Solution to In-Text Exercise 12.5:
If a player chooses every alternative (Rock, Paper, and Scissors) with positive probability, he
must be indifferent between all three of them. Let’s use and to stand for the probabilities
that his opponent chooses Rock and Paper, respectively (so that the probability of choosing
scissors is
). If he chooses Rock, his expected payoff is:
(
)
If he chooses Paper, his expected payoff is:
(
)
If he chooses Scissors, his expected payoff is:
(
)
Because his expected payoffs from Rock and Paper must be equal, we have:
Rearranging that equation, we discover that
.
Because his expected payoff from Rock and Scissors must also be equal, we have:
Rearranging that equation, we discover that
.
Combining the facts that
and , we have
. It follows that if a player is
indifferent between choosing all three alternatives, his opponent must be playing all three with
probability . Therefore, in a mixed strategy equilibrium, both players choose each alternative
(Rock, Paper, and Scissors) with probability
.