Arthur and Bertha are asked by their boss to vote on a company policy.Each of them will be allowed to vote for one of three possible policies, A, B, and C.Arthur likes A best, B second best, and C least.Bertha likes B best, A second best, and C least.The money value to Arthur of outcome C is $0, outcome B is $1, and outcome A is $4.The money value to Bertha of outcome C is $0, outcome B is $4, and outcome A is $1.The boss likes outcome C best, but if Arthur and Bertha both vote for one of the other outcomes, he will pick the outcome they voted for.If Arthur and Bertha vote for different outcomes, the boss will pick C.Arthur and Bertha know this is the case.They are not allowed to communicate with each other, and each decides to use a mixed strategy in which each randomizes between voting for A or for B.What is the mixed strategy equilibrium for Arthur and Bertha in this game?
A)Arthur votes for A with probability 4/8 and for B with probability 4/8.Bertha votes for A with probability 4/8 and for B with probability 4/8.
B)Arthur votes for A with probability 4/5 and for B with probability 1/5.Bertha votes for A with probability 1/5 and for B with probability 4/5.
C)Arthur votes for A with probability 1/5 and for B with probability 4/5.Bertha votes for A with probability 4/5 and for B with probability 1/5.
D)Arthur and Bertha each votes for A with probability 1/2 and for B with probability 1/2.
E)Arthur votes for A and Bertha votes for B.

Question

Quizplus · Accepted Answer

In a mixed strategy Nash equilibrium, each player's strategy must make the other player indifferent between their options. For Arthur, the expected value of voting for A must equal the expected value of voting for B, given Bertha's strategy. Similarly, for Bertha, the expected value of voting for A must equal the expected value of voting for B, given Arthur's strategy. By solving the system of equations that equates these expected values, we find that Arthur voting for A with a probability of 4/5 and for B with a probability of 1/5, and Bertha voting for A with a probability of 1/5 and for B with a probability of 4/5, makes each other indifferent between voting for A or B, thus establishing the mixed strategy equilibrium.

Arthur and Bertha Are Asked by Their Boss to Vote