Discrete Mathmatics Application Activity Fair is fair Severa

Discrete Mathmatics

Application Activity: Fair is fair

Several communities have been chosen as finalists for a new sports complex. The complex is seen as a benefit to the community. The possible locations for the complex were recently voted on by three communities and the following preference schedules were produced:

Center City

Fair City

Middleton

Fair City

Middleton

Center City

Middleton

Center City

Fair City

10,500 votes

12,300 votes

15,000 votes

Based on these results, choose two voting methods (approval, Borda, runoff, etc.) to decide on where the recreation center should be built. Justify your reasoning. (10 points)

Use Arrow\'s Criteria to examine the fairness of the results and to examine the methods you used to determine the winner. (10 points)

How would the results be different if the premise was that the cities did NOT want the sports complex? How would this affect the fairness? (10 points)

Solution

Using Borda count Method

BS(Center City) = 2(10500)+1(15000)+0(12300) = 21000+15000 = 36000

BS(Fair City) = 2(12,300)+1(10500)+0(15000) = 24600+10500 = 35100

BS(Middelton) = 2(15000)+1(12300)+0(10500) = 30000+12300 = 42300

As we can see that Borda Score of Middelton is higher so recreation should be built in Middelton according to Borda Count Method

Now using Hare Rule:

In the first round Center City is eliminated since it has got least votes and then this groups votes are transferred to Middelton giving 10500 votes.But then in second round Fair City has least number of votes that is 12.300

So again with Hare Rule also Middelton wins

Arrow Criteria:

a) The majority of first place votes are for Middelton and so majority criteria is satisfied

b) Also Monotonicity criteria is satisfied because suporters of Fair City and Center City change their votes to support

Middelton

c) Similarly Condorcet criteria is also satisfied

So the results are fair