1041:, and the total fraction given to each agent must be 1), with strict item rankings, where there can be more items than agents (so some items may remain unallocated). Cho and Dogan prove that, in this particular case, dl-efficiency and ul-efficiency are equivalent to sd-efficiency. In particular, they prove that if an allocation X is sd/ld/ul efficient, then:
300:
NecPE is a very strong requirement, which often cannot be satisfied. For example, suppose two agents have the same item ranking. One of them, say Alice, necessarily receives the lowest-ranked item. There are consistent additive bundle-rankings in which Alices values this item at 0 while George values
58:
In contrast, it is possible that Alice prefers {x} to {y,z} and George prefers {y,z} to {x} (for example: Alice's valuations are 12,4,2 and George's valuations are 6,3,4). Then the allocation is Pareto-efficient: in any other allocation, if Alice still gets x, then George's utility is lower; if Alice
349:
If Alice's ranking is x>y>z and George's ranking is x>z>y, then the allocation is Pareto-possible. As explained in the introduction, it is Pareto-efficient e.g. when Alice's valuations for x,y,z are 12,4,2 and George's valuations are 6,3,4. Note that both these valuations are consistent
1080:
Abdulkadiroğlu and Sönmez investigate the relation between sd-efficiency and ex-post Pareto-efficiency (in the context of random assignment). They introduce a new notion of domination for sets of assignments, and show that a lottery is sd-efficient iff each subset of the support of the lottery is
304:
If we require that all items have a strictly positive value, then giving all items to a single agent is trivially NecPE, but it very unfair. If fractional allocations are allowed, then there may be no NecPE allocation which gives both agents a positive value. For example, suppose Alice and George
54:
It is possible that Alice prefers {y,z} to {x} and George prefers {x} to {y,z} (for example: Alice's valuations for x,y,z are 8,7,6 and George's valuations are 7,1,2, so the utility profile is 8,3). Then the allocation is not Pareto-efficient, since both Alice and George would be better-off by
38:
over items, but not over bundles. That is, agents rank the items from best to worst, but they do not rank the subsets of items. In particular, they do not specify a numeric value for each item. This may cause an ambiguity regarding whether certain allocations are Pareto-efficient or not. As an
305:
both have the ranking x>y. If both get a positive value, then either Alice gets some x and George gets some y, or vice-versa. In the former case, it is possible that Alice's valuations are e.g. 4,2 and George's valuations are 8,1, so Alice can exchange a small amount
180:, then any consistent bundle ranking must have {w} < {x} < {y} < {z]. Often, one makes additional assumptions on the set of allowed bundle rankings, which imposes additional restrictions on consistency. Example assumptions are:
261:
Pareto-ensuring. As explained in the introduction, it is not Pareto-efficient e.g. when Alice's valuations for x,y,z are 8,7,6 and George's valuations are 7,1,2. Note that both these valuations are consistent with the agents'
207:: the agent assigns a value to each item, and values each bundle at the sum of its contents. This assumption is stronger than responsivity. For example, if Alice ranks {x,y}<{z} then she must rank {w,x,y}<{w,z}.
1076:
Dogan, Dogan and Yildiz study a different domination relation between allocations: an allocation X dominates an allocation Y if it is Pareto-efficient for a larger set of bundle-rankings consistent with the item
284:"X is NecPE" is equivalent to "For every other allocation Y, for every consistent bundle ranking, Y does not Pareto-dominate X". Exchanging the order of "for all" quantifiers does not change the logical meaning.
95:
Since the Pareto-efficiency of an allocation depends on the rankings of bundles, it is a-priori not clear how to determine the efficiency of an allocation when only rankings of items are given.
199:: replacing an item with a better item always improves the bundle. Thus, Alice's bundle ranking must have e.g. {w,x} < {w,y} < {x,y} < {x,z}. This is stronger than consistency.
215::the agent always ranks a bundle that contains some item x above any bundle that contains only items ranked lower than x. In the above example, Alice must rank {w,x,y} < {z}.
684:(b) there exists additive bundle-rankings consistent with the agents' item-rankings for which X is fractionally-Pareto-efficient (that is, X is Pareto-possible);
940:
This means that, in particular, if X is sd-efficient in the set of all allocations that give exactly 1 unit to each agent, then X is sd-efficient in general.
384:"X is Pareto-possible" is equivalent to "There exist a consistent bundle ranking for which, for every other allocation Y, Y does not dominate X". It must be
1073:
Aziz, Gaspers, Mackenzie and Walsh study computational issues related to ordinal fairness notions. In
Section 7 they briefly study sd-Pareto-efficiency.
270:
an allocation Y if there exists some bundle rankings consistent with the agents' item rankings, for which X Pareto-dominates Y. An allocation is called
937:. In other words, a complete allocation X can be necessarily-dominated only by an allocation Y which assigns to every agent the same amount as X does.
391:"X is PosPE" is equivalent to "For every other allocation Y, there exists a consistent bundle ranking, for which Y does not dominate X". There can be
254:
If Alice's ranking is x>y>z and George's ranking is x>z>y and the allocations must be discrete, then the allocation is Pareto-ensuring.
345:
bundle rankings that are consistent with the agents' item rankings. Obviously, every Pareto-ensuring allocation is Pareto-possible. In addition:
673:
As noted above, Pareto-possible implies PosPE, but the other direction is not logically true. McLennan proves that they are equivalent in the
1176:
657:
if there no allocation that stochastically dominates it. This is similar to PosPE, but emphasizes that the bundle rankings must be based on
59:
does not get x, then Alice's utility is lower. Moreover, the allocation is Pareto-efficient even if the items are divisible (that is, it is
1064:
The equivalence does not hold if there are distributional constraints: there are allocations which are sd-efficient but not dl-efficient.
281:"X is Pareto-ensuring" is equivalent to "For every consistent bundle ranking, for every other allocation Y, Y does not Pareto-dominate X".
687:(c) there exists additive bundle-rankings consistent with the agents' item-rankings for which X maximizes the sum of agents' utilities.
773:
has at least as many items that are at least as good as z), then for every responsive bundle-ranking consistent with the item-ranking,
176:
with an item ranking if it ranks the singleton bundles in the same order as the items they contain. For example, if Alice's ranking is
50:
Consider the allocation . Whether or not this allocation is Pareto-efficient depends on the agents' numeric valuations. For example:
1674:
691:
The implications (c) → (b) → (a) are easy; the challenging part is to prove that (a) → (c). McLennan proves it using the polyhedral
1021:
In general, sd-domination implies dl-domination and ul-domination. Therefore, dl-efficiency and ul-efficiency imply sd-efficiency.
248:
If agents' valuations are assumed to be positive, then every allocation giving all items to a single agent is Pareto-ensuring.
1298:
Katta, Akshay-Kumar; Sethuraman, Jay (2006). "A solution to the random assignment problem on the full preference domain".
692:
60:
292:
additive bundle rankings, or we allow only rankings that are based on additive valuations with diminishing differences.
157:
918:
This means that, if X wsd Y and both X and Y are complete allocations (all objects are allocated), then necessarily
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1300:
210:
513:
187:
adding an item to a bundle always improves the bundle. This corresponds to the assumption that all items are
953:
1030:
949:
702:
674:
422:
194:
369:
bundle rankings consistent with the agents' item rankings, X Pareto-dominates Y. An allocation is called
677:
problem (with strict or weak item rankings). Particularly, he proves that the following are equivalent:
155:, which means that an allocation is not dominated by a discrete allocation, and the stronger concept of
1616:
1567:
1451:
1412:
1373:
1331:
1260:
803:
then there exists at least one responsive bundle-ranking consistent with the item-ranking, for which
438:
251:
If Alice's ranking is x>y and George's ranking is y>x, then the allocation is Pareto-ensuring.
1169:
Proceedings of the 2010 Conference on ECAI 2010: 19th
European Conference on Artificial Intelligence
721:
if its exchange graph has no directed cycles. Then, an allocation sd-efficient iff it is acyclic.
1548:
1520:
1489:
1463:
1452:"The vigilant eating rule: A general approach for probabilistic economic design with constraints"
1236:
1208:
1142:
75:
of z in order to keep her utility at the same level. But then George's utility would change by 6
1165:"Fair Division under Ordinal Preferences: Computing Envy-Free Allocations of Indivisible Goods"
406:
additive bundle rankings, or we allow only rankings that are based on additive valuations with
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1597:
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1351:
1280:
1228:
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225:
31:
399:
If X is Pareto-possible then it is PosPE, but the other implication is not (logically) true.
1636:
1628:
1587:
1579:
1568:"A new ex-ante efficiency criterion and implications for the probabilistic serial mechanism"
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1424:
1385:
1343:
1309:
1272:
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203:
39:
example, consider an economy with three items and two agents, with the following rankings:
35:
1196:
1130:
266:
Bouveret, Endriss and Lang. use an equivalent definition. They say that an allocation X
710:
1632:
361:
Bouveret, Endriss and Lang. use a different definition. They say that an allocation X
1663:
1493:
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1164:
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353:
If Alice's ranking is x>y and George's ranking is y>x, then the allocation is
151:
if no other allocation Pareto-dominates it. Sometimes, a distinction is made between
713:
in which the nodes are the items, and there is an arc x→y iff there exists an agent
1552:
236:
bundle rankings that are consistent with the agents' item rankings (they allow all
161:, which means that an allocation is not dominated even by a fractional allocation.
1347:
1535:
1508:
632:). In the stochastic domination relation between allocations is also written as
820:
Therefore, the following holds for dominance relations of discrete allocations:
329:, so both gains are positive. In the latter case, an analogous argument holds.
1583:
1477:
1313:
901:, then for the valuation which assigns almost the same value for all items, v(
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that prefers x and receives a positive fraction of y. Define an allocation as
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1436:
1397:
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17:
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if-and-only-if, for every bundle ranking consistent with the item ranking, X
1428:
1389:
1276:
1507:
Aziz, Haris; Gaspers, Serge; Mackenzie, Simon; Walsh, Toby (2015-10-01).
1223:
357:
Pareto-possible, since it is always Pareto-dominated by the allocation .
1592:
568:. Equivalently, for at least one item z, the "at least as large as in Y
1641:
168:(sets of items). In our setting, agents report only their rankings of
1374:"Ordinal Efficiency and the Polyhedral Separating Hyperplane Theorem"
1113:
Brams, Steven J.; Edelman, Paul H.; Fishburn, Peter C. (2003-09-01).
872:, that is, the total quantity of objects (discrete or fractional) in
701:
prove another useful characterization of sd-efficiency, for the same
1468:
1332:"Equivalence of efficiency notions for ordinal assignment problems"
1213:
1525:
1509:"Fair assignment of indivisible objects under ordinal preferences"
1195:
Segal-Halevi, Erel; Hassidim, Avinatan; Aziz, Haris (2020-03-10).
188:
402:
The Pareto-possible condition remains the same whether we allow
1163:
Bouveret, Sylvain; Endriss, Ulle; Lang, Jérôme (2010-08-04).
948:
Cho presents two other efficiency notions for the setting of
494:
means that for every item z, the number of items better than
468:
if for every item z, the total fraction of items better than
191:. Thus, Alice's bundle ranking must have e.g. {y} < {y,x}.
67:
of x to George, then George would have to give her at least 3
55:
exchanging their bundles (the utility profile would be 13,7).
301:
it at 1. Hence, giving it to Alice is not Pareto-efficient.
727:
proved the following equivalence on dominance relations of
512:). The sd relation has several equivalent definitions; see
1566:
Doğan, Battal; Doğan, Serhat; Yıldız, Kemal (2018-05-01).
373:
if no other allocation necessarily-Pareto-dominates it.
433:, and the sum of fractions given to each agent must be
164:
The above definitions depend on the agents' ranking of
1617:"Ordinal efficiency and dominated sets of assignments"
1018:
if there is no other allocation that ul-dominates it.
1007:
if there is no other allocation that dl-dominates it.
288:
The NecPE condition remains the same whether we allow
274:
if no other allocation possibly-Pareto-dominates it.
1615:
Abdulkadiroğlu, Atila; Sönmez, Tayfun (2003-09-01).
1048:
X is non-wasteful ("wasteful" means that some agent
624:, and Y≠X (equivalently: for at least one agent i, X
639:This is equivalent to necessary Pareto-domination.
705:setting but with strict item rankings. Define the
1261:"A New Solution to the Random Assignment Problem"
421:present an efficiency notion for the setting of
30:refers to several adaptations of the concept of
1259:Bogomolnaia, Anna; Moulin, Hervé (2001-10-01).
337:Brams, Edelman and Fishburn call an allocation
1197:"Fair Allocation with Diminishing Differences"
1052:, who receives a positive fraction of an item
661:utility functions, and the allocations may be
277:The two definitions are logically equivalent:
34:to settings in which the agents only express
8:
681:(a) X is sd-efficient (that is, X is PosPE);
395:bundle ranking for every other allocation Y.
257:With the above rankings, the allocation is
1450:Aziz, Haris; Brandl, Florian (2022-09-01).
1330:Cho, Wonki Jo; Doğan, Battal (2016-09-01).
1201:Journal of Artificial Intelligence Research
388:bundle ranking for all other allocations Y.
1640:
1591:
1534:
1524:
1467:
1222:
1212:
1045:The exchange graph of X is acyclic, and -
969:downward-lexicographically (dl) dominates
944:Lexicographic-dominance Pareto-efficiency
486:(if the allocations are discrete, then X
1413:"Finite Linear Qualitative Probability"
1090:
709:of a given fractional allocation as a
576:". In the ssd relation is written as
414:Stochastic-dominance Pareto-efficiency
7:
1367:
1365:
1325:
1323:
1254:
1252:
1250:
1190:
1188:
1158:
1156:
1115:"Fair Division of Indivisible Items"
1108:
1106:
1104:
1102:
1100:
1098:
1096:
1094:
1012:upward-lexicographic (ul) domination
572:" becomes "strictly larger than in Y
272:Necessarily-Pareto-efficient (NecPE)
1417:Journal of Mathematical Psychology
1131:10.1023/B:THEO.0000024421.85722.0a
1010:Similarly, based on the notion of
25:
1411:Fishburn, Peter C. (1996-03-01).
1060:which is not entirely allocated).
1033:setting (the bundle rankings are
540:strictly-stochastically dominates
371:Possibly-Pareto-efficient (PosPE)
879:must be at least as large as in
437:). It is based on the notion of
1372:McLennan, Andrew (2002-08-01).
456:weakly-stochastically dominates
425:(where the bundle rankings are
244:bundle rankings). For example:
341:if it is Pareto-efficient for
232:if it is Pareto-efficient for
63:): if Alice yields any amount
1:
1633:10.1016/S0022-0531(03)00091-7
1348:10.1016/j.econlet.2016.07.007
991:, and for at least one agent
831:necessarily Pareto-dominates
693:separating hyperplane theorem
172:. A bundle ranking is called
61:fractionally Pareto efficient
1536:10.1016/j.artint.2015.06.002
363:necessarily Pareto-dominates
158:Fractional Pareto efficiency
1456:Games and Economic Behavior
1171:. NLD: IOS Press: 387–392.
505:is at least as large as in
479:is at least as large as in
220:Necessary Pareto-efficiency
127:weakly prefers the bundle X
1691:
1621:Journal of Economic Theory
1572:Journal of Economic Theory
1378:Journal of Economic Theory
1301:Journal of Economic Theory
1265:Journal of Economic Theory
1014:, An allocation is called
1003:. An allocation is called
350:with the agents' rankings.
333:Possible Pareto-efficiency
153:discrete-Pareto-efficiency
1584:10.1016/j.jet.2018.01.011
1478:10.1016/j.geb.2022.06.002
1314:10.1016/j.jet.2005.05.001
1037:, the allocations may be
971:another allocation Y = (Y
604:another allocation Y = (Y
309:of x for a small amount 3
268:possibly Pareto-dominates
135:, and at least one agent
115:another allocation Y = (Y
28:Ordinal Pareto efficiency
1675:Random variable ordering
766:, and for every item z,
642:An allocation is called
602:stochastically dominates
528:is at least as good as Y
514:responsive set extension
376:The two definitions are
46:George: x > z > y.
1513:Artificial Intelligence
1056:, prefers another item
999:strictly-dl-dominates Y
954:lexicographic dominance
408:diminishing differences
365:an allocation Y if for
43:Alice: x > y > z.
1429:10.1006/jmps.1996.0004
1390:10.1006/jeth.2001.2864
1277:10.1006/jeth.2000.2710
1031:fair random assignment
979:), if for every agent
950:fair random assignment
886:. This is because, if
703:fair random assignment
698:Bogomolnaia and Moulin
675:fair random assignment
612:), if for every agent
429:, the allocations are
423:fair random assignment
418:Bogomolnaia and Moulin
380:logically equivalent:
178:w < x < y < z
987:weakly-dl-dominates Y
147:. An allocation X is
1224:10.1613/jair.1.11994
959:An allocation X = (X
592:An allocation X = (X
439:stochastic dominance
103:An allocation X = (X
91:, which is negative.
1207:: 471–507–471–507.
1119:Theory and Decision
648:ordinally efficient
313:of y. Alice gains 6
228:call an allocation
224:Brams, Edelman and
516:. In particular, X
321:and George gains 8
139:strictly prefers X
123:), if every agent
1670:Pareto efficiency
1336:Economics Letters
1178:978-1-60750-605-8
735:bundle rankings:
36:ordinal utilities
32:Pareto-efficiency
16:(Redirected from
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1072:
1071:
1067:
1065:
1059:
1055:
1051:
1047:
1044:
1043:
1042:
1040:
1036:
1032:
1029:Consider the
1024:
1022:
1019:
1017:
1013:
1008:
1006:
994:
982:
970:
957:
955:
951:
943:
941:
938:
936:
932:
924:
916:
914:
907:
900:
892:
885:
878:
871:
863:
856:
849:
838:
836:
834:
830:
826:
823:
816:
809:
802:
795:
788:
786:
779:
772:
765:
758:
751:
745:
738:
737:
736:
734:
730:
726:
722:
720:
716:
712:
708:
704:
700:
696:
694:
686:
683:
680:
679:
678:
676:
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664:
660:
656:
653:
649:
645:
640:
638:
635:
615:
603:
590:
589:
582:
551:
544:
541:
538:
515:
511:
504:
497:
485:
478:
471:
467:
460:
457:
454:
447:
442:
440:
436:
432:
428:
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420:
413:
411:
409:
405:
400:
394:
390:
387:
383:
382:
381:
379:
374:
372:
368:
364:
356:
352:
348:
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344:
340:
332:
330:
328:
324:
320:
316:
312:
308:
302:
295:
293:
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283:
280:
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278:
275:
273:
269:
260:
256:
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250:
247:
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219:
214:
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212:Lexicographic
209:
206:
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190:
186:
185:Monotonicity:
183:
182:
181:
175:
171:
167:
162:
160:
159:
154:
150:
138:
126:
114:
98:
96:
90:
86:
82:
78:
74:
70:
66:
62:
57:
53:
52:
51:
45:
42:
41:
40:
37:
33:
29:
19:
18:SD-efficiency
1624:
1620:
1610:
1575:
1571:
1561:
1516:
1512:
1502:
1459:
1455:
1445:
1423:(1): 64–77.
1420:
1416:
1406:
1381:
1377:
1339:
1335:
1305:
1299:
1293:
1268:
1264:
1204:
1200:
1168:
1122:
1118:
1081:undominated.
1063:
1057:
1053:
1049:
1038:
1034:
1028:
1025:Equivalences
1020:
1016:ul-efficient
1015:
1011:
1009:
1005:dl-efficient
1004:
992:
980:
968:
958:
947:
939:
934:
926:
919:
917:
909:
902:
894:
887:
880:
873:
865:
858:
851:
844:
842:
832:
828:
824:
821:
819:
810:
804:
796:
790:
780:
774:
767:
760:
753:
746:
740:
732:
728:
723:
718:
714:
706:
699:
697:
690:
672:
669:Equivalences
662:
658:
655:
651:
647:
644:sd-efficient
643:
641:
637:>> Y".
636:
633:
613:
601:
591:
583:
577:
546:
542:
539:
533:
532:. A bundle
506:
499:
495:
480:
473:
469:
462:
458:
455:
449:
445:
443:
434:
430:
426:
419:
417:
407:
403:
401:
398:
392:
385:
377:
375:
370:
366:
362:
360:
354:
342:
338:
336:
326:
322:
318:
314:
310:
306:
303:
299:
289:
287:
276:
271:
267:
265:
258:
241:
237:
233:
229:
223:
211:
202:
196:Responsivity
195:
184:
173:
169:
165:
163:
156:
152:
148:
136:
124:
112:
102:
94:
88:
84:
80:
76:
72:
68:
64:
49:
27:
26:
1593:11693/48988
1578:: 178–200.
1462:: 168–187.
952:, based on
652:O-efficient
448:, A bundle
393:a different
99:Definitions
1664:Categories
1642:10161/1940
1469:2008.08991
1308:(1): 231.
1214:1705.07993
1086:References
1039:fractional
839:Properties
825:>> Y
797:>> Y
752:(that is:
747:>> Y
733:responsive
663:fractional
584:>> Y
431:fractional
242:responsive
204:Additivity
174:consistent
1651:0022-0531
1602:0022-0531
1545:0004-3702
1526:1312.6546
1519:: 71–92.
1494:221186811
1486:0899-8256
1437:0022-2496
1398:0022-0531
1356:0165-1765
1285:0022-0531
1241:108290839
1233:1076-9757
1147:153943630
1139:1573-7187
1077:rankings.
908:) < v(
545:a bundle
461:a bundle
435:at most 1
296:Existence
262:rankings.
238:monotonic
71:of y or 6
1342:: 8–12.
1035:additive
857:, then
729:discrete
725:Fishburn
659:additive
427:additive
386:the same
226:Fishburn
1553:1408197
893:| <
719:acyclic
166:bundles
1649:
1600:
1551:
1543:
1492:
1484:
1435:
1396:
1354:
1283:
1239:
1231:
1175:
1145:
1137:
975:,...,Y
963:,...,X
608:,...,Y
596:,...,X
119:,...,Y
107:,...,X
1549:S2CID
1521:arXiv
1490:S2CID
1464:arXiv
1237:S2CID
1209:arXiv
1143:S2CID
811:<Y
791:not X
781:>Y
628:ssd Y
620:wsd Y
560:and X
556:wsd Y
543:(ssd)
459:(wsd)
170:items
1647:ISSN
1598:ISSN
1541:ISSN
1482:ISSN
1433:ISSN
1394:ISSN
1352:ISSN
1281:ISSN
1229:ISSN
1173:ISBN
1135:ISSN
925:| =
864:| ≥
850:wsd
827:iff
552:if X
520:sd Y
490:sd Y
343:some
240:and
189:good
143:to Y
83:or 6
1637:hdl
1629:doi
1625:112
1588:hdl
1580:doi
1576:175
1531:doi
1517:227
1474:doi
1460:135
1425:doi
1386:doi
1382:105
1344:doi
1340:146
1310:doi
1306:131
1273:doi
1269:100
1219:doi
1127:doi
995:, X
915:).
843:If
789:If
739:If
650:or
616:: X
564:≠ Y
498:in
472:in
404:all
378:not
367:all
355:not
290:all
259:not
234:all
87:-24
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1570:.
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1529:.
1515:.
1511:.
1488:.
1480:.
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1431:.
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1419:.
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1392:.
1380:.
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783:i
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