Question

The Borda Count is a common way of making a choice among more than two alter- natives. As defined in the text, each member of society assigns a rank to the social alternatives (1,2, . . . ), with 1 corresponding to first choice, and so on. The ranks each alternative gets are summed over all the individuals. The alternative with the lowest sum wins; the second choice has the second-lowest sum, and so forth. Set up an example to show that the Borda Count violates the independence of irrelevant alter- natives. Hint: Use a three-person society with three alternatives and rank them using a Borda Count. Then introduce a fourth alternative.

Answer 1

Let A, B, C be 3 alternatives

Let 1, 2, 3 be ranks

Let W, X, Y be the 3 persons in the society

Now, let preference of W for (A, B, C) be A > B > C such that rank of A = 1, B = 2 and C = 3 for W

And, let preference of X for (A, B, C) be B > A > C such that rank of A = 2, B = 1 and C = 3 for X

And, let preference of Y for (A, B, C) be C > B > A such that rank of A = 3, B = 2 and C = 1 for Y

Total rank count for A = 1+ 2 + 3 = 6

Total rank count for B = 2+ 1 + 2 = 5

Total rank count for C = 3+ 3 + 1 = 7

Hence, total social rank count for B’s 5 is less than A’S 6 and C’s 7 so B is the first option, A second option and C third option based on given borda count

Now, let D be the 4^th new alternative option such that ranks are 1, 2, 3, 4 for 4 options

And let preference of W for (A, B, C, D) be A > D > C > B such that rank of A = 1, B = 4, C = 3 and D = 2

And, let preference of X for (A, B, C, D) be B > A > D > C such that rank of A = 2, B = 1, C = 4 and D = 3

And, let preference of Y for (A, B, C, D) be C > B > A > D such that rank of A = 3, B = 2, C = 1 and D = 4

Total rank count for A = 1+ 2 + 3 = 6

Total rank count for B = 4 + 1 + 2 = 7

Total rank count for C = 3 + 4 + 1 = 8

Total rank count for D = 2 + 3 + 4 = 9

Hence, total social rank count for A’s 6 is less than B’s 7, C’s 8 and D’s 9

so A is the first option, B second, C third and D fourth based on given borda count

Note that addition of a new alternative D changed the relative preference of B over C by person W resulting in better aggregate social rank of alternative A over B based on boda count- thereby violating principle of independence of irrelevant alter- natives.