michigan runs to the right, and   1 − p  1-p  1 − p   the probability they run to the left. The defense will choose to defend the right with probability   q  q  q   and the left with probability   1 − q  1-q  1 − q  . For Michigan to be indifferent between running left or right, the expected gain from both must be equal. If Michigan runs right, the expected gain is   0.7 × 5 q + 5 ( 1 − q ) = 3.5 q + 5 − 5 q = 5 − 1.5 q  0.7 	imes 5q + 5(1-q) = 3.5q + 5 - 5q = 5 - 1.5q  0.7 × 5 q + 5 ( 1 − q ) = 3.5 q + 5 − 5 q = 5 − 1.5 q  . If Michigan runs left, the expected gain when the defense guesses wrong is 5, which happens with probability   q  q  q  , so the gain is   5 q  5q  5 q  . Setting these equal to make Michigan indifferent:   5 − 1.5 q = 5 q  5 - 1.5q = 5q  5 − 1.5 q = 5 q   leads to   6.5 q = 5  6.5q = 5  6.5 q = 5  , so   q = 5 6.5 = 10 13  q = \frac{5}{6.5} = \frac{10}{13}  q = 6.5 5 ​ = 13 10 ​  . Since the defense is also indifferent, Michigan's probability   p  p  p   of running right should make the defense's expected loss the same whether they defend left or right, leading to the same calculation for   p  p  p  . Thus, Michigan runs to the right with probability   10 13  \frac{10}{13}  13 10 ​  , which is approximately 0.77.

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