Question

Consider the Matching Pennies game:

	Player B- heads	Player B- tails
Player A- heads	1, -1	-1, 1
Player A- tails	-1, 1	1, -1

Suppose Player B always uses a mixed strategy with probability of 1/2 for head and 1/2 for tails. Which of the following strategies for Player A provides the highest expected payoff?

A) Mixed strategy with probability 1/4 on heads and 3/4 on tails

B) Mixed Strategy with probability 1/2 on heads and 1/2 on tails

C) Mixed Strategy with probability 3/4 on heads and 1/4 on tails

D) Pure strategy in which Player A always selects heads

E) Pure strategy in which Player A always selects tails

Answer 1

B) Mixed Strategy with probability 1/2 on heads and 1/2 on tails

Let player A choose heads with probability p and tails with probability 1-p. So, if player B best responds with a mixed strategy then player A must make him indifferent between his strategies so that his expected payoff from his two strategies is equal. That is,
-1p + 1(1-p) = 1p + (-1)(1-p)
So, -p + 1 - p = p -1 + p
So, 2p + 2p = 2
So, 4p = 2
So, p = 2/4 = 1/2

and 1-p = 1-(1/2) = 1/2

Thus, mixed Strategy with probability 1/2 on heads and 1/2 on tails would give highest expected payoff.