ECON 451 Homework 5 Each of the following questions is about

ECON 451: Homework #5 Each of the following questions is about a \"yes-no voting system (where it is assumed that every voter has strict preferences) For questions 1-2: The restaurant chain \"Captain Cook\'s Fish and Chips\" has three shareholders: Doug, Nick, and Larry. year, the restaurant had a disastrous ad campaign and lost a lot of money As a result, Doug has proposed that they fire their current vice president of marketing. Last The shareholders must vote to either fire or keep the current VP. Each shareholder gets one vote for each share that he owns. The number of shares each person owns is as follows: Shareholder Shares 1. Assume If more than half of the votes are cast for Doug\' s proposal, it is approved. Otherwise, his proposal is not approved a. Make a list of all of the winning coalitions for the system For each voter, identify the permutations (of the set of voters) where he is pivotal. c. Using your answer in part b, find the Shapley-Shubik index for each of the three voters. d. In how many of the winning coalitions from part a is Doug\' s defection critical? In how many is Nick\' s defection critica1? In how many is Larry\' s defection critical? e Usin the Banzhaf index for each the three voters. Using your answer from part d, find

Solution

1. Total no. of votes = 101+97+2 = 200

Minimum number of votes required for approval of proposal = 200 / 2 +1 = 101

Let, P1, P2 & P3 are Doug, Nick & Larry respectively.

b.

This weighted voting system is represented mathematically as {200 : 101, 97, 2}

There are total 2³-1 i.e. 7 possible coalations, as mentioned below:

{P1} = 101

{P2} = 97

{P3} = 2

{P1, P2} = 101+97 = 198

{P1, P3} = 101+ 2= 103

{P2, P3} = 97 + 2= 99

{P1, P2, P3} = 101+97+2 = 200

Out of which there are three winning coalations, as mentioned below:

{P1, P2} = 101+97 = 198

{P1, P3} = 101+2 = 103

{P1, P2, P3} = 101+97+2 = 200.

b.

List of possible sequential coalations is as below:

{P1 , P2 , P3}
{P1 , P3 , P2}
{P2 , P1 , P3}
{P2 , P3 , P1}
{P3 , P1 , P2}
{P3 , P2 , P1}

In each of these N! i.e. 6 sequential coalitions, the pivotal player is determined and underlined as below:

{P1 , P2 , P3} = 101 + 97 = 198
{P1 , P3 , P2} = 101 + 2 = 103
{P2 , P1 , P3} = 97 + 101 = 198
{P2 , P3 , P1} = 97 + 2 + 101 = 200
{P3 , P1 , P2} = 2 + 101 = 103
{P3 , P2 , P1} = 2 + 97 + 101 = 200

Thus, number of times P1 is pivotal = 4, P2 = 1 and P3 = 1.

c.

Shapley-Shubik Index for Doug = 4/6 =2/3 for Nick = 1/6 & for Larry = 1/6.

d.

In above mentioned winning coaltions, critical members (coalation losses if the member leaves) are underlined as below:

{P1, P2} = 101+97 = 198

{P1, P3} = 101+2 = 103

{P1, P2, P3} = 101+97+2 = 200.

P1 is critical 3 times, P2 is critical 1 time and P3 is critical 1 time.

e.

Thus, total number of times all shareholders are critical = 5

No. of times P1 is critical = 3 and P2 1 time and P3 1 time.

Thus, Banzhaf Index for Doug = 3/5 for Nick = 1/5 & for Larry = 1/5