Video Upload Options

References

1. Vasiljev, Sergei (April 1, 2008), Cardinal Voting: The Way to Escape the Social Choice Impossibility, SSRN eLibrary . Note that in practice, voters could change their votes depending on who is in the race (especially in cardinal voting methods). However, this possibility is ignored, because if it were accounted for, no deterministic method could possibly pass this criterion.
2. Consistency implies participation, but not vice versa. For example, range voting complies with participation and consistency, but median ratings satisfies participation and fails consistency.
3. Woodall, Douglas (December 1994), "Properties of Preferential Election Rules", Voting Matters (3), http://www.votingmatters.org.uk/ISSUE3/P5.HTM 
4. Small, Alex (August 22, 2010), Geometric construction of voting methods that protect voters' first choices, Bibcode: 2010arXiv1008.4331S  http://adsabs.harvard.edu/abs/2010arXiv1008.4331S
5. Approval fails the majority criterion because it does not always elect a candidate preferred by over half of voters; however, it always elects the candidate approved by the most voters.
6. Condorcet, Smith and Independence of Smith-dominated alternatives criteria are incompatible with Independence of irrelevant alternatives, Consistency, Participation, Later-no-harm, Later-no-help, and Favorite betrayal[clarification needed] criteria.
7. In Approval, Range, and Majority Judgment, if all voters have perfect information about each other's true preferences and use rational strategy, any Majority Condorcet or Majority winner will be strategically forced – that is, win in all of one or more strong Nash equilibria. In particular if every voter knows that "A or B are the two most-likely to win" and places their "approval threshold" between the two, then the Condorcet winner, if one exists and is in the set , will always win. These methods also satisfy the majority criterion in the weaker sense that any majority can force their candidate to win, if it so desires. Laslier, J-F (2006), "Strategic approval voting in a large electorate", IDEP Working Papers (Marseille, France: Institut D'Economie Publique) (405), http://halshs.archives-ouvertes.fr/docs/00/12/17/51/PDF/stratapproval4.pdf 
8. Approval voting, range voting, and majority judgment satisfy IIA if it is assumed that voters rate candidates individually and independently of knowing the available alternatives in the election, using their own absolute scale. For this to hold, in some elections, some voters must use less than their full voting power or even abstain, despite having meaningful preferences among the available alternatives. If this assumption is not made, these methods fail IIA, as they become more ranked than rated methods.
9. The original Independence of clones criterion applied only to ranked voting methods. (T. Nicolaus Tideman, "Independence of clones as a criterion for voting rules", Social Choice and Welfare Vol. 4, No. 3 (1987), pp. 185–206.) Tideman notes that "in the spirit of independence of clones", "if there were two or more candidates who were so similar that every voter would rank them as tied if given the chance to rank them [...], then the number of perfect clones present would have no effect on whether the perfect clones were in the set of winning candidates under approval voting". So, Approval Voting satisfies this mathematical criterion by definition. However, there is some disagreement about whether considerations of the voter in the process of making up his vote could be tactically influenced by clones (in a way that a voter would dispossess a candidate of his approval when a clone of him is introduced) and whether the definition of clones have to be extended to these considerations additionally to the handling of actual votes.
10. Later-No-Harm and Later-No-Help assert that adding a later preference to a strictly ordered preference ballot should not help or harm an earlier preference. An Approval ballot records approvals but does not record explicit relative (e.g. later) preferences between approvals (while preferences exist from a voter's perspective). Meanwhile, a voter approving a less preferred candidate harms the probability of any other approved candidate winning, but does not help.
11. If the number of candidates grows faster than the square root of the number of voters, this may not be the case, as ties at any point in the process, even between two non-viable candidates, could affect the final result. If the rule for resolving such ties involves no randomness, though, the method does pass the criterion.
12. The number of piles that can be summed from various precincts is floor ((e−1) N!) − 1.
13. Kemeny-Young does not pass the consistency criterion for winner, but the consistency criterion for full rankings, that is, if the electorate is divided in two parts and in both parts Kemeny-Young chooses the same ranking, Kemeny-Young will also choose that ranking for the combined electorate.
14. Each prospective Kemeny-Young ordering has score equal to the sum of the pairwise entries that agree with it, and so the best ordering can be found using the pairwise matrix.
15. Bucklin voting, with skipped and equal-rankings allowed, meets the same criteria as Majority Judgment; in fact, Majority Judgment may be considered a form of Bucklin voting. Without allowing equal rankings, Bucklin's criteria compliance is worse; in particular, it fails Independence of Irrelevant Alternatives, which for a ranked method like this variant is incompatible with the Majority Criterion.
16. Majority Judgment does not always elect a candidate preferred over all others by over half of voters; however, it always elects the candidate uniquely top-rated by over half of voters.
17. Majority Judgment may elect a candidate uniquely least-preferred by over half of voters, but it never elects the candidate uniquely bottom-rated by over half of voters.
18. Majority Judgment fails the mutual majority criterion, but satisfies the criterion if the majority ranks the mutually favored set above a given absolute grade and all others below that grade.
19. Balinski and Laraki, Majority Judgment's inventors, point out that it meets a weaker criterion they call "grade consistency": if two electorates give the same rating for a candidate, then so will the combined electorate. Majority Judgment explicitly requires that ratings be expressed in a "common language", that is, that each rating have an absolute meaning. They claim that this is what makes "grade consistency" significant. Balinski M, MJ; Laraki, R (2007), A theory of measuring, electing and ranking, 104, USA: National Academy of Sciences, pp. 8720–25 
20. In Majority Judgment, Ranked Pairs, and Schulze voting, there is always a regret-free semi-honest ballot for any voter, holding other ballots constant. That is, if they know enough about how others will vote (for instance, in the case of Majority Judgment, the winning candidate and their winning median score), there is always at least one way for them to participate without grading any less-preferred candidate above any more-preferred one. However, this can cease to hold if voters have insufficient information.
21. Majority judgment can actually pass or fail reversal symmetry depending on the rounding method used to find the median when there are even numbers of voters. For instance, in a two-candidate, two-voter race, if the ratings are converted to numbers and the two central ratings are averaged, then MJ meets reversal symmetry; but if the lower one is taken, it does not, because a candidate with ["fair","fair"] would beat a candidate with ["good","poor"] with or without reversal. However, for rounding methods which do not meet reversal symmetry, the odds of breaking it are comparable to the odds of an irresolvable (tied) result; that is, vanishingly small for large numbers of voters.
22. Majority Judgment is summable at order KN, where K, the number of ranking categories, is set beforehand.
23. Though Majority Judgment does not pass this or similar criteria, there are other similar median methods, such as those based on Bucklin voting, which can meet a related, weaker criterion: ranking an additional candidate below the median grade (rather than your own grade) of your favorite candidate, cannot harm your favorite. Balinski, M., and R. Laraki. “A Theory of Measuring, Electing, and Ranking.” Proceedings of the National Academy of Sciences 104, no. 21 (2007): 8720.
24. In fact, Majority Judgment ballots use ratings expressed in "common language" rather than numbers, that is, that each rating have an absolute meaning.
25. A variant of Minimax that counts only pairwise opposition, not opposition minus support, fails the Condorcet criterion and meets later-no-harm.
26. Since plurality does not allow marking later preferences on the ballot at all, it is impossible to either harm or help a favorite candidate by marking later preferences, and so it trivially passes both Later-No-Harm and Later-No-Help. However, because it forces truncation, it shares some problems with methods that merely encourage truncation by failing Later-No-Harm. Similarly, though to a lesser degree, because it doesn't allow voters to distinguish between all but one of the candidates, it shares some problems with methods which fail Later-No-Help, which encourage voters to make such distinctions dishonestly.
27. Once for each round.
28. That is, second-round votes cannot help or harm candidates already eliminated.
29. Later preferences are only possible between the two candidates who make it to the second round.
30. STAR voting will elect a majority candidate X if X is in the runoff, and X's voters can guarantee they make the runoff by strategically giving the highest score to X and the lowest score to all opponents. However, if there are two or more opponents that get any points from X's voters, these opponents could shut X out of the runoff. Thus, STAR fails the majority criterion.
31. As with the majority criterion, STAR voting fails the mutual majority criterion. However, the more candidates are in the mutual majority set, the greater the chance that at least one of them will be in the runoff, and thus be guaranteed to win.
32. STAR does not define a full outcome ordering, only a winner. With any number of candidates besides 3, the winner cannot stay the same if the ballots are reversed.
33. In STAR voting, in order for favorite betrayal to be strategically advantageous, four separate things must be true: the favorite candidate X must be in the runoff under an honest vote, X must lose the runoff under an honest vote, the betrayal beneficiary Y must not be in the runoff under an honest vote, and the Y must win the runoff under a strategic vote.
34. Sortition, uniformly randomly chosen candidate is winner. Arbitrary winner, some external entity, not a voter, chooses the winner. These methods are not, properly speaking, voting methods at all, but are included to show that even a non-voting method can still pass some of the criteria.
35. Random ballot, uniformly randomly chosen ballot determines winner. This and closely related methods are of mathematical interest because they are the only possible methods which are truly strategy-free, that is, your best vote will never depend on anything about the other voters. However, this method is not generally considered as a serious proposal for a practical method.
36. Poundstone, William (2008), Gaming the Vote: Why Elections Aren't Fair (and What We Can Do About It), New York: Hill and Young, p. 239 
37. WDS, "Appendix" (PDF), Range vote, Temple, archived from the original on March 26, 2009, https://web.archive.org/web/20090326062716/http://math.temple.edu/~wds/homepage/rangevote.pdf 
38. Poundstone 2008, p. 257: "Range voting is still largely a samizdat enterprise on the fringes of social choice theory. The most glaring example must be Smith's pivotal 2000 paper. It has never been published in a journal."
39. Poundstone 2008, p. 240.
40. Balinski, M; Laraki, R (2011) [2007], "Election by Majority Judgement: Experimental Evidence", in Dolez, Bernard; Grofman, Bernard; Laurent, Annie, In Situ and Laboratory Experiments on Electoral Law Reform: French Presidential Elections, Springer 
41. Balinski & Laraki 2007.
42. These two-dimensional graphs are called Yee diagrams after their inventor, Ka-Ping Yee. His website includes some sample graphs. http://zesty.ca/voting/sim/
43. The ballot limited designation is technically possible only if the members of the majority, dividing their votes, cannot determine, individually, the minority appointment: Buonomo, Giampiero (2001). "L''emicollegio' sceglie il rappresentante di minoranza (o di maggioranza)". Diritto&Giustizia edizione online. https://www.questia.com/projects#!/project/89289521.  (Subscription content?)