1653:
924:
59:
899:
911:
1557:
In this summary matrix, the largest ranking score equals the sum of the counts in the upper-right, triangular half of the matrix (shown here in bold, with a green background). No other possible ranking can have a summary matrix that yields a higher sum of numbers in the upper-right, triangular half.
1561:
In this summary matrix, the sum of the numbers in the lower-left, triangular half of the matrix (shown here with a red background) are a minimum. The academic papers by John Kemeny and Peyton Young refer to finding this minimum sum, which is called the Kemeny score, and which is based on how many
4665:
In
Highest median, Ranked Pairs, and Schulze voting, there is always a regret-free, semi-honest ballot for any voter, holding all other ballots constant and assuming they know enough about how others will vote. Under such circumstances, there is always at least one way for a voter to participate
988:
This method assigns a score for each possible sequence, where each sequence considers which choice might be most popular, which choice might be second-most popular, which choice might be third-most popular, and so on down to which choice might be least-popular. The sequence that has the highest
1093:
These preferences can be expressed in a tally table. A tally table, which arranges all the pairwise counts in three columns, is useful for counting (tallying) ballot preferences and calculating ranking scores. The center column tracks when a voter indicates more than one choice at the same
1425:
After the overall ranking has been calculated, the pairwise comparison counts can be arranged in a summary matrix, as shown below, in which the choices appear in the winning order from most popular (top and left) to least popular (bottom and right). This matrix layout does not include the
1049:
In order to demonstrate how an individual preference order is converted into a tally table, it is worth considering the following example. Suppose that a single voter has a choice among four candidates (i.e. Elliot, Meredith, Roland, and Selden) and has the following preference order:
3117:
sometimes allowed the computation of full rankings for votes on as many as 40 candidates in seconds. However, certain 40-candidate 5-voter Kemeny elections generated at random were not solvable on a 3 GHz
Pentium computer in a useful time bound in 2006.
1218:
Now suppose that multiple voters had voted on those four candidates. After all ballots have been counted, the same type of tally table can be used to summarize all the preferences of all the voters. Here is an example for a case that has 100 voters:
1347:
After the tally table has been completed, each possible ranking of choices is examined in turn, and its ranking score is calculated by adding the appropriate number from each row of the tally table. For example, the possible ranking:
1034:
The ranking that has the largest score is identified as the overall ranking. (If more than one ranking has the same largest score, all these possible rankings are tied, and typically the overall ranking involves one or more ties.)
5233: — A website that calculates Kemeny–Young results, and gives further explanation and examples of the concept. It also calculates the winner according to plurality, Borda count, instant-runoff and other voting methods.
3239:
In the papers by John Kemeny and Peyton Young, the Kemeny scores use counts of how many voters oppose, rather than support, each pairwise preference, but the smallest such score identifies the same overall ranking.
3223:
approach to preference aggregation: he supposed that there was an objectively 'correct', but unknown preference order over the alternatives, and voters receive noisy signals of this true preference order (cf.
1023:
on which voters rank choices according to their order of preference. A voter is allowed to rank more than one choice at the same preference level. Unranked choices are usually interpreted as least-preferred.
2068:
The ranking score for the possible ranking of
Memphis first, Nashville second, Chattanooga third, and Knoxville fourth equals (the unit-less number) 345, which is the sum of the following annotated numbers.
1365:
satisfies the preferences Elliot > Roland, Elliot > Meredith, Elliot > Selden, Roland > Meredith, Roland > Selden, and
Meredith > Selden. The respective scores, taken from the table, are
2827:
Identifies the overall order of preference for all the choices. The method does this for all possible sets of voter preferences and always produces the same result for the same set of voter preferences.
4687:
A randomly chosen ballot determines winner. This and closely related methods are of mathematical interest and included here to demonstrate that even unreasonable methods can pass voting method criteria.
1397:
After the scores for every possible ranking have been calculated, the ranking that has the largest score can be identified, and becomes the overall ranking. In this case, the overall ranking is:
2744:
are involved, the Kemeny–Young method only produces a tie at a preference level when the number of voters with one preference exactly matches the number of voters with the opposite preference.
4651:
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.
1417:
If there are cycles or ties, more than one possible ranking can have the same largest score. Cycles are resolved by producing a single overall ranking where some of the choices are tied.
3013:
If all the ballots are divided into separate races and choice X is identified as the most popular in every such race, then choice X is the most popular when all the ballots are combined.
1027:
Kemeny–Young calculations are usually done in two steps. The first step is to create a matrix or table that counts pairwise voter preferences. The second step is to test all possible
3053:
Offering a larger number of similar choices, instead of offering only a single such choice, does not change the probability that one of these choices is identified as most popular.
2729:
In this arrangement the largest ranking score (393) equals the sum of the counts in bold, which are in the upper-right, triangular half of the matrix (with a green background).
5183: — A website that calculates Kemeny–Young results. For comparison, it also calculates the winner according to plurality, Condorcet, Borda count, and other voting methods.
2903:
If all the ballots are divided into separate races and the overall ranking for the separate races are the same, then the same ranking occurs when all the ballots are combined.
952:
5258:
4642:
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.
3190:
989:
score is the winning sequence, and the first choice in the winning sequence is the most popular choice. (As explained below, ties can occur at any ranking level.)
5207: — Command-line program for fast calculation of Kemeny-Young results, as source code and compiled binaries for Windows and Linux. Open source, except uses
3151:
1641:
4996:; Fomin, Fedor V.; Koster, Arie M. C. A.; Kratsch, Dieter; Thilikos, Dimitrios M. (2012), "A note on exact algorithms for vertex ordering problems on graphs",
3196:
for computing a Kemeny-Young ranking, and there also exists a parameterized subexponential-time algorithm with running time O(2) for computing such a ranking.
4728:
5708:
3304:
2882:
5476:
4633:. For this to hold, in some elections, some voters must use less than their full voting power despite having meaningful preferences among viable candidates.
643:
3219:
and the so-called quasi-Condorcet criterion. It can also be characterized using consistency and a monotonicity property. In other papers, Young adopted an
3106:
An algorithm for computing a Kemeny-Young ranking in time polynomial in the number of candidates is not known, and unlikely to exist since the problem is
1031:, calculate a score for each such ranking, and compare the scores. Each ranking score equals the sum of the pairwise counts that apply to that ranking.
5189: — C++ program, available on GitHub under the MIT license, that calculates VoteFair ranking results, which include Condorcet-Kemeny calculations.
2763:
There are voter preferences that can yield every possible overall order-of-preference result, including ties at any combination of preference levels.
5604:
5456:
675:
537:
532:
5703:
5481:
3314:
3310:
2938:
2737:
In all cases that do not result in an exact tie, the Kemeny–Young method identifies a most-popular choice, second-most popular choice, and so on.
945:
638:
2855:
If voters increase a choice's preference level, the ranking result either does not change or the promoted choice increases in overall popularity.
2536:
The largest ranking score is 393, and this score is associated with the following possible ranking, so this ranking is also the overall ranking:
4696:
Where a winner is randomly chosen from the candidates, sortition is included to demonstrate that even non-voting methods can pass some criteria.
5466:
5251:
4678:
A variant of
Minimax that counts only pairwise opposition, not opposition minus support, fails the Condorcet criterion and meets later-no-harm.
320:
2873:, which is the smallest nonempty set of choices such that every member of the set is pairwise preferred to every choice not in the Smith set.
5061:
4629:
Approval voting, score voting, and majority judgment satisfy IIA if it is assumed that voters rate candidates independently using their own
844:
5873:
5830:
3193:
95:
5513:
4967:
4873:
Bachmeier, Georg; Brandt, Felix; Geist, Christian; Harrenstein, Paul; Kardel, Keyvan; Peters, Dominik; Seedig, Hans Georg (2019-11-01).
938:
2593:
The summary matrix below arranges the pairwise counts in order from most popular (top and left) to least popular (bottom and right):
5244:
5164:
4849:
2841:
Any pairwise preference expressed by every voter results in the preferred choice being ranked higher than the less-preferred choice.
1634:
5848:
3236:
himself was aware of the Kemeny-Young rule and its maximum-likelihood interpretation, but was unable to clearly express his ideas.
839:
5698:
5644:
3048:
2917:
If the preferences on every ballot are inverted, then the previously most popular choice must not remain the most popular choice.
2758:
829:
579:
550:
490:
5496:
5227: — Tutorial that uses a simple formulation as integer program and is adaptable to other languages with bindings to lpsolve.
5825:
561:
86:
2999:
Ranking an additional choice (that was otherwise unranked) cannot displace a choice from being identified as the most popular.
5624:
624:
3129:. As such, many methods for the computation of feedback arc sets can be applied to this problem, including a variant of the
266:
251:
236:
3095:
An algorithm is known to determine the winner using this method in a runtime that is polynomial in the number of choices.
5667:
5652:
5471:
3229:
2791:
If a majority of voters strictly prefer choice X to every other choice, then choice X is identified as the most popular.
1627:
882:
502:
425:
346:
5718:
5441:
5431:
5292:
3215:
and Arthur
Levenglick axiomatically characterized the method, showing that it is the unique neutral method satisfying
2786:
314:
296:
137:
3225:
867:
3027:
The optimal voting strategy for an individual should always include giving their favorite candidate maximum support.
5868:
5683:
5374:
4725:
4612:
4516:
3022:
758:
741:
708:
688:
472:
460:
430:
231:
189:
122:
2985:
Adding ballots that rank choice X over choice Y never cause choice Y, instead of choice X, to become most popular.
566:
5733:
5688:
5634:
5546:
3282:
3228:.) Using a simple probabilistic model for these noisy signals, Young showed that the Kemeny–Young method was the
614:
607:
91:
5723:
5609:
5567:
5486:
5411:
5347:
5305:
3277:
668:
596:
585:
448:
435:
418:
395:
373:
336:
326:
3130:
5748:
5738:
5713:
5529:
5406:
4608:
4600:
3330:
2994:
2980:
1667:
1606:
794:
648:
331:
5224:
3192:, significantly faster for many candidates than the factorial time of testing all rankings. There exists a
5789:
5728:
5591:
5534:
3325:
3320:
2971:
A voter cannot cause a choice to become the most popular by giving the choice an insincerely high ranking.
2891:, adding or withdrawing choice X does not change a result in which choice Y is identified as most popular.
2850:
1686:
823:
703:
633:
440:
5217: — Implements Davenport's algorithm with no other library dependencies. Open source, LGPL licensed.
3121:
The Kemeny–Young method can be formulated as an instance of a more abstract problem, of finding weighted
2586:
If a single winner is needed, the first choice, Nashville, is chosen. (In this example
Nashville is the
5693:
5614:
5379:
4596:
3885:
3360:
3216:
3067:
A voter cannot cause choice X to become the most popular by giving choice Y an insincerely high ranking.
3008:
2943:
Adding or withdrawing choice X does not change a result in which choice Y is identified as most popular.
1594:
731:
571:
455:
261:
240:
172:
150:
4950:"Faster Algorithms for Feedback Arc Set Tournament, Kemeny Rank Aggregation and Betweenness Tournament"
3243:
Since 1991 the method has been promoted under the name "VoteFair popularity ranking" by
Richard Fobes.
923:
789:
4874:
5763:
5332:
3725:
3347:
3233:
1692:
1680:
1039:
862:
849:
817:
81:
1652:
5743:
5337:
5198:
4592:
3937:
3777:
3465:
3114:
2957:
A voter cannot displace a choice from most popular by giving the choice an insincerely low ranking.
2822:
2772:
1663:
1020:
768:
602:
255:
4949:
5784:
5394:
5031:
4993:
4912:
4886:
4855:
3205:
1812:
1674:
1002:
978:
928:
799:
410:
194:
5352:
4832:; Brandenburg, Franz J.; Deng, Xiaotie (2005-09-12). Healy, Patrick; Nikolov, Nikola S. (eds.).
4043:
3251:
The following table compares the Kemeny-Young method with other single-winner election methods:
877:
5192:
3156:
5753:
5662:
5619:
5539:
5461:
5384:
5369:
5327:
5208:
5160:
5084:
4904:
4845:
3989:
3673:
3517:
2912:
2836:
1943:
The Kemeny–Young method arranges the pairwise comparison counts in the following tally table:
834:
804:
726:
663:
497:
224:
199:
182:
50:
3042:
these criteria (which means the described criteria do not apply to the Kemeny–Young method):
2932:
these criteria (which means the described criteria do not apply to the Kemeny–Young method):
985:
because if there is a Condorcet winner, it will always be ranked as the most popular choice.
5794:
5426:
5282:
5267:
5076:
5013:
5005:
4976:
4952:, in: Cheong, O., Chwa, K.-Y., and Park, K. (Eds.): ISAAC 2010, Part I, LNCS 6506, pp. 3-14.
4896:
4837:
4819:
C. Dwork, R. Kumar, M. Naor, D. Sivakumar. Rank Aggregation Methods for the Web, WWW10, 2001
4253:
4095:
3287:
3126:
3122:
2800:
2741:
2587:
1584:
1094:
preference level. The above preference order can be expressed as the following tally table:
982:
970:
915:
872:
763:
751:
465:
341:
167:
161:
143:
132:
127:
115:
76:
38:
5027:
5842:
5758:
5599:
5577:
5389:
5310:
5300:
5278:
5023:
4732:
4630:
3412:
3298:
3293:
3090:
3062:
2966:
2952:
2864:
903:
736:
591:
556:
477:
388:
291:
214:
156:
34:
58:
5657:
5362:
5322:
5204:
4305:
3829:
3621:
3136:
974:
773:
713:
698:
509:
378:
353:
204:
1038:
Another way to view the ordering is that it is the one which minimizes the sum of the
898:
5862:
5401:
5080:
5035:
4962:
4829:
4803:
4790:
Giuseppe Munda, "Social multi-criteria evaluation for a sustainable economy", p. 124.
4604:
4147:
3341:
3335:
782:
482:
270:
108:
71:
46:
4933:
4859:
5416:
5357:
5186:
4916:
4358:
4200:
3220:
3212:
3076:
522:
286:
279:
209:
17:
2752:
All Condorcet methods, including the Kemeny–Young method, satisfy these criteria:
4875:"k-Majority digraphs and the hardness of voting with a constant number of voters"
5799:
5562:
5501:
5421:
5317:
4836:. Lecture Notes in Computer Science. Springer Berlin Heidelberg. pp. 1–12.
4464:
3569:
1043:
1028:
400:
358:
301:
246:
4900:
4806:, "Voting schemes for which it can be difficult to tell who won the election",
5572:
5508:
5230:
5009:
5088:
4980:
4908:
5809:
5804:
5218:
4411:
2888:
2870:
2777:
If there is a choice that wins all pairwise contests, then this choice wins.
1659:
368:
363:
4666:
without grading any less-preferred candidate above any more-preferred one.
910:
5629:
5048:
4771:
3272:
619:
5117:
H. P. Young, "Optimal ranking and choice from pairwise comparisons", in
1344:
The sum of the counts in each row must equal the total number of votes.
5491:
5214:
4841:
3107:
405:
5121:
edited by B. Grofman and G. Owen (1986), JAI Press, pp. 113–122.
3081:
The choice identified as most popular is a member of the Schwartz set.
5779:
5201:
supporting multiple Condorcet methods, including Kemeny–Young method.
5018:
2740:
A tie can occur at any preference level. Except in some cases where
981:
counts to identify the most popular choices in an election. It is a
5236:
5146:
H. P. Young, "Group choice and individual judgements", Chapter 9 of
5150:, edited by Dennis Mueller (1997) Cambridge UP., pp.181 –200.
4891:
1389:
giving a total ranking score of 30 + 60 + 60 + 70 + 60 + 40 = 320.
1426:
equal-preference pairwise counts that appear in the tally table:
4588:
4586:
4584:
4582:
4580:
4578:
4576:
5240:
1668:
All voters want the capital to be as close to them as possible.
5195:
2928:
In common with all Condorcet methods, the Kemeny–Young method
4932:
Vincent Conitzer, Andrew Davenport, and Jayant Kalagnanam, "
57:
1677:, the largest city, but far from the others (42% of voters)
1666:. The population is concentrated around four major cities.
5180:
3110:
even if there are just 4 voters (even) or 7 voters (odd).
5159:
Richard Fobes, "The Creative Problem Solver's Toolbox", (
1562:
voters oppose (rather than support) each pairwise order:
4772:
A Consistent Extension of Condorcet's Election Principle
3232:
of the true preference order. Young further argues that
3113:
It has been reported that calculation methods based on
2816:
The Kemeny–Young method also satisfies these criteria:
2805:
A single voter cannot control the outcome in all cases.
4625:
4623:
4621:
3159:
3139:
5818:
5772:
5676:
5643:
5590:
5555:
5522:
5449:
5440:
5291:
5205:
C++ Program for Kemeny-Young Preference Aggregation
1601:(depending on how the second-round tie is handled)
4965:(1964), "A comment on minimum feedback arc sets",
3184:
3145:
5215:C Program for Kemeny-Young Preference Aggregation
4928:
4926:
4674:
4672:
2073:42% (of the voters) prefer Memphis over Nashville
1558:(If it did, that would be the overall ranking.)
3102:Calculation methods and computational complexity
5225:Kemeny-Young Optimal Rank Aggregation in Python
1683:, near the center of the state (26% of voters)
1662:is holding an election on the location of its
5252:
5119:Information pooling and group decision making
5101:H. P. Young, "Condorcet's Theory of Voting",
4934:Improved bounds for computing Kemeny rankings
4724:The numbers in this example are adapted from
3133:that can compute the Kemeny–Young ranking of
1699:The preferences of each region's voters are:
1635:
992:The Kemeny–Young method is also known as the
946:
8:
5709:Independence of Smith-dominated alternatives
4749:John Kemeny, "Mathematics without numbers",
2883:Independence of Smith-dominated alternatives
2748:Satisfied criteria for all Condorcet methods
3257:Comparison of single-winner voting systems
2869:The most popular choice is a member of the
5446:
5259:
5245:
5237:
3255:
1954:Number of votes with indicated preference
1642:
1628:
1230:Number of votes with indicated preference
1105:Number of votes with indicated preference
953:
939:
29:
5148:Perspectives on public choice: a handbook
5017:
4890:
4661:
4659:
4657:
3204:The Kemeny–Young method was developed by
3173:
3158:
3138:
2924:Failed criteria for all Condorcet methods
2093:This table lists all the ranking scores:
1811:This matrix summarizes the corresponding
5060:Can, Burak; Storcken, Ton (2013-03-01).
4745:
4743:
4741:
2595:
2538:
2095:
1945:
1817:
1701:
1564:
1428:
1221:
1096:
1052:
5704:Independence of irrelevant alternatives
5482:Sequential proportional approval voting
4944:
4942:
4879:Journal of Computer and System Sciences
4766:
4764:
4762:
4726:Sample election used in Knowledge (XXG)
4717:
4572:
4566:
2939:Independence of irrelevant alternatives
45:
5049:http://cs.brown.edu/~claire/stoc07.pdf
1695:, far to the northeast (17% of voters)
5130:H. P. Young, "Optimal Voting Rules",
4798:
4796:
2088:83% prefer Chattanooga over Knoxville
2082:68% prefer Nashville over Chattanooga
7:
5108:, no. 2 (1988), pp. 1231–1244.
4810:, Vol. 6, No. 2 (1989), pp. 157–165.
3194:polynomial-time approximation scheme
5514:Indirect single transferable voting
5221:is also open source, LPGL licensed.
4968:IEEE Transactions on Circuit Theory
4802:J. Bartholdi III, C. A. Tovey, and
4776:SIAM Journal on Applied Mathematics
2085:68% prefer Nashville over Knoxville
2076:42% prefer Memphis over Chattanooga
5062:"Update monotone preference rules"
25:
5103:American Political Science Review
2079:42% prefer Memphis over Knoxville
5137:, no.1 (1995), pp. 51–64.
5132:Journal of Economic Perspectives
5081:10.1016/j.mathsocsci.2012.10.004
4770:H. P. Young and A. Levenglick, "
1651:
1046:distance) to the voters' lists.
922:
909:
897:
845:McKelvey–Schofield chaos theorem
491:Semi-proportional representation
123:First preference plurality (FPP)
5047:"How to Rank with Few Errors".
2967:Invulnerability to compromising
1948:
1689:, somewhat east (15% of voters)
1393:Calculating the overall ranking
5625:Mixed ballot transferable vote
4948:Karpinski, M. and Schudy, W.,
3179:
3163:
883:Harsanyi's utilitarian theorem
840:Moulin's impossibility theorem
805:Conflicting majorities paradox
27:Single-winner electoral system
1:
5219:A Ruby binding to the library
3038:The Kemeny–Young method also
2812:Additional satisfied criteria
1414:with a ranking score of 370.
1019:The Kemeny–Young method uses
709:Frustrated majorities paradox
5826:Comparison of voting systems
5668:Satisfaction approval voting
5653:Single non-transferable vote
5472:Proportional approval voting
5069:Mathematical Social Sciences
4781:, no. 2 (1978), pp. 285–300.
3230:maximum likelihood estimator
3063:Invulnerability to push-over
878:Condorcet dominance theorems
818:Social and collective choice
5874:Monotonic Condorcet methods
5432:Graduated majority judgment
5167:), 1993, pp. 223–225.
4998:Theory of Computing Systems
998:VoteFair popularity ranking
544:By mechanism of combination
315:Proportional representation
5890:
5684:Condorcet winner criterion
5375:First-past-the-post voting
4901:10.1016/j.jcss.2019.04.005
4834:Crossings and Permutations
4756:(1959), pp. 577–591.
4570:
3034:Additional failed criteria
3023:Sincere favorite criterion
2953:Invulnerability to burying
2887:If choice X is not in the
2060:
2057:
2054:
2049:
2044:
2041:
2038:
2033:
2028:
2025:
2022:
2017:
2012:
2009:
2006:
2001:
1996:
1993:
1990:
1985:
1980:
1977:
1974:
1969:
1336:
1333:
1330:
1325:
1320:
1317:
1314:
1309:
1304:
1301:
1298:
1293:
1288:
1285:
1282:
1277:
1272:
1269:
1266:
1261:
1256:
1253:
1250:
1245:
1211:
1208:
1205:
1200:
1195:
1192:
1189:
1184:
1179:
1176:
1173:
1168:
1163:
1160:
1157:
1152:
1147:
1144:
1141:
1136:
1131:
1128:
1125:
1120:
742:Multiple districts paradox
473:Fractional approval voting
461:Interactive representation
5839:
5831:Voting systems by country
5734:Mutual majority criterion
5689:Condorcet loser criterion
5635:Vote linkage mixed system
5547:Largest remainders method
5274:
5010:10.1007/s00224-011-9312-0
4808:Social Choice and Welfare
4595:is incompatible with the
3185:{\displaystyle O(n2^{n})}
1953:
1725:
1718:
1711:
1704:
1229:
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1104:
1099:
689:Paradoxes and pathologies
538:Mixed-member proportional
533:Mixed-member majoritarian
528:By results of combination
419:Approval-based committees
5724:Majority loser criterion
5610:Additional member system
5568:Hagenbach-Bischoff quota
5487:Single transferable vote
5412:Positional voting system
5348:Minimax Condorcet method
5306:Combined approval voting
4981:10.1109/tct.1964.1082291
3226:Condorcet's jury theorem
1385:Meredith > Selden: 40
1379:Roland > Meredith: 70
1373:Elliot > Meredith: 60
868:Condorcet's jury theorem
669:Double simultaneous vote
644:Rural–urban proportional
639:Dual-member proportional
601:
590:
557:Parallel (superposition)
449:Fractional social choice
436:Expanding approvals rule
265:
250:
235:
166:
155:
131:
5749:Resolvability criterion
5739:Participation criterion
5714:Later-no-harm criterion
5530:Highest averages method
795:Tyranny of the majority
572:Fusion (majority bonus)
389:Quota-remainder methods
5790:First-preference votes
5729:Monotonicity criterion
5699:Independence of clones
5402:Simple majoritarianism
3186:
3147:
3049:Independence of clones
1382:Roland > Selden: 60
1376:Elliot > Selden: 60
1370:Elliot > Roland: 30
929:Mathematics portal
835:Majority impossibility
824:Impossibility theorems
620:Negative vote transfer
441:Method of equal shares
62:
5694:Consistency criterion
5615:Alternative vote plus
5380:Instant-runoff voting
4593:Condorcet's criterion
3187:
3148:
1595:Instant runoff voting
1040:Kendall tau distances
732:Best-is-worst paradox
721:Pathological response
456:Direct representation
109:Single-winner methods
61:
5764:Seats-to-votes ratio
5535:Webster/Sainte-Laguë
5187:VoteFair_Ranking.cpp
3348:No favorite betrayal
3157:
3137:
2742:circular ambiguities
1021:preferential ballots
916:Economics portal
863:Median voter theorem
82:Comparative politics
5744:Plurality criterion
5343:Kemeny–Young method
4994:Bodlaender, Hans L.
4517:Tideman alternative
3258:
3153:candidates in time
3131:Held–Karp algorithm
3115:integer programming
2823:Unrestricted domain
2773:Condorcet criterion
1813:pairwise comparison
1571:First-place winner
1087:(equal preference)
979:pairwise comparison
967:Kemeny–Young method
904:Politics portal
615:Vote linkage system
586:Seat linkage system
173:Ranked-choice (RCV)
18:Kemeny-Young method
5785:Election threshold
5719:Majority criterion
5395:Supplementary vote
4842:10.1007/11618058_1
4731:2017-03-30 at the
3256:
3182:
3143:
3091:Polynomial runtime
2787:Majority criterion
1949:All possible pairs
1225:All possible pairs
1100:All possible pairs
1085:Meredith or Selden
1003:maximum likelihood
800:Discursive dilemma
759:Lesser evil voting
634:Supermixed systems
337:Largest remainders
195:Round-robin voting
63:
5869:Electoral systems
5856:
5855:
5754:Reversal symmetry
5663:Cumulative voting
5645:Semi-proportional
5620:Mixed single vote
5586:
5585:
5462:Mixed single vote
5370:Exhaustive ballot
5333:Copeland's method
5328:Condorcet methods
5268:Electoral systems
5209:Numerical Recipes
4704:
4703:
3146:{\displaystyle n}
3127:tournament graphs
3123:feedback arc sets
2913:Reversal symmetry
2837:Pareto efficiency
2726:
2725:
2583:
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1962:Equal preference
1940:
1939:
1808:
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1730:
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1670:The options are:
1615:
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1340:
1238:Equal preference
1216:
1215:
1113:Equal preference
1091:
1090:
963:
962:
850:Gibbard's theorem
790:Dominance paradox
727:Perverse response
431:Phragmen's method
297:Majority judgment
225:Positional voting
183:Condorcet methods
51:electoral systems
16:(Redirected from
5881:
5795:Liquid democracy
5447:
5427:Two-round system
5338:Dodgson's method
5261:
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4613:sincere favorite
4590:
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3868:
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3460:Approvals
3439:
3350:
3344:
3338:
3321:Cloneproof
3317:
3307:
3301:
3290:
3288:Condorcet winner
3259:
3247:Comparison table
3191:
3189:
3188:
3183:
3178:
3177:
3152:
3150:
3149:
3144:
2801:Non-dictatorship
2596:
2588:Condorcet winner
2539:
2096:
2020:Y = Chattanooga
1988:Y = Chattanooga
1965:Prefer Y over X
1959:Prefer X over Y
1951:of choice names
1946:
1818:
1728:
1721:
1714:
1707:
1702:
1655:
1644:
1637:
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1599:Elliot or Selden
1565:
1429:
1241:Prefer Y over X
1235:Prefer X over Y
1227:of choice names
1222:
1116:Prefer Y over X
1110:Prefer X over Y
1102:of choice names
1097:
1053:
983:Condorcet method
971:electoral system
955:
948:
941:
927:
926:
914:
913:
902:
901:
857:Positive results
752:Strategic voting
649:Majority jackpot
606:
595:
466:Liquid democracy
342:National remnant
332:Highest averages
269:
254:
239:
171:
162:Alternative vote
160:
144:Partisan primary
136:
77:Mechanism design
30:
21:
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5759:Smith criterion
5672:
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5600:Parallel voting
5582:
5578:Imperiali quota
5551:
5518:
5436:
5390:Contingent vote
5353:Nanson's method
5311:Unified primary
5301:Approval voting
5287:
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5265:
5193:Condorcet Class
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4733:Wayback Machine
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3294:Condorcet loser
3283:Mutual majority
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2733:Characteristics
2728:
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2050:X = Chattanooga
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1009:median relation
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830:Arrow's theorem
820:
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737:No-show paradox
718:
704:Cloning paradox
694:Spoiler effects
691:
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655:
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478:Maximal lottery
445:
426:Thiele's method
415:
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292:Approval voting
280:Cardinal voting
276:
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215:Maximal lottery
179:
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5658:Limited voting
5655:
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5385:Coombs' method
5377:
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5366:
5365:
5363:Schulze method
5360:
5355:
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5345:
5340:
5335:
5325:
5323:Bucklin voting
5320:
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5175:External links
5173:
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5123:
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5075:(2): 136–149.
5052:
5040:
5004:(3): 420–432,
4985:
4975:(2): 296–297,
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4830:Biedl, Therese
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4631:absolute scale
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4563:Ranking
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4353:Ranking
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4248:Ranking
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4090:Ranking
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4038:Ranking
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3984:Ranking
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3932:Ranking
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3886:Instant-runoff
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3875:
3872:
3869:
3865:
3862:
3859:
3855:
3852:
3849:
3846:
3843:
3839:
3835:
3832:
3830:Highest median
3826:
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3824:Ranking
3822:
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3807:
3804:
3801:
3798:
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3792:
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3772:Ranking
3770:
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3720:Ranking
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3668:Ranking
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3648:
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3642:
3639:
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3627:
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3616:Ranking
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3590:
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3578:
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3564:Ranking
3562:
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3547:
3544:
3541:
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3526:
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3512:Ranking
3510:
3507:
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3501:
3498:
3495:
3492:
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3486:
3483:
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3474:
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3409:
3408:
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3399:
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3393:
3390:
3387:
3384:
3381:
3378:
3375:
3372:
3369:
3366:
3363:
3361:Anti-plurality
3357:
3356:
3351:
3345:
3339:
3333:
3328:
3326:Monotone
3323:
3318:
3308:
3302:
3296:
3291:
3285:
3280:
3278:Majority loser
3275:
3270:
3265:
3262:
3254:
3253:
3248:
3245:
3201:
3198:
3181:
3176:
3172:
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3017:
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2792:
2789:
2781:
2780:
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2778:
2775:
2767:
2766:
2765:
2764:
2761:
2759:Non-imposition
2749:
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2724:
2723:
2720:
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2703:
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2343:
2340:
2337:
2334:
2331:
2327:
2326:
2323:
2320:
2317:
2314:
2310:
2309:
2306:
2303:
2300:
2297:
2293:
2292:
2289:
2286:
2283:
2280:
2276:
2275:
2272:
2269:
2266:
2263:
2259:
2258:
2255:
2252:
2249:
2246:
2242:
2241:
2238:
2235:
2232:
2229:
2225:
2224:
2221:
2218:
2215:
2212:
2208:
2207:
2204:
2201:
2198:
2195:
2191:
2190:
2187:
2184:
2181:
2178:
2174:
2173:
2170:
2167:
2164:
2161:
2157:
2156:
2153:
2150:
2147:
2144:
2140:
2139:
2136:
2133:
2130:
2127:
2123:
2122:
2117:
2112:
2107:
2102:
2090:
2089:
2086:
2083:
2080:
2077:
2074:
2063:
2062:
2059:
2056:
2053:
2052:Y = Knoxville
2047:
2046:
2043:
2040:
2037:
2036:Y = Knoxville
2031:
2030:
2027:
2024:
2021:
2015:
2014:
2011:
2008:
2005:
2004:Y = Knoxville
1999:
1998:
1995:
1992:
1989:
1983:
1982:
1979:
1976:
1973:
1972:Y = Nashville
1967:
1966:
1963:
1960:
1956:
1955:
1952:
1938:
1937:
1934:
1931:
1928:
1925:
1916:
1915:
1912:
1909:
1906:
1903:
1894:
1893:
1890:
1887:
1884:
1881:
1872:
1871:
1868:
1865:
1862:
1859:
1850:
1849:
1842:
1835:
1828:
1821:
1806:
1805:
1804:
1803:
1800:
1797:
1794:
1787:
1786:
1785:
1782:
1779:
1776:
1769:
1768:
1767:
1764:
1761:
1758:
1751:
1750:
1749:
1746:
1743:
1740:
1732:
1731:
1724:
1717:
1710:
1697:
1696:
1690:
1684:
1678:
1647:
1646:
1639:
1632:
1624:
1623:
1619:
1616:
1613:
1612:
1609:
1603:
1602:
1597:
1591:
1590:
1587:
1581:
1580:
1577:
1573:
1572:
1569:
1553:
1552:
1549:
1546:
1543:
1540:
1532:
1531:
1526:
1523:
1520:
1517:
1509:
1508:
1503:
1498:
1495:
1492:
1484:
1483:
1478:
1473:
1468:
1465:
1457:
1456:
1450:
1444:
1438:
1432:
1422:
1421:Summary matrix
1419:
1412:
1411:
1408:
1405:
1402:
1394:
1391:
1387:
1386:
1383:
1380:
1377:
1374:
1371:
1363:
1362:
1359:
1356:
1353:
1339:
1338:
1335:
1332:
1329:
1323:
1322:
1319:
1316:
1313:
1307:
1306:
1303:
1300:
1297:
1291:
1290:
1287:
1284:
1281:
1275:
1274:
1271:
1268:
1265:
1259:
1258:
1255:
1252:
1249:
1243:
1242:
1239:
1236:
1232:
1231:
1228:
1214:
1213:
1210:
1207:
1204:
1198:
1197:
1194:
1191:
1188:
1182:
1181:
1178:
1175:
1172:
1166:
1165:
1162:
1159:
1156:
1150:
1149:
1146:
1143:
1140:
1134:
1133:
1130:
1127:
1124:
1118:
1117:
1114:
1111:
1107:
1106:
1103:
1089:
1088:
1083:
1079:
1078:
1075:
1071:
1070:
1067:
1063:
1062:
1059:
1016:
1013:
975:ranked ballots
961:
960:
958:
957:
950:
943:
935:
932:
931:
919:
918:
906:
893:
890:
889:
886:
885:
880:
875:
870:
865:
853:
852:
847:
842:
837:
832:
821:
816:
815:
812:
811:
808:
807:
802:
797:
792:
777:
776:
774:Turkey-raising
771:
766:
761:
747:
746:
745:
744:
734:
729:
717:
716:
714:Center squeeze
711:
706:
701:
699:Spoiler effect
692:
687:
686:
683:
682:
679:
678:
673:
672:
671:
658:By ballot type
654:
653:
652:
651:
646:
641:
631:
630:
629:
628:
627:
622:
612:
611:
610:
599:
576:
575:
574:
569:
564:
559:
541:
540:
535:
526:
521:
520:
517:
516:
513:
512:
510:Limited voting
507:
506:
505:
486:
485:
480:
475:
470:
469:
468:
463:
444:
443:
438:
433:
428:
414:
413:
408:
403:
398:
384:
383:
382:
381:
379:Localized list
376:
371:
366:
361:
351:
350:
349:
347:Biproportional
344:
339:
334:
318:
313:
312:
309:
308:
305:
304:
299:
294:
289:
275:
274:
259:
244:
220:
219:
218:
217:
212:
207:
202:
192:
178:
177:
176:
175:
164:
151:Instant-runoff
148:
147:
146:
138:Jungle primary
125:
114:Single vote -
112:
107:
106:
103:
102:
100:
99:
89:
84:
79:
74:
68:
65:
64:
54:
53:
43:
42:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5886:
5875:
5872:
5870:
5867:
5866:
5864:
5851:
5850:
5845:
5844:
5838:
5832:
5829:
5827:
5824:
5823:
5821:
5817:
5811:
5808:
5806:
5803:
5801:
5798:
5796:
5793:
5791:
5788:
5786:
5783:
5781:
5778:
5777:
5775:
5771:
5765:
5762:
5760:
5757:
5755:
5752:
5750:
5747:
5745:
5742:
5740:
5737:
5735:
5732:
5730:
5727:
5725:
5722:
5720:
5717:
5715:
5712:
5710:
5707:
5705:
5702:
5700:
5697:
5695:
5692:
5690:
5687:
5685:
5682:
5681:
5679:
5675:
5669:
5666:
5664:
5661:
5659:
5656:
5654:
5651:
5650:
5648:
5646:
5642:
5636:
5633:
5631:
5628:
5626:
5623:
5621:
5618:
5616:
5613:
5611:
5608:
5606:
5603:
5601:
5598:
5597:
5595:
5593:
5589:
5579:
5576:
5574:
5571:
5569:
5566:
5564:
5561:
5560:
5558:
5554:
5548:
5545:
5541:
5538:
5536:
5533:
5532:
5531:
5528:
5527:
5525:
5521:
5515:
5512:
5510:
5507:
5503:
5500:
5498:
5495:
5493:
5490:
5489:
5488:
5485:
5483:
5480:
5478:
5475:
5473:
5470:
5468:
5465:
5463:
5460:
5458:
5455:
5454:
5452:
5448:
5445:
5443:
5439:
5433:
5430:
5428:
5425:
5423:
5420:
5418:
5415:
5413:
5410:
5408:
5405:
5403:
5400:
5396:
5393:
5391:
5388:
5386:
5383:
5382:
5381:
5378:
5376:
5373:
5371:
5368:
5364:
5361:
5359:
5356:
5354:
5351:
5349:
5346:
5344:
5341:
5339:
5336:
5334:
5331:
5330:
5329:
5326:
5324:
5321:
5319:
5316:
5312:
5309:
5307:
5304:
5303:
5302:
5299:
5298:
5296:
5294:
5293:Single-winner
5290:
5286:
5284:
5280:
5273:
5269:
5262:
5257:
5255:
5250:
5248:
5243:
5242:
5239:
5232:
5229:
5226:
5223:
5220:
5216:
5213:
5210:
5206:
5203:
5200:
5197:
5194:
5191:
5188:
5185:
5182:
5179:
5178:
5174:
5166:
5165:0-9632-2210-4
5162:
5156:
5153:
5149:
5143:
5140:
5136:
5133:
5127:
5124:
5120:
5114:
5111:
5107:
5104:
5098:
5095:
5090:
5086:
5082:
5078:
5074:
5070:
5063:
5056:
5053:
5050:
5044:
5041:
5037:
5033:
5029:
5025:
5020:
5015:
5011:
5007:
5003:
4999:
4995:
4989:
4986:
4982:
4978:
4974:
4970:
4969:
4964:
4958:
4955:
4951:
4945:
4943:
4939:
4935:
4929:
4927:
4923:
4918:
4914:
4910:
4906:
4902:
4898:
4893:
4888:
4884:
4880:
4876:
4869:
4866:
4861:
4857:
4853:
4851:9783540314257
4847:
4843:
4839:
4835:
4831:
4825:
4822:
4816:
4813:
4809:
4805:
4799:
4797:
4793:
4787:
4784:
4780:
4777:
4773:
4767:
4765:
4763:
4759:
4755:
4752:
4746:
4744:
4742:
4738:
4734:
4730:
4727:
4721:
4718:
4711:
4709:
4693:
4690:
4684:
4681:
4675:
4673:
4669:
4662:
4660:
4658:
4654:
4648:
4645:
4639:
4636:
4632:
4626:
4624:
4622:
4618:
4614:
4610:
4609:later-no-help
4606:
4605:later-no-harm
4602:
4601:participation
4598:
4594:
4589:
4587:
4585:
4583:
4581:
4579:
4577:
4573:
4567:
4562:
4559:
4556:
4553:
4550:
4547:
4544:
4541:
4538:
4535:
4532:
4529:
4526:
4523:
4520:
4518:
4515:
4514:
4510:
4507:
4504:
4501:
4498:
4495:
4492:
4489:
4486:
4483:
4480:
4477:
4474:
4471:
4468:
4466:
4463:
4462:
4458:
4455:
4452:
4449:
4446:
4443:
4440:
4437:
4434:
4431:
4428:
4425:
4422:
4419:
4416:
4413:
4410:
4409:
4405:
4402:
4399:
4396:
4393:
4390:
4387:
4383:
4380:
4377:
4374:
4371:
4368:
4365:
4362:
4360:
4357:
4356:
4352:
4349:
4346:
4343:
4339:
4336:
4333:
4330:
4327:
4324:
4321:
4318:
4315:
4312:
4309:
4307:
4304:
4303:
4299:
4296:
4293:
4290:
4287:
4284:
4281:
4278:
4275:
4272:
4269:
4266:
4263:
4260:
4257:
4255:
4252:
4251:
4247:
4244:
4241:
4238:
4234:
4231:
4228:
4225:
4222:
4219:
4216:
4213:
4210:
4207:
4204:
4202:
4199:
4198:
4194:
4191:
4188:
4185:
4182:
4179:
4176:
4173:
4170:
4167:
4164:
4161:
4158:
4155:
4152:
4149:
4148:Random ballot
4146:
4145:
4141:
4138:
4135:
4132:
4129:
4126:
4123:
4120:
4117:
4114:
4111:
4108:
4105:
4102:
4099:
4097:
4094:
4093:
4089:
4086:
4083:
4080:
4077:
4074:
4071:
4068:
4065:
4062:
4059:
4056:
4053:
4050:
4047:
4045:
4042:
4041:
4037:
4034:
4031:
4027:
4024:
4021:
4018:
4015:
4012:
4009:
4006:
4002:
3999:
3996:
3993:
3991:
3988:
3987:
3983:
3980:
3977:
3974:
3971:
3968:
3965:
3962:
3959:
3956:
3953:
3950:
3947:
3944:
3941:
3939:
3936:
3935:
3931:
3928:
3925:
3922:
3919:
3916:
3913:
3910:
3907:
3904:
3901:
3898:
3895:
3892:
3889:
3887:
3884:
3883:
3879:
3876:
3873:
3870:
3866:
3863:
3860:
3856:
3853:
3850:
3847:
3844:
3840:
3836:
3833:
3831:
3828:
3827:
3823:
3820:
3817:
3814:
3811:
3808:
3805:
3802:
3799:
3796:
3793:
3790:
3787:
3784:
3781:
3779:
3776:
3775:
3771:
3768:
3765:
3762:
3759:
3756:
3753:
3750:
3747:
3744:
3741:
3738:
3735:
3732:
3729:
3727:
3724:
3723:
3719:
3716:
3713:
3710:
3707:
3704:
3701:
3698:
3695:
3692:
3689:
3686:
3683:
3680:
3677:
3675:
3672:
3671:
3667:
3664:
3661:
3658:
3655:
3652:
3649:
3646:
3643:
3640:
3637:
3634:
3631:
3628:
3625:
3623:
3620:
3619:
3615:
3612:
3609:
3606:
3603:
3600:
3597:
3594:
3591:
3588:
3585:
3582:
3579:
3576:
3573:
3571:
3568:
3567:
3563:
3560:
3557:
3554:
3551:
3548:
3545:
3542:
3539:
3536:
3533:
3530:
3527:
3524:
3521:
3519:
3516:
3515:
3511:
3508:
3505:
3502:
3499:
3496:
3493:
3490:
3487:
3484:
3481:
3478:
3475:
3472:
3469:
3467:
3464:
3463:
3459:
3456:
3453:
3450:
3447:
3444:
3441:
3437:
3434:
3431:
3428:
3425:
3422:
3419:
3416:
3414:
3411:
3410:
3406:
3403:
3400:
3397:
3394:
3391:
3388:
3385:
3382:
3379:
3376:
3373:
3370:
3367:
3364:
3362:
3359:
3358:
3352:
3349:
3346:
3343:
3342:Later-no-help
3340:
3337:
3336:Later-no-harm
3334:
3332:
3331:Participation
3329:
3327:
3324:
3322:
3319:
3316:
3312:
3309:
3306:
3303:
3300:
3297:
3295:
3292:
3289:
3286:
3284:
3281:
3279:
3276:
3274:
3271:
3261:
3260:
3252:
3246:
3244:
3241:
3237:
3235:
3231:
3227:
3222:
3218:
3214:
3209:
3207:
3199:
3197:
3195:
3174:
3170:
3166:
3160:
3140:
3132:
3128:
3124:
3119:
3116:
3111:
3109:
3101:
3094:
3092:
3089:
3088:
3087:
3086:
3080:
3078:
3075:
3074:
3073:
3072:
3066:
3064:
3061:
3060:
3059:
3058:
3052:
3050:
3047:
3046:
3045:
3044:
3043:
3041:
3033:
3026:
3024:
3021:
3020:
3019:
3018:
3012:
3010:
3007:
3006:
3005:
3004:
2998:
2996:
2995:Later-no-harm
2993:
2992:
2991:
2990:
2984:
2982:
2981:Participation
2979:
2978:
2977:
2976:
2970:
2968:
2965:
2964:
2963:
2962:
2956:
2954:
2951:
2950:
2949:
2948:
2942:
2940:
2937:
2936:
2935:
2934:
2933:
2931:
2923:
2916:
2914:
2911:
2910:
2909:
2908:
2902:
2900:Reinforcement
2899:
2898:
2897:
2896:
2890:
2886:
2884:
2881:
2880:
2879:
2878:
2872:
2868:
2866:
2863:
2862:
2861:
2860:
2854:
2852:
2849:
2848:
2847:
2846:
2840:
2838:
2835:
2834:
2833:
2832:
2826:
2824:
2821:
2820:
2819:
2818:
2817:
2811:
2804:
2802:
2799:
2798:
2797:
2796:
2790:
2788:
2785:
2784:
2783:
2782:
2776:
2774:
2771:
2770:
2769:
2768:
2762:
2760:
2757:
2756:
2755:
2754:
2753:
2747:
2745:
2743:
2738:
2732:
2730:
2721:
2718:
2715:
2712:
2709:
2705:
2704:
2701:
2698:
2695:
2692:
2689:
2686:
2682:
2681:
2678:
2675:
2673:
2670:
2667:
2664:
2661:
2657:
2656:
2653:
2650:
2648:
2645:
2643:
2640:
2637:
2634:
2630:
2629:
2625:
2621:
2618:
2614:
2611:
2607:
2604:
2600:
2598:
2597:
2594:
2591:
2589:
2578:
2575:
2574:
2570:
2567:
2566:
2562:
2559:
2558:
2554:
2551:
2550:
2546:
2541:
2540:
2537:
2528:
2525:
2522:
2519:
2516:
2515:
2511:
2508:
2505:
2502:
2499:
2498:
2494:
2491:
2488:
2485:
2482:
2481:
2477:
2474:
2471:
2468:
2465:
2464:
2460:
2457:
2454:
2451:
2448:
2447:
2443:
2440:
2437:
2434:
2431:
2430:
2426:
2423:
2420:
2417:
2414:
2413:
2409:
2406:
2403:
2400:
2397:
2396:
2392:
2389:
2386:
2383:
2380:
2379:
2375:
2372:
2369:
2366:
2363:
2362:
2358:
2355:
2352:
2349:
2346:
2345:
2341:
2338:
2335:
2332:
2329:
2328:
2324:
2321:
2318:
2315:
2312:
2311:
2307:
2304:
2301:
2298:
2295:
2294:
2290:
2287:
2284:
2281:
2278:
2277:
2273:
2270:
2267:
2264:
2261:
2260:
2256:
2253:
2250:
2247:
2244:
2243:
2239:
2236:
2233:
2230:
2227:
2226:
2222:
2219:
2216:
2213:
2210:
2209:
2205:
2202:
2199:
2196:
2193:
2192:
2188:
2185:
2182:
2179:
2176:
2175:
2171:
2168:
2165:
2162:
2159:
2158:
2154:
2151:
2148:
2145:
2142:
2141:
2137:
2134:
2131:
2128:
2125:
2124:
2118:
2113:
2108:
2103:
2098:
2097:
2094:
2087:
2084:
2081:
2078:
2075:
2072:
2071:
2070:
2048:
2034:X = Nashville
2032:
2018:X = Nashville
2016:
2000:
1984:
1968:
1964:
1961:
1958:
1957:
1947:
1944:
1935:
1932:
1929:
1926:
1923:
1918:
1917:
1913:
1910:
1907:
1904:
1901:
1896:
1895:
1891:
1888:
1885:
1882:
1879:
1874:
1873:
1869:
1866:
1863:
1860:
1857:
1852:
1851:
1848:
1843:
1841:
1836:
1834:
1829:
1827:
1822:
1820:
1819:
1816:
1814:
1801:
1798:
1795:
1793:
1790:
1789:
1788:
1783:
1780:
1777:
1775:
1772:
1771:
1770:
1765:
1762:
1759:
1757:
1754:
1753:
1752:
1747:
1744:
1741:
1739:
1736:
1735:
1734:
1733:
1726:17% of voters
1719:15% of voters
1712:26% of voters
1705:42% of voters
1703:
1700:
1694:
1691:
1688:
1685:
1682:
1679:
1676:
1673:
1672:
1671:
1669:
1665:
1661:
1658:Suppose that
1656:
1654:
1645:
1640:
1638:
1633:
1631:
1626:
1625:
1622:
1617:
1610:
1608:
1605:
1604:
1598:
1596:
1593:
1592:
1588:
1586:
1583:
1582:
1578:
1576:Kemeny–Young
1575:
1574:
1570:
1567:
1566:
1563:
1559:
1550:
1547:
1544:
1541:
1538:
1534:
1533:
1530:
1527:
1524:
1521:
1518:
1515:
1511:
1510:
1507:
1504:
1502:
1499:
1496:
1493:
1490:
1486:
1485:
1482:
1479:
1477:
1474:
1472:
1469:
1466:
1463:
1459:
1458:
1455:
1451:
1449:
1445:
1443:
1439:
1437:
1433:
1431:
1430:
1427:
1420:
1418:
1415:
1409:
1406:
1403:
1400:
1399:
1398:
1392:
1390:
1384:
1381:
1378:
1375:
1372:
1369:
1368:
1367:
1360:
1357:
1354:
1351:
1350:
1349:
1345:
1324:
1308:
1292:
1276:
1260:
1248:Y = Meredith
1244:
1240:
1237:
1234:
1233:
1223:
1220:
1199:
1183:
1167:
1151:
1135:
1123:Y = Meredith
1119:
1115:
1112:
1109:
1108:
1098:
1095:
1084:
1081:
1080:
1076:
1073:
1072:
1068:
1065:
1064:
1060:
1055:
1054:
1051:
1047:
1045:
1041:
1036:
1032:
1030:
1025:
1022:
1014:
1012:
1010:
1006:
1004:
999:
995:
990:
986:
984:
980:
976:
972:
968:
956:
951:
949:
944:
942:
937:
936:
934:
933:
930:
925:
917:
912:
907:
905:
900:
895:
894:
892:
891:
884:
881:
879:
876:
874:
873:May's theorem
871:
869:
866:
864:
861:
860:
859:
858:
851:
848:
846:
843:
841:
838:
836:
833:
831:
828:
827:
826:
825:
819:
814:
813:
806:
803:
801:
798:
796:
793:
791:
788:
787:
786:
785:
784:
783:majority rule
781:Paradoxes of
775:
772:
770:
767:
765:
762:
760:
757:
756:
755:
754:
753:
743:
740:
739:
738:
735:
733:
730:
728:
725:
724:
723:
722:
715:
712:
710:
707:
705:
702:
700:
697:
696:
695:
690:
685:
684:
677:
674:
670:
667:
666:
665:
662:
661:
660:
659:
650:
647:
645:
642:
640:
637:
636:
635:
632:
626:
623:
621:
618:
617:
616:
613:
609:
604:
600:
598:
593:
589:
588:
587:
584:
583:
582:
581:
577:
573:
570:
568:
565:
563:
560:
558:
555:
554:
553:
552:
547:
546:
545:
539:
536:
534:
531:
530:
529:
524:
523:Mixed systems
519:
518:
511:
508:
504:
501:
500:
499:
496:
495:
494:
493:
492:
484:
483:Random ballot
481:
479:
476:
474:
471:
467:
464:
462:
459:
458:
457:
454:
453:
452:
451:
450:
442:
439:
437:
434:
432:
429:
427:
424:
423:
422:
421:
420:
412:
409:
407:
404:
402:
399:
397:
394:
393:
392:
391:
390:
380:
377:
375:
372:
370:
367:
365:
362:
360:
357:
356:
355:
352:
348:
345:
343:
340:
338:
335:
333:
330:
329:
328:
327:Apportionment
325:
324:
323:
322:
316:
311:
310:
303:
300:
298:
295:
293:
290:
288:
285:
284:
283:
282:
281:
272:
268:
263:
262:Antiplurality
260:
257:
253:
248:
245:
242:
238:
233:
230:
229:
228:
227:
226:
216:
213:
211:
208:
206:
203:
201:
198:
197:
196:
193:
191:
190:Condorcet-IRV
188:
187:
186:
185:
184:
174:
169:
165:
163:
158:
154:
153:
152:
149:
145:
142:
141:
139:
134:
129:
126:
124:
121:
120:
119:
117:
110:
105:
104:
97:
93:
90:
88:
85:
83:
80:
78:
75:
73:
72:Social choice
70:
69:
67:
66:
60:
56:
55:
52:
48:
47:Social choice
44:
40:
36:
32:
31:
19:
5847:
5841:
5457:Mixed-member
5442:Proportional
5417:Score voting
5358:Ranked pairs
5342:
5277:Part of the
5276:
5181:VoteFair.org
5155:
5147:
5142:
5134:
5131:
5126:
5118:
5113:
5105:
5102:
5097:
5072:
5068:
5055:
5043:
5001:
4997:
4988:
4972:
4966:
4957:
4882:
4878:
4868:
4833:
4824:
4815:
4807:
4786:
4778:
4775:
4753:
4750:
4720:
4707:
4692:
4683:
4647:
4638:
4568:Table Notes
4300:Single mark
4201:Ranked pairs
4195:Single mark
4142:Single mark
3938:Kemeny–Young
3407:Single mark
3250:
3242:
3238:
3213:Peyton Young
3210:
3203:
3120:
3112:
3105:
3039:
3037:
2929:
2927:
2851:Monotonicity
2815:
2751:
2739:
2736:
2727:
2707:
2699:
2684:
2676:
2671:
2659:
2651:
2646:
2641:
2632:
2623:
2616:
2609:
2602:
2592:
2584:
2563:Chattanooga
2534:
2091:
2066:
1941:
1921:
1899:
1877:
1855:
1846:
1839:
1832:
1825:
1809:
1796:Chattanooga
1791:
1773:
1760:Chattanooga
1755:
1745:Chattanooga
1737:
1698:
1657:
1650:
1621:
1560:
1556:
1536:
1528:
1513:
1505:
1500:
1488:
1480:
1475:
1470:
1461:
1453:
1447:
1441:
1435:
1424:
1416:
1413:
1396:
1388:
1364:
1346:
1342:
1310:X = Meredith
1294:X = Meredith
1217:
1185:X = Meredith
1169:X = Meredith
1092:
1048:
1037:
1033:
1026:
1018:
1008:
1001:
997:
993:
991:
987:
966:
964:
856:
855:
822:
780:
779:
764:Exaggeration
750:
749:
720:
719:
693:
657:
656:
625:Mixed ballot
580:Compensatory
578:
551:compensatory
548:
543:
527:
489:
488:
447:
446:
417:
416:
387:
386:
374:List-free PR
319:
287:Score voting
278:
277:
223:
222:
210:Ranked pairs
181:
180:
113:
5800:Spoilt vote
5563:Droop quota
5502:Schulze STV
5477:Rural–urban
5422:STAR voting
5318:Borda count
4885:: 130–157.
4804:M. A. Trick
4597:consistency
3217:consistency
3206:John Kemeny
3009:Consistency
2660:Chattanooga
2610:Chattanooga
2520:Chattanooga
2503:Chattanooga
2489:Chattanooga
2475:Chattanooga
2455:Chattanooga
2441:Chattanooga
2415:Chattanooga
2398:Chattanooga
2381:Chattanooga
2364:Chattanooga
2347:Chattanooga
2330:Chattanooga
2319:Chattanooga
2305:Chattanooga
2282:Chattanooga
2265:Chattanooga
2254:Chattanooga
2234:Chattanooga
2217:Chattanooga
2203:Chattanooga
2180:Chattanooga
2163:Chattanooga
2152:Chattanooga
2132:Chattanooga
2002:X = Memphis
1986:X = Memphis
1970:X = Memphis
1900:Chattanooga
1840:Chattanooga
1774:Chattanooga
1722:Center-East
1687:Chattanooga
1328:Y = Roland
1312:Y = Roland
1296:Y = Elliot
1280:Y = Roland
1264:Y = Elliot
1203:Y = Roland
1187:Y = Roland
1171:Y = Elliot
1155:Y = Roland
1139:Y = Elliot
1044:bubble sort
1015:Description
994:Kemeny rule
664:Single vote
567:Conditional
562:Coexistence
411:Quota Borda
401:Schulze STV
359:Closed list
302:STAR voting
247:Borda count
5863:Categories
5819:Comparison
5573:Hare quota
5523:Allocation
5509:Spare vote
5497:Hare-Clark
5467:Party-list
4963:Lawler, E.
4892:1704.06304
4226:LIIA Only
3963:LIIA Only
2571:Knoxville
2555:Nashville
2542:Preference
1799:Nashville
1781:Nashville
1778:Knoxville
1763:Knoxville
1748:Knoxville
1742:Nashville
1326:X = Elliot
1278:X = Selden
1262:X = Selden
1246:X = Selden
1201:X = Elliot
1153:X = Selden
1137:X = Selden
1121:X = Selden
1056:Preference
1007:, and the
973:that uses
769:Truncation
498:Cumulative
321:Party-list
96:By country
87:Comparison
5810:Unseating
5805:Sortition
5407:Plurality
5283:Economics
5231:QuickVote
5089:0165-4896
5036:253742611
5019:1956/4556
4936:" (2006).
4909:0022-0000
4615:criteria.
4412:Sortition
4096:Plurality
3305:Smith-IIA
3263:Criterion
3234:Condorcet
3221:epistemic
3211:In 1978,
3208:in 1959.
2889:Smith set
2871:Smith set
2685:Knoxville
2633:Nashville
2622:... over
2617:Knoxville
2615:... over
2608:... over
2603:Nashville
2601:... over
2523:Nashville
2517:Knoxville
2509:Nashville
2500:Knoxville
2486:Nashville
2483:Knoxville
2469:Nashville
2466:Knoxville
2458:Nashville
2449:Knoxville
2438:Nashville
2432:Knoxville
2421:Nashville
2418:Knoxville
2407:Nashville
2401:Knoxville
2387:Knoxville
2384:Nashville
2373:Knoxville
2367:Nashville
2356:Nashville
2353:Knoxville
2339:Knoxville
2336:Nashville
2316:Knoxville
2313:Nashville
2299:Knoxville
2296:Nashville
2285:Knoxville
2279:Nashville
2271:Knoxville
2262:Nashville
2251:Knoxville
2245:Nashville
2237:Knoxville
2228:Nashville
2220:Nashville
2214:Knoxville
2200:Nashville
2197:Knoxville
2186:Nashville
2183:Knoxville
2169:Knoxville
2166:Nashville
2149:Knoxville
2146:Nashville
2135:Knoxville
2129:Nashville
1922:Knoxville
1878:Nashville
1847:Knoxville
1833:Nashville
1792:Knoxville
1756:Nashville
1693:Knoxville
1681:Nashville
1660:Tennessee
1607:Plurality
1585:Condorcet
1452:... over
1446:... over
1440:... over
1434:... over
676:Dual-vote
369:Panachage
364:Open list
354:List type
232:Plurality
128:Two-round
116:plurality
39:Economics
5677:Criteria
5630:Scorporo
5279:politics
4860:11189107
4751:Daedalus
4729:Archived
3726:Copeland
3413:Approval
3273:Majority
3077:Schwartz
2579:Memphis
1844:... over
1837:... over
1830:... over
1823:... over
1815:counts:
1802:Memphis
1784:Memphis
1766:Memphis
1729:Far-East
1708:Far-West
1537:Meredith
1454:Meredith
1410:Meredith
1358:Meredith
1206:+1 vote
1196:+1 vote
1180:+1 vote
1164:+1 vote
1148:+1 vote
1129:+1 vote
1029:rankings
396:Hare STV
35:Politics
33:A joint
5849:Project
5540:D'Hondt
5492:CPO-STV
5450:Systems
5199:library
5028:2885638
4917:2357131
4511:Scores
4406:Scores
4306:Schulze
3990:Minimax
3880:Scores
3778:Dodgson
3622:Bucklin
3466:Baldwin
3353:Ballot
3200:History
3108:NP-hard
2708:Memphis
2706:Prefer
2683:Prefer
2658:Prefer
2631:Prefer
2624:Memphis
2576:Fourth
2560:Second
2547:Choice
2526:Memphis
2506:Memphis
2492:Memphis
2472:Memphis
2452:Memphis
2435:Memphis
2424:Memphis
2404:Memphis
2390:Memphis
2370:Memphis
2350:Memphis
2333:Memphis
2322:Memphis
2302:Memphis
2288:Memphis
2268:Memphis
2248:Memphis
2231:Memphis
2211:Memphis
2194:Memphis
2177:Memphis
2160:Memphis
2143:Memphis
2126:Memphis
2119:Ranking
1856:Memphis
1826:Memphis
1738:Memphis
1675:Memphis
1664:capital
1618:Example
1611:Selden
1589:Roland
1579:Roland
1568:Method
1535:Prefer
1512:Prefer
1487:Prefer
1460:Prefer
1077:Roland
1074:Second
1069:Elliot
1061:Choice
406:CPO-STV
256:Baldwin
205:Schulze
200:Minimax
118:methods
5843:Portal
5780:Ballot
5556:Quotas
5285:series
5163:
5087:
5034:
5026:
4915:
4907:
4858:
4848:
4611:, and
4254:Runoff
4044:Nanson
3674:Coombs
3268:Method
2568:Third
2552:First
2544:order
2121:score
2116:choice
2114:Fourth
2111:choice
2106:choice
2104:Second
2101:choice
1919:Prefer
1897:Prefer
1875:Prefer
1853:Prefer
1715:Center
1514:Selden
1489:Elliot
1462:Roland
1448:Selden
1442:Elliot
1436:Roland
1407:Selden
1404:Elliot
1401:Roland
1361:Selden
1355:Roland
1352:Elliot
1082:Third
1066:First
1058:order
1005:method
1000:, the
969:is an
271:Coombs
41:series
5773:Other
5592:Mixed
5065:(PDF)
5032:S2CID
4913:S2CID
4887:arXiv
4856:S2CID
4712:Notes
4459:None
4359:Score
3570:Borda
3518:Black
3355:type
3299:Smith
3040:fails
2930:fails
2109:Third
2099:First
608:'MMP'
597:'AMS'
5281:and
5161:ISBN
5085:ISSN
4905:ISSN
4846:ISBN
4545:Yes
4539:Yes
4536:Yes
4533:Yes
4530:Yes
4527:Yes
4524:Yes
4521:Yes
4496:Yes
4481:Yes
4472:Yes
4465:STAR
4456:Yes
4453:Yes
4450:Yes
4447:Yes
4444:Yes
4438:Yes
4403:Yes
4400:Yes
4394:Yes
4391:Yes
4388:Yes
4337:Yes
4334:Yes
4328:Yes
4325:Yes
4322:Yes
4319:Yes
4316:Yes
4313:Yes
4310:Yes
4294:Yes
4291:Yes
4270:Yes
4261:Yes
4258:Yes
4232:Yes
4229:Yes
4223:Yes
4220:Yes
4217:Yes
4214:Yes
4211:Yes
4208:Yes
4205:Yes
4192:Yes
4189:Yes
4186:Yes
4183:Yes
4180:Yes
4177:Yes
4174:Yes
4136:Yes
4133:Yes
4130:Yes
4127:Yes
4100:Yes
4063:Yes
4060:Yes
4057:Yes
4054:Yes
4051:Yes
4048:Yes
4022:Yes
3994:Yes
3969:Yes
3960:Yes
3957:Yes
3954:Yes
3951:Yes
3948:Yes
3945:Yes
3942:Yes
3926:Yes
3923:Yes
3914:Yes
3902:Yes
3896:Yes
3893:Yes
3890:Yes
3877:Yes
3874:Yes
3864:Yes
3861:Yes
3834:Yes
3791:Yes
3782:Yes
3757:Yes
3748:Yes
3745:Yes
3742:Yes
3739:Yes
3736:Yes
3733:Yes
3730:Yes
3717:Yes
3690:Yes
3684:Yes
3681:Yes
3678:Yes
3662:Yes
3653:Yes
3632:Yes
3629:Yes
3626:Yes
3610:Yes
3604:Yes
3601:Yes
3586:Yes
3577:Yes
3549:Yes
3534:Yes
3531:Yes
3525:Yes
3522:Yes
3485:Yes
3482:Yes
3479:Yes
3476:Yes
3473:Yes
3470:Yes
3457:Yes
3454:Yes
3448:Yes
3445:Yes
3442:Yes
3417:Yes
3404:Yes
3395:Yes
3392:Yes
3368:Yes
3315:LIIA
2719:42%
2716:42%
2713:42%
2710:...
2693:17%
2690:32%
2687:...
2665:32%
2662:...
2635:...
2626:...
2619:...
2612:...
2605:...
2529:255
2512:239
2495:291
2478:275
2461:223
2444:259
2427:321
2410:305
2393:357
2376:341
2359:289
2342:325
2325:327
2308:311
2291:393
2274:377
2257:295
2240:361
2223:207
2206:243
2189:273
2172:309
2155:279
2138:345
2061:17%
2055:83%
2045:32%
2039:68%
2029:32%
2023:68%
2013:58%
2007:42%
1997:58%
1991:42%
1981:58%
1975:42%
1914:83%
1892:68%
1870:42%
1539:...
1516:...
1491:...
1464:...
977:and
965:The
549:Non-
503:SNTV
92:List
49:and
37:and
5605:MMP
5196:PHP
5077:doi
5014:hdl
5006:doi
4977:doi
4897:doi
4883:105
4838:doi
4774:",
4560:No
4557:No
4554:No
4551:No
4548:No
4542:No
4508:No
4505:No
4502:No
4499:No
4493:No
4490:No
4487:No
4484:No
4478:No
4475:No
4469:No
4441:No
4435:No
4432:No
4429:No
4426:No
4423:No
4420:No
4417:No
4397:No
4384:Yes
4381:No
4378:No
4375:No
4372:No
4369:No
4366:No
4363:No
4350:No
4347:No
4344:No
4331:No
4297:No
4288:No
4285:No
4282:No
4279:No
4276:No
4273:No
4267:No
4264:No
4245:No
4242:No
4239:No
4171:No
4168:No
4165:No
4162:No
4159:No
4156:No
4153:No
4139:No
4124:No
4121:No
4118:No
4115:No
4112:No
4109:No
4106:No
4103:No
4087:No
4084:No
4081:No
4078:No
4075:No
4072:No
4069:No
4066:No
4035:No
4032:No
4025:No
4019:No
4016:No
4013:No
4010:No
4007:No
4003:Yes
4000:No
3997:No
3981:No
3978:No
3975:No
3972:No
3966:No
3929:No
3920:No
3917:No
3911:No
3908:No
3905:No
3899:No
3871:No
3857:Yes
3854:No
3851:No
3848:No
3845:No
3837:Yes
3821:No
3818:No
3815:No
3812:No
3809:No
3806:No
3803:No
3800:No
3797:No
3794:No
3788:No
3785:No
3769:No
3766:No
3763:No
3760:No
3754:No
3751:No
3714:No
3711:No
3708:No
3705:No
3702:No
3699:No
3696:No
3693:No
3687:No
3665:No
3659:No
3656:No
3650:No
3647:No
3644:No
3641:No
3638:No
3635:No
3613:No
3607:No
3598:No
3595:No
3592:No
3589:No
3583:No
3580:No
3574:No
3561:No
3558:No
3555:No
3552:No
3546:No
3543:No
3540:No
3537:No
3528:No
3509:No
3506:No
3503:No
3500:No
3497:No
3494:No
3491:No
3488:No
3451:No
3438:Yes
3435:No
3432:No
3429:No
3426:No
3423:No
3420:No
3401:No
3398:No
3389:No
3386:No
3383:No
3380:No
3377:No
3374:No
3371:No
3365:No
3311:IIA
3125:in
2700:58%
2677:58%
2672:83%
2652:58%
2647:68%
2642:68%
2590:.)
1933:17%
1930:32%
1927:58%
1924:...
1908:32%
1905:58%
1902:...
1889:68%
1883:58%
1880:...
1867:42%
1864:42%
1858:...
1548:40
1545:40
1542:30
1522:40
1519:40
1494:30
1337:70
1331:30
1321:70
1315:30
1305:60
1299:40
1289:60
1283:40
1273:60
1267:40
1257:40
1254:10
1251:50
267:el.
252:el.
241:IRV
237:el.
5865::
5846:—
5106:82
5083:.
5073:65
5071:.
5067:.
5030:,
5024:MR
5022:,
5012:,
5002:50
5000:,
4973:11
4971:,
4941:^
4925:^
4911:.
4903:.
4895:.
4881:.
4877:.
4854:.
4844:.
4795:^
4779:35
4761:^
4754:88
4740:^
4671:^
4656:^
4620:^
4607:,
4603:,
4599:,
4575:^
4340:No
4235:No
4028:No
3867:No
3841:No
2722:-
2696:-
2668:-
2638:-
2058:0
2042:0
2026:0
2010:0
1994:0
1978:0
1936:-
1551:-
1529:50
1525:-
1506:60
1501:60
1497:-
1481:70
1476:60
1471:70
1467:-
1334:0
1318:0
1302:0
1286:0
1270:0
1212:0
1209:0
1193:0
1190:0
1177:0
1174:0
1161:0
1158:0
1145:0
1142:0
1132:0
1126:0
1011:.
996:,
603:NZ
592:UK
168:US
157:UK
140:)
133:US
5260:e
5253:t
5246:v
5211:.
5135:9
5091:.
5079::
5016::
5008::
4979::
4919:.
4899::
4889::
4862:.
4840::
4735:.
3313:/
3180:)
3175:n
3171:2
3167:n
3164:(
3161:O
3141:n
1911:-
1886:-
1861:-
1643:e
1636:t
1629:v
1042:(
954:e
947:t
940:v
605::
594::
273:)
264:(
258:)
249:(
243:)
234:(
170::
159::
135::
130:(
98:)
94:(
20:)
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