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 
1^{st} Choice 
I 
P 
I 
S 
S 
2^{nd} Choice 
S 
I 
P 
P 
I 
3^{rd} 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 pluralitywithelimination 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
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.
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.
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(n1)/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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