Asked by Peyton Davis on Mar 10, 2024
Verified
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 $5.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 and Bertha each votes for A with probability 1/2 and for B with probability 1/2.
B) Arthur votes for A with probability 5/6 and for B with probability 1/6.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 5/6 and for B with probability 1/6.
D) Arthur votes for A with probability 5/9 and for B with probability 4/9.Bertha votes for A with probability 4/9 and for B with probability 5/9.
E) Arthur votes for A and Bertha votes for B.
Mixed Strategy Equilibrium
A Nash equilibrium where at least one player in a game adopts a probabilistic approach to choosing among two or more strategies.
Company Policy
Guidelines and rules that dictate how various situations should be handled within a business context, directing the operations and decisions of a company.
Money Value
The value or purchasing power of money, often considered in terms of its ability to buy goods and services.
- Master the idea of Nash equilibrium across various strategic frameworks.
- Execute mixed strategy equilibria across different game theory contexts.
- Analyze the composition of payoff matrices and envisage behavioral strategies.
Verified Answer
LZ
Leigh ZurekMar 10, 2024
Final Answer :
B
Explanation :
In a mixed strategy Nash equilibrium, each player's strategy must make the other player indifferent between their options. For Arthur, the expected payoff of voting for A must equal the expected payoff of voting for B, and the same for Bertha. By setting up the equations based on their payoffs and solving, we find that Arthur should vote for A with a probability of 5/6 and for B with a probability of 1/6, while Bertha should vote for A with a probability of 1/5 and for B with a probability of 4/5. This makes each indifferent to the other's choice, satisfying the conditions for a mixed strategy Nash equilibrium.
Learning Objectives
- Master the idea of Nash equilibrium across various strategic frameworks.
- Execute mixed strategy equilibria across different game theory contexts.
- Analyze the composition of payoff matrices and envisage behavioral strategies.