My friends and I are trying to decide where to go on vacation this summer. Paris (P), Italy (I), or Spain (S). Below is a table of our votes:  RachelJulieToniPatSam1st ChoiceIPISS2nd ChoiceSIPPI3rd ChoicePSSIP  Show how {P, I or S} wins using the Borda count methodShow how {P, I or S} wins using the plurality-with-elimination methodShow how {P, I or S} wins using the pairwise comparison methodShow how {P, I or S} wins using the majority vote method

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Asked Jun 18, 2019
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My friends and I are trying to decide where to go on vacation this summer. Paris (P), Italy (I), or Spain (S). Below is a table of our votes:

 

 

Rachel

Julie

Toni

Pat

Sam

1st Choice

I

P

I

S

S

2nd Choice

S

I

P

P

I

3rd Choice

P

S

S

I

P

  

Show how {P, I or S} wins using the Borda count method

Show how {P, I or S} wins using the plurality-with-elimination method

Show how {P, I or S} wins using the pairwise comparison method

Show how {P, I or S} wins using the majority vote method

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Expert Answer

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Step 1

Given table of votes of four friends about place to go on vacation Spain (S), Italy (I), Paris (P) as shown below

According to Borda count method 
There are total 3 destiantions and one has to be chossed among them, so n = 3
All first choice are given  3 points
Second choice are given 2 points
Third choice are given 1 points
The destination with more number of points will be the elected destination.

The total number of points for Paris = 3+2+2+1+1 = 9
The total number of points for Italy = 3+2+2+1+3 = 11
The total number of points for Spain = 3+3+1+1+2 = 10
Among the destinations Italyhave highest points by borda count method. So according to Borda count method the destination for this summer vacation is Italy.

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Rachel Julie Toni Pat Sam 1st Choice S S 3 points P 2 points 2nd Choice S P P I 1 point 3rd Choice P S S P

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Step 2

Plurality with elimination method check first choice and if there is any majority then it is declared as winner or else eliminate the destination with least first choice and go to second choice and if there is any majority then declare it as winner or else eliminate destination with least votes and  go to third choice and so on.

Here we can see in first choice for destination clearly Spain and Italy has equal preferences ( Spain first choice of Pat and Sam, while Italy first choice of Rachel and Toni). So we eliminate Paris as it has least preferences as first choice.
Now in second choice among Spain and Italy, Italy has majority as Julie and Sam prefer Italy while only Rachel prefer Spain. So according to plurality with elimination method Italy will be the destination for this summer vacation.

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Rachel Julie Toni Pat Sam 1s choice S S P 2nd choice S P PI 3'd Choice P S S I P

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Step 3

In pairwise comparison method we conduct two destinations against each other head to head.

There are total three destinations. So n = 3
The total number of comparisons = n(n-1)/2 = 3(2)/2 = 3

Paris vs Spain
Rachel, Pat and Sam prefer Spain over Paris where only Julie and Toni prefer Paris over Spain. So between Paris and Spain, Spain has more preferences than Paris. So Spain get one point.

Paris vs Italy
Rachel, Toni and Sam prefer Italy over Paris, where only Julie and Pat prefer ...

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Rachel Julie Toni Pat Sam s S 1st choice P Italy Spain Paris 1 0 2nd choice S P PI 3rd Choice P SI S

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