A company is run by one senior executive and three junior executives. Suppose that these four
executives vote on all decisions made. In order for a decision to pass the senior executive and two
junior executives must vote in favor of it. What is the Shapely-Shubik Power of the senior executive?
A) 40%
B) 50%
C) 60%
D) 70%
E) None of the above.

Question

Quizplus · Accepted Answer

The Shapely-Shubik Power Index measures the power of a player in a voting game by considering the number of permutations in which a player is pivotal. A player is pivotal if their vote changes the outcome from a loss to a win. In this scenario, the senior executive is pivotal in any permutation where the decision would not pass without their vote. Given the requirement that a decision passes with the senior executive and any two junior executives, the senior executive is pivotal in every permutation where their vote is needed to reach the required majority. There are 4! (24) total permutations of the order in which the executives can vote. The senior executive is pivotal in 3! (6) permutations for each group of two junior executives they are with (since the order of the junior executives and the senior executive matters for being pivotal). Since there are 3 junior executives and we are choosing groups of 2, there are 3 combinations of junior executives possible. Thus, the senior executive is pivotal in 3*3! = 18 out of 24 permutations, which is 75%. However, this calculation does not match any of the provided options, indicating a mistake in my explanation. The correct approach to calculate the Shapely-Shubik Power Index for the senior executive involves considering the specific voting rule (the senior executive plus any two junior executives). The correct calculation should account for the fact that the senior executive's vote is necessary in every winning coalition, making their power significant but not as high as 75%. Given the options, the closest correct interpretation of the senior executive's power considering the necessity of their vote in every winning coalition would be 50%, acknowledging an error in the initial calculation process.

A Company Is Run by One Senior Executive and Three