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It is possible that several such maximal lotteries exist, as a result of ties. However, the maximal lottery is unique whenever there the number of voters is odd. By the same argument, the bipartisan set is uniquely-defined by taking the support of the unique maximal lottery that solves a tournament
1739:{\displaystyle {\begin{matrix}{\begin{matrix}&&a\quad &b\quad &c\quad \\\end{matrix}}\\{\begin{matrix}a\\b\\c\\\end{matrix}}{\begin{pmatrix}0&1&-1\\-1&0&3\\1&-3&0\\\end{pmatrix}}\end{matrix}}}
1156:
The input to this voting system consists of the agents' ordinal preferences over outcomes (not lotteries over alternatives), but a relation on the set of lotteries can be constructed in the following way: if
1913:. Many preference profiles admit a Condorcet winner, in which case the unique maximal lottery will assign probability 1 to the Condorcet winner. If the last voter in the example above swaps alternatives
1034:
in probabilities, i.e. it is possible that the probability of an alternative decreases when a voter ranks this alternative up. However, they satisfy relative monotonicity, i.e., the probability of
1353:
given by the pairwise majority margins. As such, they have a natural interpretation in terms of electoral competition between two political parties and be computer in polynomial-time via .
2229:. In Mathematics and Social Sciences I: Proceedings of the seminars of Menthon-Saint-Bernard, France (1–27 July 1960) and of Gösing, Austria (3–27 July 1962), pages 73–79, 1965.
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1007:. The probabilistic voting rule that returns all maximal lotteries is the only rule satisfying reinforcement, Condorcet-consistency, and independence of clones. The
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1145:. Since then, they have been rediscovered multiple times by economists, mathematicians, political scientists, philosophers, and computer scientists.
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1909:. By definition, this is also the unique maximal lottery of the preference profile above. The example was carefully chosen not to have a
938:
1148:
Several natural dynamics that converge to maximal lotteries have been observed in biology, physics, chemistry, and machine learning.
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Maximal lotteries were first proposed by the French mathematician and social scientist
Germain Kreweras in 1965 and popularized by
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than vice versa. While this relation is not necessarily transitive, it does always admit at least one maximal element.
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2271:. In Proceedings of 3rd International Workshop on Computational Social Choice, pages 399–410, 2010.
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if it is more likely that a randomly selected voter will prefer the alternatives sampled from
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Suppose there are five voters who have the following preferences over three alternatives:
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if the expected value of the margin of victory of an outcome selected with distribution
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Reinforcement learning from comparisons: Three alternatives are enough, two are not
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can satisfy strategyproofness and ex post efficiency. Maximal lotteries are also
2293:
Jacopo Grilli, György Barabás, Matthew J. Michalska-Smith and
Stefano Allesina.
400:
358:
301:
246:
2135:
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that top-ranks maximal lotteries has been uniquely characterized using Arrow's
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Maximal lotteries satisfy a wide range of desirable properties: they elect the
17:
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Maximal lotteries do not satisfy the standard notion of strategyproofness, as
2153:
2119:
2096:
984:) of candidates that a majority of voters would weakly prefer to any other.
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The pairwise preferences of the voters can be represented in the following
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Higher-order interactions stabilize dynamics in competitive network models
910:
619:
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An optimal single-winner preferential voting system based on game theory
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2087:
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After two centuries should
Condorcet’s voting procedure be implemented?
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2062:
1342:
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with probability 1 if it exists and never elect candidates outside the
405:
2120:"An analytical and experimental comparison of maximal lottery schemes"
1973:
becomes the
Condorcet winner and will be selected with probability 1.
1243:
in a head-to-head vote against an outcome selected with distribution
2078:
2118:
Brandl, Florian; Brandt, Felix; Stricker, Christian (2022-01-01).
2002:
Probabilistic social choice based on simple voting comparisons
2319:
Gilbert
Laffond, Jean-François Laslier and Michel Le Breton
57:
2308:
A Natural
Adaptive Process for Collective Decision-Making
2284:. Annals of Applied Probability 27(5): 2907–2925, 2017.
2035:. Games and Economic Behavior. 14, pages 308-314, 2019.
2212:
F. Brandt, M. Brill, H. G. Seedig, and W. Suksompong.
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1606:
1602:
2063:"Manipulation of Schemes that Mix Voting with Chance"
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1040:
2351:
2338:. Social Choice and Welfare 17: pages 283–292, 2000.
2188:. Games and Economic Behavior, 5(1):182–201, 1993.
2004:. Review of Economic Studies, 51(4):683–692, 1984.
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2321:A theorem on two–player symmetric zero–sum games
2323:. Journal of Economic Theory 72: 426–431, 1997.
2242:. Journal of Graph Theory, 19(2):217–236, 1995.
2214:On the structure of stable tournament solutions
2170:. Social Choice and Welfare, 16: 513–532, 1999.
2168:Comparison functions and choice correspondences
2310:. Theoretical Economics 19(2): 667–703, 2024.
2184:G. Laffond, J.-F. Laslier, and M. Le Breton.
2022:. Econometrica. 84(5), pages 1839-1880, 2016.
946:
8:
2336:Interpretation of electoral mixed strategies
1121:of maximal lotteries, which is known as the
2051:. Econometrica. 88(2), pages 799-844, 2020.
2049:Arrovian Aggregation of Convex Preferences
2033:Welfare Maximization Entices Participation
1341:Maximal lotteries are equivalent to mixed
953:
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29:
2354:(website for computing maximal lotteries)
2240:Tournament games and positive tournaments
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2255:Behavioral Science, 37(4):250–274, 1992.
2201:Tournament solutions and majority voting
2180:
2178:
2176:
2018:F. Brandl, F. Brandt, and H. G. Seedig.
1512:denotes the number of voters who prefer
2216:. Economic Theory, 65(2):483–507, 2018.
2186:The bipartisan set of a tournament game
2031:F. Brandl, F. Brandt, and J. Hofbauer.
1982:
1013:independence of irrelevant alternatives
45:
2020:Consistent probabilistic social choice
1552:minus the number of voters who prefer
1152:Collective preferences over lotteries
7:
1748:This matrix can be interpreted as a
2227:Aggregation of preference orderings
2251:D. S. Felsenthal and M. Machover.
25:
1197:are lotteries over alternatives,
1309:to the alternative sampled from
922:
909:
897:
845:McKelvey–Schofield chaos theorem
491:Semi-proportional representation
123:First preference plurality (FPP)
2373:Single-winner electoral systems
1626:
1620:
1614:
1458:{\displaystyle c\succ a\succ b}
1424:{\displaystyle b\succ c\succ a}
1390:{\displaystyle a\succ b\succ c}
2368:Preferential electoral systems
2280:B. Laslier and J.-F. Laslier.
1882:
1876:
1839:
1833:
1796:
1790:
1133:, has been studied in detail.
883:Harsanyi's utilitarian theorem
840:Moulin's impossibility theorem
805:Conflicting majorities paradox
27:Probabilistic Condorcet method
1:
1263:is positive. In other words,
709:Frustrated majorities paradox
2297:. Nature 548: 210-214, 2017.
2166:B. Dutta and J.-F. Laslier.
1953:in his preference relation,
878:Condorcet dominance theorems
818:Social and collective choice
544:By mechanism of combination
315:Proportional representation
2389:
2238:D. C. Fisher and J. Ryan.
2136:10.1007/s00355-021-01326-x
1472:, where the entry for row
742:Multiple districts paradox
473:Fractional approval voting
461:Interactive representation
2306:F. Brandl and F. Brandt.
2124:Social Choice and Welfare
2047:F. Brandl and F. Brandt.
995:. Moreover, they satisfy
970:probabilistic voting rule
689:Paradoxes and pathologies
538:Mixed-member proportional
533:Mixed-member majoritarian
528:By results of combination
419:Approval-based committees
1902:{\displaystyle p(c)=1/5}
1859:{\displaystyle p(b)=1/5}
1816:{\displaystyle p(a)=3/5}
1337:Strategic interpretation
1282:{\displaystyle p\succ q}
1216:{\displaystyle p\succ q}
978:probability distribution
868:Condorcet's jury theorem
669:Double simultaneous vote
644:Rural–urban proportional
639:Dual-member proportional
601:
590:
557:Parallel (superposition)
449:Fractional social choice
436:Expanding approvals rule
265:
250:
235:
166:
155:
131:
2061:Gibbard, Allan (1977).
1074:does not decrease when
1009:social welfare function
795:Tyranny of the majority
572:Fusion (majority bonus)
389:Quota-remainder methods
2203:Springer-Verlag, 1997.
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1947:
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1005:independence of clones
929:Mathematics portal
835:Majority impossibility
824:Impossibility theorems
620:Negative vote transfer
441:Method of equal shares
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1470:skew-symmetric matrix
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732:Best-is-worst paradox
721:Pathological response
456:Direct representation
109:Single-winner methods
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1752:and admits a unique
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1054:relative to that of
1038:
1028:random dictatorships
1026:has shown that only
974:preferential ballots
916:Economics portal
863:Median voter theorem
82:Comparative politics
1349:) of the symmetric
904:Politics portal
615:Vote linkage system
586:Seat linkage system
173:Ranked-choice (RCV)
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1343:maximin strategies
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982:linear combination
972:. The method uses
800:Discursive dilemma
759:Lesser evil voting
634:Supermixed systems
337:Largest remainders
195:Round-robin voting
63:
1966:{\displaystyle a}
1946:{\displaystyle c}
1926:{\displaystyle a}
1773:{\displaystyle p}
1585:{\displaystyle x}
1565:{\displaystyle y}
1545:{\displaystyle y}
1525:{\displaystyle x}
1505:{\displaystyle y}
1485:{\displaystyle x}
1322:{\displaystyle q}
1302:{\displaystyle p}
1256:{\displaystyle q}
1236:{\displaystyle p}
1190:{\displaystyle q}
1170:{\displaystyle p}
1107:{\displaystyle y}
1094:is improved over
1087:{\displaystyle x}
1067:{\displaystyle y}
1047:{\displaystyle x}
1017:Pareto efficiency
966:Maximal lotteries
963:
962:
850:Gibbard's theorem
790:Dominance paradox
727:Perverse response
431:Phragmen's method
297:Majority judgment
225:Positional voting
183:Condorcet methods
51:electoral systems
16:(Redirected from
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2000:P. C. Fishburn.
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1754:Nash equilibrium
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857:Positive results
752:Strategic voting
649:Majority jackpot
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466:Liquid democracy
342:National remnant
332:Highest averages
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1347:Nash equilibria
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2073:(3): 665–681.
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1143:Peter Fishburn
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1135:
1129:bipartisan set
1103:
1083:
1063:
1043:
976:and returns a
961:
960:
958:
957:
950:
943:
935:
932:
931:
919:
918:
906:
893:
890:
889:
886:
885:
880:
875:
870:
865:
853:
852:
847:
842:
837:
832:
821:
816:
815:
812:
811:
808:
807:
802:
797:
792:
777:
776:
774:Turkey-raising
771:
766:
761:
747:
746:
745:
744:
734:
729:
717:
716:
714:Center squeeze
711:
706:
701:
699:Spoiler effect
692:
687:
686:
683:
682:
679:
678:
673:
672:
671:
658:By ballot type
654:
653:
652:
651:
646:
641:
631:
630:
629:
628:
627:
622:
612:
611:
610:
599:
576:
575:
574:
569:
564:
559:
541:
540:
535:
526:
521:
520:
517:
516:
513:
512:
510:Limited voting
507:
506:
505:
486:
485:
480:
475:
470:
469:
468:
463:
444:
443:
438:
433:
428:
414:
413:
408:
403:
398:
384:
383:
382:
381:
379:Localized list
376:
371:
366:
361:
351:
350:
349:
347:Biproportional
344:
339:
334:
318:
313:
312:
309:
308:
305:
304:
299:
294:
289:
275:
274:
259:
244:
220:
219:
218:
217:
212:
207:
202:
192:
178:
177:
176:
175:
164:
151:Instant-runoff
148:
147:
146:
138:Jungle primary
125:
114:Single vote -
112:
107:
106:
103:
102:
100:
99:
89:
84:
79:
74:
68:
65:
64:
54:
53:
43:
42:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2385:
2374:
2371:
2369:
2366:
2365:
2363:
2353:
2350:
2349:
2345:
2337:
2334:
2329:
2326:
2322:
2316:
2313:
2309:
2303:
2300:
2296:
2290:
2287:
2283:
2277:
2274:
2270:
2267:and E. Shen.
2266:
2261:
2258:
2254:
2248:
2245:
2241:
2235:
2232:
2228:
2225:G. Kreweras.
2222:
2219:
2215:
2209:
2206:
2202:
2199:
2194:
2191:
2187:
2181:
2179:
2177:
2173:
2169:
2163:
2160:
2155:
2151:
2146:
2141:
2137:
2133:
2129:
2125:
2121:
2114:
2111:
2106:
2102:
2098:
2094:
2089:
2084:
2080:
2076:
2072:
2068:
2064:
2057:
2054:
2050:
2044:
2042:
2038:
2034:
2028:
2025:
2021:
2015:
2013:
2011:
2007:
2003:
1997:
1995:
1993:
1991:
1989:
1987:
1983:
1976:
1974:
1960:
1940:
1920:
1912:
1896:
1892:
1888:
1885:
1879:
1873:
1853:
1849:
1845:
1842:
1836:
1830:
1810:
1806:
1802:
1799:
1793:
1787:
1767:
1759:
1755:
1751:
1750:zero-sum game
1746:
1727:
1721:
1716:
1713:
1708:
1701:
1696:
1691:
1688:
1681:
1678:
1673:
1668:
1662:
1653:
1646:
1639:
1623:
1617:
1611:
1593:
1579:
1559:
1539:
1519:
1499:
1479:
1471:
1452:
1449:
1446:
1443:
1440:
1432:
1418:
1415:
1412:
1409:
1406:
1398:
1384:
1381:
1378:
1375:
1372:
1364:
1363:
1362:
1356:
1354:
1352:
1351:zero-sum game
1348:
1344:
1336:
1334:
1330:
1316:
1296:
1276:
1273:
1270:
1250:
1230:
1210:
1207:
1204:
1184:
1164:
1151:
1149:
1146:
1144:
1136:
1134:
1132:
1124:
1123:essential set
1120:
1115:
1101:
1081:
1061:
1041:
1033:
1029:
1025:
1024:Allan Gibbard
1020:
1018:
1014:
1010:
1006:
1002:
1001:participation
998:
997:reinforcement
994:
990:
985:
983:
979:
975:
971:
967:
956:
951:
949:
944:
942:
937:
936:
934:
933:
930:
925:
917:
912:
907:
905:
900:
895:
894:
892:
891:
884:
881:
879:
876:
874:
873:May's theorem
871:
869:
866:
864:
861:
860:
859:
858:
851:
848:
846:
843:
841:
838:
836:
833:
831:
828:
827:
826:
825:
819:
814:
813:
806:
803:
801:
798:
796:
793:
791:
788:
787:
786:
785:
784:
783:majority rule
781:Paradoxes of
775:
772:
770:
767:
765:
762:
760:
757:
756:
755:
754:
753:
743:
740:
739:
738:
735:
733:
730:
728:
725:
724:
723:
722:
715:
712:
710:
707:
705:
702:
700:
697:
696:
695:
690:
685:
684:
677:
674:
670:
667:
666:
665:
662:
661:
660:
659:
650:
647:
645:
642:
640:
637:
636:
635:
632:
626:
623:
621:
618:
617:
616:
613:
609:
604:
600:
598:
593:
589:
588:
587:
584:
583:
582:
581:
577:
573:
570:
568:
565:
563:
560:
558:
555:
554:
553:
552:
547:
546:
545:
539:
536:
534:
531:
530:
529:
524:
523:Mixed systems
519:
518:
511:
508:
504:
501:
500:
499:
496:
495:
494:
493:
492:
484:
483:Random ballot
481:
479:
476:
474:
471:
467:
464:
462:
459:
458:
457:
454:
453:
452:
451:
450:
442:
439:
437:
434:
432:
429:
427:
424:
423:
422:
421:
420:
412:
409:
407:
404:
402:
399:
397:
394:
393:
392:
391:
390:
380:
377:
375:
372:
370:
367:
365:
362:
360:
357:
356:
355:
352:
348:
345:
343:
340:
338:
335:
333:
330:
329:
328:
327:Apportionment
325:
324:
323:
322:
316:
311:
310:
303:
300:
298:
295:
293:
290:
288:
285:
284:
283:
282:
281:
272:
268:
263:
262:Antiplurality
260:
257:
253:
248:
245:
242:
238:
233:
230:
229:
228:
227:
226:
216:
213:
211:
208:
206:
203:
201:
198:
197:
196:
193:
191:
190:Condorcet-IRV
188:
187:
186:
185:
184:
174:
169:
165:
163:
158:
154:
153:
152:
149:
145:
142:
141:
139:
134:
129:
126:
124:
121:
120:
119:
117:
110:
105:
104:
97:
93:
90:
88:
85:
83:
80:
78:
75:
73:
72:Social choice
70:
69:
67:
66:
60:
56:
55:
52:
48:
47:Social choice
44:
40:
36:
32:
31:
19:
2335:
2328:
2320:
2315:
2302:
2294:
2289:
2281:
2276:
2268:
2265:R. L. Rivest
2260:
2252:
2247:
2239:
2234:
2226:
2221:
2213:
2208:
2200:
2193:
2185:
2167:
2162:
2145:10419/286729
2127:
2123:
2113:
2088:10419/220534
2070:
2067:Econometrica
2066:
2056:
2027:
2001:
1747:
1594:
1467:
1360:
1340:
1331:
1155:
1147:
1140:
1126:
1122:
1116:
1032:nonmonotonic
1021:
986:
968:refers to a
965:
964:
856:
855:
822:
780:
779:
764:Exaggeration
750:
749:
720:
719:
693:
657:
656:
625:Mixed ballot
580:Compensatory
578:
551:compensatory
548:
543:
527:
489:
488:
447:
446:
417:
416:
387:
386:
374:List-free PR
319:
287:Score voting
278:
277:
223:
222:
210:Ranked pairs
181:
180:
113:
2130:(1): 5–38.
1492:and column
664:Single vote
567:Conditional
562:Coexistence
411:Quota Borda
401:Schulze STV
359:Closed list
302:STAR voting
247:Borda count
2362:Categories
1977:References
1399:2 voters:
1365:2 voters:
769:Truncation
498:Cumulative
321:Party-list
96:By country
87:Comparison
2352:voting.ml
2154:1432-217X
2097:0012-9682
1714:−
1689:−
1679:−
1450:≻
1444:≻
1433:1 voter:
1416:≻
1410:≻
1382:≻
1376:≻
1274:≻
1208:≻
993:Smith set
676:Dual-vote
369:Panachage
364:Open list
354:List type
232:Plurality
128:Two-round
116:plurality
39:Economics
396:Hare STV
35:Politics
33:A joint
2105:1911681
1357:Example
1137:History
1125:or the
1119:support
406:CPO-STV
256:Baldwin
205:Schulze
200:Minimax
118:methods
2152:
2103:
2095:
1780:where
1333:game.
1003:, and
271:Coombs
41:series
2101:JSTOR
608:'MMP'
597:'AMS'
2150:ISSN
2093:ISSN
1933:and
1756:(or
1345:(or
1177:and
1117:The
1015:and
980:(or
549:Non-
503:SNTV
92:List
49:and
37:and
2140:hdl
2132:doi
2083:hdl
2075:doi
1572:to
1532:to
267:el.
252:el.
241:IRV
237:el.
2364::
2175:^
2148:.
2138:.
2128:58
2126:.
2122:.
2099:.
2091:.
2081:.
2071:45
2069:.
2065:.
2040:^
2009:^
1985:^
1866:,
1823:,
1760:)
1592:.
1114:.
1019:.
999:,
603:NZ
592:UK
168:US
157:UK
140:)
133:US
2156:.
2142::
2134::
2107:.
2085::
2077::
1961:a
1941:c
1921:a
1897:5
1893:/
1889:1
1886:=
1883:)
1880:c
1877:(
1874:p
1854:5
1850:/
1846:1
1843:=
1840:)
1837:b
1834:(
1831:p
1811:5
1807:/
1803:3
1800:=
1797:)
1794:a
1791:(
1788:p
1768:p
1728:)
1722:0
1717:3
1709:1
1702:3
1697:0
1692:1
1682:1
1674:1
1669:0
1663:(
1654:c
1647:b
1640:a
1624:c
1618:b
1612:a
1580:x
1560:y
1540:y
1520:x
1500:y
1480:x
1453:b
1447:a
1441:c
1419:a
1413:c
1407:b
1385:c
1379:b
1373:a
1317:q
1297:p
1277:q
1271:p
1251:q
1231:p
1211:q
1205:p
1185:q
1165:p
1102:y
1082:x
1062:y
1042:x
954:e
947:t
940:v
605::
594::
273:)
264:(
258:)
249:(
243:)
234:(
170::
159::
135::
130:(
98:)
94:(
20:)
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