Consider the Matching Pennies game: Suppose Player B always uses a mixed strategy with probability of 3/4 for head and 1/4 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

Question

Consider the Matching Pennies game:  Suppose Player B always uses a mixed strategy with probability of 3/4 for head and 1/4 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

Quizplus · Accepted Answer

To determine the best strategy for Player A, we need to calculate the expected payoffs for each option given Player B's mixed strategy.

A) Mixed strategy with probability 1/4 on heads and 3/4 on tails: 
- If A plays heads (H): (0) * (3/4) + (-1) * (1/4) = -1/4 
- If A plays tails (T): (-1) * (3/4) + (0) * (1/4) = -3/4 
Expected payoff: (-1/4)(1/4) + (-3/4)(3/4) = -5/8

B) Mixed strategy with probability 1/2 on heads and 1/2 on tails: 
- If A plays H: (0) * (1/2) + (-1) * (1/2) = -1/2 
- If A plays T: (-1) * (1/2) + (0) * (1/2) = -1/2 
Expected payoff: (-1/2)(1/2) + (-1/2)(1/2) = -1/2

C) Mixed strategy with probability 3/4 on heads and 1/4 on tails: 
- If A plays H: (0) * (3/4) + (-1) * (1/4) = -1/4 
- If A plays T: (-1) * (3/4) + (0) * (1/4) = -3/4 
Expected payoff: (-1/4)(3/4) + (-3/4)(1/4) = -1/2

D) Pure strategy in which Player A always selects heads: 
- If A plays H: (0) * (3/4) + (-1) * (1/4) = -1/4 
Expected payoff: (-1/4)(1) + (0)(0) = -1/4

Therefore, the option that provides the highest expected payoff is D, with an expected payoff of -1/4.

Consider the Matching Pennies Game: Suppose Player B Always