3266:. Responsiveness constrains only the relation between bundles of the same size with one item replaced, or bundles of different sizes where the small is contained in the large. It says nothing about bundles of different sizes that are not subsets of each other. So, for example, a responsive order can have both
894:) is defined not only on discrete bundles but also on fractional bundles (bundles that contains fractions of items). Informally, a bundle Y is SD-preferred to a bundle X if, for each item z, the bundle Y contains at least as many objects, that are at least as good as z, as the bundle X.
2923:
if it is contains the responsive-set-extension of some total order on items. I.e., it contains all the relations that are implied by the underlying ordering of the items, and adds some more relations that are not implied nor contradicted.
3223:
On the other hand, a total order may responsive but not additive: it may contain the AU extension which is consistent with all additive functions, but may also contain other relations that are inconsistent with a single additive
3049:
1262:
1846:
1711:
1571:
1444:
1030:
697:
3218:
602:
3500:
2008:
3426:
3364:
3264:
1315:
343:
136:
248:
196:
3308:
2658:
2472:
2432:
2396:
2358:
2322:
2284:
2248:
2117:
1141:
931:
756:
302:
3120:
3090:
2909:
2879:
2849:
2819:
2785:
2755:
2722:
2565:
2204:
880:
499:
631:
551:
3610:
2605:
2504:
2157:
438:
395:
2076:
2044:
1913:
1881:
91:
2530:
845:
525:
465:
1576:
Assuming the items are independent, the utility function on bundles is additive, so the utility of a bundle is the sum of the utilities of its items, for example:
2678:
2585:
2137:
1937:
1161:
1102:
1082:
1062:
951:
819:
799:
779:
415:
375:
2954:
1169:
1717:
1582:
3543:
Aziz, Haris; Gaspers, Serge; MacKenzie, Simon; Walsh, Toby (2015). "Fair assignment of indivisible objects under ordinal preferences".
3741:
3664:
1450:
1323:
959:
636:
349:
on the bundles of items, that includes all relations that can be deduced from the item-ranking and the independence assumption.
3792:
3125:
3624:
Katta, Akshay-Kumar; Sethuraman, Jay (2006). "A solution to the random assignment problem on the full preference domain".
3589:
3626:
559:
3431:
1945:
3369:
3313:
3231:
1282:
310:
103:
203:
151:
44:
3269:
2610:
2441:
2401:
2365:
2327:
2291:
2253:
2217:
2086:
1110:
900:
891:
725:
257:
3739:; Talgam-Cohen, Inbal (2021). "Competitive equilibrium with indivisible goods and generic budgets".
3095:
3065:
2884:
2854:
2824:
2794:
2760:
2730:
2683:
2535:
2165:
850:
478:
3750:
3717:
3604:
3570:
3552:
759:
704:
610:
530:
3709:
3660:
3059:
2590:
2477:
2142:
1279:
Many different utility functions are compatible with a given ordering. For example, the order
423:
380:
142:
2049:
2017:
1886:
1854:
58:
3760:
3701:
3635:
3562:
3516:
2509:
1273:
824:
504:
250:– the person prefers his best and third-best items to his second-best and fourth-best items.
3772:
3686:
3653:
Brandt, Felix; Conitzer, Vincent; Endriss, Ulle; Lang, Jérôme; Procaccia, Ariel D. (2016).
443:
3768:
3511:
2663:
2570:
2122:
1922:
1146:
1087:
1067:
1038:
936:
804:
784:
764:
400:
360:
28:
3786:
346:
17:
3721:
3366:. But this is incompatible with additivity: there is no additive function for which
3574:
3654:
3566:
3672:
94:
3736:
3705:
3639:
3713:
3685:
Kyropoulou, Maria; Suksompong, Warut; Voudouris, Alexandros A. (2020-11-12).
3764:
141:(i.e., z is his best item, then y, then x, then w). Assuming the items are
3122:, so it is responsive. Similarly, if a utility function is additive, then
2931:
if it induces a responsive order. To be more explicit, a utility function
3044:{\displaystyle u(y)\geq u(z)\implies u(X\cup \{y\})\geq u(X\cup \{z\})}
47:
on individual items, to a partial preference-relation of item-bundles.
475:
The original RS extension is constructed as follows. For every bundle
93:. A person states that he ranks the items according to the following
3755:
1257:{\displaystyle |\{x\in X|x\succeq z\}|\leq |\{y\in Y|y\succeq z\}|}
3557:
1841:{\displaystyle u_{2}(\{w,x\})=2,u_{2}(\{w,z\})=5,u_{2}(\{x,y\})=6}
1706:{\displaystyle u_{1}(\{w,x\})=2,u_{1}(\{w,z\})=7,u_{1}(\{x,y\})=6}
1107:
If the bundles are discrete, the definition has a simpler form.
719:
of the items in one bundle with the items in the other bundle.
198:– the person prefers his two best items to his two worst items;
3054:
Responsiveness is implied by additivity, but not vice versa:
699:(- replacing an item with a better item improves the bundle).
1566:{\displaystyle u_{2}(w)=0,u_{2}(x)=2,u_{2}(y)=4,u_{2}(z)=5}
1439:{\displaystyle u_{1}(w)=0,u_{1}(x)=2,u_{1}(y)=4,u_{1}(z)=7}
1025:{\displaystyle \sum _{x\succeq z}X\leq \sum _{y\succeq z}Y}
692:{\displaystyle X\preceq ^{RS}(X\setminus \{x\})\cup \{y\}}
3213:{\displaystyle u(X\cup \{y\})-u(X\cup \{z\})=u(y)-u(z)}
3434:
3372:
3316:
3272:
3234:
3128:
3098:
3068:
2957:
2887:
2857:
2827:
2797:
2763:
2733:
2686:
2666:
2613:
2593:
2573:
2538:
2512:
2480:
2444:
2404:
2368:
2330:
2294:
2256:
2220:
2168:
2145:
2125:
2089:
2052:
2020:
1948:
1925:
1889:
1857:
1720:
1585:
1453:
1326:
1285:
1172:
1149:
1113:
1090:
1070:
1041:
962:
939:
903:
853:
827:
807:
787:
767:
728:
639:
613:
562:
533:
507:
481:
446:
426:
403:
383:
363:
313:
260:
206:
154:
106:
61:
1317:
is compatible with the following utility functions:
3687:"Almost envy-freeness in group resource allocation"
3588:Barberà, S., Bossert, W., Pattanaik, P. K. (2004).
2927:Similarly, a utility function on bundles is called
1915:according to both utility functions. Moreover, for
304:; we do not know which of them the person prefers.
3494:
3420:
3358:
3302:
3258:
3212:
3114:
3084:
3062:) then by definition it contains the AU extension
3043:
2903:
2873:
2843:
2813:
2779:
2749:
2716:
2672:
2652:
2599:
2579:
2559:
2524:
2498:
2466:
2426:
2390:
2352:
2316:
2278:
2242:
2198:
2151:
2131:
2111:
2070:
2038:
2002:
1931:
1907:
1875:
1840:
1705:
1565:
1438:
1309:
1256:
1155:
1135:
1096:
1076:
1056:
1024:
945:
925:
874:
839:
813:
793:
773:
750:
691:
625:
596:
545:
519:
493:
459:
432:
409:
389:
369:
337:
296:
254:But, one cannot deduce anything about the bundles
242:
190:
130:
85:
3058:If a total order is additive (represented by an
467:. It can be defined in several equivalent ways.
3228:For example, suppose there are four items with
2046:can be either less or more than the utility of
1272:The AU extension is based on the notion of an
8:
3609:: CS1 maint: multiple names: authors list (
3486:
3468:
3453:
3441:
3412:
3400:
3385:
3379:
3353:
3335:
3329:
3317:
3297:
3285:
3279:
3273:
3174:
3168:
3147:
3141:
3035:
3029:
3008:
3002:
2065:
2053:
2033:
2021:
1994:
1982:
1967:
1955:
1902:
1890:
1870:
1858:
1826:
1814:
1786:
1774:
1746:
1734:
1691:
1679:
1651:
1639:
1611:
1599:
1246:
1220:
1204:
1178:
686:
680:
671:
665:
597:{\displaystyle X\setminus \{x\}\prec ^{RS}X}
575:
569:
291:
279:
273:
261:
237:
225:
219:
207:
185:
173:
167:
155:
3495:{\displaystyle u(\{w,z\})>u(\{w,x,y\})}
2989:
2985:
3754:
3556:
3433:
3371:
3315:
3271:
3233:
3127:
3103:
3097:
3073:
3067:
2956:
2892:
2886:
2862:
2856:
2832:
2826:
2802:
2796:
2768:
2762:
2738:
2732:
2685:
2665:
2612:
2592:
2572:
2537:
2511:
2479:
2452:
2443:
2412:
2403:
2376:
2367:
2338:
2329:
2302:
2293:
2264:
2255:
2228:
2219:
2167:
2144:
2124:
2119:iff, for every additive utility function
2097:
2088:
2081:This motivates the following definition:
2051:
2019:
1947:
1924:
1888:
1856:
1805:
1765:
1725:
1719:
1670:
1630:
1590:
1584:
1542:
1514:
1486:
1458:
1452:
1415:
1387:
1359:
1331:
1325:
1284:
1249:
1232:
1215:
1207:
1190:
1173:
1171:
1148:
1121:
1112:
1089:
1069:
1040:
998:
967:
961:
938:
911:
902:
852:
826:
806:
786:
766:
736:
727:
647:
638:
612:
582:
561:
532:
506:
480:
451:
445:
425:
402:
382:
362:
312:
259:
205:
153:
105:
60:
3538:
3536:
3534:
3532:
2567:. Therefore, for every utility function
2003:{\displaystyle u(\{w,x\})<u(\{w,z\})}
3656:Handbook of Computational Social Choice
3528:
2014:In contrast, the utility of the bundle
662:
566:
3602:
3421:{\displaystyle u(\{z\})<u(\{x,y\})}
3359:{\displaystyle \{w,z\}\succ \{w,x,y\}}
3259:{\displaystyle w\prec x\prec y\prec z}
1310:{\displaystyle w\prec x\prec y\prec z}
604:(- adding an item improves the bundle)
338:{\displaystyle w\prec x\prec y\prec z}
131:{\displaystyle w\prec x\prec y\prec z}
7:
243:{\displaystyle \{w,y\}\prec \{x,z\}}
191:{\displaystyle \{w,x\}\prec \{y,z\}}
2919:A total order on bundles is called
1939:compatible with the above ranking:
3742:Mathematics of Operations Research
3303:{\displaystyle \{z\}\prec \{x,y\}}
2935:is responsive if for every bundle
25:
3220:, so responsiveness is satisfied.
2915:Responsive orders and valuations
2653:{\displaystyle u(x)\leq u(f(x))}
553:, take the following relations:
307:The RS extension of the ranking
2791:Therefore, the four extensions
2467:{\displaystyle X\preceq ^{PD}Y}
2427:{\displaystyle X\preceq ^{AU}Y}
2391:{\displaystyle X\preceq ^{PD}Y}
2353:{\displaystyle X\preceq ^{PD}Y}
2317:{\displaystyle X\preceq ^{RS}Y}
2279:{\displaystyle X\preceq ^{RS}Y}
2243:{\displaystyle X\preceq ^{SD}Y}
2112:{\displaystyle X\preceq ^{AU}Y}
1136:{\displaystyle X\preceq ^{SD}Y}
926:{\displaystyle X\preceq ^{SD}Y}
758:if-and-only-if there exists an
751:{\displaystyle X\preceq ^{PD}Y}
715:The PD extension is based on a
297:{\displaystyle \{w,z\},\{x,y\}}
3659:. Cambridge University Press.
3489:
3465:
3456:
3438:
3415:
3397:
3388:
3376:
3207:
3201:
3192:
3186:
3177:
3159:
3150:
3132:
3038:
3020:
3011:
2993:
2986:
2982:
2976:
2967:
2961:
2711:
2705:
2696:
2690:
2647:
2644:
2638:
2632:
2623:
2617:
2554:
2548:
2490:
2193:
2187:
2178:
2172:
1997:
1979:
1970:
1952:
1829:
1811:
1789:
1771:
1749:
1731:
1694:
1676:
1654:
1636:
1614:
1596:
1554:
1548:
1526:
1520:
1498:
1492:
1470:
1464:
1427:
1421:
1399:
1393:
1371:
1365:
1343:
1337:
1250:
1233:
1216:
1208:
1191:
1174:
1051:
1045:
1019:
1013:
988:
982:
890:The SD extension (named after
869:
863:
674:
656:
55:Suppose there are four items:
1:
3115:{\displaystyle \preceq ^{RS}}
3085:{\displaystyle \preceq ^{AU}}
2904:{\displaystyle \preceq ^{AU}}
2874:{\displaystyle \preceq ^{SD}}
2844:{\displaystyle \preceq ^{PD}}
2814:{\displaystyle \preceq ^{RS}}
2780:{\displaystyle \preceq ^{SD}}
2750:{\displaystyle \preceq ^{AU}}
2717:{\displaystyle u(X)\leq u(Y)}
2560:{\displaystyle x\preceq f(x)}
2474:, then there is an injection
2199:{\displaystyle u(X)\leq u(Y)}
875:{\displaystyle x\preceq f(x)}
3694:Theoretical Computer Science
3567:10.1016/j.artint.2015.06.002
494:{\displaystyle X\subseteq O}
3809:
3627:Journal of Economic Theory
3597:Handbook of utility theory
3590:"Ranking sets of objects."
626:{\displaystyle x\preceq y}
3706:10.1016/j.tcs.2020.07.008
3640:10.1016/j.jet.2005.05.001
3092:, which is equivalent to
886:Stochastic dominance (SD)
546:{\displaystyle y\notin X}
2787:are equivalent, see e.g.
2600:{\displaystyle \preceq }
2499:{\displaystyle f:X\to Y}
2152:{\displaystyle \preceq }
1064:is the fraction of item
703:The RS extension is the
433:{\displaystyle \preceq }
390:{\displaystyle \preceq }
377:be a set of objects and
3545:Artificial Intelligence
2071:{\displaystyle \{x,y\}}
2039:{\displaystyle \{w,z\}}
1908:{\displaystyle \{w.z\}}
1876:{\displaystyle \{w,x\}}
711:Pairwise dominance (PD)
145:, one can deduce that:
86:{\displaystyle w,x,y,z}
3793:Utility function types
3765:10.1287/moor.2020.1062
3496:
3422:
3360:
3304:
3260:
3214:
3116:
3086:
3045:
2905:
2875:
2845:
2815:
2781:
2751:
2718:
2674:
2654:
2601:
2581:
2561:
2526:
2525:{\displaystyle x\in X}
2500:
2468:
2428:
2392:
2354:
2318:
2280:
2244:
2200:
2153:
2133:
2113:
2072:
2040:
2004:
1933:
1909:
1883:has less utility than
1877:
1842:
1707:
1567:
1440:
1311:
1258:
1157:
1137:
1098:
1078:
1058:
1026:
947:
927:
876:
841:
840:{\displaystyle x\in X}
815:
795:
775:
752:
693:
627:
598:
547:
521:
520:{\displaystyle x\in X}
495:
461:
440:is a partial order on
434:
411:
391:
371:
339:
298:
244:
192:
132:
87:
3497:
3423:
3361:
3305:
3261:
3215:
3117:
3087:
3046:
2906:
2876:
2846:
2816:
2782:
2752:
2719:
2675:
2655:
2602:
2582:
2562:
2527:
2501:
2469:
2429:
2393:
2355:
2319:
2281:
2245:
2201:
2154:
2134:
2114:
2073:
2041:
2005:
1934:
1910:
1878:
1843:
1708:
1568:
1441:
1312:
1268:Additive utility (AU)
1259:
1158:
1138:
1099:
1079:
1059:
1027:
948:
928:
877:
842:
816:
796:
776:
753:
694:
628:
599:
548:
522:
496:
462:
460:{\displaystyle 2^{O}}
435:
412:
392:
372:
340:
299:
245:
193:
133:
88:
43:is an extension of a
18:Responsive valuations
3432:
3370:
3314:
3270:
3232:
3126:
3096:
3066:
2955:
2939:and every two items
2911:are all equivalent.
2885:
2855:
2825:
2795:
2761:
2731:
2684:
2664:
2611:
2591:
2571:
2536:
2510:
2478:
2442:
2402:
2366:
2328:
2292:
2254:
2218:
2166:
2143:
2123:
2087:
2050:
2018:
1946:
1923:
1887:
1855:
1718:
1583:
1451:
1324:
1283:
1170:
1147:
1143:iff, for every item
1111:
1088:
1068:
1039:
960:
937:
933:iff, for every item
901:
892:stochastic dominance
851:
825:
821:such that, for each
805:
785:
765:
726:
707:of these relations.
637:
611:
560:
531:
505:
479:
444:
424:
420:The RS extension of
401:
381:
361:
311:
258:
204:
152:
104:
59:
3673:free online version
2506:such that, for all
471:Responsive set (RS)
45:preference-relation
3492:
3418:
3356:
3300:
3256:
3210:
3112:
3082:
3041:
2901:
2871:
2841:
2811:
2777:
2747:
2714:
2680:is additive, then
2670:
2650:
2597:
2577:
2557:
2522:
2496:
2464:
2424:
2388:
2350:
2314:
2276:
2240:
2196:
2149:
2129:
2109:
2068:
2036:
2000:
1929:
1905:
1873:
1838:
1703:
1563:
1436:
1307:
1254:
1153:
1133:
1094:
1074:
1054:
1022:
1009:
978:
943:
923:
872:
837:
811:
791:
771:
760:Injective function
748:
705:transitive closure
689:
623:
594:
543:
517:
491:
457:
430:
407:
387:
367:
335:
294:
240:
188:
128:
83:
3735:Babaioff, Moshe;
3060:additive function
2727:It is known that
2673:{\displaystyle u}
2580:{\displaystyle u}
2132:{\displaystyle u}
1932:{\displaystyle u}
1919:utility function
1156:{\displaystyle z}
1097:{\displaystyle X}
1077:{\displaystyle x}
1057:{\displaystyle X}
994:
963:
946:{\displaystyle z}
814:{\displaystyle Y}
794:{\displaystyle X}
774:{\displaystyle f}
410:{\displaystyle O}
397:a total order on
370:{\displaystyle O}
143:independent goods
16:(Redirected from
3800:
3777:
3776:
3758:
3732:
3726:
3725:
3691:
3682:
3676:
3670:
3650:
3644:
3643:
3621:
3615:
3614:
3608:
3600:
3594:
3585:
3579:
3578:
3560:
3540:
3517:Picking sequence
3501:
3499:
3498:
3493:
3427:
3425:
3424:
3419:
3365:
3363:
3362:
3357:
3309:
3307:
3306:
3301:
3265:
3263:
3262:
3257:
3219:
3217:
3216:
3211:
3121:
3119:
3118:
3113:
3111:
3110:
3091:
3089:
3088:
3083:
3081:
3080:
3050:
3048:
3047:
3042:
2947:that are not in
2910:
2908:
2907:
2902:
2900:
2899:
2880:
2878:
2877:
2872:
2870:
2869:
2850:
2848:
2847:
2842:
2840:
2839:
2820:
2818:
2817:
2812:
2810:
2809:
2786:
2784:
2783:
2778:
2776:
2775:
2756:
2754:
2753:
2748:
2746:
2745:
2723:
2721:
2720:
2715:
2679:
2677:
2676:
2671:
2660:. Therefore, if
2659:
2657:
2656:
2651:
2606:
2604:
2603:
2598:
2587:compatible with
2586:
2584:
2583:
2578:
2566:
2564:
2563:
2558:
2531:
2529:
2528:
2523:
2505:
2503:
2502:
2497:
2473:
2471:
2470:
2465:
2460:
2459:
2433:
2431:
2430:
2425:
2420:
2419:
2397:
2395:
2394:
2389:
2384:
2383:
2359:
2357:
2356:
2351:
2346:
2345:
2323:
2321:
2320:
2315:
2310:
2309:
2285:
2283:
2282:
2277:
2272:
2271:
2249:
2247:
2246:
2241:
2236:
2235:
2205:
2203:
2202:
2197:
2158:
2156:
2155:
2150:
2139:compatible with
2138:
2136:
2135:
2130:
2118:
2116:
2115:
2110:
2105:
2104:
2077:
2075:
2074:
2069:
2045:
2043:
2042:
2037:
2009:
2007:
2006:
2001:
1938:
1936:
1935:
1930:
1914:
1912:
1911:
1906:
1882:
1880:
1879:
1874:
1847:
1845:
1844:
1839:
1810:
1809:
1770:
1769:
1730:
1729:
1712:
1710:
1709:
1704:
1675:
1674:
1635:
1634:
1595:
1594:
1572:
1570:
1569:
1564:
1547:
1546:
1519:
1518:
1491:
1490:
1463:
1462:
1445:
1443:
1442:
1437:
1420:
1419:
1392:
1391:
1364:
1363:
1336:
1335:
1316:
1314:
1313:
1308:
1274:additive utility
1263:
1261:
1260:
1255:
1253:
1236:
1219:
1211:
1194:
1177:
1162:
1160:
1159:
1154:
1142:
1140:
1139:
1134:
1129:
1128:
1103:
1101:
1100:
1095:
1083:
1081:
1080:
1075:
1063:
1061:
1060:
1055:
1031:
1029:
1028:
1023:
1008:
977:
952:
950:
949:
944:
932:
930:
929:
924:
919:
918:
881:
879:
878:
873:
846:
844:
843:
838:
820:
818:
817:
812:
800:
798:
797:
792:
780:
778:
777:
772:
757:
755:
754:
749:
744:
743:
698:
696:
695:
690:
655:
654:
632:
630:
629:
624:
603:
601:
600:
595:
590:
589:
552:
550:
549:
544:
526:
524:
523:
518:
500:
498:
497:
492:
466:
464:
463:
458:
456:
455:
439:
437:
436:
431:
416:
414:
413:
408:
396:
394:
393:
388:
376:
374:
373:
368:
344:
342:
341:
336:
303:
301:
300:
295:
249:
247:
246:
241:
197:
195:
194:
189:
137:
135:
134:
129:
92:
90:
89:
84:
21:
3808:
3807:
3803:
3802:
3801:
3799:
3798:
3797:
3783:
3782:
3781:
3780:
3734:
3733:
3729:
3689:
3684:
3683:
3679:
3667:
3652:
3651:
3647:
3623:
3622:
3618:
3601:
3592:
3587:
3586:
3582:
3542:
3541:
3530:
3525:
3512:Weakly additive
3508:
3430:
3429:
3368:
3367:
3312:
3311:
3268:
3267:
3230:
3229:
3124:
3123:
3099:
3094:
3093:
3069:
3064:
3063:
2953:
2952:
2917:
2888:
2883:
2882:
2858:
2853:
2852:
2828:
2823:
2822:
2798:
2793:
2792:
2764:
2759:
2758:
2734:
2729:
2728:
2682:
2681:
2662:
2661:
2609:
2608:
2589:
2588:
2569:
2568:
2534:
2533:
2508:
2507:
2476:
2475:
2448:
2440:
2439:
2408:
2400:
2399:
2372:
2364:
2363:
2360:are equivalent.
2334:
2326:
2325:
2298:
2290:
2289:
2260:
2252:
2251:
2224:
2216:
2215:
2212:
2164:
2163:
2141:
2140:
2121:
2120:
2093:
2085:
2084:
2048:
2047:
2016:
2015:
1944:
1943:
1921:
1920:
1885:
1884:
1853:
1852:
1801:
1761:
1721:
1716:
1715:
1666:
1626:
1586:
1581:
1580:
1538:
1510:
1482:
1454:
1449:
1448:
1411:
1383:
1355:
1327:
1322:
1321:
1281:
1280:
1270:
1168:
1167:
1145:
1144:
1117:
1109:
1108:
1086:
1085:
1066:
1065:
1037:
1036:
958:
957:
935:
934:
907:
899:
898:
888:
849:
848:
823:
822:
803:
802:
783:
782:
763:
762:
732:
724:
723:
713:
643:
635:
634:
609:
608:
578:
558:
557:
529:
528:
527:and every item
503:
502:
477:
476:
473:
447:
442:
441:
422:
421:
399:
398:
379:
378:
359:
358:
355:
309:
308:
256:
255:
202:
201:
150:
149:
102:
101:
57:
56:
53:
23:
22:
15:
12:
11:
5:
3806:
3804:
3796:
3795:
3785:
3784:
3779:
3778:
3749:(1): 382–403.
3727:
3677:
3665:
3645:
3616:
3599:. Springer US.
3580:
3527:
3526:
3524:
3521:
3520:
3519:
3514:
3507:
3504:
3491:
3488:
3485:
3482:
3479:
3476:
3473:
3470:
3467:
3464:
3461:
3458:
3455:
3452:
3449:
3446:
3443:
3440:
3437:
3417:
3414:
3411:
3408:
3405:
3402:
3399:
3396:
3393:
3390:
3387:
3384:
3381:
3378:
3375:
3355:
3352:
3349:
3346:
3343:
3340:
3337:
3334:
3331:
3328:
3325:
3322:
3319:
3299:
3296:
3293:
3290:
3287:
3284:
3281:
3278:
3275:
3255:
3252:
3249:
3246:
3243:
3240:
3237:
3226:
3225:
3221:
3209:
3206:
3203:
3200:
3197:
3194:
3191:
3188:
3185:
3182:
3179:
3176:
3173:
3170:
3167:
3164:
3161:
3158:
3155:
3152:
3149:
3146:
3143:
3140:
3137:
3134:
3131:
3109:
3106:
3102:
3079:
3076:
3072:
3040:
3037:
3034:
3031:
3028:
3025:
3022:
3019:
3016:
3013:
3010:
3007:
3004:
3001:
2998:
2995:
2992:
2988:
2984:
2981:
2978:
2975:
2972:
2969:
2966:
2963:
2960:
2916:
2913:
2898:
2895:
2891:
2868:
2865:
2861:
2838:
2835:
2831:
2808:
2805:
2801:
2789:
2788:
2774:
2771:
2767:
2744:
2741:
2737:
2725:
2713:
2710:
2707:
2704:
2701:
2698:
2695:
2692:
2689:
2669:
2649:
2646:
2643:
2640:
2637:
2634:
2631:
2628:
2625:
2622:
2619:
2616:
2596:
2576:
2556:
2553:
2550:
2547:
2544:
2541:
2521:
2518:
2515:
2495:
2492:
2489:
2486:
2483:
2463:
2458:
2455:
2451:
2447:
2423:
2418:
2415:
2411:
2407:
2387:
2382:
2379:
2375:
2371:
2361:
2349:
2344:
2341:
2337:
2333:
2313:
2308:
2305:
2301:
2297:
2287:
2275:
2270:
2267:
2263:
2259:
2239:
2234:
2231:
2227:
2223:
2211:
2208:
2207:
2206:
2195:
2192:
2189:
2186:
2183:
2180:
2177:
2174:
2171:
2148:
2128:
2108:
2103:
2100:
2096:
2092:
2067:
2064:
2061:
2058:
2055:
2035:
2032:
2029:
2026:
2023:
2012:
2011:
1999:
1996:
1993:
1990:
1987:
1984:
1981:
1978:
1975:
1972:
1969:
1966:
1963:
1960:
1957:
1954:
1951:
1928:
1904:
1901:
1898:
1895:
1892:
1872:
1869:
1866:
1863:
1860:
1849:
1848:
1837:
1834:
1831:
1828:
1825:
1822:
1819:
1816:
1813:
1808:
1804:
1800:
1797:
1794:
1791:
1788:
1785:
1782:
1779:
1776:
1773:
1768:
1764:
1760:
1757:
1754:
1751:
1748:
1745:
1742:
1739:
1736:
1733:
1728:
1724:
1713:
1702:
1699:
1696:
1693:
1690:
1687:
1684:
1681:
1678:
1673:
1669:
1665:
1662:
1659:
1656:
1653:
1650:
1647:
1644:
1641:
1638:
1633:
1629:
1625:
1622:
1619:
1616:
1613:
1610:
1607:
1604:
1601:
1598:
1593:
1589:
1574:
1573:
1562:
1559:
1556:
1553:
1550:
1545:
1541:
1537:
1534:
1531:
1528:
1525:
1522:
1517:
1513:
1509:
1506:
1503:
1500:
1497:
1494:
1489:
1485:
1481:
1478:
1475:
1472:
1469:
1466:
1461:
1457:
1446:
1435:
1432:
1429:
1426:
1423:
1418:
1414:
1410:
1407:
1404:
1401:
1398:
1395:
1390:
1386:
1382:
1379:
1376:
1373:
1370:
1367:
1362:
1358:
1354:
1351:
1348:
1345:
1342:
1339:
1334:
1330:
1306:
1303:
1300:
1297:
1294:
1291:
1288:
1269:
1266:
1265:
1264:
1252:
1248:
1245:
1242:
1239:
1235:
1231:
1228:
1225:
1222:
1218:
1214:
1210:
1206:
1203:
1200:
1197:
1193:
1189:
1186:
1183:
1180:
1176:
1152:
1132:
1127:
1124:
1120:
1116:
1093:
1084:in the bundle
1073:
1053:
1050:
1047:
1044:
1033:
1032:
1021:
1018:
1015:
1012:
1007:
1004:
1001:
997:
993:
990:
987:
984:
981:
976:
973:
970:
966:
942:
922:
917:
914:
910:
906:
887:
884:
871:
868:
865:
862:
859:
856:
836:
833:
830:
810:
790:
770:
747:
742:
739:
735:
731:
712:
709:
701:
700:
688:
685:
682:
679:
676:
673:
670:
667:
664:
661:
658:
653:
650:
646:
642:
622:
619:
616:
605:
593:
588:
585:
581:
577:
574:
571:
568:
565:
542:
539:
536:
516:
513:
510:
490:
487:
484:
472:
469:
454:
450:
429:
406:
386:
366:
354:
351:
334:
331:
328:
325:
322:
319:
316:
293:
290:
287:
284:
281:
278:
275:
272:
269:
266:
263:
252:
251:
239:
236:
233:
230:
227:
224:
221:
218:
215:
212:
209:
199:
187:
184:
181:
178:
175:
172:
169:
166:
163:
160:
157:
139:
138:
127:
124:
121:
118:
115:
112:
109:
82:
79:
76:
73:
70:
67:
64:
52:
49:
33:responsive set
29:utility theory
24:
14:
13:
10:
9:
6:
4:
3:
2:
3805:
3794:
3791:
3790:
3788:
3774:
3770:
3766:
3762:
3757:
3752:
3748:
3744:
3743:
3738:
3731:
3728:
3723:
3719:
3715:
3711:
3707:
3703:
3699:
3695:
3688:
3681:
3678:
3674:
3668:
3666:9781107060432
3662:
3658:
3657:
3649:
3646:
3641:
3637:
3633:
3629:
3628:
3620:
3617:
3612:
3606:
3598:
3591:
3584:
3581:
3576:
3572:
3568:
3564:
3559:
3554:
3550:
3546:
3539:
3537:
3535:
3533:
3529:
3522:
3518:
3515:
3513:
3510:
3509:
3505:
3503:
3483:
3480:
3477:
3474:
3471:
3462:
3459:
3450:
3447:
3444:
3435:
3409:
3406:
3403:
3394:
3391:
3382:
3373:
3350:
3347:
3344:
3341:
3338:
3332:
3326:
3323:
3320:
3294:
3291:
3288:
3282:
3276:
3253:
3250:
3247:
3244:
3241:
3238:
3235:
3222:
3204:
3198:
3195:
3189:
3183:
3180:
3171:
3165:
3162:
3156:
3153:
3144:
3138:
3135:
3129:
3107:
3104:
3100:
3077:
3074:
3070:
3061:
3057:
3056:
3055:
3052:
3032:
3026:
3023:
3017:
3014:
3005:
2999:
2996:
2990:
2979:
2973:
2970:
2964:
2958:
2950:
2946:
2942:
2938:
2934:
2930:
2925:
2922:
2914:
2912:
2896:
2893:
2889:
2866:
2863:
2859:
2836:
2833:
2829:
2806:
2803:
2799:
2772:
2769:
2765:
2742:
2739:
2735:
2726:
2708:
2702:
2699:
2693:
2687:
2667:
2641:
2635:
2629:
2626:
2620:
2614:
2594:
2574:
2551:
2545:
2542:
2539:
2519:
2516:
2513:
2493:
2487:
2484:
2481:
2461:
2456:
2453:
2449:
2445:
2437:
2421:
2416:
2413:
2409:
2405:
2385:
2380:
2377:
2373:
2369:
2362:
2347:
2342:
2339:
2335:
2331:
2311:
2306:
2303:
2299:
2295:
2288:
2273:
2268:
2265:
2261:
2257:
2237:
2232:
2229:
2225:
2221:
2214:
2213:
2209:
2190:
2184:
2181:
2175:
2169:
2162:
2161:
2160:
2146:
2126:
2106:
2101:
2098:
2094:
2090:
2082:
2079:
2062:
2059:
2056:
2030:
2027:
2024:
1991:
1988:
1985:
1976:
1973:
1964:
1961:
1958:
1949:
1942:
1941:
1940:
1926:
1918:
1899:
1896:
1893:
1867:
1864:
1861:
1835:
1832:
1823:
1820:
1817:
1806:
1802:
1798:
1795:
1792:
1783:
1780:
1777:
1766:
1762:
1758:
1755:
1752:
1743:
1740:
1737:
1726:
1722:
1714:
1700:
1697:
1688:
1685:
1682:
1671:
1667:
1663:
1660:
1657:
1648:
1645:
1642:
1631:
1627:
1623:
1620:
1617:
1608:
1605:
1602:
1591:
1587:
1579:
1578:
1577:
1560:
1557:
1551:
1543:
1539:
1535:
1532:
1529:
1523:
1515:
1511:
1507:
1504:
1501:
1495:
1487:
1483:
1479:
1476:
1473:
1467:
1459:
1455:
1447:
1433:
1430:
1424:
1416:
1412:
1408:
1405:
1402:
1396:
1388:
1384:
1380:
1377:
1374:
1368:
1360:
1356:
1352:
1349:
1346:
1340:
1332:
1328:
1320:
1319:
1318:
1304:
1301:
1298:
1295:
1292:
1289:
1286:
1277:
1275:
1267:
1243:
1240:
1237:
1229:
1226:
1223:
1212:
1201:
1198:
1195:
1187:
1184:
1181:
1166:
1165:
1164:
1150:
1130:
1125:
1122:
1118:
1114:
1105:
1091:
1071:
1048:
1042:
1016:
1010:
1005:
1002:
999:
995:
991:
985:
979:
974:
971:
968:
964:
956:
955:
954:
940:
920:
915:
912:
908:
904:
895:
893:
885:
883:
866:
860:
857:
854:
834:
831:
828:
808:
788:
768:
761:
745:
740:
737:
733:
729:
720:
718:
710:
708:
706:
683:
677:
668:
659:
651:
648:
644:
640:
620:
617:
614:
606:
591:
586:
583:
579:
572:
563:
556:
555:
554:
540:
537:
534:
514:
511:
508:
501:, every item
488:
485:
482:
470:
468:
452:
448:
427:
418:
404:
384:
364:
352:
350:
348:
347:partial order
332:
329:
326:
323:
320:
317:
314:
305:
288:
285:
282:
276:
270:
267:
264:
234:
231:
228:
222:
216:
213:
210:
200:
182:
179:
176:
170:
164:
161:
158:
148:
147:
146:
144:
125:
122:
119:
116:
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110:
107:
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96:
80:
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71:
68:
65:
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3737:Nisan, Noam
3700:: 110–123.
2210:Equivalence
1851:The bundle
353:Definitions
95:total order
3756:1703.08150
3634:(1): 231.
3523:References
2929:responsive
2921:responsive
1276:function.
897:Formally,
722:Formally,
3714:0304-3975
3605:cite book
3558:1312.6546
3551:: 71–92.
3333:≻
3283:≺
3251:≺
3245:≺
3239:≺
3224:function.
3196:−
3166:∪
3154:−
3139:∪
3101:⪯
3071:⪯
3027:∪
3015:≥
3000:∪
2987:⟹
2971:≥
2890:⪯
2860:⪯
2830:⪯
2800:⪯
2766:⪯
2736:⪯
2700:≤
2627:≤
2595:⪯
2543:⪯
2517:∈
2491:→
2450:⪯
2410:⪯
2374:⪯
2336:⪯
2300:⪯
2262:⪯
2226:⪯
2182:≤
2147:⪯
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1290:≺
1241:⪰
1227:∈
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1199:⪰
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996:∑
992:≤
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965:∑
909:⪯
858:⪯
832:∈
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678:∪
663:∖
645:⪯
618:⪯
580:≺
567:∖
538:∉
512:∈
486:⊆
428:⪯
385:⪯
330:≺
324:≺
318:≺
223:≺
171:≺
123:≺
117:≺
111:≺
41:extension
3787:Category
3722:59222796
3506:See also
2398:implies
2250:implies
3773:4224433
3575:1408197
717:pairing
51:Example
3771:
3720:
3712:
3663:
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3428:while
1035:where
31:, the
3751:arXiv
3718:S2CID
3690:(PDF)
3593:(PDF)
3571:S2CID
3553:arXiv
2438:: If
2436:Proof
1917:every
781:from
633:then
345:is a
3710:ISSN
3661:ISBN
3611:link
3460:>
3392:<
3310:and
2881:and
2851:and
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2757:and
2324:and
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357:Let
3761:doi
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607:If
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