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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
1728:{\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}}}
1145:
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
1902:. 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
1023:
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
1342:
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 .
871:
2218:. 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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996:. 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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1134:. Since then, they have been rediscovered multiple times by economists, mathematicians, political scientists, philosophers, and computer scientists.
1898:. By definition, this is also the unique maximal lottery of the preference profile above. The example was carefully chosen not to have a
632:
1137:
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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2260:. 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
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145:
23:
2282:
Jacopo Grilli, György Barabás, Matthew J. Michalska-Smith and
Stefano Allesina.
2124:
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that top-ranks maximal lotteries has been uniquely characterized using Arrow's
976:
Maximal lotteries satisfy a wide range of desirable properties: they elect the
762:
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Maximal lotteries do not satisfy the standard notion of strategyproofness, as
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973:) 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
47:
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An optimal single-winner preferential voting system based on game theory
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887:
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After two centuries should
Condorcet’s voting procedure be implemented?
2093:
2051:
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with probability 1 if it exists and never elect candidates outside the
2109:"An analytical and experimental comparison of maximal lottery schemes"
1962:
becomes the
Condorcet winner and will be selected with probability 1.
1232:
in a head-to-head vote against an outcome selected with distribution
357:
352:
2067:
608:
2107:
Brandl, Florian; Brandt, Felix; Stricker, Christian (2022-01-01).
1331:
394:
899:
1991:
Probabilistic social choice based on simple voting comparisons
2308:
Gilbert
Laffond, Jean-François Laslier and Michel Le Breton
2297:
A Natural
Adaptive Process for Collective Decision-Making
1140:
2273:. Annals of Applied Probability 27(5): 2907–2925, 2017.
2024:. Games and Economic Behavior. 14, pages 308-314, 2019.
2201:
F. Brandt, M. Brill, H. G. Seedig, and W. Suksompong.
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1625:
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2052:"Manipulation of Schemes that Mix Voting with Chance"
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2327:. Social Choice and Welfare 17: pages 283–292, 2000.
2106:
2177:. Games and Economic Behavior, 5(1):182–201, 1993.
1993:. Review of Economic Studies, 51(4):683–692, 1984.
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1035:
2348:
2310:A theorem on two–player symmetric zero–sum games
2312:. Journal of Economic Theory 72: 426–431, 1997.
2231:. Journal of Graph Theory, 19(2):217–236, 1995.
2203:On the structure of stable tournament solutions
2159:. Social Choice and Welfare, 16: 513–532, 1999.
2157:Comparison functions and choice correspondences
2299:. Theoretical Economics 19(2): 667–703, 2024.
2173:G. Laffond, J.-F. Laslier, and M. Le Breton.
2011:. Econometrica. 84(5), pages 1839-1880, 2016.
935:
2325:Interpretation of electoral mixed strategies
1110:of maximal lotteries, which is known as the
2315:
2040:. Econometrica. 88(2), pages 799-844, 2020.
1325:
2038:Arrovian Aggregation of Convex Preferences
2022:Welfare Maximization Entices Participation
1330:Maximal lotteries are equivalent to mixed
942:
928:
2343:(website for computing maximal lotteries)
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2229:Tournament games and positive tournaments
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2030:
2003:
2001:
1999:
1985:
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2244:Behavioral Science, 37(4):250–274, 1992.
2190:Tournament solutions and majority voting
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2007:F. Brandl, F. Brandt, and H. G. Seedig.
1501:denotes the number of voters who prefer
2289:
2247:
2234:
2221:
2208:
2205:. Economic Theory, 65(2):483–507, 2018.
2175:The bipartisan set of a tournament game
2049:
2020:F. Brandl, F. Brandt, and J. Hofbauer.
1002:independence of irrelevant alternatives
2349:
2195:
2027:
2014:
2009:Consistent probabilistic social choice
1996:
1972:
1541:minus the number of voters who prefer
2162:
1141:Collective preferences over lotteries
2263:
1737:This matrix can be interpreted as a
2216:Aggregation of preference orderings
2149:
13:
2240:D. S. Felsenthal and M. Machover.
46:
14:
2373:
2334:
1186:are lotteries over alternatives,
1298:to the alternative sampled from
911:
898:
886:
834:McKelvey–Schofield chaos theorem
480:Semi-proportional representation
112:First preference plurality (FPP)
2362:Single-winner electoral systems
2302:
1615:
1609:
1603:
1447:{\displaystyle c\succ a\succ b}
1413:{\displaystyle b\succ c\succ a}
1379:{\displaystyle a\succ b\succ c}
2357:Preferential electoral systems
2269:B. Laslier and J.-F. Laslier.
2100:
2043:
1871:
1865:
1828:
1822:
1785:
1779:
1122:, has been studied in detail.
872:Harsanyi's utilitarian theorem
829:Moulin's impossibility theorem
794:Conflicting majorities paradox
16:Probabilistic Condorcet method
1:
1965:
1252:is positive. In other words,
698:Frustrated majorities paradox
2286:. Nature 548: 210-214, 2017.
2155:B. Dutta and J.-F. Laslier.
1942:in his preference relation,
867:Condorcet dominance theorems
807:Social and collective choice
7:
533:By mechanism of combination
304:Proportional representation
10:
2378:
2227:D. C. Fisher and J. Ryan.
2125:10.1007/s00355-021-01326-x
1461:, where the entry for row
1345:
1125:
731:Multiple districts paradox
462:Fractional approval voting
450:Interactive representation
2295:F. Brandl and F. Brandt.
2113:Social Choice and Welfare
2036:F. Brandl and F. Brandt.
984:. Moreover, they satisfy
959:probabilistic voting rule
678:Paradoxes and pathologies
527:Mixed-member proportional
522:Mixed-member majoritarian
517:By results of combination
408:Approval-based committees
1891:{\displaystyle p(c)=1/5}
1848:{\displaystyle p(b)=1/5}
1805:{\displaystyle p(a)=3/5}
1326:Strategic interpretation
1271:{\displaystyle p\succ q}
1205:{\displaystyle p\succ q}
967:probability distribution
857:Condorcet's jury theorem
658:Double simultaneous vote
633:Rural–urban proportional
628:Dual-member proportional
590:
579:
546:Parallel (superposition)
438:Fractional social choice
425:Expanding approvals rule
254:
239:
224:
155:
144:
120:
2050:Gibbard, Allan (1977).
1063:does not decrease when
998:social welfare function
784:Tyranny of the majority
561:Fusion (majority bonus)
378:Quota-remainder methods
2192:Springer-Verlag, 1997.
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994:independence of clones
918:Mathematics portal
824:Majority impossibility
813:Impossibility theorems
609:Negative vote transfer
430:Method of equal shares
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1459:skew-symmetric matrix
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721:Best-is-worst paradox
710:Pathological response
445:Direct representation
98:Single-winner methods
50:
1946:
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1741:and admits a unique
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1043:relative to that of
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1017:random dictatorships
1015:has shown that only
963:preferential ballots
905:Economics portal
852:Median voter theorem
71:Comparative politics
1338:) of the symmetric
893:Politics portal
604:Vote linkage system
575:Seat linkage system
162:Ranked-choice (RCV)
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1332:maximin strategies
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971:linear combination
961:. The method uses
789:Discursive dilemma
748:Lesser evil voting
623:Supermixed systems
326:Largest remainders
184:Round-robin voting
52:
1955:{\displaystyle a}
1935:{\displaystyle c}
1915:{\displaystyle a}
1762:{\displaystyle p}
1574:{\displaystyle x}
1554:{\displaystyle y}
1534:{\displaystyle y}
1514:{\displaystyle x}
1494:{\displaystyle y}
1474:{\displaystyle x}
1311:{\displaystyle q}
1291:{\displaystyle p}
1245:{\displaystyle q}
1225:{\displaystyle p}
1179:{\displaystyle q}
1159:{\displaystyle p}
1096:{\displaystyle y}
1083:is improved over
1076:{\displaystyle x}
1056:{\displaystyle y}
1036:{\displaystyle x}
1006:Pareto efficiency
955:Maximal lotteries
952:
951:
839:Gibbard's theorem
779:Dominance paradox
716:Perverse response
420:Phragmen's method
286:Majority judgment
214:Positional voting
172:Condorcet methods
40:electoral systems
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1989:P. C. Fishburn.
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1900:Condorcet winner
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1743:Nash equilibrium
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846:Positive results
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638:Majority jackpot
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455:Liquid democracy
331:National remnant
321:Highest averages
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151:Alternative vote
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133:Partisan primary
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1336:Nash equilibria
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693:Cloning paradox
683:Spoiler effects
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467:Maximal lottery
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415:Thiele's method
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269:Cardinal voting
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204:Maximal lottery
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2187:Laslier, J.-F.
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2062:(3): 665–681.
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1267:
1264:
1261:
1241:
1221:
1201:
1198:
1195:
1175:
1155:
1142:
1139:
1132:Peter Fishburn
1127:
1124:
1118:bipartisan set
1092:
1072:
1052:
1032:
965:and returns a
950:
949:
947:
946:
939:
932:
924:
921:
920:
908:
907:
895:
882:
879:
878:
875:
874:
869:
864:
859:
854:
842:
841:
836:
831:
826:
821:
810:
805:
804:
801:
800:
797:
796:
791:
786:
781:
766:
765:
763:Turkey-raising
760:
755:
750:
736:
735:
734:
733:
723:
718:
706:
705:
703:Center squeeze
700:
695:
690:
688:Spoiler effect
681:
676:
675:
672:
671:
668:
667:
662:
661:
660:
647:By ballot type
643:
642:
641:
640:
635:
630:
620:
619:
618:
617:
616:
611:
601:
600:
599:
588:
565:
564:
563:
558:
553:
548:
530:
529:
524:
515:
510:
509:
506:
505:
502:
501:
499:Limited voting
496:
495:
494:
475:
474:
469:
464:
459:
458:
457:
452:
433:
432:
427:
422:
417:
403:
402:
397:
392:
387:
373:
372:
371:
370:
368:Localized list
365:
360:
355:
350:
340:
339:
338:
336:Biproportional
333:
328:
323:
307:
302:
301:
298:
297:
294:
293:
288:
283:
278:
264:
263:
248:
233:
209:
208:
207:
206:
201:
196:
191:
181:
167:
166:
165:
164:
153:
140:Instant-runoff
137:
136:
135:
127:Jungle primary
114:
103:Single vote -
101:
96:
95:
92:
91:
89:
88:
78:
73:
68:
63:
57:
54:
53:
43:
42:
32:
31:
15:
9:
6:
4:
3:
2:
2374:
2363:
2360:
2358:
2355:
2354:
2352:
2342:
2339:
2338:
2326:
2323:
2318:
2311:
2305:
2298:
2292:
2285:
2279:
2272:
2266:
2259:
2256:and E. Shen.
2255:
2250:
2243:
2237:
2230:
2224:
2217:
2214:G. Kreweras.
2211:
2204:
2198:
2191:
2188:
2183:
2176:
2170:
2168:
2166:
2158:
2152:
2144:
2140:
2135:
2130:
2126:
2122:
2118:
2114:
2110:
2103:
2095:
2091:
2087:
2083:
2078:
2073:
2069:
2065:
2061:
2057:
2053:
2046:
2039:
2033:
2031:
2023:
2017:
2010:
2004:
2002:
2000:
1992:
1986:
1984:
1982:
1980:
1978:
1976:
1971:
1963:
1949:
1929:
1909:
1901:
1885:
1881:
1877:
1874:
1868:
1862:
1842:
1838:
1834:
1831:
1825:
1819:
1799:
1795:
1791:
1788:
1782:
1776:
1756:
1748:
1744:
1740:
1739:zero-sum game
1735:
1716:
1710:
1705:
1702:
1697:
1690:
1685:
1680:
1677:
1670:
1667:
1662:
1657:
1651:
1642:
1635:
1628:
1612:
1606:
1600:
1582:
1568:
1548:
1528:
1508:
1488:
1468:
1460:
1441:
1438:
1435:
1432:
1429:
1421:
1407:
1404:
1401:
1398:
1395:
1387:
1373:
1370:
1367:
1364:
1361:
1353:
1352:
1351:
1343:
1341:
1340:zero-sum game
1337:
1333:
1323:
1319:
1305:
1285:
1265:
1262:
1259:
1239:
1219:
1199:
1196:
1193:
1173:
1153:
1138:
1135:
1133:
1123:
1121:
1113:
1112:essential set
1109:
1104:
1090:
1070:
1050:
1030:
1022:
1018:
1014:
1013:Allan Gibbard
1009:
1007:
1003:
999:
995:
991:
990:participation
987:
986:reinforcement
983:
979:
974:
972:
968:
964:
960:
956:
945:
940:
938:
933:
931:
926:
925:
923:
922:
919:
914:
906:
901:
896:
894:
889:
884:
883:
881:
880:
873:
870:
868:
865:
863:
862:May's theorem
860:
858:
855:
853:
850:
849:
848:
847:
840:
837:
835:
832:
830:
827:
825:
822:
820:
817:
816:
815:
814:
808:
803:
802:
795:
792:
790:
787:
785:
782:
780:
777:
776:
775:
774:
773:
772:majority rule
770:Paradoxes of
764:
761:
759:
756:
754:
751:
749:
746:
745:
744:
743:
742:
732:
729:
728:
727:
724:
722:
719:
717:
714:
713:
712:
711:
704:
701:
699:
696:
694:
691:
689:
686:
685:
684:
679:
674:
673:
666:
663:
659:
656:
655:
654:
651:
650:
649:
648:
639:
636:
634:
631:
629:
626:
625:
624:
621:
615:
612:
610:
607:
606:
605:
602:
598:
593:
589:
587:
582:
578:
577:
576:
573:
572:
571:
570:
566:
562:
559:
557:
554:
552:
549:
547:
544:
543:
542:
541:
536:
535:
534:
528:
525:
523:
520:
519:
518:
513:
512:Mixed systems
508:
507:
500:
497:
493:
490:
489:
488:
485:
484:
483:
482:
481:
473:
472:Random ballot
470:
468:
465:
463:
460:
456:
453:
451:
448:
447:
446:
443:
442:
441:
440:
439:
431:
428:
426:
423:
421:
418:
416:
413:
412:
411:
410:
409:
401:
398:
396:
393:
391:
388:
386:
383:
382:
381:
380:
379:
369:
366:
364:
361:
359:
356:
354:
351:
349:
346:
345:
344:
341:
337:
334:
332:
329:
327:
324:
322:
319:
318:
317:
316:Apportionment
314:
313:
312:
311:
305:
300:
299:
292:
289:
287:
284:
282:
279:
277:
274:
273:
272:
271:
270:
261:
257:
252:
251:Antiplurality
249:
246:
242:
237:
234:
231:
227:
222:
219:
218:
217:
216:
215:
205:
202:
200:
197:
195:
192:
190:
187:
186:
185:
182:
180:
179:Condorcet-IRV
177:
176:
175:
174:
173:
163:
158:
154:
152:
147:
143:
142:
141:
138:
134:
131:
130:
128:
123:
118:
115:
113:
110:
109:
108:
106:
99:
94:
93:
86:
82:
79:
77:
74:
72:
69:
67:
64:
62:
61:Social choice
59:
58:
56:
55:
49:
45:
44:
41:
37:
36:Social choice
34:
33:
29:
25:
21:
20:
2324:
2317:
2309:
2304:
2291:
2283:
2278:
2270:
2265:
2257:
2254:R. L. Rivest
2249:
2241:
2236:
2228:
2223:
2215:
2210:
2202:
2197:
2189:
2182:
2174:
2156:
2151:
2134:10419/286729
2116:
2112:
2102:
2077:10419/220534
2059:
2056:Econometrica
2055:
2045:
2016:
1990:
1736:
1583:
1456:
1349:
1329:
1320:
1144:
1136:
1129:
1115:
1111:
1105:
1021:nonmonotonic
1010:
975:
957:refers to a
954:
953:
845:
844:
811:
769:
768:
753:Exaggeration
739:
738:
709:
708:
682:
646:
645:
614:Mixed ballot
569:Compensatory
567:
540:compensatory
537:
532:
516:
478:
477:
436:
435:
406:
405:
376:
375:
363:List-free PR
308:
276:Score voting
267:
266:
212:
211:
199:Ranked pairs
170:
169:
102:
2119:(1): 5–38.
1481:and column
653:Single vote
556:Conditional
551:Coexistence
400:Quota Borda
390:Schulze STV
348:Closed list
291:STAR voting
236:Borda count
2351:Categories
1966:References
1388:2 voters:
1354:2 voters:
758:Truncation
487:Cumulative
310:Party-list
85:By country
76:Comparison
2341:voting.ml
2143:1432-217X
2086:0012-9682
1703:−
1678:−
1668:−
1439:≻
1433:≻
1422:1 voter:
1405:≻
1399:≻
1371:≻
1365:≻
1263:≻
1197:≻
982:Smith set
665:Dual-vote
358:Panachage
353:Open list
343:List type
221:Plurality
117:Two-round
105:plurality
28:Economics
385:Hare STV
24:Politics
22:A joint
2094:1911681
1346:Example
1126:History
1114:or the
1108:support
395:CPO-STV
245:Baldwin
194:Schulze
189:Minimax
107:methods
2141:
2092:
2084:
1769:where
1322:game.
992:, and
260:Coombs
30:series
2090:JSTOR
597:'MMP'
586:'AMS'
2139:ISSN
2082:ISSN
1922:and
1745:(or
1334:(or
1166:and
1106:The
1004:and
969:(or
538:Non-
492:SNTV
81:List
38:and
26:and
2129:hdl
2121:doi
2072:hdl
2064:doi
1561:to
1521:to
256:el.
241:el.
230:IRV
226:el.
2353::
2164:^
2137:.
2127:.
2117:58
2115:.
2111:.
2088:.
2080:.
2070:.
2060:45
2058:.
2054:.
2029:^
1998:^
1974:^
1855:,
1812:,
1749:)
1581:.
1103:.
1008:.
988:,
592:NZ
581:UK
157:US
146:UK
129:)
122:US
2145:.
2131::
2123::
2096:.
2074::
2066::
1950:a
1930:c
1910:a
1886:5
1882:/
1878:1
1875:=
1872:)
1869:c
1866:(
1863:p
1843:5
1839:/
1835:1
1832:=
1829:)
1826:b
1823:(
1820:p
1800:5
1796:/
1792:3
1789:=
1786:)
1783:a
1780:(
1777:p
1757:p
1717:)
1711:0
1706:3
1698:1
1691:3
1686:0
1681:1
1671:1
1663:1
1658:0
1652:(
1643:c
1636:b
1629:a
1613:c
1607:b
1601:a
1569:x
1549:y
1529:y
1509:x
1489:y
1469:x
1442:b
1436:a
1430:c
1408:a
1402:c
1396:b
1374:c
1368:b
1362:a
1306:q
1286:p
1266:q
1260:p
1240:q
1220:p
1200:q
1194:p
1174:q
1154:p
1091:y
1071:x
1051:y
1031:x
943:e
936:t
929:v
594::
583::
262:)
253:(
247:)
238:(
232:)
223:(
159::
148::
124::
119:(
87:)
83:(
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