1 1 Explain why the plurality method satisfies the monotonic

1.

1) Explain why the plurality method satisfies the monotonicity criterion.

2) Explain why the Borda count method satisfies the monotonicity criterion.

3) Explain why the method of pairwise comparisons satisfies the Condorcet criterion.

2.

Consider a variation of the Borda count method in which a first-place vote in an election with N candidates is worth F points (where F > N) and all other places in the ballot are the same as in the ordinary Borda count: N ? 1 points for second place, N ? 2 points for third place,..., 1 point for last place. Let M denote the number of voters in this election. By choosing F large enough (depending on M and N) we can make this variation of the Borda count method satisfy the majority criterion. Find the smallest value of F (expressed in terms of M and N) for which this happens.

Solution

1) Explain why the plurality method satisfies the monotonicity criterion:

Let us assume that Candidate X receives more first place votes than any other candidate. We also know that in a re-election, changes on ballots are the changes in which X’s position gets better. In particular, in the re-election X receives as many first place votes as he did in the original election, and the other candidates would not receive more first place votes than they did in the original election, so X would receives more first place votes than any other candidate, hence X still wins using Plurality. Hence, Plurality satisfies the Monotonicity Criterion.

2) Explain why the Borda count method satisfies the monotonicity criterion:

Let us assume candidate X receives the highest Borda point total in the original election, than what other candidates receive. We know that in a re-election, the only changes on ballots are changes in which X’s position will get better than it\'s previous position. We also know that, in the re-election, X’s Borda point total would be at least as high as in the original election. Hence the rest of the candidates either stay in the same position as before, or drop a position due to being surpassed by X. Thus, in the re-election, the remaining candidates’ point total are no greater than they were in the original election. Therefore, X still has the highest Borda point total, so X wins the re-election. This suggests that Borda Count satisfies the Monotonicty Criterion.

3) Explain why the method of pairwise comparisons satisfies the Condorcet criterion:

A Condorcet candidate wins all its head to head comparisons with other candidates and thus have the highest total.