1803:. 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
2510:
36:
78:
268:: 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.
2020:
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
1832:
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
1823:
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
1819:
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
2039:
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
1820:
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.
1853:
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
2443:. 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.
2021:
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.
2489:
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
1484:
1388:
2003:
1319:
1212:
2362:
2342:
2239:
2179:
2133:
1715:
1631:
1050:
628:
2035:
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
1126:
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.
2062:
1952:
1924:
1268:
1240:
1156:
1561:
2454:
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.
2159:
1588:
1531:
2386:
2322:
2299:
2279:
2259:
2219:
2199:
2106:
1892:
1872:
1851:
1762:
1738:
1675:
1651:
1504:
1411:
1110:
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1070:
982:
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916:
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827:
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738:
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555:
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513:
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466:
445:
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403:
377:
356:
335:
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262:
242:
222:
202:
2462:
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.
3344:
2670:(April 1975). "Strategy-proofness and Arrow's conditions: Existence and correspondence theorems for voting procedures and social welfare functions".
2494:
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
1270:: 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
96:
1772:. If the dictator has several equally most-liked alternatives among the possible outcomes, then the winning alternative is simply one of them.
2344:
satisfies independence of irrelevant alternatives. Arrow's impossibility theorem says that, when there are three or more alternatives, such a
3277:
Duggan, John; Schwartz, Thomas (2000). "Strategic manipulability without resoluteness or shared beliefs: Gibbard-Satterthwaite generalized".
2477:
The
Gibbard–Satterthwaite theorem is generally presented as a result about voting systems, but it can also be seen as an important result of
2068:, i.e. every alternative is a possible outcome. The assumption of being onto is sometimes even replaced with the assumption that the rule is
3070:
653:
But Alice can vote strategically and change the result. Assume that she modifies her ballot, in order to produce the following situation.
52:
3000:
139:
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:
3105:
3080:
2881:
172:
are even more general and extend these results to non-deterministic processes, where the outcome may depend partly on chance; the
2523:
2495:
2436:
2081:
1420:
1324:
2566:
Gibbard's theorem does not imply that cardinal methods necessarily incentivize reversing one's relative rank of two candidates.
1961:
1277:
1791:
A variety of "counterexamples" to the
Gibbard-Satterthwaite theorem exist when the conditions of the theorem do not apply.
3349:
1166:, even if they are not necessarily persons: they can also be several possible decisions about a given issue. We denote by
2948:
Barberá, Salvador (1983). "Strategy-Proofness and
Pivotal Voters: A Direct Proof of the Gibbard-Satterthwaite Theorem".
1833:
alternatives reach the same number of points, the tie is broken in an arbitrary but deterministic manner, e.g. outcome
2533:
2466:
173:
3339:
1169:
2469:
extend this result in another direction, by dealing with deterministic voting rules that choose multiple winners.
2411:
This principle of voting makes an election more of a game of skill than a real test of the wishes of the electors.
2031:
If a strict voting rule has at least 3 possible outcomes, it is non-manipulable if and only if it is dictatorial.
1902:
We now consider the case where by assumption, a voter cannot be indifferent between two candidates. We denote by
2455:
165:
2085:
2347:
2327:
2224:
2164:
2118:
1782:
If an ordinal voting rule has at least 3 possible outcomes and is non-dictatorial, then it is manipulable.
1680:
1596:
3334:
2868:
2738:
2679:
2365:
184:
Consider three voters named Alice, Bob and Carol, who wish to select a winner among four candidates named
2423:, he conjectures that deterministic voting rules with at least three outcomes are never straightforward
987:
565:
2729:
Reny, Philip J. (2001). "Arrow's
Theorem and the Gibbard-Satterthwaite Theorem: A Unified Approach".
2538:
2459:
2451:
2440:
2403:, a pioneer in social choice theory. His quote (about a particular voting system) was made famous by
1804:
1768:, in the sense that the winning alternative is always her most-liked one among the possible outcomes
169:
157:
87:
2743:
2684:
2043:
1933:
1905:
1249:
1221:
1137:
92:
2897:
Gärdenfors, Peter (1977). "A Concise Proof of
Theorem on Manipulation of Social Choice Functions".
150:
3310:
3302:
3259:
3251:
3216:
3181:
3122:
3048:
3040:
2973:
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2922:
2667:
2646:
2605:
2432:
132:
3294:
3155:
3076:
2996:
2965:
2914:
2877:
2831:
2808:
2786:
2416:
2138:
143:
The rule is dictatorial, i.e. there exists a distinguished voter who can choose the winner; or
124:
3018:"An interview with Michael Dummett: From analytical philosophy to voting analysis and beyond"
2583:"An Interview with Michael Dummett: From analytical philosophy to voting analysis and beyond"
1119:: there exists situations where a sincere ballot does not defend a voter's preferences best.
160:
is more general and covers processes of collective decision that may not be ordinal, such as
17:
3286:
3243:
3208:
3171:
3114:
3032:
2957:
2906:
2823:
2778:
2748:
2689:
2638:
2597:
2548:
2478:
2399:
The strategic aspect of voting is already noticed in 1876 by
Charles Dodgson, also known as
2109:
1536:
1566:
1509:
1072:
is elected. Alice is satisfied by her ballot modification, because she prefers the outcome
2860:
2528:
2424:
2420:
161:
120:
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2708:
3176:
3159:
2852:
2515:
2371:
2307:
2284:
2264:
2244:
2204:
2184:
2091:
2072:, in the sense that if all voters prefer the same candidate, then she must be elected.
1877:
1857:
1836:
1747:
1723:
1660:
1636:
1489:
1396:
1095:
1075:
1055:
967:
947:
922:
901:
880:
859:
833:
812:
791:
770:
744:
723:
702:
681:
633:
540:
519:
498:
477:
451:
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299:
247:
227:
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187:
2827:
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2428:
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1243:
1123:
136:
128:
116:
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3052:
2609:
2404:
1800:
3314:
2769:
Benoît, Jean-Pierre (2000). "The
Gibbard-Satterthwaite Theorem: A Simple Proof".
1927:
265:
3247:
3036:
3017:
2856:
2601:
2582:
2505:
2482:
3298:
2969:
2918:
2835:
2790:
3199:
Dummett, Michael; Farquharson, Robin (January 1961). "Stability in voting".
2509:
2088:. We give a sketch of proof in the simplified case where some voting rule
3306:
3255:
3044:
2926:
2709:"Alternatives vs. Outcomes: A Note on the Gibbard-Satterthwaite Theorem"
3290:
3220:
3185:
3126:
2977:
2910:
2650:
1677:
has at least three possible outcomes if and only if the cardinality of
3103:
Taylor, Alan D. (April 2002). "The manipulability of voting systems".
3212:
3118:
2961:
2642:
2419:
published influential articles on voting theory. In an article with
153:(one that does not depend on other voters' preferences or behavior).
3234:
Dummett, Michael (2005). "The work and life of Robin
Farquharson".
146:
The rule limits the possible outcomes to two alternatives only; or
1112:, which is the outcome she would obtain if she voted sincerely.
71:
29:
2064:
contains at least three elements and that the voting rule is
2629:(1973). "Manipulation of voting schemes: A general result".
2049:
1990:
1974:
1939:
1911:
1693:
1609:
1465:
1369:
1306:
1290:
1255:
1227:
1175:
1143:
2809:"Another Direct Proof of the Gibbard-Satterthwaite Theorem"
1479:{\displaystyle (P_{1},\ldots ,P_{n})\in {\mathcal {P}}^{n}}
1383:{\displaystyle (P_{1},\ldots ,P_{n})\in {\mathcal {P}}^{n}}
49:
inadequate description of theorem and practical importance.
2391:
Later authors have developed other variants of the proof.
562:
If the voters cast sincere ballots, then the scores are:
149:
The rule is not straightforward, i.e. there is no single
127:
in 1961 and then proved independently by the philosopher
176:
extends these results to multiwinner electoral systems.
2221:
are moved to the top of all voters' preferences. Then,
1563:, can get an outcome that she prefers (in the sense of
2481:, which deals with a broader class of decision rules.
2080:
The Gibbard–Satterthwaite theorem can be proved using
1998:{\displaystyle f:{\mathcal {L}}^{n}\to {\mathcal {A}}}
1314:{\displaystyle f:{\mathcal {P}}^{n}\to {\mathcal {A}}}
2374:
2350:
2330:
2310:
2287:
2267:
2247:
2227:
2207:
2187:
2167:
2141:
2121:
2094:
2046:
1964:
1936:
1908:
1880:
1860:
1839:
1750:
1726:
1683:
1663:
1639:
1599:
1569:
1539:
1512:
1492:
1423:
1399:
1390:
and it yields the identity of the winning candidate.
1327:
1280:
1252:
1224:
1172:
1140:
1098:
1078:
1058:
990:
970:
950:
925:
904:
883:
862:
836:
815:
794:
773:
747:
726:
705:
684:
636:
568:
543:
522:
501:
480:
454:
433:
412:
391:
365:
344:
323:
302:
250:
230:
210:
190:
27:
Impossibility result for ranked-choice voting systems
2427:. This conjecture was later proven independently by
1122:
The Gibbard–Satterthwaite theorem states that every
2380:
2356:
2336:
2316:
2293:
2273:
2253:
2233:
2213:
2193:
2173:
2153:
2127:
2115:It is possible to build a social ranking function
2100:
2056:
1997:
1946:
1918:
1886:
1866:
1845:
1756:
1732:
1709:
1669:
1645:
1625:
1582:
1555:
1525:
1498:
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1405:
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1313:
1262:
1234:
1206:
1150:
1104:
1084:
1064:
1044:
976:
956:
931:
910:
889:
868:
842:
821:
800:
779:
753:
732:
711:
690:
642:
622:
549:
528:
507:
486:
460:
439:
418:
397:
371:
350:
329:
308:
256:
236:
216:
196:
1799:Consider a three-candidate election conducted by
2581:Rudolf Farra and Maurice Salles (October 2006).
2713:Technical Report (University Library of Munich)
2662:
2660:
1207:{\displaystyle {\mathcal {N}}=\{1,\ldots ,n\}}
271:Assume that their preferences are as follows.
119:. It was first conjectured by the philosopher
2876:. Cambridge, UK: Cambridge University Press.
2439:from 1951 to prove the result we now know as
1874:wins if it gets two thirds of the votes, and
1770:regardless of the preferences of other voters
45:needs attention from an expert in game theory
8:
2017:have natural adaptations to this framework.
1201:
1183:
2025:
1776:
1657:for the election. For example, we say that
944:Alice has strategically upgraded candidate
3160:"Straightforwardness in voting procedures"
2181:function creates new preferences in which
3175:
2742:
2683:
2621:
2619:
2373:
2349:
2329:
2309:
2286:
2266:
2246:
2226:
2206:
2186:
2166:
2140:
2135:, as follows: in order to decide whether
2120:
2093:
2048:
2047:
2045:
1989:
1988:
1979:
1973:
1972:
1963:
1938:
1937:
1935:
1910:
1909:
1907:
1879:
1859:
1838:
1749:
1725:
1698:
1692:
1691:
1682:
1662:
1638:
1614:
1608:
1607:
1598:
1574:
1568:
1544:
1538:
1517:
1511:
1491:
1470:
1464:
1463:
1450:
1431:
1422:
1398:
1374:
1368:
1367:
1354:
1335:
1326:
1305:
1304:
1295:
1289:
1288:
1279:
1254:
1253:
1251:
1226:
1225:
1223:
1174:
1173:
1171:
1142:
1141:
1139:
1097:
1077:
1057:
989:
969:
949:
924:
903:
882:
861:
835:
814:
793:
772:
746:
725:
704:
683:
635:
567:
542:
521:
500:
479:
453:
432:
411:
390:
364:
343:
322:
301:
249:
229:
209:
189:
2764:
2762:
2324:is non-manipulable and non-dictatorial,
1321:. Its input is a profile of preferences
655:
273:
2573:
2559:
1162:(which is assumed finite), also called
3142:The theory of committees and elections
3016:Fara, Rudolf; Salles, Maurice (2006).
2724:
2722:
2435:. In a 1973 article, Gibbard exploits
1417:if and only if there exists a profile
55:may be able to help recruit an expert.
3072:Axioms of Cooperative Decision Making
2847:
2845:
2802:
2800:
2357:{\displaystyle \operatorname {Rank} }
2337:{\displaystyle \operatorname {Rank} }
2234:{\displaystyle \operatorname {Rank} }
2174:{\displaystyle \operatorname {Rank} }
2128:{\displaystyle \operatorname {Rank} }
1710:{\displaystyle f({\mathcal {P}}^{n})}
1626:{\displaystyle f({\mathcal {P}}^{n})}
135:in 1975. It deals with deterministic
7:
1744:if and only if there exists a voter
3177:10.1093/oxfordjournals.oep.a042255
25:
3106:The American Mathematical Monthly
2304:It is possible to prove that, if
1045:{\displaystyle (a:2,b:7,c:6,d:3)}
623:{\displaystyle (a:3,b:6,c:7,d:2)}
3345:Theorems in discrete mathematics
2508:
650:will be elected, with 7 points.
76:
34:
1115:We say that the Borda count is
3144:. Cambridge: University Press.
3075:. Cambridge University Press.
2057:{\displaystyle {\mathcal {A}}}
1985:
1947:{\displaystyle {\mathcal {A}}}
1919:{\displaystyle {\mathcal {L}}}
1704:
1687:
1620:
1603:
1456:
1424:
1360:
1328:
1301:
1263:{\displaystyle {\mathcal {A}}}
1235:{\displaystyle {\mathcal {P}}}
1151:{\displaystyle {\mathcal {A}}}
1039:
991:
617:
569:
158:Gibbard's proof of the theorem
1:
2950:International Economic Review
2828:10.1016/S0165-1765(00)00362-1
2783:10.1016/S0165-1765(00)00312-8
2753:10.1016/S0165-1765(00)00332-3
2524:Arrow's impossibility theorem
2496:Arrow's impossibility theorem
2437:Arrow's impossibility theorem
2388:must also be a dictatorship.
2082:Arrow's impossibility theorem
1787:Counterexamples and loopholes
1777:Gibbard–Satterthwaite theorem
113:Gibbard–Satterthwaite theorem
102:Proposed since November 2023.
18:Gibbard-Satterthwaite theorem
2694:10.1016/0022-0531(75)90050-2
2368:. Hence, such a voting rule
2995:. Oxford University Press.
264:. Assume that they use the
85:It has been suggested that
47:. The specific problem is:
3366:
2672:Journal of Economic Theory
1506:, by replacing her ballot
3279:Social Choice and Welfare
3248:10.1007/s00355-005-0014-x
3236:Social Choice and Welfare
3037:10.1007/s00355-006-0128-9
3025:Social Choice and Welfare
2991:Dummett, Michael (1984).
2668:Satterthwaite, Mark Allen
2602:10.1007/s00355-006-0128-9
2590:Social Choice and Welfare
964:and downgraded candidate
137:ordinal electoral systems
2485:describes this relation:
2154:{\displaystyle a\prec b}
2086:social ranking functions
2870:Algorithmic Game Theory
2534:Duggan–Schwartz theorem
2467:Duggan–Schwartz theorem
2040:sometimes assumed that
984:. Now, the scores are:
174:Duggan–Schwartz theorem
53:WikiProject Game theory
3164:Oxford Economic Papers
3140:Black, Duncan (1958).
2492:
2456:Gibbard's 1978 theorem
2413:
2382:
2358:
2338:
2318:
2295:
2275:
2255:
2235:
2215:
2195:
2175:
2155:
2129:
2102:
2058:
1999:
1948:
1920:
1888:
1868:
1847:
1758:
1734:
1711:
1671:
1647:
1627:
1584:
1557:
1556:{\displaystyle P_{i}'}
1527:
1500:
1480:
1407:
1384:
1315:
1264:
1236:
1208:
1152:
1106:
1086:
1066:
1046:
978:
958:
933:
912:
891:
870:
844:
823:
802:
781:
755:
734:
713:
692:
644:
624:
551:
530:
509:
488:
462:
441:
420:
399:
373:
352:
331:
310:
258:
238:
218:
198:
166:Gibbard's 1978 theorem
131:in 1973 and economist
123:and the mathematician
2807:Sen, Arunava (2001).
2707:Weber, Tjark (2009).
2487:
2409:
2383:
2359:
2339:
2319:
2296:
2276:
2256:
2236:
2216:
2196:
2176:
2156:
2130:
2103:
2059:
2005:. The definitions of
2000:
1949:
1921:
1889:
1869:
1848:
1759:
1735:
1712:
1672:
1648:
1628:
1585:
1583:{\displaystyle P_{i}}
1558:
1528:
1526:{\displaystyle P_{i}}
1501:
1481:
1408:
1385:
1316:
1265:
1237:
1209:
1153:
1107:
1087:
1067:
1047:
979:
959:
934:
913:
892:
871:
845:
824:
803:
782:
756:
735:
714:
693:
645:
625:
552:
531:
510:
489:
463:
442:
421:
400:
374:
353:
332:
311:
259:
239:
219:
199:
3350:Social choice theory
2555:Notes and references
2372:
2348:
2328:
2308:
2285:
2265:
2245:
2225:
2205:
2185:
2165:
2139:
2119:
2092:
2044:
1962:
1934:
1906:
1878:
1858:
1837:
1828:Simple majority vote
1748:
1724:
1681:
1661:
1637:
1597:
1567:
1537:
1533:with another ballot
1510:
1490:
1421:
1397:
1325:
1278:
1250:
1222:
1170:
1138:
1124:ranked-choice voting
1096:
1076:
1056:
988:
968:
948:
923:
902:
881:
860:
834:
813:
792:
771:
745:
724:
703:
682:
634:
566:
541:
520:
499:
478:
452:
431:
410:
389:
363:
342:
321:
300:
248:
228:
208:
188:
180:Informal description
151:always-best strategy
95:into this article. (
2364:function must be a
2029: —
1928:strict total orders
1817:serial dictatorship
1811:Serial dictatorship
1780: —
1552:
630:. Hence, candidate
3340:Economics theorems
3291:10.1007/PL00007177
3156:Farquharson, Robin
2911:10.1007/bf01718676
2853:Vazirani, Vijay V.
2433:Mark Satterthwaite
2415:During the 1950s,
2378:
2354:
2334:
2314:
2291:
2271:
2251:
2231:
2211:
2191:
2171:
2151:
2125:
2098:
2054:
2027:
1995:
1956:strict voting rule
1944:
1916:
1884:
1864:
1843:
1778:
1754:
1730:
1707:
1667:
1653:, i.e. the set of
1643:
1623:
1580:
1553:
1540:
1523:
1496:
1476:
1403:
1380:
1311:
1260:
1244:strict weak orders
1232:
1204:
1148:
1102:
1082:
1062:
1042:
974:
954:
929:
908:
887:
866:
840:
819:
798:
777:
751:
730:
709:
688:
640:
620:
547:
526:
505:
484:
458:
437:
416:
395:
369:
348:
327:
306:
254:
234:
214:
194:
133:Mark Satterthwaite
3158:(February 1956).
2993:Voting Procedures
2816:Economics Letters
2771:Economics Letters
2731:Economics Letters
2539:Gibbard's theorem
2460:Hylland's theorem
2452:Gibbard's theorem
2441:Gibbard's theorem
2417:Robin Farquharson
2381:{\displaystyle f}
2317:{\displaystyle f}
2294:{\displaystyle b}
2274:{\displaystyle a}
2254:{\displaystyle f}
2241:examines whether
2214:{\displaystyle b}
2194:{\displaystyle a}
2108:is assumed to be
2101:{\displaystyle f}
2037:possible outcomes
2007:possible outcomes
1887:{\displaystyle b}
1867:{\displaystyle a}
1846:{\displaystyle a}
1805:Gibbard's theorem
1757:{\displaystyle i}
1733:{\displaystyle f}
1670:{\displaystyle f}
1655:possible outcomes
1646:{\displaystyle f}
1499:{\displaystyle i}
1486:where some voter
1406:{\displaystyle f}
1105:{\displaystyle c}
1085:{\displaystyle b}
1065:{\displaystyle b}
977:{\displaystyle c}
957:{\displaystyle b}
942:
941:
932:{\displaystyle a}
911:{\displaystyle d}
890:{\displaystyle b}
869:{\displaystyle c}
843:{\displaystyle a}
822:{\displaystyle d}
801:{\displaystyle b}
780:{\displaystyle c}
754:{\displaystyle c}
733:{\displaystyle d}
712:{\displaystyle a}
691:{\displaystyle b}
643:{\displaystyle c}
560:
559:
550:{\displaystyle a}
529:{\displaystyle d}
508:{\displaystyle b}
487:{\displaystyle c}
461:{\displaystyle a}
440:{\displaystyle d}
419:{\displaystyle b}
398:{\displaystyle c}
372:{\displaystyle d}
351:{\displaystyle c}
330:{\displaystyle b}
309:{\displaystyle a}
257:{\displaystyle d}
237:{\displaystyle c}
217:{\displaystyle b}
197:{\displaystyle a}
170:Hylland's theorem
125:Robin Farquharson
109:
108:
104:
88:Gibbard's theorem
70:
69:
16:(Redirected from
3357:
3319:
3318:
3274:
3268:
3267:
3231:
3225:
3224:
3196:
3190:
3189:
3179:
3152:
3146:
3145:
3137:
3131:
3130:
3100:
3094:
3093:
3091:
3089:
3063:
3057:
3056:
3022:
3013:
3007:
3006:
2988:
2982:
2981:
2945:
2939:
2938:
2894:
2888:
2887:
2875:
2861:Roughgarden, Tim
2849:
2840:
2839:
2813:
2804:
2795:
2794:
2766:
2757:
2756:
2746:
2726:
2717:
2716:
2704:
2698:
2697:
2687:
2664:
2655:
2654:
2623:
2614:
2613:
2587:
2578:
2567:
2564:
2549:Strategic voting
2518:
2513:
2512:
2479:mechanism design
2387:
2385:
2384:
2379:
2363:
2361:
2360:
2355:
2343:
2341:
2340:
2335:
2323:
2321:
2320:
2315:
2300:
2298:
2297:
2292:
2280:
2278:
2277:
2272:
2260:
2258:
2257:
2252:
2240:
2238:
2237:
2232:
2220:
2218:
2217:
2212:
2200:
2198:
2197:
2192:
2180:
2178:
2177:
2172:
2160:
2158:
2157:
2152:
2134:
2132:
2131:
2126:
2110:Pareto-efficient
2107:
2105:
2104:
2099:
2063:
2061:
2060:
2055:
2053:
2052:
2030:
2004:
2002:
2001:
1996:
1994:
1993:
1984:
1983:
1978:
1977:
1954:and we define a
1953:
1951:
1950:
1945:
1943:
1942:
1925:
1923:
1922:
1917:
1915:
1914:
1894:wins otherwise.
1893:
1891:
1890:
1885:
1873:
1871:
1870:
1865:
1852:
1850:
1849:
1844:
1781:
1763:
1761:
1760:
1755:
1739:
1737:
1736:
1731:
1716:
1714:
1713:
1708:
1703:
1702:
1697:
1696:
1676:
1674:
1673:
1668:
1652:
1650:
1649:
1644:
1632:
1630:
1629:
1624:
1619:
1618:
1613:
1612:
1589:
1587:
1586:
1581:
1579:
1578:
1562:
1560:
1559:
1554:
1548:
1532:
1530:
1529:
1524:
1522:
1521:
1505:
1503:
1502:
1497:
1485:
1483:
1482:
1477:
1475:
1474:
1469:
1468:
1455:
1454:
1436:
1435:
1412:
1410:
1409:
1404:
1389:
1387:
1386:
1381:
1379:
1378:
1373:
1372:
1359:
1358:
1340:
1339:
1320:
1318:
1317:
1312:
1310:
1309:
1300:
1299:
1294:
1293:
1269:
1267:
1266:
1261:
1259:
1258:
1241:
1239:
1238:
1233:
1231:
1230:
1213:
1211:
1210:
1205:
1179:
1178:
1157:
1155:
1154:
1149:
1147:
1146:
1130:Formal statement
1111:
1109:
1108:
1103:
1091:
1089:
1088:
1083:
1071:
1069:
1068:
1063:
1051:
1049:
1048:
1043:
983:
981:
980:
975:
963:
961:
960:
955:
938:
936:
935:
930:
917:
915:
914:
909:
896:
894:
893:
888:
875:
873:
872:
867:
849:
847:
846:
841:
828:
826:
825:
820:
807:
805:
804:
799:
786:
784:
783:
778:
760:
758:
757:
752:
739:
737:
736:
731:
718:
716:
715:
710:
697:
695:
694:
689:
656:
649:
647:
646:
641:
629:
627:
626:
621:
556:
554:
553:
548:
535:
533:
532:
527:
514:
512:
511:
506:
493:
491:
490:
485:
467:
465:
464:
459:
446:
444:
443:
438:
425:
423:
422:
417:
404:
402:
401:
396:
378:
376:
375:
370:
357:
355:
354:
349:
336:
334:
333:
328:
315:
313:
312:
307:
274:
263:
261:
260:
255:
243:
241:
240:
235:
223:
221:
220:
215:
203:
201:
200:
195:
115:is a theorem in
100:
80:
79:
72:
65:
62:
56:
38:
37:
30:
21:
3365:
3364:
3360:
3359:
3358:
3356:
3355:
3354:
3325:
3324:
3323:
3322:
3276:
3275:
3271:
3233:
3232:
3228:
3213:10.2307/1907685
3198:
3197:
3193:
3154:
3153:
3149:
3139:
3138:
3134:
3119:10.2307/2695497
3102:
3101:
3097:
3087:
3085:
3083:
3065:
3064:
3060:
3020:
3015:
3014:
3010:
3003:
2990:
2989:
2985:
2962:10.2307/2648754
2947:
2946:
2942:
2896:
2895:
2891:
2884:
2873:
2851:
2850:
2843:
2811:
2806:
2805:
2798:
2768:
2767:
2760:
2744:10.1.1.130.1704
2728:
2727:
2720:
2706:
2705:
2701:
2685:10.1.1.471.9842
2666:
2665:
2658:
2643:10.2307/1914083
2625:
2624:
2617:
2585:
2580:
2579:
2575:
2570:
2565:
2561:
2557:
2529:Condorcet cycle
2514:
2507:
2504:
2475:
2449:
2447:Related results
2425:tactical voting
2421:Michael Dummett
2397:
2370:
2369:
2346:
2345:
2326:
2325:
2306:
2305:
2283:
2282:
2263:
2262:
2243:
2242:
2223:
2222:
2203:
2202:
2183:
2182:
2163:
2162:
2137:
2136:
2117:
2116:
2090:
2089:
2078:
2076:Sketch of proof
2042:
2041:
2033:
2028:
1971:
1960:
1959:
1932:
1931:
1904:
1903:
1900:
1876:
1875:
1856:
1855:
1835:
1834:
1830:
1813:
1797:
1795:Cardinal voting
1789:
1784:
1779:
1746:
1745:
1722:
1721:
1690:
1679:
1678:
1659:
1658:
1635:
1634:
1606:
1595:
1594:
1570:
1565:
1564:
1535:
1534:
1513:
1508:
1507:
1488:
1487:
1462:
1446:
1427:
1419:
1418:
1395:
1394:
1366:
1350:
1331:
1323:
1322:
1287:
1276:
1275:
1248:
1247:
1220:
1219:
1168:
1167:
1136:
1135:
1132:
1094:
1093:
1074:
1073:
1054:
1053:
986:
985:
966:
965:
946:
945:
921:
920:
900:
899:
879:
878:
858:
857:
832:
831:
811:
810:
790:
789:
769:
768:
743:
742:
722:
721:
701:
700:
680:
679:
632:
631:
564:
563:
539:
538:
518:
517:
497:
496:
476:
475:
450:
449:
429:
428:
408:
407:
387:
386:
361:
360:
340:
339:
319:
318:
298:
297:
246:
245:
226:
225:
206:
205:
186:
185:
182:
162:cardinal voting
121:Michael Dummett
105:
81:
77:
66:
60:
57:
51:
39:
35:
28:
23:
22:
15:
12:
11:
5:
3363:
3361:
3353:
3352:
3347:
3342:
3337:
3327:
3326:
3321:
3320:
3269:
3242:(2): 475–483.
3226:
3191:
3166:. New Series.
3147:
3132:
3113:(4): 321–337.
3095:
3081:
3058:
3031:(2): 347–364.
3008:
3002:978-0198761884
3001:
2983:
2956:(2): 413–417.
2940:
2889:
2882:
2841:
2822:(3): 381–385.
2796:
2777:(3): 319–322.
2758:
2718:
2699:
2678:(2): 187–217.
2656:
2637:(4): 587–601.
2627:Gibbard, Allan
2615:
2596:(2): 347–364.
2572:
2571:
2569:
2568:
2558:
2556:
2553:
2552:
2551:
2546:
2541:
2536:
2531:
2526:
2520:
2519:
2516:Economy portal
2503:
2500:
2474:
2471:
2448:
2445:
2396:
2393:
2377:
2353:
2333:
2313:
2290:
2270:
2250:
2230:
2210:
2190:
2170:
2150:
2147:
2144:
2124:
2097:
2077:
2074:
2051:
2023:
1992:
1987:
1982:
1976:
1970:
1967:
1958:as a function
1941:
1913:
1899:
1896:
1883:
1863:
1842:
1829:
1826:
1812:
1809:
1796:
1793:
1788:
1785:
1774:
1753:
1729:
1717:is 3 or more.
1706:
1701:
1695:
1689:
1686:
1666:
1642:
1622:
1617:
1611:
1605:
1602:
1577:
1573:
1551:
1547:
1543:
1520:
1516:
1495:
1473:
1467:
1461:
1458:
1453:
1449:
1445:
1442:
1439:
1434:
1430:
1426:
1402:
1377:
1371:
1365:
1362:
1357:
1353:
1349:
1346:
1343:
1338:
1334:
1330:
1308:
1303:
1298:
1292:
1286:
1283:
1274:is a function
1257:
1242:be the set of
1229:
1203:
1200:
1197:
1194:
1191:
1188:
1185:
1182:
1177:
1158:be the set of
1145:
1131:
1128:
1101:
1081:
1061:
1041:
1038:
1035:
1032:
1029:
1026:
1023:
1020:
1017:
1014:
1011:
1008:
1005:
1002:
999:
996:
993:
973:
953:
940:
939:
928:
918:
907:
897:
886:
876:
865:
855:
851:
850:
839:
829:
818:
808:
797:
787:
776:
766:
762:
761:
750:
740:
729:
719:
708:
698:
687:
677:
673:
672:
669:
666:
663:
660:
639:
619:
616:
613:
610:
607:
604:
601:
598:
595:
592:
589:
586:
583:
580:
577:
574:
571:
558:
557:
546:
536:
525:
515:
504:
494:
483:
473:
469:
468:
457:
447:
436:
426:
415:
405:
394:
384:
380:
379:
368:
358:
347:
337:
326:
316:
305:
295:
291:
290:
287:
284:
281:
278:
253:
233:
213:
193:
181:
178:
155:
154:
147:
144:
107:
106:
84:
82:
75:
68:
67:
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3362:
3351:
3348:
3346:
3343:
3341:
3338:
3336:
3335:Voting theory
3333:
3332:
3330:
3316:
3312:
3308:
3304:
3300:
3296:
3292:
3288:
3284:
3280:
3273:
3270:
3265:
3261:
3257:
3253:
3249:
3245:
3241:
3237:
3230:
3227:
3222:
3218:
3214:
3210:
3206:
3202:
3195:
3192:
3187:
3183:
3178:
3173:
3169:
3165:
3161:
3157:
3151:
3148:
3143:
3136:
3133:
3128:
3124:
3120:
3116:
3112:
3108:
3107:
3099:
3096:
3084:
3082:9780521424585
3078:
3074:
3073:
3068:
3067:Moulin, Hervé
3062:
3059:
3054:
3050:
3046:
3042:
3038:
3034:
3030:
3026:
3019:
3012:
3009:
3004:
2998:
2994:
2987:
2984:
2979:
2975:
2971:
2967:
2963:
2959:
2955:
2951:
2944:
2941:
2936:
2932:
2928:
2924:
2920:
2916:
2912:
2908:
2904:
2900:
2899:Public Choice
2893:
2890:
2885:
2883:0-521-87282-0
2879:
2872:
2871:
2866:
2862:
2858:
2854:
2848:
2846:
2842:
2837:
2833:
2829:
2825:
2821:
2817:
2810:
2803:
2801:
2797:
2792:
2788:
2784:
2780:
2776:
2772:
2765:
2763:
2759:
2754:
2750:
2745:
2740:
2737:(1): 99–105.
2736:
2732:
2725:
2723:
2719:
2714:
2710:
2703:
2700:
2695:
2691:
2686:
2681:
2677:
2673:
2669:
2663:
2661:
2657:
2652:
2648:
2644:
2640:
2636:
2632:
2628:
2622:
2620:
2616:
2611:
2607:
2603:
2599:
2595:
2591:
2584:
2577:
2574:
2563:
2560:
2554:
2550:
2547:
2545:
2544:Ranked voting
2542:
2540:
2537:
2535:
2532:
2530:
2527:
2525:
2522:
2521:
2517:
2511:
2506:
2501:
2499:
2497:
2491:
2486:
2484:
2480:
2472:
2470:
2468:
2463:
2461:
2457:
2453:
2446:
2444:
2442:
2438:
2434:
2430:
2429:Allan Gibbard
2426:
2422:
2418:
2412:
2408:
2406:
2402:
2401:Lewis Carroll
2394:
2392:
2389:
2375:
2367:
2351:
2331:
2311:
2302:
2288:
2268:
2248:
2228:
2208:
2188:
2168:
2148:
2145:
2142:
2122:
2113:
2111:
2095:
2087:
2083:
2075:
2073:
2071:
2067:
2038:
2032:
2022:
2018:
2016:
2012:
2008:
1980:
1968:
1965:
1957:
1929:
1897:
1895:
1881:
1861:
1840:
1827:
1825:
1821:
1818:
1810:
1808:
1806:
1802:
1794:
1792:
1786:
1783:
1773:
1771:
1767:
1751:
1743:
1727:
1718:
1699:
1684:
1664:
1656:
1640:
1633:the image of
1615:
1600:
1593:We denote by
1591:
1575:
1571:
1549:
1545:
1541:
1518:
1514:
1493:
1471:
1459:
1451:
1447:
1443:
1440:
1437:
1432:
1428:
1416:
1400:
1391:
1375:
1363:
1355:
1351:
1347:
1344:
1341:
1336:
1332:
1296:
1284:
1281:
1273:
1245:
1217:
1198:
1195:
1192:
1189:
1186:
1180:
1165:
1161:
1129:
1127:
1125:
1120:
1118:
1113:
1099:
1079:
1059:
1036:
1033:
1030:
1027:
1024:
1021:
1018:
1015:
1012:
1009:
1006:
1003:
1000:
997:
994:
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43:This article
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3285:(1): 85–93.
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3207:(1): 33–43.
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3201:Econometrica
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3194:
3170:(1): 80–89.
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3086:. Retrieved
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2631:Econometrica
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2410:
2405:Duncan Black
2398:
2390:
2366:dictatorship
2303:
2114:
2079:
2069:
2065:
2036:
2034:
2024:
2019:
2014:
2010:
2006:
1955:
1901:
1831:
1824:among them.
1822:
1816:
1814:
1801:score voting
1798:
1790:
1775:
1769:
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1741:
1720:We say that
1719:
1654:
1592:
1414:
1393:We say that
1392:
1271:
1215:
1163:
1160:alternatives
1159:
1133:
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1116:
1114:
943:
652:
561:
270:
183:
156:
112:
110:
101:
86:
58:
48:
44:
2905:: 137–142.
2865:Tardos, Éva
2857:Nisan, Noam
2015:dictatorial
2011:manipulable
1926:the set of
1742:dictatorial
1415:manipulable
1272:voting rule
1214:the set of
1117:manipulable
266:Borda count
3329:Categories
3088:10 January
2483:Noam Nisan
2473:Importance
1164:candidates
3299:0176-1714
2970:0020-6598
2935:153421058
2919:0048-5829
2836:0165-1765
2791:0165-1765
2739:CiteSeerX
2680:CiteSeerX
2146:≺
2070:unanimous
1986:→
1898:Corollary
1764:who is a
1460:∈
1441:…
1364:∈
1345:…
1302:→
1193:…
1052:. Hence,
671:Choice 4
289:Choice 4
61:June 2024
3307:41106341
3264:27639067
3256:41106711
3069:(1991).
3053:46164353
3045:41106783
2927:30023000
2867:(2007).
2610:46164353
2502:See also
2261:chooses
1766:dictator
1550:′
668:Choice 3
665:Choice 2
662:Choice 1
286:Choice 3
283:Choice 2
280:Choice 1
3221:1907685
3186:2662065
3127:2695497
2978:2648754
2651:1914083
2395:History
2026:Theorem
97:Discuss
3315:271833
3313:
3305:
3297:
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3254:
3219:
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2608:
2490:model.
2161:, the
1218:. Let
1216:voters
93:merged
3311:S2CID
3303:JSTOR
3260:S2CID
3252:JSTOR
3217:JSTOR
3182:JSTOR
3123:JSTOR
3049:S2CID
3041:JSTOR
3021:(PDF)
2974:JSTOR
2931:S2CID
2923:JSTOR
2874:(PDF)
2812:(PDF)
2647:JSTOR
2606:S2CID
2586:(PDF)
1930:over
1246:over
854:Carol
676:Alice
659:Voter
472:Carol
294:Alice
277:Voter
3295:ISSN
3090:2016
3077:ISBN
2997:ISBN
2966:ISSN
2915:ISSN
2878:ISBN
2832:ISSN
2787:ISSN
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2123:Rank
2084:for
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244:and
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111:The
3287:doi
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