1792:. It is always optimal for a voter to give the best candidate the highest possible score, and the worst candidate the lowest possible score. Then, no matter which score the voter assigns to the middle candidate, it will always fall (non-strictly) between the first and last scores; this implies the voter's score ballot will be weakly consistent with that voter's honest ranking. However, the actual optimal score may depend on the other ballots cast, as indicated by
2516:
25:
67:
257:: each voter communicates his or her preference order over the candidates. For each ballot, 3 points are assigned to the top candidate, 2 points to the second candidate, 1 point to the third one and 0 points to the last one. Once all ballots have been counted, the candidate with the most points is declared the winner.
2009:
For a strict voting rule, the converse of the
Gibbard–Satterthwaite theorem is true. Indeed, a strict voting rule is dictatorial if and only if it always selects the most-liked candidate of the dictator among the possible outcomes; in particular, it does not depend on the other voters' ballots. As a
1821:
If there are only 2 possible outcomes, a voting rule may be non-manipulable without being dictatorial. For example, it is the case of the simple majority vote: each voter assigns 1 point to her top alternative and 0 to the other, and the alternative with most points is declared the winner. (If both
1812:
This voting rule is not manipulable: a voter is always better off communicating his or her sincere preferences. It is also dictatorial, and its dictator is voter 1: the winning alternative is always that specific voter's most-liked one or, if there are several most-liked alternatives, it is chosen
1808:
is defined as follows. If voter 1 has a unique most-liked candidate, then this candidate is elected. Otherwise, possible outcomes are restricted to the most-liked candidates, whereas the other candidates are eliminated. Then voter 2's ballot is examined: if there is a unique best-liked candidate
2028:
of the voting rule. It is possible that some other alternatives can be elected in no circumstances: the theorem and the corollary still apply. However, the corollary is sometimes presented under a less general form: instead of assuming that the rule has at least three possible outcomes, it is
1809:
among the non-eliminated ones, then this candidate is elected. Otherwise, the list of possible outcomes is reduced again, etc. If there are still several non-eliminated candidates after all ballots have been examined, then an arbitrary tie-breaking rule is used.
1842:
wins.) This voting rule is not manipulable because a voter is always better off communicating his or her sincere preferences; and it is clearly not dictatorial. Many other rules are neither manipulable nor dictatorial: for example, assume that the alternative
2449:. Independently, Satterthwaite proved the same result in his PhD dissertation in 1973, then published it in a 1975 article. This proof is also based on Arrow's impossibility theorem, but does not involve the more general version given by Gibbard's theorem.
2010:
consequence, it is not manipulable: the dictator is perfectly defended by her sincere ballot, and the other voters have no impact on the outcome, hence they have no incentive to deviate from sincere voting. Thus, we obtain the following equivalence.
2495:
The GS theorem seems to quash any hope of designing incentive-compatible social-choice functions. The whole field of
Mechanism Design attempts escaping from this impossibility result using various modifications in the
1473:
1377:
1992:
1308:
2392:
1201:
2351:
2331:
2228:
2168:
2122:
1704:
1620:
1039:
617:
2024:
In the theorem, as well as in the corollary, it is not needed to assume that any alternative can be elected. It is only assumed that at least three of them can win, i.e. are
1115:
is manipulable, except possibly in two cases: if there is a distinguished voter who has a dictatorial power, or if the rule limits the possible outcomes to two options only.
2051:
1941:
1913:
1257:
1229:
1145:
1550:
2460:
deals with processes of collective choice that may not be ordinal, i.e. where a voter's action may not consist in communicating a preference order over the candidates.
2148:
1577:
1520:
2375:
2311:
2288:
2268:
2248:
2208:
2188:
2095:
1881:
1861:
1840:
1751:
1727:
1664:
1640:
1493:
1400:
1099:
1079:
1059:
971:
951:
926:
905:
884:
863:
837:
816:
795:
774:
748:
727:
706:
685:
637:
544:
523:
502:
481:
455:
434:
413:
392:
366:
345:
324:
303:
251:
231:
211:
191:
2468:
extend these results to non-deterministic mechanisms, i.e. where the outcome may not only depend on the ballots but may also involve a part of chance.
3350:
2676:(April 1975). "Strategy-proofness and Arrow's conditions: Existence and correspondence theorems for voting procedures and social welfare functions".
2500:
The main idea of these "escape routes" is that they allow for a broader class of mechanisms than ranked voting, similarly to the escape routes from
1259:: an element of this set can represent the preferences of a voter, where a voter may be indifferent regarding the ordering of some alternatives. A
85:
1761:. If the dictator has several equally most-liked alternatives among the possible outcomes, then the winning alternative is simply one of them.
2333:
satisfies independence of irrelevant alternatives. Arrow's impossibility theorem says that, when there are three or more alternatives, such a
3283:
Duggan, John; Schwartz, Thomas (2000). "Strategic manipulability without resoluteness or shared beliefs: Gibbard-Satterthwaite generalized".
2483:
The
Gibbard–Satterthwaite theorem is generally presented as a result about voting systems, but it can also be seen as an important result of
2057:, i.e. every alternative is a possible outcome. The assumption of being onto is sometimes even replaced with the assumption that the rule is
3076:
642:
But Alice can vote strategically and change the result. Assume that she modifies her ballot, in order to produce the following situation.
41:
2387:
3006:
128:
that choose a single winner, and shows that for every voting rule of this form, at least one of the following three things must hold:
3111:
3086:
2887:
161:
are even more general and extend these results to non-deterministic processes, where the outcome may depend partly on chance; the
2529:
2501:
2442:
2070:
1409:
1313:
2572:
Gibbard's theorem does not imply that cardinal methods necessarily incentivize reversing one's relative rank of two candidates.
1950:
1266:
1780:
A variety of "counterexamples" to the
Gibbard-Satterthwaite theorem exist when the conditions of the theorem do not apply.
3355:
1155:, even if they are not necessarily persons: they can also be several possible decisions about a given issue. We denote by
2954:
Barberá, Salvador (1983). "Strategy-Proofness and
Pivotal Voters: A Direct Proof of the Gibbard-Satterthwaite Theorem".
1822:
alternatives reach the same number of points, the tie is broken in an arbitrary but deterministic manner, e.g. outcome
2539:
2472:
162:
3345:
1158:
2475:
extend this result in another direction, by dealing with deterministic voting rules that choose multiple winners.
2417:
This principle of voting makes an election more of a game of skill than a real test of the wishes of the electors.
2020:
If a strict voting rule has at least 3 possible outcomes, it is non-manipulable if and only if it is dictatorial.
1891:
We now consider the case where by assumption, a voter cannot be indifferent between two candidates. We denote by
2461:
154:
2074:
2336:
2316:
2213:
2153:
2107:
1771:
If an ordinal voting rule has at least 3 possible outcomes and is non-dictatorial, then it is manipulable.
1669:
1585:
3340:
2874:
2744:
2685:
2354:
173:
Consider three voters named Alice, Bob and Carol, who wish to select a winner among four candidates named
2429:, he conjectures that deterministic voting rules with at least three outcomes are never straightforward
976:
554:
2735:
Reny, Philip J. (2001). "Arrow's
Theorem and the Gibbard-Satterthwaite Theorem: A Unified Approach".
2544:
2465:
2457:
2446:
2409:, a pioneer in social choice theory. His quote (about a particular voting system) was made famous by
1793:
1757:, in the sense that the winning alternative is always her most-liked one among the possible outcomes
158:
146:
76:
2749:
2690:
2032:
1922:
1894:
1238:
1210:
1126:
81:
2903:
Gärdenfors, Peter (1977). "A Concise Proof of
Theorem on Manipulation of Social Choice Functions".
139:
3316:
3308:
3265:
3257:
3222:
3187:
3128:
3054:
3046:
2979:
2936:
2928:
2673:
2652:
2611:
2438:
121:
3300:
3161:
3082:
3002:
2971:
2920:
2883:
2837:
2814:
2792:
2422:
2127:
132:
The rule is dictatorial, i.e. there exists a distinguished voter who can choose the winner; or
113:
3024:"An interview with Michael Dummett: From analytical philosophy to voting analysis and beyond"
2589:"An Interview with Michael Dummett: From analytical philosophy to voting analysis and beyond"
1108:: there exists situations where a sincere ballot does not defend a voter's preferences best.
149:
is more general and covers processes of collective decision that may not be ordinal, such as
3292:
3249:
3214:
3177:
3120:
3038:
2963:
2912:
2829:
2784:
2754:
2695:
2644:
2603:
2554:
2484:
2405:
The strategic aspect of voting is already noticed in 1876 by
Charles Dodgson, also known as
2098:
1525:
1555:
1498:
1061:
is elected. Alice is satisfied by her ballot modification, because she prefers the outcome
2866:
2534:
2430:
2426:
150:
109:
3072:
2714:
3182:
3165:
2858:
2521:
2360:
2296:
2273:
2253:
2233:
2193:
2173:
2080:
2061:, in the sense that if all voters prefer the same candidate, then she must be elected.
1866:
1846:
1825:
1736:
1712:
1649:
1625:
1478:
1385:
1084:
1064:
1044:
956:
936:
911:
890:
869:
848:
822:
801:
780:
759:
733:
712:
691:
670:
622:
529:
508:
487:
466:
440:
419:
398:
377:
351:
330:
309:
288:
236:
216:
196:
176:
2833:
2788:
2758:
3334:
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2870:
2699:
2632:
2549:
2434:
2406:
2383:
1232:
1112:
125:
117:
105:
3269:
3058:
2615:
2410:
1789:
3320:
2775:
Benoît, Jean-Pierre (2000). "The
Gibbard-Satterthwaite Theorem: A Simple Proof".
1916:
254:
3253:
3042:
3023:
2862:
2607:
2588:
2511:
2488:
3304:
2975:
2924:
2841:
2796:
3205:
Dummett, Michael; Farquharson, Robin (January 1961). "Stability in voting".
2515:
2077:. We give a sketch of proof in the simplified case where some voting rule
3312:
3261:
3050:
2932:
2715:"Alternatives vs. Outcomes: A Note on the Gibbard-Satterthwaite Theorem"
3296:
3226:
3191:
3132:
2983:
2916:
2656:
1666:
has at least three possible outcomes if and only if the cardinality of
3109:
Taylor, Alan D. (April 2002). "The manipulability of voting systems".
3218:
3124:
2967:
2648:
2425:
published influential articles on voting theory. In an article with
142:(one that does not depend on other voters' preferences or behavior).
3240:
Dummett, Michael (2005). "The work and life of Robin
Farquharson".
135:
The rule limits the possible outcomes to two alternatives only; or
1101:, which is the outcome she would obtain if she voted sincerely.
60:
18:
2053:
contains at least three elements and that the voting rule is
2635:(1973). "Manipulation of voting schemes: A general result".
2038:
1979:
1963:
1928:
1900:
1682:
1598:
1454:
1358:
1295:
1279:
1244:
1216:
1164:
1132:
2815:"Another Direct Proof of the Gibbard-Satterthwaite Theorem"
1468:{\displaystyle (P_{1},\ldots ,P_{n})\in {\mathcal {P}}^{n}}
1372:{\displaystyle (P_{1},\ldots ,P_{n})\in {\mathcal {P}}^{n}}
38:
inadequate description of theorem and practical importance.
2380:
Later authors have developed other variants of the proof.
551:
If the voters cast sincere ballots, then the scores are:
138:
The rule is not straightforward, i.e. there is no single
116:
in 1961 and then proved independently by the philosopher
165:
extends these results to multiwinner electoral systems.
2210:
are moved to the top of all voters' preferences. Then,
1552:, can get an outcome that she prefers (in the sense of
2487:, which deals with a broader class of decision rules.
2069:
The Gibbard–Satterthwaite theorem can be proved using
1987:{\displaystyle f:{\mathcal {L}}^{n}\to {\mathcal {A}}}
1303:{\displaystyle f:{\mathcal {P}}^{n}\to {\mathcal {A}}}
2363:
2339:
2319:
2299:
2276:
2256:
2236:
2216:
2196:
2176:
2156:
2130:
2110:
2083:
2035:
1953:
1925:
1897:
1869:
1849:
1828:
1739:
1715:
1672:
1652:
1628:
1588:
1558:
1528:
1501:
1481:
1412:
1388:
1379:
and it yields the identity of the winning candidate.
1316:
1269:
1241:
1213:
1161:
1129:
1087:
1067:
1047:
979:
959:
939:
914:
893:
872:
851:
825:
804:
783:
762:
736:
715:
694:
673:
625:
557:
532:
511:
490:
469:
443:
422:
401:
380:
354:
333:
312:
291:
239:
219:
199:
179:
16:
Impossibility result for ranked-choice voting systems
2433:. This conjecture was later proven independently by
1111:
The Gibbard–Satterthwaite theorem states that every
2369:
2345:
2325:
2305:
2282:
2262:
2242:
2222:
2202:
2182:
2162:
2142:
2116:
2104:It is possible to build a social ranking function
2089:
2045:
1986:
1935:
1907:
1875:
1855:
1834:
1745:
1721:
1698:
1658:
1634:
1614:
1571:
1544:
1514:
1487:
1467:
1394:
1371:
1302:
1251:
1223:
1195:
1139:
1093:
1073:
1053:
1033:
965:
945:
920:
899:
878:
857:
831:
810:
789:
768:
742:
721:
700:
679:
631:
611:
538:
517:
496:
475:
449:
428:
407:
386:
360:
339:
318:
297:
245:
225:
205:
185:
1788:Consider a three-candidate election conducted by
2587:Rudolf Farra and Maurice Salles (October 2006).
2719:Technical Report (University Library of Munich)
2668:
2666:
1196:{\displaystyle {\mathcal {N}}=\{1,\ldots ,n\}}
260:Assume that their preferences are as follows.
108:. It was first conjectured by the philosopher
2882:. Cambridge, UK: Cambridge University Press.
2445:from 1951 to prove the result we now know as
1863:wins if it gets two thirds of the votes, and
1759:regardless of the preferences of other voters
34:needs attention from an expert in game theory
8:
2006:have natural adaptations to this framework.
1190:
1172:
2014:
1765:
1646:for the election. For example, we say that
933:Alice has strategically upgraded candidate
3166:"Straightforwardness in voting procedures"
2170:function creates new preferences in which
3181:
2748:
2689:
2627:
2625:
2362:
2338:
2318:
2298:
2275:
2255:
2235:
2215:
2195:
2175:
2155:
2129:
2124:, as follows: in order to decide whether
2109:
2082:
2037:
2036:
2034:
1978:
1977:
1968:
1962:
1961:
1952:
1927:
1926:
1924:
1899:
1898:
1896:
1868:
1848:
1827:
1738:
1714:
1687:
1681:
1680:
1671:
1651:
1627:
1603:
1597:
1596:
1587:
1563:
1557:
1533:
1527:
1506:
1500:
1480:
1459:
1453:
1452:
1439:
1420:
1411:
1387:
1363:
1357:
1356:
1343:
1324:
1315:
1294:
1293:
1284:
1278:
1277:
1268:
1243:
1242:
1240:
1215:
1214:
1212:
1163:
1162:
1160:
1131:
1130:
1128:
1086:
1066:
1046:
978:
958:
938:
913:
892:
871:
850:
824:
803:
782:
761:
735:
714:
693:
672:
624:
556:
531:
510:
489:
468:
442:
421:
400:
379:
353:
332:
311:
290:
238:
218:
198:
178:
2770:
2768:
2313:is non-manipulable and non-dictatorial,
1310:. Its input is a profile of preferences
644:
262:
2579:
2565:
1151:(which is assumed finite), also called
3148:The theory of committees and elections
3022:Fara, Rudolf; Salles, Maurice (2006).
2730:
2728:
2441:. In a 1973 article, Gibbard exploits
1406:if and only if there exists a profile
44:may be able to help recruit an expert.
3078:Axioms of Cooperative Decision Making
2853:
2851:
2808:
2806:
2346:{\displaystyle \operatorname {Rank} }
2326:{\displaystyle \operatorname {Rank} }
2223:{\displaystyle \operatorname {Rank} }
2163:{\displaystyle \operatorname {Rank} }
2117:{\displaystyle \operatorname {Rank} }
1699:{\displaystyle f({\mathcal {P}}^{n})}
1615:{\displaystyle f({\mathcal {P}}^{n})}
124:in 1975. It deals with deterministic
7:
1733:if and only if there exists a voter
3183:10.1093/oxfordjournals.oep.a042255
14:
3112:The American Mathematical Monthly
2293:It is possible to prove that, if
1034:{\displaystyle (a:2,b:7,c:6,d:3)}
612:{\displaystyle (a:3,b:6,c:7,d:2)}
3351:Theorems in discrete mathematics
2514:
639:will be elected, with 7 points.
65:
23:
1104:We say that the Borda count is
3150:. Cambridge: University Press.
3081:. Cambridge University Press.
2046:{\displaystyle {\mathcal {A}}}
1974:
1936:{\displaystyle {\mathcal {A}}}
1908:{\displaystyle {\mathcal {L}}}
1693:
1676:
1609:
1592:
1445:
1413:
1349:
1317:
1290:
1252:{\displaystyle {\mathcal {A}}}
1224:{\displaystyle {\mathcal {P}}}
1140:{\displaystyle {\mathcal {A}}}
1028:
980:
606:
558:
147:Gibbard's proof of the theorem
1:
2956:International Economic Review
2834:10.1016/S0165-1765(00)00362-1
2789:10.1016/S0165-1765(00)00312-8
2759:10.1016/S0165-1765(00)00332-3
2530:Arrow's impossibility theorem
2502:Arrow's impossibility theorem
2443:Arrow's impossibility theorem
2377:must also be a dictatorship.
2071:Arrow's impossibility theorem
1776:Counterexamples and loopholes
1766:Gibbard–Satterthwaite theorem
102:Gibbard–Satterthwaite theorem
91:Proposed since November 2023.
2700:10.1016/0022-0531(75)90050-2
2357:. Hence, such a voting rule
3001:. Oxford University Press.
253:. Assume that they use the
74:It has been suggested that
36:. The specific problem is:
3372:
2678:Journal of Economic Theory
2388:Excessive citations inline
1495:, by replacing her ballot
3285:Social Choice and Welfare
3254:10.1007/s00355-005-0014-x
3242:Social Choice and Welfare
3043:10.1007/s00355-006-0128-9
3031:Social Choice and Welfare
2997:Dummett, Michael (1984).
2674:Satterthwaite, Mark Allen
2608:10.1007/s00355-006-0128-9
2596:Social Choice and Welfare
953:and downgraded candidate
126:ordinal electoral systems
2491:describes this relation:
2143:{\displaystyle a\prec b}
2075:social ranking functions
2876:Algorithmic Game Theory
2540:Duggan–Schwartz theorem
2473:Duggan–Schwartz theorem
2393:considered for deletion
2029:sometimes assumed that
973:. Now, the scores are:
163:Duggan–Schwartz theorem
42:WikiProject Game theory
3170:Oxford Economic Papers
3146:Black, Duncan (1958).
2498:
2462:Gibbard's 1978 theorem
2419:
2371:
2347:
2327:
2307:
2284:
2264:
2244:
2224:
2204:
2184:
2164:
2144:
2118:
2091:
2047:
1988:
1937:
1909:
1877:
1857:
1836:
1747:
1723:
1700:
1660:
1636:
1616:
1573:
1546:
1545:{\displaystyle P_{i}'}
1516:
1489:
1469:
1396:
1373:
1304:
1253:
1225:
1197:
1141:
1095:
1075:
1055:
1035:
967:
947:
922:
901:
880:
859:
833:
812:
791:
770:
744:
723:
702:
681:
633:
613:
540:
519:
498:
477:
451:
430:
409:
388:
362:
341:
320:
299:
247:
227:
207:
187:
155:Gibbard's 1978 theorem
120:in 1973 and economist
112:and the mathematician
2813:Sen, Arunava (2001).
2713:Weber, Tjark (2009).
2493:
2415:
2372:
2348:
2328:
2308:
2285:
2265:
2245:
2225:
2205:
2185:
2165:
2145:
2119:
2092:
2048:
1994:. The definitions of
1989:
1938:
1910:
1878:
1858:
1837:
1748:
1724:
1701:
1661:
1637:
1617:
1574:
1572:{\displaystyle P_{i}}
1547:
1517:
1515:{\displaystyle P_{i}}
1490:
1470:
1397:
1374:
1305:
1254:
1226:
1198:
1142:
1096:
1076:
1056:
1036:
968:
948:
923:
902:
881:
860:
834:
813:
792:
771:
745:
724:
703:
682:
634:
614:
541:
520:
499:
478:
452:
431:
410:
389:
363:
342:
321:
300:
248:
228:
208:
188:
3356:Social choice theory
2561:Notes and references
2361:
2337:
2317:
2297:
2274:
2254:
2234:
2214:
2194:
2174:
2154:
2128:
2108:
2081:
2033:
1951:
1923:
1895:
1867:
1847:
1826:
1817:Simple majority vote
1737:
1713:
1670:
1650:
1626:
1586:
1556:
1526:
1522:with another ballot
1499:
1479:
1410:
1386:
1314:
1267:
1239:
1211:
1159:
1127:
1113:ranked-choice voting
1085:
1065:
1045:
977:
957:
937:
912:
891:
870:
849:
823:
802:
781:
760:
734:
713:
692:
671:
623:
555:
530:
509:
488:
467:
441:
420:
399:
378:
352:
331:
310:
289:
237:
217:
197:
177:
169:Informal description
140:always-best strategy
84:into this article. (
2353:function must be a
2018: —
1917:strict total orders
1806:serial dictatorship
1800:Serial dictatorship
1769: —
1541:
619:. Hence, candidate
3346:Economics theorems
3297:10.1007/PL00007177
3162:Farquharson, Robin
2917:10.1007/bf01718676
2859:Vazirani, Vijay V.
2439:Mark Satterthwaite
2421:During the 1950s,
2367:
2343:
2323:
2303:
2280:
2260:
2240:
2220:
2200:
2180:
2160:
2140:
2114:
2087:
2043:
2016:
1984:
1945:strict voting rule
1933:
1905:
1873:
1853:
1832:
1767:
1743:
1719:
1696:
1656:
1642:, i.e. the set of
1632:
1612:
1569:
1542:
1529:
1512:
1485:
1465:
1392:
1369:
1300:
1249:
1233:strict weak orders
1221:
1193:
1137:
1091:
1071:
1051:
1031:
963:
943:
918:
897:
876:
855:
829:
808:
787:
766:
740:
719:
698:
677:
629:
609:
536:
515:
494:
473:
447:
426:
405:
384:
358:
337:
316:
295:
243:
223:
203:
183:
122:Mark Satterthwaite
3164:(February 1956).
2999:Voting Procedures
2822:Economics Letters
2777:Economics Letters
2737:Economics Letters
2545:Gibbard's theorem
2466:Hylland's theorem
2458:Gibbard's theorem
2447:Gibbard's theorem
2423:Robin Farquharson
2370:{\displaystyle f}
2306:{\displaystyle f}
2283:{\displaystyle b}
2263:{\displaystyle a}
2243:{\displaystyle f}
2230:examines whether
2203:{\displaystyle b}
2183:{\displaystyle a}
2097:is assumed to be
2090:{\displaystyle f}
2026:possible outcomes
1996:possible outcomes
1876:{\displaystyle b}
1856:{\displaystyle a}
1835:{\displaystyle a}
1794:Gibbard's theorem
1746:{\displaystyle i}
1722:{\displaystyle f}
1659:{\displaystyle f}
1644:possible outcomes
1635:{\displaystyle f}
1488:{\displaystyle i}
1475:where some voter
1395:{\displaystyle f}
1094:{\displaystyle c}
1074:{\displaystyle b}
1054:{\displaystyle b}
966:{\displaystyle c}
946:{\displaystyle b}
931:
930:
921:{\displaystyle a}
900:{\displaystyle d}
879:{\displaystyle b}
858:{\displaystyle c}
832:{\displaystyle a}
811:{\displaystyle d}
790:{\displaystyle b}
769:{\displaystyle c}
743:{\displaystyle c}
722:{\displaystyle d}
701:{\displaystyle a}
680:{\displaystyle b}
632:{\displaystyle c}
549:
548:
539:{\displaystyle a}
518:{\displaystyle d}
497:{\displaystyle b}
476:{\displaystyle c}
450:{\displaystyle a}
429:{\displaystyle d}
408:{\displaystyle b}
387:{\displaystyle c}
361:{\displaystyle d}
340:{\displaystyle c}
319:{\displaystyle b}
298:{\displaystyle a}
246:{\displaystyle d}
226:{\displaystyle c}
206:{\displaystyle b}
186:{\displaystyle a}
159:Hylland's theorem
114:Robin Farquharson
98:
97:
93:
77:Gibbard's theorem
59:
58:
3363:
3325:
3324:
3280:
3274:
3273:
3237:
3231:
3230:
3202:
3196:
3195:
3185:
3158:
3152:
3151:
3143:
3137:
3136:
3106:
3100:
3099:
3097:
3095:
3069:
3063:
3062:
3028:
3019:
3013:
3012:
2994:
2988:
2987:
2951:
2945:
2944:
2900:
2894:
2893:
2881:
2867:Roughgarden, Tim
2855:
2846:
2845:
2819:
2810:
2801:
2800:
2772:
2763:
2762:
2752:
2732:
2723:
2722:
2710:
2704:
2703:
2693:
2670:
2661:
2660:
2629:
2620:
2619:
2593:
2584:
2573:
2570:
2555:Strategic voting
2524:
2519:
2518:
2485:mechanism design
2396:
2376:
2374:
2373:
2368:
2352:
2350:
2349:
2344:
2332:
2330:
2329:
2324:
2312:
2310:
2309:
2304:
2289:
2287:
2286:
2281:
2269:
2267:
2266:
2261:
2249:
2247:
2246:
2241:
2229:
2227:
2226:
2221:
2209:
2207:
2206:
2201:
2189:
2187:
2186:
2181:
2169:
2167:
2166:
2161:
2149:
2147:
2146:
2141:
2123:
2121:
2120:
2115:
2099:Pareto-efficient
2096:
2094:
2093:
2088:
2052:
2050:
2049:
2044:
2042:
2041:
2019:
1993:
1991:
1990:
1985:
1983:
1982:
1973:
1972:
1967:
1966:
1943:and we define a
1942:
1940:
1939:
1934:
1932:
1931:
1914:
1912:
1911:
1906:
1904:
1903:
1883:wins otherwise.
1882:
1880:
1879:
1874:
1862:
1860:
1859:
1854:
1841:
1839:
1838:
1833:
1770:
1752:
1750:
1749:
1744:
1728:
1726:
1725:
1720:
1705:
1703:
1702:
1697:
1692:
1691:
1686:
1685:
1665:
1663:
1662:
1657:
1641:
1639:
1638:
1633:
1621:
1619:
1618:
1613:
1608:
1607:
1602:
1601:
1578:
1576:
1575:
1570:
1568:
1567:
1551:
1549:
1548:
1543:
1537:
1521:
1519:
1518:
1513:
1511:
1510:
1494:
1492:
1491:
1486:
1474:
1472:
1471:
1466:
1464:
1463:
1458:
1457:
1444:
1443:
1425:
1424:
1401:
1399:
1398:
1393:
1378:
1376:
1375:
1370:
1368:
1367:
1362:
1361:
1348:
1347:
1329:
1328:
1309:
1307:
1306:
1301:
1299:
1298:
1289:
1288:
1283:
1282:
1258:
1256:
1255:
1250:
1248:
1247:
1230:
1228:
1227:
1222:
1220:
1219:
1202:
1200:
1199:
1194:
1168:
1167:
1146:
1144:
1143:
1138:
1136:
1135:
1119:Formal statement
1100:
1098:
1097:
1092:
1080:
1078:
1077:
1072:
1060:
1058:
1057:
1052:
1040:
1038:
1037:
1032:
972:
970:
969:
964:
952:
950:
949:
944:
927:
925:
924:
919:
906:
904:
903:
898:
885:
883:
882:
877:
864:
862:
861:
856:
838:
836:
835:
830:
817:
815:
814:
809:
796:
794:
793:
788:
775:
773:
772:
767:
749:
747:
746:
741:
728:
726:
725:
720:
707:
705:
704:
699:
686:
684:
683:
678:
645:
638:
636:
635:
630:
618:
616:
615:
610:
545:
543:
542:
537:
524:
522:
521:
516:
503:
501:
500:
495:
482:
480:
479:
474:
456:
454:
453:
448:
435:
433:
432:
427:
414:
412:
411:
406:
393:
391:
390:
385:
367:
365:
364:
359:
346:
344:
343:
338:
325:
323:
322:
317:
304:
302:
301:
296:
263:
252:
250:
249:
244:
232:
230:
229:
224:
212:
210:
209:
204:
192:
190:
189:
184:
104:is a theorem in
89:
69:
68:
61:
54:
51:
45:
27:
26:
19:
3371:
3370:
3366:
3365:
3364:
3362:
3361:
3360:
3331:
3330:
3329:
3328:
3282:
3281:
3277:
3239:
3238:
3234:
3219:10.2307/1907685
3204:
3203:
3199:
3160:
3159:
3155:
3145:
3144:
3140:
3125:10.2307/2695497
3108:
3107:
3103:
3093:
3091:
3089:
3071:
3070:
3066:
3026:
3021:
3020:
3016:
3009:
2996:
2995:
2991:
2968:10.2307/2648754
2953:
2952:
2948:
2902:
2901:
2897:
2890:
2879:
2857:
2856:
2849:
2817:
2812:
2811:
2804:
2774:
2773:
2766:
2750:10.1.1.130.1704
2734:
2733:
2726:
2712:
2711:
2707:
2691:10.1.1.471.9842
2672:
2671:
2664:
2649:10.2307/1914083
2631:
2630:
2623:
2591:
2586:
2585:
2581:
2576:
2571:
2567:
2563:
2535:Condorcet cycle
2520:
2513:
2510:
2481:
2455:
2453:Related results
2431:tactical voting
2427:Michael Dummett
2403:
2381:
2359:
2358:
2335:
2334:
2315:
2314:
2295:
2294:
2272:
2271:
2252:
2251:
2232:
2231:
2212:
2211:
2192:
2191:
2172:
2171:
2152:
2151:
2126:
2125:
2106:
2105:
2079:
2078:
2067:
2065:Sketch of proof
2031:
2030:
2022:
2017:
1960:
1949:
1948:
1921:
1920:
1893:
1892:
1889:
1865:
1864:
1845:
1844:
1824:
1823:
1819:
1802:
1786:
1784:Cardinal voting
1778:
1773:
1768:
1735:
1734:
1711:
1710:
1679:
1668:
1667:
1648:
1647:
1624:
1623:
1595:
1584:
1583:
1559:
1554:
1553:
1524:
1523:
1502:
1497:
1496:
1477:
1476:
1451:
1435:
1416:
1408:
1407:
1384:
1383:
1355:
1339:
1320:
1312:
1311:
1276:
1265:
1264:
1237:
1236:
1209:
1208:
1157:
1156:
1125:
1124:
1121:
1083:
1082:
1063:
1062:
1043:
1042:
975:
974:
955:
954:
935:
934:
910:
909:
889:
888:
868:
867:
847:
846:
821:
820:
800:
799:
779:
778:
758:
757:
732:
731:
711:
710:
690:
689:
669:
668:
621:
620:
553:
552:
528:
527:
507:
506:
486:
485:
465:
464:
439:
438:
418:
417:
397:
396:
376:
375:
350:
349:
329:
328:
308:
307:
287:
286:
235:
234:
215:
214:
195:
194:
175:
174:
171:
151:cardinal voting
110:Michael Dummett
94:
70:
66:
55:
49:
46:
40:
28:
24:
17:
12:
11:
5:
3369:
3367:
3359:
3358:
3353:
3348:
3343:
3333:
3332:
3327:
3326:
3275:
3248:(2): 475–483.
3232:
3197:
3172:. New Series.
3153:
3138:
3119:(4): 321–337.
3101:
3087:
3064:
3037:(2): 347–364.
3014:
3008:978-0198761884
3007:
2989:
2962:(2): 413–417.
2946:
2895:
2888:
2847:
2828:(3): 381–385.
2802:
2783:(3): 319–322.
2764:
2724:
2705:
2684:(2): 187–217.
2662:
2643:(4): 587–601.
2633:Gibbard, Allan
2621:
2602:(2): 347–364.
2578:
2577:
2575:
2574:
2564:
2562:
2559:
2558:
2557:
2552:
2547:
2542:
2537:
2532:
2526:
2525:
2522:Economy portal
2509:
2506:
2480:
2477:
2454:
2451:
2402:
2399:
2366:
2342:
2322:
2302:
2279:
2259:
2239:
2219:
2199:
2179:
2159:
2139:
2136:
2133:
2113:
2086:
2066:
2063:
2040:
2012:
1981:
1976:
1971:
1965:
1959:
1956:
1947:as a function
1930:
1902:
1888:
1885:
1872:
1852:
1831:
1818:
1815:
1801:
1798:
1785:
1782:
1777:
1774:
1763:
1742:
1718:
1706:is 3 or more.
1695:
1690:
1684:
1678:
1675:
1655:
1631:
1611:
1606:
1600:
1594:
1591:
1566:
1562:
1540:
1536:
1532:
1509:
1505:
1484:
1462:
1456:
1450:
1447:
1442:
1438:
1434:
1431:
1428:
1423:
1419:
1415:
1391:
1366:
1360:
1354:
1351:
1346:
1342:
1338:
1335:
1332:
1327:
1323:
1319:
1297:
1292:
1287:
1281:
1275:
1272:
1263:is a function
1246:
1231:be the set of
1218:
1192:
1189:
1186:
1183:
1180:
1177:
1174:
1171:
1166:
1147:be the set of
1134:
1120:
1117:
1090:
1070:
1050:
1030:
1027:
1024:
1021:
1018:
1015:
1012:
1009:
1006:
1003:
1000:
997:
994:
991:
988:
985:
982:
962:
942:
929:
928:
917:
907:
896:
886:
875:
865:
854:
844:
840:
839:
828:
818:
807:
797:
786:
776:
765:
755:
751:
750:
739:
729:
718:
708:
697:
687:
676:
666:
662:
661:
658:
655:
652:
649:
628:
608:
605:
602:
599:
596:
593:
590:
587:
584:
581:
578:
575:
572:
569:
566:
563:
560:
547:
546:
535:
525:
514:
504:
493:
483:
472:
462:
458:
457:
446:
436:
425:
415:
404:
394:
383:
373:
369:
368:
357:
347:
336:
326:
315:
305:
294:
284:
280:
279:
276:
273:
270:
267:
242:
222:
202:
182:
170:
167:
144:
143:
136:
133:
96:
95:
73:
71:
64:
57:
56:
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
3368:
3357:
3354:
3352:
3349:
3347:
3344:
3342:
3341:Voting theory
3339:
3338:
3336:
3322:
3318:
3314:
3310:
3306:
3302:
3298:
3294:
3290:
3286:
3279:
3276:
3271:
3267:
3263:
3259:
3255:
3251:
3247:
3243:
3236:
3233:
3228:
3224:
3220:
3216:
3212:
3208:
3201:
3198:
3193:
3189:
3184:
3179:
3175:
3171:
3167:
3163:
3157:
3154:
3149:
3142:
3139:
3134:
3130:
3126:
3122:
3118:
3114:
3113:
3105:
3102:
3090:
3088:9780521424585
3084:
3080:
3079:
3074:
3073:Moulin, Hervé
3068:
3065:
3060:
3056:
3052:
3048:
3044:
3040:
3036:
3032:
3025:
3018:
3015:
3010:
3004:
3000:
2993:
2990:
2985:
2981:
2977:
2973:
2969:
2965:
2961:
2957:
2950:
2947:
2942:
2938:
2934:
2930:
2926:
2922:
2918:
2914:
2910:
2906:
2905:Public Choice
2899:
2896:
2891:
2889:0-521-87282-0
2885:
2878:
2877:
2872:
2868:
2864:
2860:
2854:
2852:
2848:
2843:
2839:
2835:
2831:
2827:
2823:
2816:
2809:
2807:
2803:
2798:
2794:
2790:
2786:
2782:
2778:
2771:
2769:
2765:
2760:
2756:
2751:
2746:
2743:(1): 99–105.
2742:
2738:
2731:
2729:
2725:
2720:
2716:
2709:
2706:
2701:
2697:
2692:
2687:
2683:
2679:
2675:
2669:
2667:
2663:
2658:
2654:
2650:
2646:
2642:
2638:
2634:
2628:
2626:
2622:
2617:
2613:
2609:
2605:
2601:
2597:
2590:
2583:
2580:
2569:
2566:
2560:
2556:
2553:
2551:
2550:Ranked voting
2548:
2546:
2543:
2541:
2538:
2536:
2533:
2531:
2528:
2527:
2523:
2517:
2512:
2507:
2505:
2503:
2497:
2492:
2490:
2486:
2478:
2476:
2474:
2469:
2467:
2463:
2459:
2452:
2450:
2448:
2444:
2440:
2436:
2435:Allan Gibbard
2432:
2428:
2424:
2418:
2414:
2412:
2408:
2407:Lewis Carroll
2400:
2398:
2394:
2390:
2389:
2385:
2378:
2364:
2356:
2340:
2320:
2300:
2291:
2277:
2257:
2237:
2217:
2197:
2177:
2157:
2137:
2134:
2131:
2111:
2102:
2100:
2084:
2076:
2072:
2064:
2062:
2060:
2056:
2027:
2021:
2011:
2007:
2005:
2001:
1997:
1969:
1957:
1954:
1946:
1918:
1886:
1884:
1870:
1850:
1829:
1816:
1814:
1810:
1807:
1799:
1797:
1795:
1791:
1783:
1781:
1775:
1772:
1762:
1760:
1756:
1740:
1732:
1716:
1707:
1688:
1673:
1653:
1645:
1629:
1622:the image of
1604:
1589:
1582:We denote by
1580:
1564:
1560:
1538:
1534:
1530:
1507:
1503:
1482:
1460:
1448:
1440:
1436:
1432:
1429:
1426:
1421:
1417:
1405:
1389:
1380:
1364:
1352:
1344:
1340:
1336:
1333:
1330:
1325:
1321:
1285:
1273:
1270:
1262:
1234:
1206:
1187:
1184:
1181:
1178:
1175:
1169:
1154:
1150:
1118:
1116:
1114:
1109:
1107:
1102:
1088:
1068:
1048:
1025:
1022:
1019:
1016:
1013:
1010:
1007:
1004:
1001:
998:
995:
992:
989:
986:
983:
960:
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106:voting theory
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32:This article
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3291:(1): 85–93.
3288:
3284:
3278:
3245:
3241:
3235:
3213:(1): 33–43.
3210:
3207:Econometrica
3206:
3200:
3176:(1): 80–89.
3173:
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3104:
3092:. Retrieved
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2637:Econometrica
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2470:
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2416:
2411:Duncan Black
2404:
2386:
2379:
2355:dictatorship
2292:
2103:
2068:
2058:
2054:
2025:
2023:
2013:
2008:
2003:
1999:
1995:
1944:
1890:
1820:
1813:among them.
1811:
1805:
1803:
1790:score voting
1787:
1779:
1764:
1758:
1754:
1730:
1709:We say that
1708:
1643:
1581:
1403:
1382:We say that
1381:
1260:
1204:
1152:
1149:alternatives
1148:
1122:
1110:
1105:
1103:
932:
641:
550:
259:
172:
145:
101:
99:
90:
75:
47:
37:
33:
2911:: 137–142.
2871:Tardos, Éva
2863:Nisan, Noam
2004:dictatorial
2000:manipulable
1915:the set of
1731:dictatorial
1404:manipulable
1261:voting rule
1203:the set of
1106:manipulable
255:Borda count
3335:Categories
3094:10 January
2489:Noam Nisan
2479:Importance
1153:candidates
3305:0176-1714
2976:0020-6598
2941:153421058
2925:0048-5829
2842:0165-1765
2797:0165-1765
2745:CiteSeerX
2686:CiteSeerX
2391:is being
2135:≺
2059:unanimous
1975:→
1887:Corollary
1753:who is a
1449:∈
1430:…
1353:∈
1334:…
1291:→
1182:…
1041:. Hence,
660:Choice 4
278:Choice 4
50:June 2024
3313:41106341
3270:27639067
3262:41106711
3075:(1991).
3059:46164353
3051:41106783
2933:30023000
2873:(2007).
2616:46164353
2508:See also
2384:template
2250:chooses
1755:dictator
1539:′
657:Choice 3
654:Choice 2
651:Choice 1
275:Choice 3
272:Choice 2
269:Choice 1
3227:1907685
3192:2662065
3133:2695497
2984:2648754
2657:1914083
2401:History
2397:
2015:Theorem
86:Discuss
3321:271833
3319:
3311:
3303:
3268:
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3225:
3190:
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2614:
2496:model.
2150:, the
1207:. Let
1205:voters
82:merged
3317:S2CID
3309:JSTOR
3266:S2CID
3258:JSTOR
3223:JSTOR
3188:JSTOR
3129:JSTOR
3055:S2CID
3047:JSTOR
3027:(PDF)
2980:JSTOR
2937:S2CID
2929:JSTOR
2880:(PDF)
2818:(PDF)
2653:JSTOR
2612:S2CID
2592:(PDF)
2382:‹The
1919:over
1235:over
843:Carol
665:Alice
648:Voter
461:Carol
283:Alice
266:Voter
3301:ISSN
3096:2016
3083:ISBN
3003:ISBN
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