Borda count Consider an election in which there are two candidates, A and B. Suppose the voters have the following preferences: Candidate A would receive 66% Borda points (66%×1 + 34%×0) and B would receive 34% (66%×0 + 34%×1). Thus candidate A would win by a 66% landslide. Now suppose supporters of B nominate an additional candidate, B2, that is very similar to B but considered inferior by all voters. For the 66% who prefer A, B continues to be their second choice. For the 34% who prefer B, A continues to be their least preferred candidate. Now the voters' preferences are as follows: Candidate A now has 132% Borda points (66%×2 + 34%×0). B has 134% (66%×1 + 34%×2). B2 has 34% (66%×0 + 34%×1). The nomination of B2 changes the winner from A to B, overturning the landslide, even though the additional information about voters' preferences is redundant due to the similarity of B2 to B. Similar examples can be constructed to show that
given the Borda count, any arbitrarily large landslide can be overturned by adding enough candidates (assuming at least one voter prefers the landslide loser). For example, to overturn a 90% landslide preference for A over B, add 9 alternatives similar/inferior to B. Then A's score would be 900% (90%×10 + 10%×0) and B's score would be 910% (90%×9 + 10%×10). No knowledge of the voters' preferences is needed to exploit this strategy. Factions could simply nominate as many alternatives as possible that are similar to their preferred alternative. In typical elections, game theory suggests this manipulability of Borda can be expected to be a serious problem, particularly when a significant number of voters can be expected to vote their sincere order of preference (as in public elections, where many voters are not strategically sophisticated; cite Michael R. Alvarez of Caltech). Small minorities typically have the power to nominate additional candidates, and typically it is easy to find additional candidates that are similar. In the context of people running for office, people can take similar positions on the issues, and in the context of voting on proposals, it is easy to construct similar proposals. Game theory suggests that all factions would seek to nominate as many similar candidates as possible since the winner would depend on the number of similar candidates, regardless of the voters' preferences.
Copeland These examples show that Copeland's method violates the Independence of clones criterion.
Crowding Copeland's method is vulnerable to crowding, that is the outcome of the election is changed by adding (non-winning) clones of a non-winning candidate. Assume five candidates A, B, B2, B3 and C and 4 voters with the following preferences: Note, that B, B2 and B3 form a clone set.
Clones not nominated If only one of the clones would compete, preferences would be as follows: The results would be tabulated as follows: • [X] indicates voters who preferred the candidate listed in the column caption to the candidate listed in the row caption • [Y] indicates voters who preferred the candidate listed in the row caption to the candidate listed in the column caption
Result: C has one win and no defeats, A has one win and one defeat. Thus,
C is elected Copeland winner.
Clones nominated Assume, all three clones would compete. The preferences would be the following: The results would be tabulated as follows:
Result: Still, C has one win and no defeat, but now A has three wins and one defeat. Thus,
A is elected Copeland winner.
Conclusion A benefits from the clones of the candidate he defeats, while C cannot benefit from the clones because C ties with all of them. Thus, by adding two clones of the non-winning candidate B, the winner has changed. Thus, Copeland's method is vulnerable against crowding and fails the independence of clones criterion.
Teaming Copeland's method is also vulnerable against teaming, that is adding clones raises the winning chances of the set of clones. Again, assume five candidates A, B, B2, B3 and C and 2 voters with the following preferences: Note, that B, B2 and B3 form a clone set.
Clones not nominated Assume that only one of the clones would compete. The preferences would be as follows: The results would be tabulated as follows:
Result: A has one win and no defeats, B has no wins or defeats so
A is elected Copeland winner.
Clones nominated If all three clones competed, the preferences would be as follows: The results would be tabulated as follows:
Result: A has one win and no defeat, but now B has two wins and no defeat. Thus,
B is elected Copeland winner.
Conclusion B benefits from adding inferior clones, while A cannot benefit from the clones because he ties with all of them. So, by adding two clones of B, B changed from loser to winner. Thus, Copeland's method is vulnerable against Teaming and fails the Independence of clones criterion.
Plurality voting Suppose there are two candidates, A and B, and 55% of the voters prefer A over B. A would win the election, 55% to 45%. But suppose the supporters of B also nominate an alternative similar to A, named A2. Assume a significant number of the voters who prefer A over B also prefer A2 over A. When they vote for A2, this reduces A's total below 45%, causing B to win.
Range voting Range voting satisfies the independence of clones criterion under the conditions that it satisfies
independence of irrelevant alternatives. Whenever the voters use an absolute scale that does not depend on the candidates running, range voting satisfies IIA and thus is also clone-independent. However, if the voters use relative judgments, then their ratings of different candidates can change as clones drop out, which can lead range voting to fail clone independence. This can be seen by a simple example: In range voting, the voter can give the maximum possible score to their most preferred alternative and the minimum possible score to their least preferred alternative. This can be done strategically or just as a natural way of anchoring one's ratings to the candidates that matter in the election. Begin by supposing there are 3 alternatives: A, B and B2, where B2 is similar to B but considered inferior by the supporters of A and B. The voters supporting A would have the order of preference "A>B>B2" so that they give A the maximum possible score, they give B2 the minimum possible score, and they give B a score that's somewhere in between (greater than the minimum). The supporters of B would have the order of preference "B>B2>A", so they give B the maximum possible score, A the minimum score and B2 a score somewhere in between. Assume B narrowly wins the election. Now suppose B2 isn't nominated. The voters supporting A who would have given B a score somewhere in between would now give B the minimum score while the supporters of B will still give B the maximum score, changing the winner to A. This teaming effect violates the criterion. Note, that if the voters that support B would prefer B2 to B, this result would not hold, since removing B2 would raise the score B receives from his supporters in an analogous way as the score he receives from the supporters of A would decrease. The conclusion that can be drawn is that considering all voters voting in a certain relative way, range voting creates an incentive to nominate additional alternatives that are similar to one you prefer, but considered clearly inferior by his voters and by the voters of his opponent, since this can be expected to cause the voters supporting the opponent to raise their score of the one you prefer (because it looks better by comparison to the inferior ones), but not his own voters to lower their score.
Approval voting The analysis of
approval voting is more difficult, since the independence of clones criterion involves rankings and approval ballots contain less information than ranked ones.
Clones nominated The finalists are Amy and her clone, and Amy's clone wins. ==See also==