966:
924:
59:
899:
911:
1321:
1256:
1278:
1213:
1429:
estimated at least 15% of IRV elections would be nonmonotonic in the best-case scenario (with only three equally-competitive candidates). The researchers concluded that "three-way competitive races will exhibit unacceptably frequent monotonicity failures" and "In light of these results, those seeking
1341:
have difficulties electing the majority-preferred candidate, because the first-round vote is split between an extremist and a moderate. Here, the loss of support for Bottom policies makes the Top party more popular, allowing it to defeat the Center party in the first round.
1421:
model estimate about 15% of elections with 3 candidates; however, the true probability may be much higher, especially when restricting observation to close elections. For moderate numbers of candidates, the probability of a less-is-more paradoxes quickly approaches 100%.
1031:
Systems that violate positive response can create situations where a voter's ballot has a reversed effect on the election, making it "less than worthless". As a result, perversity is often described by social choice theorists as an exceptionally severe
1404:
For electoral methods failing positive value, the frequency of less-is-more paradoxes will depend on the electoral method, the candidates, and the distribution of outcomes. Negative voting weights tend to be most common with
973:
under IRV. The diagram shows who would win an IRV election if the electorate is centered at a particular point. Moving the electorate to the left can cause a right-wing candidate to win, and vice versa. Black lines show the
1021:. Positive response rules out cases where a candidate loses an election as a result of receiving too much support from voters (i.e. being "too popular to win"). Rules that violate positive response are said to show
1585:
Apart from majority judgment, these systems satisfy an even stronger form of positive responsiveness: if there is a tie, any increase in a candidate's rating will break the tie in that candidate's favor.
882:
1443:
1346:
566:
585:
1973:
Ornstein, Joseph T.; Norman, Robert Z. (2014-10-01). "Frequency of monotonicity failure under
Instant Runoff Voting: estimates based on a spatial model of elections".
1886:
Ornstein, Joseph T.; Norman, Robert Z. (2014-10-01). "Frequency of monotonicity failure under
Instant Runoff Voting: estimates based on a spatial model of elections".
2018:"How an Ideologically Concentrated Minority Can Trump a Dispersed Majority: Nonmedian Voter Results for Plurality, Run-off, and Sequential Elimination Elections"
952:
789:
2282:
1523:
1389:
2732:
1151:
An example with three parties (Top, Center, Bottom) is shown below. In this scenario, the Bottom party initially loses. However, they are
2500:
643:
1490:
elections where no candidate was ranked first by a majority of voters found five additional elections containing monotonicity failures.
1467:
2628:
2480:
1033:
675:
537:
532:
2727:
2505:
945:
638:
2490:
2275:
320:
2068:
Graham-Squire, Adam T.; McCune, David (2023-06-12). "An
Examination of Ranked-Choice Voting in the United States, 2004–2022".
1837:
1671:
1507:
1373:
2854:
844:
95:
2537:
2892:
938:
2268:
839:
2722:
2668:
1447:
1137:
829:
579:
550:
490:
19:
This article is about the voting system criterion. For the mathematical notion of an order preserving mapping, see
1518:, a strategy that allowed the party to earn an additional seat, causing substantial controversy. As a result, the
2849:
2520:
1561:
1515:
1381:
561:
86:
2102:
1854:
2872:
2648:
624:
688:
266:
251:
236:
2691:
2676:
2495:
1519:
1503:
1385:
1369:
1037:
502:
425:
346:
2485:
2742:
2465:
2455:
2316:
1357:
1068:
867:
314:
296:
137:
2707:
2398:
758:
741:
708:
472:
460:
430:
231:
189:
122:
1788:
2757:
2712:
2658:
2570:
2226:
614:
607:
91:
2187:
2747:
2633:
2591:
2510:
2435:
2371:
2329:
1365:
1361:
1083:
668:
596:
448:
435:
418:
395:
373:
336:
326:
877:
2772:
2762:
2737:
2553:
2430:
2366:
1550:
1539:
1426:
1106:
794:
648:
331:
24:
763:
2813:
2615:
2167:
1109:
is a closely-related, but different, concept. While positive responsiveness deals with a voter
823:
703:
633:
440:
2558:
2717:
2638:
2403:
1487:
1406:
1163:
popular platform, which pushes their supporters away from the party and into the Top party.
1141:
1133:
1126:
1079:
731:
571:
455:
261:
240:
172:
150:
2135:
1789:"Interacting double monotonicity failure with direction of impact under five voting methods"
2787:
2356:
1555:
1499:
1091:
1087:
1010:
862:
849:
817:
81:
1474:
was re-elected, despite losing in a head-to-head matchup with
Democrat Andy Montroll (the
8:
2767:
2361:
1566:
768:
602:
255:
2808:
2418:
2376:
2073:
2045:
1998:
1911:
1635:
1608:"A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decision"
1430:
to implement a fairer multi-candidate election system should be wary of adopting IRV."
1113:
their opinion (or vote), participation deals with situations where a voter choosing to
1056:
997:
928:
799:
410:
194:
20:
1827:
1661:
1478:). However, if Kiss had gained more support from Wright voters, Kiss would have lost.
2777:
2686:
2643:
2563:
2408:
2393:
2351:
2246:
2207:
2110:
2037:
1990:
1950:
1903:
1862:
1833:
1808:
1804:
1769:
1708:
1667:
1627:
1527:
1418:
1393:
1041:
923:
834:
804:
726:
663:
497:
224:
199:
182:
50:
2002:
1915:
2818:
2450:
2306:
2291:
2238:
2199:
2083:
2029:
1982:
1942:
1895:
1800:
1761:
1700:
1619:
1475:
1338:
1095:
979:
915:
872:
751:
465:
341:
167:
161:
143:
132:
127:
115:
76:
38:
2242:
2087:
2866:
2782:
2623:
2601:
2413:
2334:
2324:
2302:
1409:, with what some researchers have described as an "unacceptably high" frequency.
1064:
975:
903:
736:
591:
556:
477:
388:
291:
214:
156:
34:
2136:"Monotonicity Failure in IRV Elections with Three Candidates: Closeness Matters"
1729:
2681:
2386:
2346:
1829:
Proportional representation : apportionment methods and their applications
1734:
1663:
Proportional representation : apportionment methods and their applications
1334:
1099:
773:
713:
698:
509:
378:
353:
204:
2203:
1986:
1899:
2886:
2425:
2250:
2211:
2114:
2041:
1994:
1954:
1907:
1866:
1812:
1773:
1712:
1631:
1545:
1048:
782:
482:
270:
108:
71:
46:
1931:"Single Transferrable Vote: An Example of a Perverse Social Choice Function"
1750:"Single Transferrable Vote: An Example of a Perverse Social Choice Function"
1689:"Single Transferrable Vote: An Example of a Perverse Social Choice Function"
2440:
2381:
1455:
1072:
1060:
1040:
has previously struck down nonmonotonic systems for violating the right to
983:
522:
286:
279:
209:
58:
2823:
2586:
2525:
2445:
2341:
1451:
1384:, a strategy that allowed the party an additional seat. As a result, the
1052:
400:
358:
301:
246:
898:
2596:
2532:
2049:
2017:
1930:
1749:
1688:
1639:
1607:
1148:
won the election as a result of 750 ballots ranking him in last place.
2833:
2828:
1498:
A negative voting weight event famously resulted in the abolition of
970:
368:
363:
2260:
2033:
1946:
1765:
1704:
1623:
2653:
2078:
1471:
1145:
619:
1368:
do not pass the positive response criterion. This happened in the
2515:
1511:
1377:
405:
2803:
2016:
McGann, Anthony J.; Koetzle, William; Grofman, Bernard (2002).
1136:
fail the positive response criterion. A notable example is the
910:
1345:
A famous example of a less-is-more paradox can be seen in the
23:. For the concept of population or voter monotonicity, see
2169:
MONOTONICITY FAILURE IN IRV ELECTIONS WITH THREE ANDIDATES
965:
1017:
a candidate's ranking or rating should not cause them to
1724:
1722:
2015:
1832:. Internet Archive. Cham; New York : Springer.
1787:
Felsenthal, Dan S.; Tideman, Nicolaus (2014-01-01).
1719:
1666:. Internet Archive. Cham; New York : Springer.
2224:
2067:
1481:
1493:
2225:Graham-Squire, Adam; McCune, David (2024-01-02).
1786:
2884:
2174:Impartial Culture Profiles: All, Total MF: 15.0%
1337:, a class of elections where instant-runoff and
1117:a ballot has a reversed effect on the election.
1522:ruled that negative voting weights violate the
1388:ruled that negative voting weights violate the
1972:
1885:
1514:were instructed to strategically vote for the
1059:) satisfy positive response, as do all common
2276:
1730:"Monotonicity and Single-Seat Election Rules"
946:
2733:Independence of Smith-dominated alternatives
1928:
1747:
1686:
2188:"Anomalous Outcomes in Preferential Voting"
2154:Impartial Culture Profiles: All, TMF: 15.1%
1446:resulted in negative vote weights for many
1444:Alaska's first-ever instant-runoff election
2283:
2269:
1825:
1659:
1399:
953:
939:
2077:
1929:Doron, Gideon; Kronick, Richard (1977).
1748:Doron, Gideon; Kronick, Richard (1977).
1687:Doron, Gideon; Kronick, Richard (1977).
1433:
964:
2728:Independence of irrelevant alternatives
2506:Sequential proportional approval voting
2143:University of Maryland Baltimore County
1502:for apportionment in Germany after the
1458:by placing her first on their ballots.
2885:
2165:
2133:
2103:"Bundestag beschließt neues Wahlrecht"
1855:"Bundestag beschließt neues Wahlrecht"
1461:
1100:the elimination of moderate candidates
1090:. The paradox is especially common in
1078:However, the criterion is violated by
2290:
2264:
2063:
2061:
2059:
2022:American Journal of Political Science
1968:
1966:
1964:
1935:American Journal of Political Science
1754:American Journal of Political Science
1693:American Journal of Political Science
1655:
1653:
1651:
1649:
1425:A 2013 study using a two-dimensional
1412:
1347:2022 Alaska at-large special election
1134:ranked choice voting (instant-runoff)
1132:Runoff-based voting systems, such as
2185:
2009:
1159:successful campaign and adopting an
2538:Indirect single transferable voting
2227:"Ranked Choice Wackiness in Alaska"
2100:
1879:
1852:
1819:
1605:
1038:German Federal Constitutional Court
13:
2094:
2056:
1961:
1846:
1646:
57:
14:
2904:
1333:This election is an example of a
1805:10.1016/j.mathsocsci.2013.08.001
1482:Survey of nonmonotonic elections
1468:Burlington's second IRV election
1380:were instructed to vote for the
1319:
1276:
1254:
1211:
1138:2009 Burlington mayoral election
1125:
922:
909:
897:
845:McKelvey–Schofield chaos theorem
491:Semi-proportional representation
123:First preference plurality (FPP)
2218:
2179:
2159:
2127:
1922:
1562:Monotonicity (mechanism design)
1494:2005 German Election in Dresden
1102:and tends to favor extremists.
1098:, a behavior which can lead to
1088:Hamilton's apportionment method
2649:Mixed ballot transferable vote
1826:Pukelsheim, Friedrich (2014).
1780:
1741:
1680:
1660:Pukelsheim, Friedrich (2014).
1599:
1579:
1438:
1352:
1092:ranked-choice voting (RCV-IRV)
883:Harsanyi's utilitarian theorem
840:Moulin's impossibility theorem
805:Conflicting majorities paradox
1:
2243:10.1080/10724117.2023.2224675
2088:10.1080/00344893.2023.2221689
1592:
709:Frustrated majorities paradox
16:Property of electoral systems
2850:Comparison of voting systems
2692:Satisfaction approval voting
2677:Single non-transferable vote
2496:Proportional approval voting
2186:Quas, Anthony (2004-03-01).
2166:Miller, Nicholas R. (2012).
2134:Miller, Nicholas R. (2016).
1793:Mathematical Social Sciences
1520:Federal Constitutional Court
1386:Federal Constitutional Court
1370:2005 German federal election
1140:, the United States' second
1120:
878:Condorcet dominance theorems
818:Social and collective choice
7:
2456:Graduated majority judgment
1542:, a closely-related concept
1533:
1358:Proportional representation
544:By mechanism of combination
315:Proportional representation
10:
2909:
2708:Condorcet winner criterion
2399:First-past-the-post voting
2172:(PowerPoint). p. 23.
1454:, who could have defeated
742:Multiple districts paradox
473:Fractional approval voting
461:Interactive representation
18:
2893:Electoral system criteria
2863:
2855:Voting systems by country
2842:
2796:
2758:Mutual majority criterion
2713:Condorcet loser criterion
2700:
2667:
2659:Vote linkage mixed system
2614:
2579:
2571:Largest remainders method
2546:
2473:
2464:
2315:
2298:
2204:10.1142/S0219493704000912
1987:10.1007/s11127-013-0118-2
1900:10.1007/s11127-013-0118-2
1528:equal and direct suffrage
1486:A survey of 185 American
1394:equal and direct suffrage
1175:
1170:
1144:in the modern era, where
1042:equal and direct suffrage
689:Paradoxes and pathologies
538:Mixed-member proportional
533:Mixed-member majoritarian
528:By results of combination
419:Approval-based committees
2748:Majority loser criterion
2634:Additional member system
2592:Hagenbach-Bischoff quota
2511:Single transferable vote
2436:Positional voting system
2372:Minimax Condorcet method
2330:Combined approval voting
2192:Stochastics and Dynamics
1606:May, Kenneth O. (1952).
1572:
1084:single transferable vote
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:
2773:Resolvability criterion
2763:Participation criterion
2738:Later-no-harm criterion
2554:Highest averages method
2145:(2nd ed.). Table 2
1551:Voting system criterion
1540:Participation criterion
1427:spatial model of voting
1400:Frequency of violations
1142:instant-runoff election
1107:participation criterion
795:Tyranny of the majority
572:Fusion (majority bonus)
389:Quota-remainder methods
25:Participation criterion
2814:First-preference votes
2753:Monotonicity criterion
2723:Independence of clones
2426:Simple majoritarianism
987:
929:Mathematics portal
835:Majority impossibility
824:Impossibility theorems
620:Negative vote transfer
441:Method of equal shares
62:
2718:Consistency criterion
2639:Alternative vote plus
2404:Instant-runoff voting
1504:2005 federal election
1434:Real-world situations
1080:instant-runoff voting
968:
732:Best-is-worst paradox
721:Pathological response
456:Direct representation
109:Single-winner methods
61:
2788:Seats-to-votes ratio
2559:Webster/Sainte-Laguë
1556:Monotone preferences
916:Economics portal
863:Median voter theorem
82:Comparative politics
2768:Plurality criterion
2367:Kemeny–Young method
1567:Maskin monotonicity
1524:German constitution
1462:Burlington, Vermont
1390:German constitution
1167:
1063:methods (including
904:Politics portal
615:Vote linkage system
586:Seat linkage system
173:Ranked-choice (RCV)
2809:Election threshold
2743:Majority criterion
2419:Supplementary vote
2101:dpa (2013-02-22).
1853:dpa (2013-02-22).
1558:in consumer theory
1417:Results using the
1413:Theoretical models
1362:largest remainders
1166:
1009:is a principle of
988:
800:Discursive dilemma
759:Lesser evil voting
634:Supermixed systems
337:Largest remainders
195:Round-robin voting
63:
21:monotonic function
2880:
2879:
2778:Reversal symmetry
2687:Cumulative voting
2669:Semi-proportional
2644:Mixed single vote
2610:
2609:
2486:Mixed single vote
2394:Exhaustive ballot
2357:Copeland's method
2352:Condorcet methods
2292:Electoral systems
1839:978-3-319-03855-1
1673:978-3-319-03855-1
1500:Hamilton's method
1419:impartial culture
1331:
1330:
1176:Unpopular Bottom
1155:after running an
1057:round-robin rules
992:positive response
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
2900:
2819:Liquid democracy
2471:
2470:
2451:Two-round system
2362:Dodgson's method
2285:
2278:
2271:
2262:
2261:
2255:
2254:
2222:
2216:
2215:
2183:
2177:
2176:
2163:
2157:
2156:
2151:
2150:
2140:
2131:
2125:
2124:
2122:
2121:
2098:
2092:
2091:
2081:
2065:
2054:
2053:
2013:
2007:
2006:
1970:
1959:
1958:
1926:
1920:
1919:
1883:
1877:
1876:
1874:
1873:
1850:
1844:
1843:
1823:
1817:
1816:
1784:
1778:
1777:
1745:
1739:
1726:
1717:
1716:
1684:
1678:
1677:
1657:
1644:
1643:
1603:
1586:
1583:
1526:'s guarantee of
1476:Condorcet winner
1392:'s guarantee of
1326:
1323:
1322:
1283:
1280:
1279:
1261:
1258:
1257:
1218:
1215:
1214:
1168:
1165:
1096:two-round system
976:optimal solution
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:
29:
2908:
2907:
2903:
2902:
2901:
2899:
2898:
2897:
2883:
2882:
2881:
2876:
2859:
2838:
2792:
2783:Smith criterion
2696:
2663:
2624:Parallel voting
2606:
2602:Imperiali quota
2575:
2542:
2460:
2414:Contingent vote
2377:Nanson's method
2335:Unified primary
2325:Approval voting
2311:
2294:
2289:
2259:
2258:
2223:
2219:
2184:
2180:
2164:
2160:
2148:
2146:
2138:
2132:
2128:
2119:
2117:
2099:
2095:
2066:
2057:
2034:10.2307/3088418
2014:
2010:
1971:
1962:
1947:10.2307/2110496
1927:
1923:
1884:
1880:
1871:
1869:
1851:
1847:
1840:
1824:
1820:
1785:
1781:
1766:10.2307/2110496
1746:
1742:
1738:, Issue 6, 1996
1727:
1720:
1705:10.2307/2110496
1685:
1681:
1674:
1658:
1647:
1624:10.2307/1907651
1604:
1600:
1595:
1590:
1589:
1584:
1580:
1575:
1536:
1496:
1484:
1464:
1441:
1436:
1415:
1402:
1355:
1324:
1320:
1281:
1277:
1259:
1255:
1216:
1212:
1171:Popular Bottom
1130:
1123:
1069:highest medians
1055:and all common
959:
921:
920:
908:
896:
888:
887:
854:
830:Arrow's theorem
820:
810:
809:
778:
748:
737:No-show paradox
718:
704:Cloning paradox
694:Spoiler effects
691:
681:
680:
655:
542:
525:
515:
514:
487:
478:Maximal lottery
445:
426:Thiele's method
415:
385:
317:
307:
306:
292:Approval voting
280:Cardinal voting
276:
221:
215:Maximal lottery
179:
111:
101:
28:
17:
12:
11:
5:
2906:
2896:
2895:
2878:
2877:
2864:
2861:
2860:
2858:
2857:
2852:
2846:
2844:
2840:
2839:
2837:
2836:
2831:
2826:
2821:
2816:
2811:
2806:
2800:
2798:
2794:
2793:
2791:
2790:
2785:
2780:
2775:
2770:
2765:
2760:
2755:
2750:
2745:
2740:
2735:
2730:
2725:
2720:
2715:
2710:
2704:
2702:
2698:
2697:
2695:
2694:
2689:
2684:
2682:Limited voting
2679:
2673:
2671:
2665:
2664:
2662:
2661:
2656:
2651:
2646:
2641:
2636:
2631:
2626:
2620:
2618:
2612:
2611:
2608:
2607:
2605:
2604:
2599:
2594:
2589:
2583:
2581:
2577:
2576:
2574:
2573:
2568:
2567:
2566:
2561:
2550:
2548:
2544:
2543:
2541:
2540:
2535:
2530:
2529:
2528:
2523:
2518:
2508:
2503:
2498:
2493:
2488:
2483:
2477:
2475:
2468:
2462:
2461:
2459:
2458:
2453:
2448:
2443:
2438:
2433:
2428:
2423:
2422:
2421:
2416:
2411:
2409:Coombs' method
2401:
2396:
2391:
2390:
2389:
2387:Schulze method
2384:
2379:
2374:
2369:
2364:
2359:
2349:
2347:Bucklin voting
2344:
2339:
2338:
2337:
2332:
2321:
2319:
2313:
2312:
2299:
2296:
2295:
2288:
2287:
2280:
2273:
2265:
2257:
2256:
2217:
2178:
2158:
2126:
2093:
2070:Representation
2055:
2028:(1): 134–147.
2008:
1960:
1941:(2): 303–311.
1921:
1878:
1845:
1838:
1818:
1779:
1760:(2): 303–311.
1740:
1735:Voting matters
1718:
1699:(2): 303–311.
1679:
1672:
1645:
1618:(4): 680–684.
1597:
1596:
1594:
1591:
1588:
1587:
1577:
1576:
1574:
1571:
1570:
1569:
1564:
1559:
1553:
1548:
1543:
1535:
1532:
1495:
1492:
1488:instant-runoff
1483:
1480:
1463:
1460:
1450:supporters of
1440:
1437:
1435:
1432:
1414:
1411:
1407:instant-runoff
1401:
1398:
1360:systems using
1354:
1351:
1335:center-squeeze
1329:
1328:
1315:
1312:
1305:
1302:
1299:
1296:
1288:
1287:
1285:
1272:
1265:
1262:
1249:
1246:
1238:
1237:
1234:
1231:
1224:
1221:
1219:
1206:
1198:
1197:
1194:
1191:
1189:
1187:
1184:
1181:
1178:
1177:
1174:
1172:
1129:
1124:
1122:
1119:
1049:ranked methods
969:A 4-candidate
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:
15:
9:
6:
4:
3:
2:
2905:
2894:
2891:
2890:
2888:
2875:
2874:
2869:
2868:
2862:
2856:
2853:
2851:
2848:
2847:
2845:
2841:
2835:
2832:
2830:
2827:
2825:
2822:
2820:
2817:
2815:
2812:
2810:
2807:
2805:
2802:
2801:
2799:
2795:
2789:
2786:
2784:
2781:
2779:
2776:
2774:
2771:
2769:
2766:
2764:
2761:
2759:
2756:
2754:
2751:
2749:
2746:
2744:
2741:
2739:
2736:
2734:
2731:
2729:
2726:
2724:
2721:
2719:
2716:
2714:
2711:
2709:
2706:
2705:
2703:
2699:
2693:
2690:
2688:
2685:
2683:
2680:
2678:
2675:
2674:
2672:
2670:
2666:
2660:
2657:
2655:
2652:
2650:
2647:
2645:
2642:
2640:
2637:
2635:
2632:
2630:
2627:
2625:
2622:
2621:
2619:
2617:
2613:
2603:
2600:
2598:
2595:
2593:
2590:
2588:
2585:
2584:
2582:
2578:
2572:
2569:
2565:
2562:
2560:
2557:
2556:
2555:
2552:
2551:
2549:
2545:
2539:
2536:
2534:
2531:
2527:
2524:
2522:
2519:
2517:
2514:
2513:
2512:
2509:
2507:
2504:
2502:
2499:
2497:
2494:
2492:
2489:
2487:
2484:
2482:
2479:
2478:
2476:
2472:
2469:
2467:
2463:
2457:
2454:
2452:
2449:
2447:
2444:
2442:
2439:
2437:
2434:
2432:
2429:
2427:
2424:
2420:
2417:
2415:
2412:
2410:
2407:
2406:
2405:
2402:
2400:
2397:
2395:
2392:
2388:
2385:
2383:
2380:
2378:
2375:
2373:
2370:
2368:
2365:
2363:
2360:
2358:
2355:
2354:
2353:
2350:
2348:
2345:
2343:
2340:
2336:
2333:
2331:
2328:
2327:
2326:
2323:
2322:
2320:
2318:
2317:Single-winner
2314:
2310:
2308:
2304:
2297:
2293:
2286:
2281:
2279:
2274:
2272:
2267:
2266:
2263:
2252:
2248:
2244:
2240:
2236:
2232:
2231:Math Horizons
2228:
2221:
2213:
2209:
2205:
2201:
2198:(1): 95–105.
2197:
2193:
2189:
2182:
2175:
2171:
2170:
2162:
2155:
2144:
2137:
2130:
2116:
2112:
2109:(in German).
2108:
2104:
2097:
2089:
2085:
2080:
2075:
2071:
2064:
2062:
2060:
2051:
2047:
2043:
2039:
2035:
2031:
2027:
2023:
2019:
2012:
2004:
2000:
1996:
1992:
1988:
1984:
1980:
1976:
1975:Public Choice
1969:
1967:
1965:
1956:
1952:
1948:
1944:
1940:
1936:
1932:
1925:
1917:
1913:
1909:
1905:
1901:
1897:
1893:
1889:
1888:Public Choice
1882:
1868:
1864:
1861:(in German).
1860:
1856:
1849:
1841:
1835:
1831:
1830:
1822:
1814:
1810:
1806:
1802:
1798:
1794:
1790:
1783:
1775:
1771:
1767:
1763:
1759:
1755:
1751:
1744:
1737:
1736:
1731:
1728:D R Woodall,
1725:
1723:
1714:
1710:
1706:
1702:
1698:
1694:
1690:
1683:
1675:
1669:
1665:
1664:
1656:
1654:
1652:
1650:
1641:
1637:
1633:
1629:
1625:
1621:
1617:
1613:
1609:
1602:
1598:
1582:
1578:
1568:
1565:
1563:
1560:
1557:
1554:
1552:
1549:
1547:
1546:Voting system
1544:
1541:
1538:
1537:
1531:
1529:
1525:
1521:
1517:
1513:
1509:
1505:
1501:
1491:
1489:
1479:
1477:
1473:
1469:
1459:
1457:
1453:
1449:
1445:
1431:
1428:
1423:
1420:
1410:
1408:
1397:
1395:
1391:
1387:
1383:
1379:
1375:
1371:
1367:
1366:apportionment
1363:
1359:
1350:
1348:
1343:
1340:
1336:
1327:
1316:
1313:
1311:
1310:
1306:
1303:
1300:
1297:
1295:
1294:
1290:
1289:
1286:
1284:
1273:
1271:
1270:
1266:
1263:
1253:
1250:
1247:
1245:
1244:
1240:
1239:
1235:
1232:
1230:
1229:
1225:
1222:
1220:
1210:
1207:
1205:
1204:
1200:
1199:
1195:
1192:
1190:
1188:
1185:
1182:
1180:
1179:
1173:
1169:
1164:
1162:
1158:
1154:
1149:
1147:
1143:
1139:
1135:
1128:
1127:Runoff voting
1118:
1116:
1112:
1108:
1103:
1101:
1097:
1093:
1089:
1085:
1081:
1076:
1074:
1070:
1066:
1062:
1058:
1054:
1050:
1045:
1043:
1039:
1035:
1029:
1027:
1024:
1020:
1016:
1012:
1011:social choice
1008:
1005:
1004:nonperversity
1001:
1000:
996:
993:
985:
981:
978:(achieved by
977:
972:
967:
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
45:
44:
40:
36:
32:
31:
26:
22:
2871:
2865:
2752:
2481:Mixed-member
2466:Proportional
2441:Score voting
2382:Ranked pairs
2301:Part of the
2300:
2237:(1): 24–27.
2234:
2230:
2220:
2195:
2191:
2181:
2173:
2168:
2161:
2153:
2147:. Retrieved
2142:
2129:
2118:. Retrieved
2106:
2096:
2069:
2025:
2021:
2011:
1981:(1–2): 1–9.
1978:
1974:
1938:
1934:
1924:
1894:(1–2): 1–9.
1891:
1887:
1881:
1870:. Retrieved
1858:
1848:
1828:
1821:
1796:
1792:
1782:
1757:
1753:
1743:
1733:
1696:
1692:
1682:
1662:
1615:
1612:Econometrica
1611:
1601:
1581:
1497:
1485:
1470:, incumbent
1465:
1456:Mary Peltola
1442:
1424:
1416:
1403:
1356:
1344:
1332:
1317:
1308:
1307:
1292:
1291:
1274:
1268:
1267:
1251:
1242:
1241:
1227:
1226:
1208:
1202:
1201:
1160:
1156:
1152:
1150:
1131:
1114:
1110:
1104:
1077:
1061:rated voting
1046:
1030:
1025:
1022:
1018:
1014:
1006:
1003:
999:monotonicity
998:
995:
991:
989:
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:
2824:Spoilt vote
2587:Droop quota
2526:Schulze STV
2501:Rural–urban
2446:STAR voting
2342:Borda count
1452:Sarah Palin
1439:Alaska 2022
1353:Quota rules
1051:(including
971:Yee diagram
664:Single vote
567:Conditional
562:Coexistence
411:Quota Borda
401:Schulze STV
359:Closed list
302:STAR voting
247:Borda count
2843:Comparison
2597:Hare quota
2547:Allocation
2533:Spare vote
2521:Hare-Clark
2491:Party-list
2149:2020-07-26
2120:2024-05-02
2079:2301.12075
1872:2024-05-02
1593:References
1510:voters in
1448:Republican
1376:voters in
1036:, and the
1015:increasing
1013:that says
769:Truncation
498:Cumulative
321:Party-list
96:By country
87:Comparison
2834:Unseating
2829:Sortition
2431:Plurality
2307:Economics
2251:1072-4117
2212:0219-4937
2115:0044-2070
2042:0092-5853
1995:0048-5829
1955:0092-5853
1908:0048-5829
1867:0044-2070
1813:0165-4896
1799:: 57–66.
1774:0092-5853
1713:0092-5853
1632:0012-9682
1339:plurality
1121:By method
1034:pathology
1007:criterion
980:Condorcet
676:Dual-vote
369:Panachage
364:Open list
354:List type
232:Plurality
128:Two-round
116:plurality
39:Economics
2887:Category
2701:Criteria
2654:Scorporo
2303:politics
2107:Die Zeit
2072:: 1–19.
2003:30833409
1916:30833409
1859:Die Zeit
1534:See also
1472:Bob Kiss
1196:Round 2
1193:Round 1
1186:Round 2
1183:Round 1
1146:Bob Kiss
1111:changing
1094:and the
1065:approval
1026:response
1023:perverse
986:voting).
396:Hare STV
35:Politics
33:A joint
2873:Project
2564:D'Hondt
2516:CPO-STV
2474:Systems
2050:3088418
1640:1907651
1512:Dresden
1378:Dresden
1372:, when
1153:elected
406:CPO-STV
256:Baldwin
205:Schulze
200:Minimax
118:methods
2867:Portal
2804:Ballot
2580:Quotas
2309:series
2249:
2210:
2113:
2048:
2040:
2001:
1993:
1953:
1914:
1906:
1865:
1836:
1811:
1772:
1711:
1670:
1638:
1630:
1309:Bottom
1293:Bottom
1269:Center
1243:Center
1086:, and
1082:, the
1071:, and
271:Coombs
41:series
2797:Other
2616:Mixed
2139:(PDF)
2074:arXiv
2046:JSTOR
1999:S2CID
1912:S2CID
1636:JSTOR
1573:Notes
1073:score
1053:Borda
1047:Most
1002:, or
984:score
608:'MMP'
597:'AMS'
2305:and
2247:ISSN
2208:ISSN
2111:ISSN
2038:ISSN
1991:ISSN
1951:ISSN
1904:ISSN
1863:ISSN
1834:ISBN
1809:ISSN
1770:ISSN
1709:ISSN
1668:ISBN
1628:ISSN
1364:for
1318:54%
1314:39%
1304:-6%
1301:45%
1298:45%
1275:30%
1248:30%
1236:46%
1233:31%
1223:+6%
1115:cast
1105:The
1019:lose
990:The
549:Non-
503:SNTV
92:List
49:and
37:and
2629:MMP
2239:doi
2200:doi
2084:doi
2030:doi
1983:doi
1979:161
1943:doi
1896:doi
1892:161
1801:doi
1762:doi
1701:doi
1620:doi
1516:FDP
1508:CDU
1466:In
1382:FDP
1374:CDU
1252:55%
1228:Top
1209:25%
1203:Top
1075:).
982:or
267:el.
252:el.
241:IRV
237:el.
2889::
2870:—
2245:.
2235:31
2233:.
2229:.
2206:.
2196:04
2194:.
2190:.
2152:.
2141:.
2105:.
2082:.
2058:^
2044:.
2036:.
2026:46
2024:.
2020:.
1997:.
1989:.
1977:.
1963:^
1949:.
1939:21
1937:.
1933:.
1910:.
1902:.
1890:.
1857:.
1807:.
1797:67
1795:.
1791:.
1768:.
1758:21
1756:.
1752:.
1732:,
1721:^
1707:.
1697:21
1695:.
1691:.
1648:^
1634:.
1626:.
1616:20
1614:.
1610:.
1530:.
1506:.
1396:.
1349:.
1264:↗
1161:un
1157:un
1067:,
1044:.
1028:.
603:NZ
592:UK
168:US
157:UK
140:)
133:US
2284:e
2277:t
2270:v
2253:.
2241::
2214:.
2202::
2123:.
2090:.
2086::
2076::
2052:.
2032::
2005:.
1985::
1957:.
1945::
1918:.
1898::
1875:.
1842:.
1815:.
1803::
1776:.
1764::
1715:.
1703::
1676:.
1642:.
1622::
1325:Y
1282:N
1260:Y
1217:N
994:,
954:e
947:t
940:v
605::
594::
273:)
264:(
258:)
249:(
243:)
234:(
170::
159::
135::
130:(
98:)
94:(
27:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
