The Borda and Condorcet Methods:- The following example was presented by Condorcet, in a critique of the Borda Method. A committee composed of 81 members is to choose a winner from among three candidates, A, B, and C. The rankings of the committee members appear in the following table:
|
No of voters
|
First Candidate
|
Second Candidate
|
Third Candidate
|
|
30
|
A
|
B
|
C
|
|
1
|
A
|
C
|
B
|
|
29
|
B
|
A
|
C
|
|
10
|
B
|
C
|
A
|
|
10
|
C
|
A
|
B
|
|
1
|
C
|
B
|
A
|
(a) Is there a Condorcet winner? If yes, who is the Condorcet winner?
(b) What is the Borda ranking, and who is the Borda winner?
(c) Based on your answers above, what is Condorcet's critique of the Borda Method?
A counterclaim to Condorcet's critique might be given by analyzing the preference profile in the following way. A Condorcet component is a set of 3n individuals whose strict preference relations are as follows:
|
No of voters
|
First Candidate
|
Second Candidate
|
Third Candidate
|
|
n
|
a1
|
a2
|
a3
|
|
n
|
a3
|
a1
|
a2
|
|
n
|
a2
|
a3
|
a1
|
The variables a1, a2, a3 are three distinct alternatives (this example can be generalized to an arbitrary number of alternatives). In a certain sense, these voters "neutralize" each other, and they can therefore be removed from the list of voters.
(a) What are all the Condorcet components in this example?
(b) Remove all the individuals appearing in one of the Condorcet components. From among the remaining individuals, find another Condorcet component and remove all the individuals in that component. Repeat this process until there remain no Condorcet components. Is there a Condorcet winner according to the strict preference profile that remains at the end of this process? If so, which candidate is the Condorcet winner?
(c) Given the above two items, elucidate a counterclaim to Condorcet's critique of the Borda Method.