Asked by Alexis Klein on May 24, 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 $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.
Mixed Strategy Equilibrium
A concept in game theory where players use a random mixture of actions in strategic games, ensuring that no player has an incentive to deviate from their strategy given the strategies of the other players.
Company Policy
A set of rules and guidelines created by a company to outline its operations, decision-making processes, and ethical standards.
Money Value
Refers to the purchasing power of money, indicating how much goods or services a unit of money can buy.
- Comprehend the notion of Nash equilibrium within different strategic contexts.
- Implement mixed strategy equilibria within the scope of game theory scenarios.
- Comprehend the structure of payoff matrices and anticipate strategic behaviors.
Verified Answer
NV
Naresh VeeravalliMay 30, 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 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.
Learning Objectives
- Comprehend the notion of Nash equilibrium within different strategic contexts.
- Implement mixed strategy equilibria within the scope of game theory scenarios.
- Comprehend the structure of payoff matrices and anticipate strategic behaviors.