2090:
The maxsum rule gives region i to agent i, but it is not EF since Carl envies Alice. Using a linear program, it is possible to find the unique maxsum-EF allocation, and show that it must share both region 1 and region 2 between Alice and Bob. However, such allocation cannot be PO since Alice and Bob
1633:
variables: each (agent, region) pair has a variable that determines the fraction of the region given to the agent. For each region, there is a constraint saying that the sum of all fractions from this region is 1; for each (agent, agent) pair, there is a constraint saying that the first agent does
1449:: A finite algorithm has value-data only about a finite number of pieces. I.e. there is only a finite number of subsets of the cake, for which the algorithm knows the valuations of the partners. Suppose the algorithm has stopped after having value-data about
2004:(EQ) cake divisions, and relate them to maxsum and Pareto-optimality (PO). As explained above, maxsum allocations are always PO. However, when maxsum is constrained by fairness, this is not necessarily true. They prove the following:
1570:
One approach to this conflict is to bound the "price of fairness" - calculate upper and lower bounds on the amount of decrease in the sum of utilities, that is required for fairness. For more details, see
965:
448:
The utilitarian rule gives each part to the partner with the highest utility. In this case, the utilitarian rule gives the entire chocolate to Alice and the entire
Vanilla to George. The maxsum is 13.
855:
751:
1351:
1260:
1966:
514:
1533:
When the cake is 1-dimensional and the pieces must be connected, the simple algorithm of assigning each piece to the agent that values it the most no longer works, even with
2102:, the utility-sum of a maxsum-EF allocation is at least as large as a maxsum-EQ allocation. This result extends to general valuations up to an additive approximation (i.e.,
1883:
1710:
1375:
When the value functions are additive, maxsum divisions always exist. Intuitively, we can give each fraction of the cake to the partner that values it the most, as in the
1743:
1419:
1776:
1926:
1903:
1857:
1830:
1803:
1590:
with maximum sum-of-utilities, for a cake which is a 1-dimensional interval, when each person may receive disconnected pieces and the value functions are additive:
995:
693:
584:
1523:
1989:
1655:
1612:
1491:
1467:
1107:
1087:
1067:
1047:
1027:
878:
774:
666:
646:
626:
604:
557:
537:
485:
1473:
value measure. In this case, the largest possible utilitarian value that the algorithm can achieve is 1. However, it is possible that deep inside one of the
1186:: whenever the utility functions of the partners are multiplied by constants (a possibly different constant to each partner), the recommendations given by
487:. It is usually assumed to be either a finite 1-dimensional segment, a 2-dimensional polygon or a finite subset of the multidimensional Euclidean plane
1578:
Another approach to combining efficiency and fairness is to find, among all possible fair divisions, a fair division with a highest sum-of-utilities:
356:
1431:
When the cake is not piecewise-homogeneous, the above algorithm does not work since there is an infinite number of different "pieces" to consider.
1289:
are utilitarian-maximal (in other words, all divisions must be WUM and all agents must have equal weights. This follows from the PTE property).
2161:
2036:, there may be even no maxsum-EF allocations that are PO. For example, consider a cake with three regions and three agents with values:
2276:
1991:
partners with general valuations: additive approximation to envy and efficiency, based on the piecewise-constant-valuations algorithm.
2192:
Barbanel, Julius B.; Zwicker, William S. (1997). "Two applications of a theorem of
Dvoretsky, Wald, and Wolfovitz to cake division".
1428:, i.e., the cake can be divided to a finite number of pieces such that the value-density of each piece is constant for all partners.
890:
2131:. The relative-utilitarian rule (maximizing the sum of normalized utilities) is population-monotonic but not resource-monotonic.
230:
73:
1128:
385:
240:
1928:), and attains a sum that is at least the maximum sum of an EF allocation. Its run-time is polynomial in the input and in
400:
Consider a cake with two parts: chocolate and vanilla, and two partners: Alice and George, with the following valuations:
376:) is a rule for dividing a heterogeneous resource, such as a cake or a land-estate, among several partners with different
2030:
1615:
1534:
349:
1525:, so the sum of utilities is strictly more than 1. Hence, the division returned by the finite algorithm is not maxsum.
1435:
2012:
1494:
2096:
1658:
794:
698:
58:
2019:
1549:
452:
170:
1832:
cannot be calculated because it might be irrational, but in practice, when the valuations are piecewise-linear,
155:
1379:. Similarly, WUM divisions can be found by giving each fraction of the cake to the partner for whom the ratio
1208:
The following is proved for partners that assign positive utility to every piece of cake with positive size:
1124:
2501:
2166:
2128:
342:
275:
260:
140:
20:
2146:
1587:
456:
310:
300:
165:
2352:
1439:
2496:
2124:
1634:
not envy the second one. Note that the allocation produced by this procedure might be highly fractioned.
1310:
1219:
1201:
265:
1931:
490:
2434:
2223:
160:
92:
1862:
2219:
2156:
1667:
270:
145:
2357:. Proceedings of the 26th AAAI Conference on Artificial Intelligence (AAAI-12). pp. 1285–1291
2472:
2446:
2415:
2389:
2293:
2209:
2171:
2001:
1715:
1626:
1382:
1196:: for a fixed piece of cake, the set of utility profiles which map to a specific allocation is a
1152:: whenever a cake is partitioned to several sub-cakes and each cake is divided according to rule
320:
235:
150:
1748:
1469:
subsets. Now, it may be the case that all partners answered all the queries as if they have the
388:. Utilitarian cake-cutting is often not "fair"; hence, utilitarianism is often in conflict with
1911:
1888:
2464:
2407:
2272:
2151:
1572:
1131:
to the WUM rule. The characterization is based on the following properties of a division rule
389:
332:
315:
2008:
When there are two agents, maxsum-EF, maximum-EQ and maximum-EF-EQ allocations are always PO.
1493:
pieces, there is a subset which two partners value differently. In this case, there exists a
860:
The concept is often generalized by assigning a different weight to each partner. A division
2456:
2399:
2245:
2201:
1555:
For every set of positive weights, a WUM division exists and can be found in a similar way.
1006:
377:
225:
105:
1835:
1808:
1781:
973:
671:
562:
280:
63:
1500:
2348:
1974:
1640:
1597:
1476:
1452:
1092:
1072:
1052:
1032:
1012:
863:
759:
651:
631:
611:
589:
542:
522:
470:
175:
110:
100:
68:
28:
2222:. For a similar result related to the problem of homogeneous resource allocation, see
384:
of the utilities of the partners is as large as possible. It is a special case of the
2490:
2460:
2213:
1170:, the result does not depend on the utilities of the partners in the other sub-cakes.
78:
2476:
2123:, the absolute-utilitarian rule (maximizing the sum of non-normalized utilities) is
120:
2419:
2330:
Cohler, Yuga Julian; Lai, John Kwang; Parkes, David C; Procaccia, Ariel (2011).
305:
1859:
can be approximated by an "irrational search" approximation algorithm. For any
1180:
recommends at least one division that gives a positive utility to each partner.
1156:, the result is the same as if the original cake were partitioned according to
2403:
2249:
2205:
1197:
220:
115:
53:
48:
2468:
2435:"Resource-monotonicity and population-monotonicity in connected cake-cutting"
2411:
2000:
Brams, Feldman, Lai, Morgenstern and
Procaccia study both envy-free (EF) and
2377:
2314:
215:
200:
2331:
1266:
are WUM with these weights (it is known that every PE division is WUM with
1664:: for each point in the cake, calculate the ratio between the utilities:
455:
since George receives less than half the total cake value, and it is not
205:
2022:, which is preserved under Pareto improvements). However, there may be
1805:
is a threshold calculated so that the division is envy-free. In general
2236:
Chambers, Christopher P. (2005). "Allocation rules for land division".
1905:-EF (the value of each agent is at least the value of each other agent
1538:
648:
disjoint pieces, one piece per partner. The piece allocated to partner
210:
2106:-EF allocations have a utility-sum of at least EQ allocations −
1300:
is a dictatorial rule - it gives the entire cake to a single partner.
2451:
2394:
2298:
1542:
2018:, maxsum-EF allocations are always PO (since EF is equivalent to
1116:
Pareto-efficient division is WUM for some selection of weights.
195:
43:
2351:; John K. Lai; Jamie Morgenstern; Ariel D. Procaccia (2012).
1307:
is PE DI IIL and CO, then there exists a sequence of weights
960:{\displaystyle \sum _{i=1}^{n}{\frac {V_{i}(X_{i})}{w_{i}}}}
2091:
could both gain by swapping their shares in these regions.
2378:"Monotonicity and competitive equilibrium in cake-cutting"
1361:
recommends all and only WUM divisions with these weights).
1216:
is PE DI and IIL, then there exists a sequence of weights
2313:
Aumann, Yonatan; Dombb, Yair; Hassidim, Avinatan (2013).
1552:
algorithm which is exponential in the number of players.
1445:
However, no finite algorithm can find a maxsum division.
1434:
Maxsum divisions still exist. This is a corollary of the
2292:
Ianovski, Egor (2012-03-01). "Cake
Cutting Mechanisms".
1537:. In this case, the problem of finding a UM division is
1285:
is PE DI IIL and PTE, then all divisions recommended by
1270:
weights; the news are that all divisions recommended by
1176:: whenever all partners have the same utility function,
1497:, in which each partner receives a value of more than
1005:
Every WUM division with positive weights is obviously
2433:
Segal-Halevi, Erel; Sziklai, Balázs R. (2018-09-01).
2376:
Segal-Halevi, Erel; Sziklai, Balázs R. (2019-09-01).
1977:
1934:
1914:
1891:
1865:
1838:
1811:
1784:
1751:
1718:
1670:
1643:
1600:
1503:
1479:
1455:
1385:
1313:
1222:
1095:
1075:
1055:
1035:
1015:
976:
893:
866:
797:
762:
701:
674:
654:
634:
614:
592:
565:
545:
525:
493:
473:
1548:
There is an 8-factor approximation algorithm, and a
2174:- the utilitarian principle in a different context.
2115:
Monotonicity properties of utilitarian cake-cutting
2026:
maxsum-EQ and maxsum-EQ-EF allocations that are PO.
1567:. Similarly, a fair division is not always maxsum.
1983:
1960:
1920:
1897:
1877:
1851:
1824:
1797:
1770:
1737:
1704:
1649:
1606:
1517:
1485:
1461:
1413:
1345:
1254:
1146:returns only divisions which are Pareto-efficient.
1101:
1081:
1061:
1041:
1021:
989:
959:
872:
849:
768:
745:
687:
660:
640:
620:
598:
578:
551:
531:
508:
479:
1586:The following algorithms can be used to find an
884:(WUM) if it maximizes the following expression:
451:The utilitarian division is not fair: it is not
1424:This process is easy to carry out when cake is
1563:A maxsum division is not always fair; see the
1166:: whenever a sub-cake is divided according to
850:{\displaystyle \sum _{i=1}^{n}{V_{i}(X_{i})}}
746:{\displaystyle C=X_{1}\sqcup ...\sqcup X_{n}}
350:
8:
1278:weights. This follows from the DI property).
2316:Computing Socially-Efficient Cake Divisions
1885:, The algorithm find an allocation that is
2263:Brams, Steven J.; Taylor, Alan D. (1996).
1996:Properties of utilitarian-fair allocations
788:if it maximizes the following expression:
357:
343:
15:
2450:
2393:
2297:
2134:This no longer holds when the pieces are
2029:When there are three or more agents with
2011:When there are three or more agents with
1976:
1947:
1933:
1913:
1890:
1864:
1843:
1837:
1816:
1810:
1789:
1783:
1762:
1750:
1729:
1717:
1696:
1687:
1681:
1669:
1642:
1599:
1507:
1502:
1478:
1454:
1405:
1396:
1390:
1384:
1337:
1318:
1312:
1246:
1227:
1221:
1094:
1074:
1054:
1034:
1014:
981:
975:
949:
935:
922:
915:
909:
898:
892:
865:
837:
824:
819:
813:
802:
796:
761:
737:
712:
700:
679:
673:
653:
633:
613:
591:
570:
564:
544:
524:
500:
496:
495:
492:
472:
2041:
1120:Characterization of the utilitarian rule
1049:, then the weighted sum-of-utilities in
402:
2269:From cake-cutting to dispute resolution
2184:
1357:is a WUM rule with these weights (i.e.
1262:such that all divisions recommended by
27:
1164:Independence of infeasible land (IIL)
7:
1582:Finding utilitarian-fair allocations
1438:and it can also be proved using the
2162:Pareto-efficient envy-free division
1625:totally-constant regions. Solve a
1436:Dubins–Spanier compactness theorem
1346:{\displaystyle w_{1},\dots ,w_{n}}
1255:{\displaystyle w_{1},\dots ,w_{n}}
1174:Positive treatment of equals (PTE)
14:
1961:{\displaystyle \log(1/\epsilon )}
1712:. Give partner 1 the points with
1009:. This is because, if a division
2461:10.1016/j.mathsocsci.2018.07.001
509:{\displaystyle \mathbb {R} ^{d}}
1112:What's more surprising is that
2333:Optimal Envy-Free Cake Cutting
1955:
1941:
1878:{\displaystyle \epsilon >0}
1745:and partner 2 the points with
997:are given positive constants.
941:
928:
843:
830:
559:has a personal value function
386:utilitarian social choice rule
241:Utilitarian social choice rule
1:
2354:On Maxsum Fair Cake Divisions
1705:{\displaystyle r=u_{1}/u_{2}}
1535:piecewise-constant valuations
1366:Finding utilitarian divisions
2439:Mathematical Social Sciences
1564:
1376:
1029:Pareto-dominates a division
1001:Maxsum and Pareto-efficiency
882:weighted-utilitarian-maximal
1738:{\displaystyle r\geq r^{*}}
1495:super-proportional division
1414:{\displaystyle V_{i}/w_{i}}
1069:is strictly larger than in
459:since George envies Alice.
2518:
2238:Journal of Economic Theory
1771:{\displaystyle r<r^{*}}
1296:is PE DI IIL and SI, then
1150:Division independence (DI)
1109:cannot be a WUM division.
2404:10.1007/s00199-018-1128-6
2250:10.1016/j.jet.2004.04.008
1921:{\displaystyle \epsilon }
1898:{\displaystyle \epsilon }
1550:fixed-parameter tractable
1545:is possible unless P=NP.
380:functions, such that the
370:Utilitarian cake-cutting
2206:10.1023/a:1004966624893
2167:Rank-maximal allocation
2119:When the pieces may be
1621:: divide the cake into
1125:Christopher P. Chambers
606:("pieces") to numbers.
539:partners. Each partner
276:Replaceability argument
261:Demandingness objection
134:Types of utilitarianism
59:Claude Adrien Helvétius
2147:Efficient cake-cutting
1985:
1962:
1922:
1899:
1879:
1853:
1826:
1799:
1772:
1739:
1706:
1651:
1608:
1588:envy-free cake-cutting
1519:
1487:
1463:
1415:
1347:
1256:
1140:Pareto-efficiency (PE)
1103:
1083:
1063:
1043:
1023:
991:
961:
914:
874:
851:
818:
770:
747:
689:
662:
642:
622:
600:
586:which maps subsets of
580:
553:
533:
510:
481:
311:Neoclassical economics
301:Rational choice theory
2095:When all agents have
1986:
1963:
1923:
1900:
1880:
1854:
1852:{\displaystyle r^{*}}
1827:
1825:{\displaystyle r^{*}}
1800:
1798:{\displaystyle r^{*}}
1773:
1740:
1707:
1652:
1609:
1541:, and furthermore no
1520:
1488:
1464:
1426:piecewise-homogeneous
1416:
1348:
1257:
1202:pointwise convergence
1184:Scale-invariance (SI)
1104:
1084:
1064:
1044:
1024:
992:
990:{\displaystyle w_{i}}
962:
894:
875:
852:
798:
771:
748:
690:
688:{\displaystyle X_{i}}
663:
643:
628:has to be divided to
623:
601:
581:
579:{\displaystyle V_{i}}
554:
534:
511:
482:
266:Mere addition paradox
2129:population-monotonic
1975:
1932:
1912:
1889:
1863:
1836:
1809:
1782:
1749:
1716:
1668:
1641:
1598:
1501:
1477:
1453:
1383:
1311:
1220:
1093:
1073:
1053:
1033:
1013:
974:
891:
864:
795:
760:
699:
672:
652:
632:
612:
590:
563:
543:
523:
491:
471:
2271:]. p. 48.
2194:Theory and Decision
1559:Maxsum and fairness
1518:{\displaystyle 1/n}
1371:Disconnected pieces
782:utilitarian-maximal
467:The cake is called
374:maxsum cake-cutting
271:Paradox of hedonism
231:Equal consideration
2172:Utilitarian voting
2125:resource-monotonic
2032:piecewise-constant
1981:
1958:
1918:
1895:
1875:
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1822:
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1768:
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1647:
1617:piecewise-constant
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743:
685:
658:
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618:
596:
576:
549:
529:
506:
477:
321:Effective altruism
236:Felicific calculus
2347:Steven J. Brams;
2224:Varian's theorems
2152:Fair cake-cutting
2086:
2085:
2014:piecewise-uniform
1984:{\displaystyle n}
1650:{\displaystyle 2}
1607:{\displaystyle n}
1573:price of fairness
1486:{\displaystyle k}
1462:{\displaystyle k}
1440:Radon–Nikodym set
1102:{\displaystyle X}
1082:{\displaystyle X}
1062:{\displaystyle Y}
1042:{\displaystyle X}
1022:{\displaystyle Y}
955:
873:{\displaystyle X}
769:{\displaystyle X}
661:{\displaystyle i}
641:{\displaystyle n}
621:{\displaystyle C}
599:{\displaystyle C}
552:{\displaystyle i}
532:{\displaystyle n}
480:{\displaystyle C}
446:
445:
390:fair cake-cutting
367:
366:
333:Philosophy portal
316:Population ethics
74:Francis Hutcheson
2509:
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2301:
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2260:
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2253:
2233:
2227:
2220:Weller's theorem
2217:
2189:
2157:Weller's theorem
2109:
2105:
2098:piecewise-linear
2042:
1990:
1988:
1987:
1982:
1967:
1965:
1964:
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1951:
1927:
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1919:
1904:
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1700:
1691:
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1685:
1660:piecewise-linear
1656:
1654:
1653:
1648:
1613:
1611:
1610:
1605:
1529:Connected pieces
1524:
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1232:
1231:
1129:characterization
1108:
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1068:
1066:
1065:
1060:
1048:
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1028:
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1020:
1007:Pareto-efficient
996:
994:
993:
988:
986:
985:
966:
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953:
944:
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403:
378:cardinal utility
359:
352:
345:
226:Consequentialism
106:John Stuart Mill
16:
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2508:
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2382:Economic Theory
2375:
2374:
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2257:
2235:
2234:
2230:
2191:
2190:
2186:
2181:
2143:
2117:
2107:
2103:
2020:proportionality
1998:
1973:
1972:
1930:
1929:
1910:
1909:
1887:
1886:
1861:
1860:
1839:
1834:
1833:
1812:
1807:
1806:
1785:
1780:
1779:
1758:
1747:
1746:
1725:
1714:
1713:
1692:
1677:
1666:
1665:
1639:
1638:
1596:
1595:
1584:
1561:
1531:
1499:
1498:
1475:
1474:
1451:
1450:
1401:
1386:
1381:
1380:
1373:
1368:
1333:
1314:
1309:
1308:
1242:
1223:
1218:
1217:
1194:Continuity (CO)
1122:
1091:
1090:
1071:
1070:
1051:
1050:
1031:
1030:
1011:
1010:
1003:
977:
972:
971:
945:
931:
918:
917:
889:
888:
862:
861:
833:
820:
793:
792:
758:
757:
733:
708:
697:
696:
675:
670:
669:
650:
649:
630:
629:
610:
609:
588:
587:
566:
561:
560:
541:
540:
521:
520:
494:
489:
488:
469:
468:
465:
442:
437:
427:
422:
398:
363:
327:
326:
325:
295:
287:
286:
285:
281:Utility monster
255:
247:
246:
245:
190:
182:
181:
180:
135:
127:
126:
125:
95:
85:
84:
83:
64:Cesare Beccaria
38:
12:
11:
5:
2515:
2513:
2505:
2504:
2502:Utilitarianism
2499:
2489:
2488:
2483:
2482:
2425:
2388:(2): 363–401.
2368:
2349:Michal Feldman
2339:
2322:
2305:
2284:
2278:978-0521556446
2277:
2255:
2244:(2): 236–258.
2228:
2183:
2182:
2180:
2177:
2176:
2175:
2169:
2164:
2159:
2154:
2149:
2142:
2139:
2116:
2113:
2112:
2111:
2088:
2087:
2084:
2083:
2080:
2077:
2074:
2070:
2069:
2066:
2063:
2060:
2056:
2055:
2052:
2049:
2046:
2038:
2037:
2027:
2009:
1997:
1994:
1993:
1992:
1980:
1969:
1957:
1954:
1950:
1946:
1943:
1940:
1937:
1917:
1894:
1874:
1871:
1868:
1846:
1842:
1819:
1815:
1792:
1788:
1765:
1761:
1757:
1754:
1732:
1728:
1724:
1721:
1699:
1695:
1690:
1684:
1680:
1676:
1673:
1657:partners with
1646:
1635:
1627:linear program
1614:partners with
1603:
1583:
1580:
1560:
1557:
1530:
1527:
1514:
1510:
1506:
1482:
1458:
1408:
1404:
1399:
1393:
1389:
1372:
1369:
1367:
1364:
1363:
1362:
1340:
1336:
1332:
1329:
1326:
1321:
1317:
1301:
1290:
1279:
1249:
1245:
1241:
1238:
1235:
1230:
1226:
1206:
1205:
1191:
1190:do not change.
1181:
1171:
1161:
1147:
1121:
1118:
1098:
1078:
1058:
1038:
1018:
1002:
999:
984:
980:
968:
967:
952:
948:
943:
938:
934:
930:
925:
921:
912:
907:
904:
901:
897:
869:
858:
857:
845:
840:
836:
832:
827:
823:
816:
811:
808:
805:
801:
765:
740:
736:
732:
729:
726:
723:
720:
715:
711:
707:
704:
682:
678:
657:
637:
617:
595:
573:
569:
548:
528:
503:
498:
476:
464:
461:
444:
443:
440:
438:
435:
433:
429:
428:
425:
423:
420:
418:
414:
413:
410:
407:
397:
394:
365:
364:
362:
361:
354:
347:
339:
336:
335:
329:
328:
324:
323:
318:
313:
308:
303:
297:
296:
294:Related topics
293:
292:
289:
288:
284:
283:
278:
273:
268:
263:
257:
256:
253:
252:
249:
248:
244:
243:
238:
233:
228:
223:
218:
213:
208:
203:
198:
192:
191:
188:
187:
184:
183:
179:
178:
173:
168:
163:
158:
153:
148:
143:
137:
136:
133:
132:
129:
128:
124:
123:
118:
113:
111:Henry Sidgwick
108:
103:
101:Jeremy Bentham
97:
96:
93:Key proponents
91:
90:
87:
86:
82:
81:
76:
71:
69:William Godwin
66:
61:
56:
51:
46:
40:
39:
36:
35:
32:
31:
29:Utilitarianism
25:
24:
13:
10:
9:
6:
4:
3:
2:
2514:
2503:
2500:
2498:
2495:
2494:
2492:
2478:
2474:
2470:
2466:
2462:
2458:
2453:
2448:
2444:
2440:
2436:
2429:
2426:
2421:
2417:
2413:
2409:
2405:
2401:
2396:
2391:
2387:
2383:
2379:
2372:
2369:
2356:
2355:
2350:
2343:
2340:
2335:
2334:
2326:
2323:
2318:
2317:
2309:
2306:
2300:
2295:
2288:
2285:
2280:
2274:
2270:
2266:
2265:Fair Division
2259:
2256:
2251:
2247:
2243:
2239:
2232:
2229:
2225:
2221:
2215:
2211:
2207:
2203:
2199:
2195:
2188:
2185:
2178:
2173:
2170:
2168:
2165:
2163:
2160:
2158:
2155:
2153:
2150:
2148:
2145:
2144:
2140:
2138:
2137:
2132:
2130:
2126:
2122:
2114:
2101:
2099:
2094:
2093:
2092:
2081:
2078:
2075:
2072:
2071:
2067:
2064:
2061:
2058:
2057:
2053:
2050:
2047:
2044:
2043:
2040:
2039:
2035:
2033:
2028:
2025:
2021:
2017:
2015:
2010:
2007:
2006:
2005:
2003:
1995:
1978:
1970:
1952:
1948:
1944:
1938:
1935:
1915:
1908:
1892:
1872:
1869:
1866:
1844:
1840:
1817:
1813:
1790:
1786:
1763:
1759:
1755:
1752:
1730:
1726:
1722:
1719:
1697:
1693:
1688:
1682:
1678:
1674:
1671:
1663:
1661:
1644:
1636:
1632:
1628:
1624:
1620:
1618:
1601:
1593:
1592:
1591:
1589:
1581:
1579:
1576:
1574:
1568:
1566:
1565:example above
1558:
1556:
1553:
1551:
1546:
1544:
1540:
1536:
1528:
1526:
1512:
1508:
1504:
1496:
1480:
1472:
1456:
1448:
1443:
1441:
1437:
1432:
1429:
1427:
1422:
1406:
1402:
1397:
1391:
1387:
1378:
1377:example above
1370:
1365:
1360:
1356:
1338:
1334:
1330:
1327:
1324:
1319:
1315:
1306:
1302:
1299:
1295:
1291:
1288:
1284:
1280:
1277:
1274:are WUM with
1273:
1269:
1265:
1247:
1243:
1239:
1236:
1233:
1228:
1224:
1215:
1211:
1210:
1209:
1203:
1199:
1195:
1192:
1189:
1185:
1182:
1179:
1175:
1172:
1169:
1165:
1162:
1159:
1155:
1151:
1148:
1145:
1141:
1138:
1137:
1136:
1134:
1130:
1126:
1119:
1117:
1115:
1110:
1096:
1076:
1056:
1036:
1016:
1008:
1000:
998:
982:
978:
950:
946:
936:
932:
923:
919:
910:
905:
902:
899:
895:
887:
886:
885:
883:
867:
838:
834:
825:
821:
814:
809:
806:
803:
799:
791:
790:
789:
787:
783:
779:
763:
754:
738:
734:
730:
727:
724:
721:
718:
713:
709:
705:
702:
680:
676:
655:
635:
615:
607:
593:
571:
567:
546:
526:
517:
501:
474:
462:
460:
458:
454:
449:
439:
434:
431:
430:
424:
419:
416:
415:
411:
408:
405:
404:
401:
395:
393:
391:
387:
383:
379:
375:
372:(also called
371:
360:
355:
353:
348:
346:
341:
340:
338:
337:
334:
331:
330:
322:
319:
317:
314:
312:
309:
307:
304:
302:
299:
298:
291:
290:
282:
279:
277:
274:
272:
269:
267:
264:
262:
259:
258:
251:
250:
242:
239:
237:
234:
232:
229:
227:
224:
222:
219:
217:
214:
212:
209:
207:
204:
202:
199:
197:
194:
193:
186:
185:
177:
174:
172:
169:
167:
164:
162:
159:
157:
154:
152:
149:
147:
144:
142:
139:
138:
131:
130:
122:
119:
117:
114:
112:
109:
107:
104:
102:
99:
98:
94:
89:
88:
80:
79:William Paley
77:
75:
72:
70:
67:
65:
62:
60:
57:
55:
52:
50:
47:
45:
42:
41:
34:
33:
30:
26:
22:
18:
17:
2497:Cake-cutting
2442:
2438:
2428:
2385:
2381:
2371:
2359:. Retrieved
2353:
2342:
2332:
2325:
2315:
2308:
2287:
2268:
2264:
2258:
2241:
2237:
2231:
2197:
2193:
2187:
2135:
2133:
2121:disconnected
2120:
2118:
2097:
2089:
2031:
2023:
2013:
1999:
1906:
1659:
1630:
1622:
1616:
1585:
1577:
1569:
1562:
1554:
1547:
1532:
1470:
1446:
1444:
1433:
1430:
1425:
1423:
1421:is largest.
1374:
1358:
1354:
1304:
1297:
1293:
1286:
1282:
1275:
1271:
1267:
1263:
1213:
1207:
1193:
1187:
1183:
1177:
1173:
1167:
1163:
1157:
1153:
1149:
1143:
1139:
1132:
1123:
1113:
1111:
1004:
969:
881:
859:
785:
781:
777:
755:
608:
518:
466:
453:proportional
450:
447:
399:
381:
373:
369:
368:
189:Key concepts
121:Peter Singer
37:Predecessors
2218:. See also
1142:: the rule
1127:suggests a
778:utilitarian
756:A division
306:Game theory
2491:Categories
2452:1703.08928
2395:1510.05229
2361:6 December
2200:(2): 203.
2179:References
2136:connected.
2100:valuations
2034:valuations
2016:valuations
1662:valuations
1619:valuations
1353:such that
1198:closed set
970:where the
880:is called
776:is called
668:is called
519:There are
221:Eudaimonia
171:Preference
116:R. M. Hare
54:David Hume
49:Shantideva
2469:0165-4896
2445:: 19–30.
2412:1432-0479
2299:1203.0100
2214:118505359
2002:equitable
1953:ϵ
1939:
1916:ϵ
1893:ϵ
1867:ϵ
1845:∗
1818:∗
1791:∗
1764:∗
1731:∗
1723:≥
1328:…
1237:…
896:∑
800:∑
731:⊔
719:⊔
457:envy-free
409:Chocolate
216:Happiness
201:Suffering
176:Classical
156:Two-level
2477:16282641
2319:. AAMAS.
2141:See also
1778:, where
1276:the same
463:Notation
412:Vanilla
254:Problems
206:Pleasure
141:Negative
21:a series
19:Part of
2336:. AAAI.
2082:50/111
1539:NP-hard
406:Partner
396:Example
211:Utility
166:Average
2475:
2467:
2420:179618
2418:
2410:
2275:
2212:
2108:ε
2104:ε
2079:10/111
2076:51/111
2065:51/101
2062:50/101
2051:50/101
2048:51/101
2045:Alice
1200:under
786:maxsum
695:, and
432:George
2473:S2CID
2447:arXiv
2416:S2CID
2390:arXiv
2294:arXiv
2267:[
2210:S2CID
2073:Carl
1907:minus
1629:with
1543:FPTAS
1447:Proof
1114:every
1089:, so
417:Alice
161:Total
2465:ISSN
2408:ISSN
2363:2015
2273:ISBN
2127:and
2059:Bob
1971:For
1870:>
1756:<
1637:For
1594:For
1471:same
1268:some
196:Pain
146:Rule
44:Mozi
2457:doi
2400:doi
2246:doi
2242:121
2202:doi
1936:log
1303:If
1292:If
1281:If
1212:If
784:or
780:or
382:sum
151:Act
2493::
2471:.
2463:.
2455:.
2443:95
2441:.
2437:.
2414:.
2406:.
2398:.
2386:68
2384:.
2380:.
2240:.
2208:.
2198:43
2196:.
2110:).
2068:0
2054:0
2024:no
1631:nm
1575:.
1442:.
1135::
753:.
516:.
392:.
23:on
2479:.
2459::
2449::
2422:.
2402::
2392::
2365:.
2302:.
2296::
2281:.
2252:.
2248::
2226:.
2216:.
2204::
1979:n
1968:.
1956:)
1949:/
1945:1
1942:(
1873:0
1841:r
1814:r
1787:r
1760:r
1753:r
1727:r
1720:r
1698:2
1694:u
1689:/
1683:1
1679:u
1675:=
1672:r
1645:2
1623:m
1602:n
1513:n
1509:/
1505:1
1481:k
1457:k
1407:i
1403:w
1398:/
1392:i
1388:V
1359:R
1355:R
1339:n
1335:w
1331:,
1325:,
1320:1
1316:w
1305:R
1298:R
1294:R
1287:R
1283:R
1272:R
1264:R
1248:n
1244:w
1240:,
1234:,
1229:1
1225:w
1214:R
1204:.
1188:R
1178:R
1168:R
1160:.
1158:R
1154:R
1144:R
1133:R
1097:X
1077:X
1057:Y
1037:X
1017:Y
983:i
979:w
951:i
947:w
942:)
937:i
933:X
929:(
924:i
920:V
911:n
906:1
903:=
900:i
868:X
844:)
839:i
835:X
831:(
826:i
822:V
815:n
810:1
807:=
804:i
764:X
739:n
735:X
728:.
725:.
722:.
714:1
710:X
706:=
703:C
681:i
677:X
656:i
636:n
616:C
594:C
572:i
568:V
547:i
527:n
502:d
497:R
475:C
441:4
436:6
426:1
421:9
358:e
351:t
344:v
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