An Example Regarding the Borda Count

Here is an example taken from “Social Choice Theory – A Quick Overview” which I found on the Internet which purportedly shows deficiencies in the Borda Count.

“In this example the Borda count does select the Condorcet winner and it is decisive (although it is not always decisive). The Borda count, however, is susceptible to insincere voting. Imagine that the five voters honestly assess their preferences for the four policies as shown in the table on the left below. The Borda count would result in policy W being selected. But, if voter II, whose first choice is policy X, insincerely switches his ordering of policies W and Z (that is W is ranked 1 while Z is ranked 3), the Borda count changes such that policy X is selected. Thus by one person changing their preference for policies that are not even his optimal choice, the social choice is changed.

"Honest" Preferences Insincere Preferences

Voters: Voters:

I II III IV V Borda I II III IV V Borda

 

W  3 3 2 3 4  15 W  3 1 2 3 4  13

X  4 4 3 1 2  14 X  4 4 3 1 2  14

Y  1 2 4 2 1  10 Y  1 2 4 2 1  10

Z  2 1 1 4 3  11 Z  2 3 1 4 3  13

Au contraire, one can dissect this example as follows:

Let us assume that voter II wants x to win really bad which is the source of the insincere voting. Then why would they switch w and z?! The proper course would be to give x a ranking of 4 and to give w, y and z a tied ranking of 1 if ties were to be allowed (which they’re not with a strict Borda count.) In other words x would be ranked first and the other candidates would be tied for last. This would accomplish voter II’s objective. To accomplish voter II’s objective in the way the example suggests, voter II would have to know how the other voters’ had voted or at least the vote totals. In any respectable election this would be impossible. Therefore, if voter II wants to go all out for x, he would vote strategically as I have suggested. However, this way of voting has its risks as well as its potential rewards. Since voter II (we may assume) does not know the outcome of the election, he takes the chance that by lowering his sincere preference for w, neither w nor x may win. In fact z, who he ranks last, could win. Does he really want to take this chance? After all w is his second choice

A better strategy might be to lower his ranking of y to a tie with z for last. However, this has the effect of elevating z in a close race between y and z, and, after all, voter II prefers y to z. So whenever a voter votes insincerely, it is not without risk, but, if he wants to take the risk, I say that is his prerogative.

The point is that this example, and others like it, in no way invalidates the Borda count as a reasonable way of conducting elections especially when the Borda count is altered to allow ties.