Each of the following questions is about a \"yes-no voting system\" (where it is

Question

Each of the following questions is about a "yes-no voting system" (where it is assumed that every voter has strict preferences). For questions 4-6: The restaurant chain "Bay Crab House" has three shareholders: Kristen, Dave and Garrett. The chain currently has all of its restaurants in Maryland. Kristen has proposed expanding the company by opening up a restaurant in Delaware. The shareholders must vote to either expand or not expand. Each shareholder gets one vote for each share that he or she owns. If more than half of the votes are cast for Kristen's proposal, it is approved. Otherwise, her proposal is not approved. 4. The number of shares each person owns is as follows: a. Find the Shapley-Shubik index for each of the three voters. b. Find the Banzhaf index for each the three voters.

Explanation / Answer

a)

Suppose decisions are made by majority rule in a body consisting of A, B, C who have 58, 36, 20 votes, respectively. The majority vote threshold is 58. There are 3! = 6 possible orders for these members to vote:

[A being Kristen, B being Dave and C being Garrett].

ABC, ACB, BAC, BCA, CAB, CBA.

For each voting sequence the pivot voter – that voter who first raises the cumulative sum to 58 or more – is bolded. Here, A is pivotal in 6 of the 6 sequences. Therefore,

A has an index of power (6/6) = 1.

B has an index of power = 0/6 = 0

C has an index of power = 0/6 = 0

b)

total no. of votes = 58+36+20 = 114

no. of votes to pass = 114/2 + 1 = 58

[58: 58, 36, 20]

The numbers in the brackets mean a measure requires 58 votes to pass, and voter A can cast 58 votes, B 36 votes, C 20, . The winning groups, with underlined swing voters, are as follows:

A, AB, AC, ABC

There are total of 4 swing votes.

Banzhaf index of a group = (no. of swing votes of that group) / (total swing votes)

Banzhaf index of A = 4/4 = 1

Banzhaf index of B = 0/4 = 0

Banzhaf index of C = 0/4 = 0

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