2481:
25:
2227:
If there are only 2 possible outcomes, a game form may be strategyproof and not dictatorial. For example, it is the case of the simple majority vote: each voter casts a ballot for her most-liked alternative (among the two possible outcomes), and the alternative with most votes is declared the winner.
2174:
is defined as follows. If voter 1 has a unique most-liked candidate, then this candidate is elected. Otherwise, possible outcomes are restricted to his ex-aequo most-liked candidates and the other candidates are eliminated. Then voter 2's ballot is examined: if he has a unique best-liked candidate
2228:
This game form is strategyproof because it is always optimal to vote for one's most-liked alternative (unless one is indifferent between them). However, it is clearly not dictatorial. Many other game forms are strategyproof and not dictatorial: for example, assume that the alternative
2296:
The terminology for this varies. Gibbard states that 'an individual "manipulates" the voting scheme if, by misrepresenting his preferences, he secures an outcome he prefers to the "honest" outcome', while Brams and
Fishburn call every ballot with an honest ordering
2281:. This game form is clearly dictatorial, because voter 1 can impose the result. However, it is not strategyproof: the other voters face the same issue of strategic voting as in the usual Borda count. Thus, Gibbard's theorem is an implication and not an equivalence.
2175:
among the non-eliminated ones, then this candidate is elected. Otherwise, the list of possible outcomes is reduced again, etc. If there is still several non-eliminated candidates after all ballots have been examined, then an arbitrary tie-breaking rule is used.
990:
Gibbard's theorem states that a deterministic process of collective decision cannot be strategyproof, except possibly in two cases: if there is a distinguished agent who has a dictatorial power, or if the process limits the outcome to two possible options only.
987:: once the voter has identified her own preferences, she does not have a ballot at her disposal that best defends her opinions in all situations; she needs to act strategically, possibly by spying over the other voters to determine how they intend to vote.
2276:
Consider the following game form. Voter 1 can vote for a candidate of her choice, or she can abstain. In the first case, the specified candidate is automatically elected. Otherwise, the other voters use a classic voting rule, for example the
1327:
2178:
This game form is strategyproof: whatever the preferences of a voter, he has a dominant strategy that consists in declaring his sincere preference order. It is also dictatorial, and its dictator is voter 1: if he wishes to see candidate
1808:
367:. Once the ballots are collected, the candidate with highest total grade is declared the winner. Ties between candidates are broken by alphabetical order: for example, if there is a tie between candidates
104:. Because of its broader scope, Gibbard's theorem makes no claim about whether voters need to reverse their ranking of candidates, only that their optimal ballots depend on the other voters' ballots.
107:
Gibbard's theorem is more general, and considers processes of collective decision that may not be ordinal: for example, voting systems where voters assign grades to or otherwise rate candidates (
1077:
2100:
1195:
1140:
2008:
1951:
1918:
1841:
1741:
1619:
1493:
1975:
1021:
126:, which extend these results to non-deterministic processes, i.e. where the outcome may not only depend on the agents' actions but may also involve an element of chance.
1240:
981:
943:
861:
803:
745:
707:
666:
588:
550:
305:
1693:
1646:
1571:
1544:
1445:
2266:
2246:
2217:
2197:
2140:
2120:
2057:
2031:
1885:
1861:
1666:
1517:
1415:
1385:
1347:
1235:
1215:
1164:
1109:
905:
881:
823:
765:
628:
608:
508:
488:
468:
448:
425:
405:
385:
365:
345:
325:
263:
243:
223:
203:
183:
163:
1749:
2512:
2450:
43:
74:
in 1973. It states that for any deterministic process of collective decision, at least one of the following three properties must hold:
2471:
97:
38:
2466:
112:
2517:
134:
2507:
1034:
1360:, but with no utilities associated to the possible outcomes: it describes the procedure only, without specifying
119:
100:
about voting rules. The key difference between the two theorems is that
Gibbard–Satterthwaite applies only to
2066:
1743:
that best defends her preferences, with no need to know or guess the strategies chosen by the other agents.
2502:
2035:
79:
2316:
1169:
1114:
1980:
1923:
1890:
1813:
2152:
If a game form is not dictatorial and has at least 3 possible outcomes, then it is not strategyproof.
1698:
1576:
1450:
129:
Gibbard's theorem assumes the collective decision results in exactly one winner and does not apply to
269:: each voter assigns to each candidate the grade 1 (approval) or 0 (withhold approval). For example,
123:
63:
2401:
1956:
1002:
33:
2451:
Strategy proofness of voting procedures with lotteries as outcomes and infinite sets of strategies
1322:{\displaystyle (s_{1},\ldots ,s_{n})\in {\mathcal {S}}_{1}\times \cdots \times {\mathcal {S}}_{n}}
510:. Which ballot will best defend her opinions? For example, consider the two following situations.
2432:
2382:
2339:
130:
2374:
1648:). This property is desirable for a democratic decision process: it means that once the agent
1388:
948:
910:
828:
770:
712:
674:
633:
555:
517:
272:
2424:
2416:
2366:
2331:
907:
faces a strategic voting dilemma: depending on the ballots that the other voters will cast,
89:
59:
1671:
1624:
1549:
1522:
1423:
266:
108:
1803:{\displaystyle {\mathcal {S}}={\mathcal {S}}_{1}\times \cdots \times {\mathcal {S}}_{n}}
983:
can be a ballot that best defends her opinions. We then say that approval voting is not
2486:
2251:
2231:
2202:
2182:
2125:
2105:
2042:
2016:
1870:
1846:
1651:
1502:
1400:
1370:
1332:
1220:
1200:
1149:
1094:
890:
866:
808:
750:
613:
593:
493:
473:
453:
433:
410:
390:
370:
350:
330:
310:
248:
228:
208:
188:
168:
148:
2496:
2312:
2167:
1418:
984:
101:
71:
1546:: there is no profile of strategies for the other agents such that another strategy
671:
However, if we assume instead that the two other voters respectively cast ballots
2278:
1354:
2476:
2378:
88:
The process is not straightforward; the optimal ballot for a voter "requires
610:
has only one ballot that leads to the election of her favorite alternative
2480:
2428:
307:
is an authorized ballot: it means that the voter approves of candidates
2436:
2386:
2343:
2420:
2370:
2335:
1887:
has at least 3 possible outcomes if and only if the cardinality of
2199:
elected, then he just has to communicate a preference order where
2357:
Brams, Steven J.; Fishburn, Peter C. (1978). "Approval Voting".
92:", i.e. it depends on their beliefs about other voters' ballots.
2122:
has a strategy at her disposal that ensures that the result is
2402:"Straightforwardness of Game Forms with Lotteries as Outcomes"
82:, i.e. there is a single voter whose vote chooses the outcome.
18:
1977:
is not assumed to be finite, the subset of possible outcomes
1353:. In other words, a game form is essentially defined like an
85:
The process limits the possible outcomes to two options only.
2084:
1992:
1962:
1935:
1902:
1825:
1789:
1766:
1755:
1308:
1285:
1176:
1121:
1040:
1008:
1621:, would lead to a strictly better outcome (in the sense of
1091:
depending on the context of application. For each agent
205:
who wish to select an option among three alternatives:
2142:, whatever the strategies chosen by the other agents.
1953:
is finite also; thus, even if the set of alternatives
1364:
the gain that each agent would get from each outcome.
2254:
2234:
2205:
2185:
2128:
2108:
2069:
2045:
2019:
1983:
1959:
1926:
1893:
1873:
1849:
1816:
1752:
1701:
1674:
1654:
1627:
1579:
1552:
1525:
1505:
1453:
1426:
1403:
1373:
1335:
1243:
1223:
1203:
1172:
1152:
1117:
1097:
1037:
1005:
951:
913:
893:
869:
831:
811:
773:
753:
715:
677:
636:
616:
596:
558:
520:
496:
476:
456:
436:
413:
393:
373:
353:
333:
313:
275:
251:
231:
211:
191:
171:
151:
2272:
A game form showing that the converse does not hold
2317:"Manipulation of voting schemes: A general result"
2260:
2240:
2211:
2191:
2134:
2114:
2094:
2051:
2025:
2002:
1969:
1945:
1920:is 3 or more. Since the strategy sets are finite,
1912:
1879:
1855:
1835:
1802:
1735:
1687:
1660:
1640:
1613:
1565:
1538:
1511:
1487:
1439:
1409:
1379:
1341:
1321:
1229:
1209:
1189:
1158:
1134:
1103:
1071:
1015:
975:
937:
899:
875:
855:
817:
797:
759:
739:
701:
660:
622:
602:
582:
544:
514:If the two other voters respectively cast ballots
502:
482:
462:
442:
419:
399:
379:
359:
339:
319:
299:
257:
237:
217:
197:
177:
157:
133:. A similar result for multi-winner voting is the
1447:over the alternatives, there exists a strategy
1072:{\displaystyle {\mathcal {N}}=\{1,\ldots ,n\}}
2248:wins if it gets two thirds of the votes, and
2063:, in the sense that for any possible outcome
8:
1066:
1048:
2146:
1867:of the game form. For example, we say that
118:Gibbard's theorem is itself generalized by
32:It has been suggested that this article be
16:Impossibility of straightforward game forms
2253:
2233:
2204:
2184:
2166:We assume that each voter communicates a
2127:
2107:
2083:
2082:
2068:
2044:
2018:
1991:
1990:
1982:
1961:
1960:
1958:
1934:
1933:
1925:
1901:
1900:
1892:
1872:
1848:
1824:
1823:
1815:
1794:
1788:
1787:
1771:
1765:
1764:
1754:
1753:
1751:
1724:
1711:
1706:
1700:
1679:
1673:
1653:
1632:
1626:
1602:
1589:
1584:
1578:
1557:
1551:
1530:
1524:
1504:
1476:
1463:
1458:
1452:
1431:
1425:
1402:
1372:
1334:
1313:
1307:
1306:
1290:
1284:
1283:
1270:
1251:
1242:
1222:
1202:
1181:
1175:
1174:
1171:
1151:
1126:
1120:
1119:
1116:
1096:
1039:
1038:
1036:
1007:
1006:
1004:
950:
912:
892:
868:
830:
810:
772:
752:
714:
676:
635:
615:
595:
557:
519:
495:
475:
455:
435:
412:
392:
372:
352:
332:
312:
274:
250:
230:
210:
190:
170:
150:
111:). Gibbard's theorem can be proven using
2304:
2289:
1142:be a set that represents the available
2095:{\displaystyle a\in g({\mathcal {S}})}
7:
2219:is the unique most-liked candidate.
1329:, maps an alternative. The function
1668:has identified her own preferences
96:A corollary of this theorem is the
1190:{\displaystyle {\mathcal {S}}_{i}}
1135:{\displaystyle {\mathcal {S}}_{i}}
347:but does not approve of candidate
70:is a result proven by philosopher
14:
2359:American Political Science Review
2003:{\displaystyle g({\mathcal {S}})}
1946:{\displaystyle g({\mathcal {S}})}
1913:{\displaystyle g({\mathcal {S}})}
1836:{\displaystyle g({\mathcal {S}})}
2513:Theorems in discrete mathematics
2479:
1736:{\displaystyle s_{i}^{*}(P_{i})}
1614:{\displaystyle s_{i}^{*}(P_{i})}
1488:{\displaystyle s_{i}^{*}(P_{i})}
23:
2089:
2079:
1997:
1987:
1970:{\displaystyle {\mathcal {A}}}
1940:
1930:
1907:
1897:
1830:
1820:
1730:
1717:
1608:
1595:
1482:
1469:
1276:
1244:
1016:{\displaystyle {\mathcal {A}}}
970:
952:
932:
914:
850:
832:
792:
774:
734:
716:
696:
678:
655:
637:
577:
559:
539:
521:
294:
276:
1:
2472:Gibbard–Satterthwaite theorem
2467:Arrow's impossibility theorem
113:Arrow's impossibility theorem
98:Gibbard–Satterthwaite theorem
49:Proposed since November 2023.
39:Gibbard–Satterthwaite theorem
1695:, she can choose a strategy
1217:be a function that, to each
1031:in a context of voting. Let
825:win; she should rather vote
1083:, which can also be called
1027:, which can also be called
2534:
2170:over the candidates. The
2039:if there exists an agent
1519:when she has preferences
2400:Gibbard, Allan (1978).
976:{\displaystyle (1,1,0)}
938:{\displaystyle (1,0,0)}
856:{\displaystyle (1,1,0)}
798:{\displaystyle (1,0,0)}
740:{\displaystyle (0,1,1)}
702:{\displaystyle (0,0,1)}
661:{\displaystyle (1,0,0)}
583:{\displaystyle (1,1,1)}
545:{\displaystyle (0,1,1)}
300:{\displaystyle (1,1,0)}
135:Duggan–Schwartz theorem
2262:
2242:
2213:
2193:
2136:
2116:
2096:
2053:
2027:
2004:
1971:
1947:
1914:
1881:
1863:, i.e. the set of the
1857:
1837:
1804:
1737:
1689:
1662:
1642:
1615:
1567:
1540:
1513:
1489:
1441:
1411:
1381:
1343:
1323:
1231:
1211:
1191:
1160:
1136:
1105:
1073:
1017:
977:
939:
901:
877:
857:
819:
799:
761:
741:
703:
662:
624:
604:
584:
546:
504:
484:
464:
444:
421:
401:
381:
361:
341:
321:
301:
259:
239:
219:
199:
179:
159:
120:Gibbard's 1978 theorem
2263:
2243:
2214:
2194:
2137:
2117:
2097:
2054:
2028:
2005:
1972:
1948:
1915:
1882:
1858:
1838:
1805:
1738:
1690:
1688:{\displaystyle P_{i}}
1663:
1643:
1641:{\displaystyle P_{i}}
1616:
1568:
1566:{\displaystyle s_{i}}
1541:
1539:{\displaystyle P_{i}}
1514:
1490:
1442:
1440:{\displaystyle P_{i}}
1412:
1382:
1344:
1324:
1237:-tuple of strategies
1232:
1212:
1192:
1161:
1137:
1106:
1074:
1018:
978:
940:
902:
878:
858:
820:
800:
762:
742:
704:
663:
625:
605:
585:
547:
505:
485:
465:
445:
422:
402:
382:
362:
342:
322:
302:
260:
240:
220:
200:
180:
160:
145:Consider some voters
2518:Social choice theory
2285:Notes and references
2252:
2232:
2223:Simple majority vote
2203:
2183:
2126:
2106:
2067:
2043:
2017:
1981:
1957:
1924:
1891:
1871:
1847:
1814:
1750:
1699:
1672:
1652:
1625:
1577:
1550:
1523:
1503:
1451:
1424:
1401:
1393:(originally called:
1371:
1333:
1241:
1221:
1201:
1170:
1150:
1115:
1095:
1035:
1003:
949:
911:
891:
867:
829:
809:
771:
751:
713:
675:
634:
614:
594:
556:
518:
494:
474:
454:
450:prefers alternative
434:
411:
391:
371:
351:
331:
311:
273:
249:
229:
209:
189:
169:
149:
64:social choice theory
2172:serial dictatorship
2162:Serial dictatorship
2150: —
2010:is necessarily so.
1716:
1594:
1468:
1397:) if for any agent
131:multi-winner voting
2508:Economics theorems
2258:
2238:
2209:
2189:
2148:
2132:
2112:
2092:
2049:
2023:
2000:
1967:
1943:
1910:
1877:
1853:
1833:
1800:
1733:
1702:
1685:
1658:
1638:
1611:
1580:
1563:
1536:
1509:
1485:
1454:
1437:
1407:
1377:
1339:
1319:
1227:
1207:
1187:
1156:
1132:
1101:
1069:
1013:
973:
935:
897:
873:
853:
815:
795:
757:
737:
699:
658:
620:
600:
580:
542:
500:
480:
460:
440:
430:Assume that voter
417:
397:
377:
357:
337:
317:
297:
265:. Assume they use
255:
235:
215:
195:
175:
155:
2449:Hylland, Aanund.
2261:{\displaystyle b}
2241:{\displaystyle a}
2212:{\displaystyle a}
2192:{\displaystyle a}
2168:strict weak order
2147:Gibbard's theorem
2135:{\displaystyle a}
2115:{\displaystyle i}
2052:{\displaystyle i}
2026:{\displaystyle g}
1880:{\displaystyle g}
1865:possible outcomes
1856:{\displaystyle g}
1661:{\displaystyle i}
1573:, different from
1512:{\displaystyle i}
1419:strict weak order
1410:{\displaystyle i}
1380:{\displaystyle g}
1342:{\displaystyle g}
1230:{\displaystyle n}
1210:{\displaystyle g}
1159:{\displaystyle i}
1104:{\displaystyle i}
900:{\displaystyle 1}
887:To sum up, voter
876:{\displaystyle b}
818:{\displaystyle c}
805:because it makes
760:{\displaystyle 1}
623:{\displaystyle a}
603:{\displaystyle 1}
503:{\displaystyle c}
483:{\displaystyle b}
463:{\displaystyle a}
443:{\displaystyle 1}
420:{\displaystyle a}
400:{\displaystyle b}
380:{\displaystyle a}
360:{\displaystyle c}
340:{\displaystyle b}
320:{\displaystyle a}
258:{\displaystyle c}
238:{\displaystyle b}
218:{\displaystyle a}
198:{\displaystyle 3}
178:{\displaystyle 2}
158:{\displaystyle 1}
124:Hylland's theorem
68:Gibbard's theorem
58:In the fields of
56:
55:
51:
2525:
2489:
2484:
2483:
2454:
2447:
2441:
2440:
2406:
2397:
2391:
2390:
2354:
2348:
2347:
2321:
2309:
2298:
2294:
2268:wins otherwise.
2267:
2265:
2264:
2259:
2247:
2245:
2244:
2239:
2218:
2216:
2215:
2210:
2198:
2196:
2195:
2190:
2151:
2141:
2139:
2138:
2133:
2121:
2119:
2118:
2113:
2101:
2099:
2098:
2093:
2088:
2087:
2058:
2056:
2055:
2050:
2032:
2030:
2029:
2024:
2009:
2007:
2006:
2001:
1996:
1995:
1976:
1974:
1973:
1968:
1966:
1965:
1952:
1950:
1949:
1944:
1939:
1938:
1919:
1917:
1916:
1911:
1906:
1905:
1886:
1884:
1883:
1878:
1862:
1860:
1859:
1854:
1842:
1840:
1839:
1834:
1829:
1828:
1809:
1807:
1806:
1801:
1799:
1798:
1793:
1792:
1776:
1775:
1770:
1769:
1759:
1758:
1742:
1740:
1739:
1734:
1729:
1728:
1715:
1710:
1694:
1692:
1691:
1686:
1684:
1683:
1667:
1665:
1664:
1659:
1647:
1645:
1644:
1639:
1637:
1636:
1620:
1618:
1617:
1612:
1607:
1606:
1593:
1588:
1572:
1570:
1569:
1564:
1562:
1561:
1545:
1543:
1542:
1537:
1535:
1534:
1518:
1516:
1515:
1510:
1494:
1492:
1491:
1486:
1481:
1480:
1467:
1462:
1446:
1444:
1443:
1438:
1436:
1435:
1416:
1414:
1413:
1408:
1386:
1384:
1383:
1378:
1348:
1346:
1345:
1340:
1328:
1326:
1325:
1320:
1318:
1317:
1312:
1311:
1295:
1294:
1289:
1288:
1275:
1274:
1256:
1255:
1236:
1234:
1233:
1228:
1216:
1214:
1213:
1208:
1196:
1194:
1193:
1188:
1186:
1185:
1180:
1179:
1165:
1163:
1162:
1157:
1141:
1139:
1138:
1133:
1131:
1130:
1125:
1124:
1110:
1108:
1107:
1102:
1078:
1076:
1075:
1070:
1044:
1043:
1022:
1020:
1019:
1014:
1012:
1011:
995:Formal statement
982:
980:
979:
974:
944:
942:
941:
936:
906:
904:
903:
898:
882:
880:
879:
874:
862:
860:
859:
854:
824:
822:
821:
816:
804:
802:
801:
796:
767:should not vote
766:
764:
763:
758:
746:
744:
743:
738:
708:
706:
705:
700:
667:
665:
664:
659:
629:
627:
626:
621:
609:
607:
606:
601:
589:
587:
586:
581:
551:
549:
548:
543:
509:
507:
506:
501:
489:
487:
486:
481:
469:
467:
466:
461:
449:
447:
446:
441:
426:
424:
423:
418:
406:
404:
403:
398:
386:
384:
383:
378:
366:
364:
363:
358:
346:
344:
343:
338:
326:
324:
323:
318:
306:
304:
303:
298:
264:
262:
261:
256:
244:
242:
241:
236:
224:
222:
221:
216:
204:
202:
201:
196:
184:
182:
181:
176:
164:
162:
161:
156:
90:strategic voting
60:mechanism design
47:
27:
26:
19:
2533:
2532:
2528:
2527:
2526:
2524:
2523:
2522:
2493:
2492:
2485:
2478:
2463:
2458:
2457:
2448:
2444:
2421:10.2307/1914235
2404:
2399:
2398:
2394:
2371:10.2307/1955105
2356:
2355:
2351:
2336:10.2307/1914083
2319:
2311:
2310:
2306:
2301:
2295:
2291:
2287:
2274:
2250:
2249:
2230:
2229:
2225:
2201:
2200:
2181:
2180:
2164:
2159:
2154:
2149:
2124:
2123:
2104:
2103:
2065:
2064:
2041:
2040:
2015:
2014:
1979:
1978:
1955:
1954:
1922:
1921:
1889:
1888:
1869:
1868:
1845:
1844:
1812:
1811:
1786:
1763:
1748:
1747:
1720:
1697:
1696:
1675:
1670:
1669:
1650:
1649:
1628:
1623:
1622:
1598:
1575:
1574:
1553:
1548:
1547:
1526:
1521:
1520:
1501:
1500:
1472:
1449:
1448:
1427:
1422:
1421:
1399:
1398:
1395:straightforward
1369:
1368:
1331:
1330:
1305:
1282:
1266:
1247:
1239:
1238:
1219:
1218:
1199:
1198:
1197:is finite. Let
1173:
1168:
1167:
1148:
1147:
1118:
1113:
1112:
1093:
1092:
1033:
1032:
1001:
1000:
997:
947:
946:
909:
908:
889:
888:
865:
864:
827:
826:
807:
806:
769:
768:
749:
748:
711:
710:
673:
672:
632:
631:
612:
611:
592:
591:
554:
553:
516:
515:
492:
491:
472:
471:
452:
451:
432:
431:
409:
408:
389:
388:
369:
368:
349:
348:
329:
328:
309:
308:
271:
270:
267:approval voting
247:
246:
227:
226:
207:
206:
187:
186:
167:
166:
147:
146:
143:
109:cardinal voting
78:The process is
52:
28:
24:
17:
12:
11:
5:
2531:
2529:
2521:
2520:
2515:
2510:
2505:
2495:
2494:
2491:
2490:
2487:Economy portal
2475:
2474:
2469:
2462:
2459:
2456:
2455:
2442:
2415:(3): 595–614.
2392:
2365:(3): 831–847.
2349:
2330:(4): 587–601.
2313:Gibbard, Allan
2303:
2302:
2300:
2299:
2288:
2286:
2283:
2273:
2270:
2257:
2237:
2224:
2221:
2208:
2188:
2163:
2160:
2158:
2155:
2144:
2131:
2111:
2091:
2086:
2081:
2078:
2075:
2072:
2048:
2022:
1999:
1994:
1989:
1986:
1964:
1942:
1937:
1932:
1929:
1909:
1904:
1899:
1896:
1876:
1852:
1832:
1827:
1822:
1819:
1810:and denote by
1797:
1791:
1785:
1782:
1779:
1774:
1768:
1762:
1757:
1732:
1727:
1723:
1719:
1714:
1709:
1705:
1682:
1678:
1657:
1635:
1631:
1610:
1605:
1601:
1597:
1592:
1587:
1583:
1560:
1556:
1533:
1529:
1508:
1484:
1479:
1475:
1471:
1466:
1461:
1457:
1434:
1430:
1406:
1376:
1338:
1316:
1310:
1304:
1301:
1298:
1293:
1287:
1281:
1278:
1273:
1269:
1265:
1262:
1259:
1254:
1250:
1246:
1226:
1206:
1184:
1178:
1166:; assume that
1155:
1129:
1123:
1100:
1079:be the set of
1068:
1065:
1062:
1059:
1056:
1053:
1050:
1047:
1042:
1023:be the set of
1010:
996:
993:
972:
969:
966:
963:
960:
957:
954:
934:
931:
928:
925:
922:
919:
916:
896:
885:
884:
872:
863:, which makes
852:
849:
846:
843:
840:
837:
834:
814:
794:
791:
788:
785:
782:
779:
776:
756:
736:
733:
730:
727:
724:
721:
718:
698:
695:
692:
689:
686:
683:
680:
669:
657:
654:
651:
648:
645:
642:
639:
619:
599:
579:
576:
573:
570:
567:
564:
561:
541:
538:
535:
532:
529:
526:
523:
499:
479:
459:
439:
416:
396:
376:
356:
336:
316:
296:
293:
290:
287:
284:
281:
278:
254:
234:
214:
194:
174:
154:
142:
139:
94:
93:
86:
83:
54:
53:
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
2530:
2519:
2516:
2514:
2511:
2509:
2506:
2504:
2503:Voting theory
2501:
2500:
2498:
2488:
2482:
2477:
2473:
2470:
2468:
2465:
2464:
2460:
2452:
2446:
2443:
2438:
2434:
2430:
2426:
2422:
2418:
2414:
2410:
2403:
2396:
2393:
2388:
2384:
2380:
2376:
2372:
2368:
2364:
2360:
2353:
2350:
2345:
2341:
2337:
2333:
2329:
2325:
2318:
2314:
2308:
2305:
2293:
2290:
2284:
2282:
2280:
2271:
2269:
2255:
2235:
2222:
2220:
2206:
2186:
2176:
2173:
2169:
2161:
2156:
2153:
2143:
2129:
2109:
2076:
2073:
2070:
2062:
2046:
2038:
2037:
2020:
2011:
1984:
1927:
1894:
1874:
1866:
1850:
1843:the range of
1817:
1795:
1783:
1780:
1777:
1772:
1760:
1744:
1725:
1721:
1712:
1707:
1703:
1680:
1676:
1655:
1633:
1629:
1603:
1599:
1590:
1585:
1581:
1558:
1554:
1531:
1527:
1506:
1498:
1477:
1473:
1464:
1459:
1455:
1432:
1428:
1420:
1404:
1396:
1392:
1391:
1390:strategyproof
1374:
1365:
1363:
1359:
1357:
1352:
1336:
1314:
1302:
1299:
1296:
1291:
1279:
1271:
1267:
1263:
1260:
1257:
1252:
1248:
1224:
1204:
1182:
1153:
1145:
1127:
1098:
1090:
1086:
1082:
1063:
1060:
1057:
1054:
1051:
1045:
1030:
1026:
994:
992:
988:
986:
985:strategyproof
967:
964:
961:
958:
955:
929:
926:
923:
920:
917:
894:
870:
847:
844:
841:
838:
835:
812:
789:
786:
783:
780:
777:
754:
747:, then voter
731:
728:
725:
722:
719:
693:
690:
687:
684:
681:
670:
652:
649:
646:
643:
640:
617:
597:
590:, then voter
574:
571:
568:
565:
562:
536:
533:
530:
527:
524:
513:
512:
511:
497:
477:
457:
437:
428:
414:
394:
374:
354:
334:
314:
291:
288:
285:
282:
279:
268:
252:
232:
212:
192:
172:
152:
140:
138:
136:
132:
127:
125:
121:
116:
114:
110:
105:
103:
102:ranked voting
99:
91:
87:
84:
81:
77:
76:
75:
73:
72:Allan Gibbard
69:
65:
61:
50:
45:
41:
40:
35:
30:
21:
20:
2445:
2429:10419/220562
2412:
2409:Econometrica
2408:
2395:
2362:
2358:
2352:
2327:
2324:Econometrica
2323:
2307:
2292:
2275:
2226:
2177:
2171:
2165:
2145:
2060:
2034:
2013:We say that
2012:
1864:
1745:
1496:
1417:and for any
1394:
1389:
1367:We say that
1366:
1361:
1358:-player game
1355:
1350:
1349:is called a
1143:
1088:
1084:
1080:
1028:
1025:alternatives
1024:
998:
989:
886:
429:
144:
128:
117:
106:
95:
67:
57:
48:
37:
2279:Borda count
2036:dictatorial
80:dictatorial
2497:Categories
2297:"sincere."
1499:for agent
1146:for agent
1144:strategies
1029:candidates
2379:0003-0554
2074:∈
2059:who is a
1784:×
1781:⋯
1778:×
1713:∗
1591:∗
1465:∗
1351:game form
1303:×
1300:⋯
1297:×
1280:∈
1261:…
1058:…
490:and then
2461:See also
2315:(1973).
2157:Examples
2102:, agent
2061:dictator
1497:dominant
1495:that is
1362:a priori
630: :
141:Overview
2453:, 1980.
2437:1914235
2387:1955105
2344:1914083
1746:We let
1089:voters,
1085:players
470:, then
407:, then
44:Discuss
2435:
2385:
2377:
2342:
1111:, let
1081:agents
427:wins.
34:merged
2433:JSTOR
2405:(PDF)
2383:JSTOR
2340:JSTOR
2320:(PDF)
36:into
2375:ISSN
999:Let
883:win.
709:and
552:and
387:and
327:and
245:and
185:and
122:and
62:and
2425:hdl
2417:doi
2367:doi
2332:doi
2033:is
1387:is
1087:or
945:or
42:. (
2499::
2431:.
2423:.
2413:46
2411:.
2407:.
2381:.
2373:.
2363:72
2361:.
2338:.
2328:41
2326:.
2322:.
225:,
165:,
137:.
115:.
66:,
2439:.
2427::
2419::
2389:.
2369::
2346:.
2334::
2256:b
2236:a
2207:a
2187:a
2130:a
2110:i
2090:)
2085:S
2080:(
2077:g
2071:a
2047:i
2021:g
1998:)
1993:S
1988:(
1985:g
1963:A
1941:)
1936:S
1931:(
1928:g
1908:)
1903:S
1898:(
1895:g
1875:g
1851:g
1831:)
1826:S
1821:(
1818:g
1796:n
1790:S
1773:1
1767:S
1761:=
1756:S
1731:)
1726:i
1722:P
1718:(
1708:i
1704:s
1681:i
1677:P
1656:i
1634:i
1630:P
1609:)
1604:i
1600:P
1596:(
1586:i
1582:s
1559:i
1555:s
1532:i
1528:P
1507:i
1483:)
1478:i
1474:P
1470:(
1460:i
1456:s
1433:i
1429:P
1405:i
1375:g
1356:n
1337:g
1315:n
1309:S
1292:1
1286:S
1277:)
1272:n
1268:s
1264:,
1258:,
1253:1
1249:s
1245:(
1225:n
1205:g
1183:i
1177:S
1154:i
1128:i
1122:S
1099:i
1067:}
1064:n
1061:,
1055:,
1052:1
1049:{
1046:=
1041:N
1009:A
971:)
968:0
965:,
962:1
959:,
956:1
953:(
933:)
930:0
927:,
924:0
921:,
918:1
915:(
895:1
871:b
851:)
848:0
845:,
842:1
839:,
836:1
833:(
813:c
793:)
790:0
787:,
784:0
781:,
778:1
775:(
755:1
735:)
732:1
729:,
726:1
723:,
720:0
717:(
697:)
694:1
691:,
688:0
685:,
682:0
679:(
668:.
656:)
653:0
650:,
647:0
644:,
641:1
638:(
618:a
598:1
578:)
575:1
572:,
569:1
566:,
563:1
560:(
540:)
537:1
534:,
531:1
528:,
525:0
522:(
498:c
478:b
458:a
438:1
415:a
395:b
375:a
355:c
335:b
315:a
295:)
292:0
289:,
286:1
283:,
280:1
277:(
253:c
233:b
213:a
193:3
173:2
153:1
46:)
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