22:
698:. Note that, with binary valuations, EF1 is equivalent to EFX, but weaker than EFX0. A unanimously-EFX0 allocation might not exist if the group sizes are (1,2); this is in contrast to the situation with individual agents, that is, group sizes (1,1), where an EFX0 allocation always exists even for monotone valuations.
1299:
mechanism orders agents uniformly at random and awards each their request as long as there is available capacity. This common mechanism might yield arbitrarily unfair and inefficient outcomes. The
Weighted Individual Lottery is an alternative mechanism in which the processing order is biased to favor
260:
requires that, in each group, a certain fraction of the agents agree that the division is fair; preferredly this fraction should be at least 1/2. A practical situation in which such requirement may be useful is when two democratic countries agree to divide a certain disputed land among them, and the
133:
of agents, rather than among individual agents. After the division, all members in each group consume the same share, but they may have different preferences; therefore, different members in the same group might disagree on whether the allocation is fair or not. Some examples of group fair division
1283:
A practical application of fair division among groups is dividing tickets to parks or other experiences with limited capacity. Often, tickets are divided at random. When people arrive on their own, a simple uniformly-random lottery among all candidates is a fair solution. But people often come in
371:
for any number of groups. Unanimous-envy-freeness is compatible with Pareto-efficiency for 2 groups, but not for 3 or more groups. 1/2-democratic envy-freeness is compatible with Pareto-efficiency for 2 groups, but not for 5 or more groups. It is open whether they are compatible for 3 or 4
172:
Some 30 people want to use the local basketball court. Each game involves 10 players with different preferences regarding which time is better. It is required to partition the time of day into 3 parts and partition the players into 3 groups and assign a group to each
1258:
in the paper) may not exist when the cost-sharing policy is equal or proportional, but always exists with free cost-sharing policy. Moreover, a unanimously-envy-free allocation with free cost-sharing that maximizes the total rent can be found in polynomial time.
1354:
is a resource that is consumed simultaneously by all members in a single group ("club"), but is excluded from members of other groups. In the group fair division problem, all allocated goods are club goods in the group they are allocated
556:-1) or (2,2) or (2,3). The positive results are attainable by polynomial-time algorithms. In all other cases, there are instances in which at least one agent with a positive MMS gets a zero value in all allocations.
233:
assigns to each group a certain aggregate function, such as: sum, product, arithmetic mean or geometric mean. It requires that the allocation be considered fair according to this aggregate function. For example:
274:
is another important criterion that is required in addition to fairness. It is defined in the usual way: no allocation is better for at least one individual agent and at least as good for all individual agents.
836:
147:
Two neighboring countries want to divide a disputed region among them. The citizens in each country differ on which parts of the region are more important. This is a common obstacle to resolving international
1292:
mechanism orders groups uniformly at random, and processes them sequentially as long as there is available capacity. This natural mechanism might be unfair and inefficient; there are some better alternatives.
167:
In all the above examples, the groups are fixed in advance. In some settings, the groups can be determined ad-hoc, that is, people can be grouped based on their preferences. An example of such a setting is:
906:
728:
is true, then a unanimously-EF1 balanced allocation exists also for group sizes (1,4), (2,3) and arbitrary monotone valuations. A unanimously-EFX allocation might not exist if the group sizes are (1,2).
141:
A partnership is dissolved, and its assets should be divided among the partners. The partners are firms; each firm has several stockholders, who might disagree regarding which asset is more important.
696:
144:
The university management wants to allocate some meeting-rooms among its departments. In each department there are several faculty members, with differing opinions about which rooms are better.
1226:. If envy-freeness is relaxed to proportionality or maximin-share, then similar guarantees can be attained using a polynomial-time algorithm. For groups with additive valuations, a variant of
138:
Several siblings inherited some houses from their parents and have to divide them. Each sibling has a family, whose members may have different opinions regarding which house is better.
974:
1092:
1159:
268:
Unanimous-fairness implies both aggregate-fairness and democratic-fairness. Aggregate-fairness and democratic fairness are independent - none of them implies the other.
1284:
families or groups of friends, who want to enter together. This leads to various considerations in how exactly to design the lottery. The following results are known:
361:>2 groups, connected 1/2-democratic fair allocations might not exist, but the number of required components is smaller than for unanimous-proportional allocations.
1203:
For two groups with general monotone valuations, there always exists a 1/2-democratic envy-free-except-1 allocation, and it can be found by an efficient algorithm.
567:-1 groups contain a single agent; in contrast, if all groups contain 2 agents and one group contains at least 5 agents, then no positive approximation is possible.
1185:
586:, a unanimously-EF1 allocation exists if the group sizes are (1,5) or (2,3), but might not exist if the group sizes are (1,6) or (2,4) or (3,3). In general, an EF
1021:(with any number of agents), there always exists a 1/2-democratic envy-free-except-1 allocation. The constant 1/2 is tight even if we allow envy-free-except-
39:
353:: 1/2-democratic proportional and 1/2-democratic envy-free allocations always exist. With two groups, there exist such allocations that are also
1924:
1878:
253:
if, for each group, the product of agents' values of the group share is at least the product of their values of the share of any other group.
768:
86:
58:
552:, a positive multiplicative approximation to MMS-fairness can be guaranteed if-and-only-if the numbers of agents in the groups are (1,
845:
343:
connected components (that is, each group may get a connected piece). However, they cannot be found using a finite algorithm in the
105:
434:
for more information). It can be proved that unanimous-proportionality is equivalent to consensus division in the following sense:
65:
1861:
Ghodsi, Mohammad; Latifian, Mohamad; Mohammadi, Arman; Moradian, Sadra; Seddighin, Masoud (2018). "Rent
Division Among Groups".
735:, with any number of agents with arbitrary monotone valuations, there exists a unanimously-EF1 allocation. There also exists a
376:
The division problem is easier when the agents can be grouped ad-hoc based on their preferences. In this case, there exists a
344:
593:
306:
43:
72:
1191:
allocation can be improved from 1/2 to 3/5; it is an open question whether the upper bound of 3/4 can always be attained.
1726:
Manurangsi, Pasin; Suksompong, Warut (2022). "Almost envy-freeness for groups: Improved bounds via discrepancy theory".
302:: Unanimous-proportional and unanimous-envy-free allocations always exist. However, they may be disconnected: at least
54:
1227:
1099:
513:-1) cuts, and that finding an approximate unanimous-proportional division is in PPA. The number of cuts is tight for
1361:
is a subset of items that is considered, by all people in a certain group, to be at least as good as its complement.
1327:. Its properties are maximal in the sense that it is impossible to improve one property without harming another one.
1033:. A different fairness notion, that can be guaranteed to more than 1/2 of the agents in each group, is the ordinal
1303:
The
Iterative Probability Maximization algorithm finds a lottery that maximizes the smallest utility (based on the
32:
1324:
163:
or different moods. Such people can be represented as a group of agents, each of whom has a different preference.
1507:
Segal-Halevi, Erel; Suksompong, Warut (2 January 2021). "How to Cut a Cake Fairly: A Generalization to Groups".
329:) components are always sufficient for a unanimous-envy-free allocation. It is an open question whether or not
1771:
Manurangsi, Pasin; Suksompong, Warut (1 September 2017). "Asymptotic existence of fair divisions for groups".
223:
if every agent in every group values his/her group's share at least as much as the share of any other group.
1994:
1816:
Segal-Halevi, Erel; Suksompong, Warut (December 2019). "Democratic fair allocation of indivisible goods".
1553:
1411:
724:), a unanimously-EF1 balanced allocation exists if the group sizes are (1,2). If a certain conjecture on
935:
716:
183:
79:
1456:
941:
200:
requires that the allocation be considered fair in the eyes of all agents in all groups. For example:
1210:-democratic envy-free-except-1 allocation; with general monotone valuations, there always exists a 1/
531:
156:
1106:
inside each group. The upper bound on the fraction of agents that can be guaranteed 1 of their best
1337:
1312:
1044:
750:, with any number of agents with additive valuations, there exists a unanimously-PROP*1 allocation.
191:
1681:
Plaut, Benjamin; Roughgarden, Tim (January 2020). "Almost Envy-Freeness with
General Valuations".
1971:
1930:
1902:
1843:
1825:
1798:
1780:
1753:
1735:
1708:
1690:
1663:
1621:
1603:
1576:
1534:
1516:
1468:
1434:
1117:
1017:
987:
909:
505:
families. In particular, it implies that exact unanimous-proportional division can be done with (
242:
if, for each group, the arithmetic mean of the agents' values to the group share is at least 1/
1920:
1874:
1486:
1385:
368:
284:
271:
908:. All bounds are asymptotically tight when the number of groups is constant. The proofs use
743:
allocation of the goods. The EF1 cannot be strengthened to EFX even with additive valuations.
1961:
1912:
1866:
1835:
1790:
1745:
1700:
1655:
1613:
1568:
1526:
1478:
1426:
1358:
1316:
1304:
1198:
to decide if a given instance admits an allocation that gives each agent a positive utility.
721:
1640:
501:
pieces implies a solution to unanimous-proportional division among n+1 agents grouped into
1999:
1274:
in the paper) always exists when the cost-sharing policy is equal or proportional or free.
1103:
1164:
466:
pieces. In particular, it implies that unanimous-proportional division requires at least
339:: Average-proportional and average-envy-free allocations always exist, and require only
1295:
If agents may request multiple tickets without identifying members of their group, the
1262:
With ad-hoc groups, unanimous envy-freeness exists even with equal cost-sharing policy.
1244:
930:
groups contain the same number of agents, and their valuations are drawn at random, an
431:
390:
160:
478:-1 cuts is FIXP-hard, and finding an approximate unanimous-proportional division with
1988:
1975:
1934:
1847:
1794:
1757:
1712:
1667:
1617:
1594:
Suksompong, Warut (1 March 2018). "Approximate maximin shares for groups of agents".
1538:
1320:
1308:
1034:
539:
187:
126:
1802:
1639:
Kyropoulou, Maria; Suksompong, Warut; Voudouris, Alexandros A. (12 November 2020).
1625:
1580:
1438:
1300:
agents with smaller requests. It is approximately fair and approximately efficient.
1098:
MMS-fair allocation. These allocations can be found efficiently using a variant of
725:
1530:
1206:
For three or more groups with binary additive valuations, there always exists a 1/
325:) components are always sufficient for a unanimous-proportional allocation, and O(
1870:
1839:
976:, and can be attained by a greedy algorithm that maximizes the sum of utilities.
194:. There are several ways to extend these criteria to fair division among groups.
1315:, and attains a 1/2-factor approximation of the maximum utilization. It is also
151:
The "group of agents" may also represent different conflicting preferences of a
21:
1572:
1430:
1749:
1659:
1482:
1389:
262:
1966:
1949:
1490:
1344:
agents. It says that, after each individual agent gets his private share, no
563:, a positive multiplicative approximation to MMS-fairness can be attained if
190:, judge the division from the point-of-view of a single agent, with a single
1916:
1351:
931:
380:
allocation for any number of groups and any number of agents in each group.
227:
Unanimous fairness is a strong requirement, and often cannot be satisfied.
1288:
For the setting in which all group members are identified in advance, the
1897:
Arnosti, Nick; Bonet, Carlos (2022). "Lotteries for Shared
Experiences".
1247:(envy-free division of rooms and rent), the following results are known.
208:
if every agent in every group values his/her group's share as at least 1/
1865:. Lecture Notes in Computer Science. Vol. 11346. pp. 577–591.
1379:
1704:
1195:
702:
1003:, then with high probability, an envy-free allocation does not exist.
1907:
1899:
Proceedings of the 23rd ACM Conference on
Economics and Computation
1830:
1785:
1740:
1695:
1608:
1521:
1473:
831:{\displaystyle O({\sqrt {n}})\geq c\geq \Omega ({\sqrt {n/k^{3}}})}
761:
groups, there always exists an allocation that is envy-free up to
705:
to decide if a given instance admits a unanimously-EF1 allocation.
426:
cuts is FIXP-hard, and finding an approximate exact division with
901:{\displaystyle O({\sqrt {n}})\geq c\geq \Omega ({\sqrt {n/k}})}
159:, people often change their preferences according to different
15:
1954:
Proceedings of the AAAI Conference on
Artificial Intelligence
474:
components), finding a unanimous-proportional division with
367:: All three variants of proportionality are compatible with
406:
pieces such that all agents value all pieces at exactly 1/
1214:-democratic envy-free-except-2 allocation. The factor 1/
458:
families implies a solution to consensus division among
691:{\displaystyle ({2c+1 \choose c+1},{2c+1 \choose c+1})}
446:, a solution to unanimous-proportional division among
1167:
1120:
1047:
944:
848:
771:
596:
129:
problems, in which the resources are allocated among
1552:
Segal-Halevi, Erel; Nitzan, Shmuel (December 2019).
1410:
Segal-Halevi, Erel; Nitzan, Shmuel (December 2019).
1348:
of agents envies another coalition of the same size.
1641:"Almost envy-freeness in group resource allocation"
1230:
can be used to find a 1/3-democratic 1-out-of-best-
1029:. The same is true also for proportionality-except-
402:agents, and the goal is to partition the cake into
46:. Unsourced material may be challenged and removed.
1381:Resource allocation and decision making for groups
1179:
1153:
1086:
968:
900:
830:
690:
590:allocation might not exist if the group sizes are
679:
647:
635:
603:
357:, and they can be found in polynomial time. With
295:is the number of agents in all groups together).
1340:is a fairness criterion for fair division among
1455:Bade, Sophie; Segal-Halevi, Erel (2023-09-01).
838:. The same is true for proportionality up to
739:partition of the agents and a unanimously-EF1
842:items. For consensus division the bounds are
8:
1948:Arbiv, Tal; Aumann, Yonatan (28 June 2022).
994:-envy-free allocation with high probability.
384:Unanimous proportionality and exact division
1863:Combinatorial Optimization and Applications
410:. It is known that an exact division with
1965:
1906:
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1784:
1739:
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1607:
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817:
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803:
778:
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678:
646:
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634:
602:
600:
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287:, the following results are known (where
106:Learn how and when to remove this message
1370:
1110:items (a property weaker than 1-out-of-
999:If the number of goods is in less than
313:components are always sufficient. With
1950:"Fair and Truthful Giveaway Lotteries"
1239:Group fair division of items and money
1892:
1890:
493:, a solution to exact-division among
418:-1) always exists. However, even for
7:
1683:SIAM Journal on Discrete Mathematics
1502:
1500:
1450:
1448:
1405:
1403:
1401:
1399:
1187:, the lower bound for 1-out-of-best-
757:agents partitioned arbitrarily into
309:might be required. With two groups,
44:adding citations to reliable sources
572:Unanimous approximate envy-freeness
534:, the following results are known.
422:=2, finding an exact division with
1554:"Fair cake-cutting among families"
1412:"Fair cake-cutting among families"
945:
874:
797:
651:
607:
261:agreement should be approved by a
182:Common fairness criteria, such as
14:
1509:The American Mathematical Monthly
1279:Fair division of ticket lotteries
1037:approximation. For every integer
333:components are always sufficient.
1795:10.1016/j.mathsocsci.2017.05.006
1618:10.1016/j.mathsocsci.2017.09.004
1457:"Fairness for multi-self agents"
969:{\displaystyle \Omega (n\log n)}
20:
979:The results can be extended to
279:Results for divisible resources
31:needs additional citations for
1218:is tight for envy-free-except-
1148:
1121:
1081:
1048:
963:
948:
895:
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862:
852:
825:
800:
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775:
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597:
1:
1531:10.1080/00029890.2021.1835338
1087:{\displaystyle (1-1/2^{c-1})}
938:if the number of goods is in
526:Results for indivisible items
378:unanimous envy-free connected
291:is the number of groups, and
1871:10.1007/978-3-030-04651-4_39
1840:10.1016/j.artint.2019.103167
1773:Mathematical Social Sciences
1728:Theoretical Computer Science
1648:Theoretical Computer Science
1596:Mathematical Social Sciences
1222:allocation for any constant
1025:allocation for any constant
55:"Fair division among groups"
1461:Games and Economic Behavior
1228:round-robin item allocation
1154:{\displaystyle (1-1/2^{c})}
1100:round-robin item allocation
2016:
1573:10.1007/s00355-019-01210-9
1431:10.1007/s00355-019-01210-9
1378:Suksompong, Warut (2018).
584:binary additive valuations
454:-1)+1 agents grouped into
430:cuts is PPA-complete (see
345:Robertson–Webb query model
212:of the total value, where
119:Fair division among groups
1750:10.1016/j.tcs.2022.07.022
1660:10.1016/j.tcs.2020.07.008
1561:Social Choice and Welfare
1483:10.1016/j.geb.2023.06.004
1419:Social Choice and Welfare
1967:10.1609/aaai.v36i5.20405
517:=2 families but not for
216:is the number of groups.
206:unanimously-proportional
1917:10.1145/3490486.3538312
1818:Artificial Intelligence
1272:aggregate envy-freeness
1252:Unanimous envy-freeness
917:Unanimous envy-freeness
365:Fairness and efficiency
155:person. As observed in
1901:. pp. 1179–1180.
1181:
1155:
1088:
970:
902:
832:
692:
538:Unanimous approximate
1268:Average envy-freeness
1182:
1156:
1094:-democratic 1-out-of-
1089:
983:with different sizes.
971:
936:with high probability
920:with high probability
903:
833:
717:responsive valuations
693:
249:A division is called
238:A division is called
221:unanimously-envy-free
219:A division is called
204:A division is called
1256:strong envy-freeness
1165:
1118:
1045:
1014:For two groups with
942:
846:
769:
594:
561:three or more groups
532:fair item allocation
482:-1 cuts is PPA-hard.
307:connected components
240:average-proportional
157:behavioral economics
40:improve this article
1338:Group envy-freeness
1313:group strategyproof
1180:{\displaystyle c=2}
1018:additive valuations
1008:Democratic fairness
722:additive valuations
351:Democratic fairness
265:in both countries.
258:Democratic fairness
246:of the total value.
192:preference relation
1705:10.1137/19M124397X
1297:Individual Lottery
1243:In the context of
1177:
1151:
1084:
988:truthful mechanism
966:
934:allocation exists
910:discrepancy theory
898:
828:
688:
530:In the context of
396:consensus division
337:Aggregate fairness
300:Unanimous fairness
283:In the context of
231:Aggregate fairness
198:Unanimous fairness
1926:978-1-4503-9150-4
1880:978-3-030-04650-7
1041:, there exists a
893:
860:
823:
783:
733:two ad-hoc groups
677:
633:
369:Pareto-efficiency
285:fair cake-cutting
272:Pareto efficiency
251:product-envy-free
178:Fairness criteria
116:
115:
108:
90:
2007:
1980:
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1969:
1960:(5): 4785–4792.
1945:
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1807:
1806:
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1768:
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1761:
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1717:
1716:
1698:
1689:(2): 1039–1068.
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1611:
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1584:
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1359:Agreeable subset
1332:Related concepts
1317:Pareto-efficient
1305:egalitarian rule
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1184:
1183:
1178:
1160:
1158:
1157:
1152:
1147:
1146:
1137:
1102:, with weighted
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1090:
1085:
1080:
1079:
1064:
990:that attains an
986:There is also a
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317:>2 groups, O(
125:) is a class of
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24:
16:
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1376:
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1334:
1281:
1241:
1163:
1162:
1138:
1116:
1115:
1104:approval voting
1065:
1043:
1042:
940:
939:
844:
843:
813:
767:
766:
748:k ad-hoc groups
720:(a superset of
714:of agents with
710:When there are
666:
652:
645:
622:
608:
601:
592:
591:
582:of agents with
578:When there are
559:When there are
548:When there are
528:
386:
281:
184:proportionality
180:
112:
101:
95:
92:
49:
47:
37:
25:
12:
11:
5:
2013:
2011:
2003:
2002:
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1981:
1940:
1925:
1886:
1879:
1853:
1808:
1763:
1718:
1673:
1631:
1586:
1567:(4): 709–740.
1544:
1496:
1444:
1425:(4): 709–740.
1395:
1369:
1367:
1364:
1363:
1362:
1356:
1349:
1333:
1330:
1329:
1328:
1301:
1293:
1280:
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1275:
1265:
1264:
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1245:rental harmony
1240:
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888:
884:
879:
876:
873:
870:
867:
864:
859:
854:
851:
827:
820:
816:
811:
807:
802:
799:
796:
793:
790:
787:
782:
777:
774:
751:
744:
729:
708:
707:
706:
687:
681:
675:
672:
669:
664:
661:
658:
655:
649:
643:
637:
631:
628:
625:
620:
617:
614:
611:
605:
599:
569:
568:
557:
527:
524:
523:
522:
483:
432:exact division
391:exact division
385:
382:
374:
373:
362:
348:
334:
280:
277:
255:
254:
247:
225:
224:
217:
179:
176:
175:
174:
165:
164:
161:frames of mind
149:
145:
142:
139:
134:settings are:
114:
113:
28:
26:
19:
13:
10:
9:
6:
4:
3:
2:
2012:
2001:
1998:
1996:
1995:Fair division
1993:
1992:
1990:
1977:
1973:
1968:
1963:
1959:
1955:
1951:
1944:
1941:
1936:
1932:
1928:
1922:
1918:
1914:
1909:
1904:
1900:
1893:
1891:
1887:
1882:
1876:
1872:
1868:
1864:
1857:
1854:
1849:
1845:
1841:
1837:
1832:
1827:
1823:
1819:
1812:
1809:
1804:
1800:
1796:
1792:
1787:
1782:
1778:
1774:
1767:
1764:
1759:
1755:
1751:
1747:
1742:
1737:
1733:
1729:
1722:
1719:
1714:
1710:
1706:
1702:
1697:
1692:
1688:
1684:
1677:
1674:
1669:
1665:
1661:
1657:
1653:
1649:
1642:
1635:
1632:
1627:
1623:
1619:
1615:
1610:
1605:
1601:
1597:
1590:
1587:
1582:
1578:
1574:
1570:
1566:
1562:
1555:
1548:
1545:
1540:
1536:
1532:
1528:
1523:
1518:
1514:
1510:
1503:
1501:
1497:
1492:
1488:
1484:
1480:
1475:
1470:
1466:
1462:
1458:
1451:
1449:
1445:
1440:
1436:
1432:
1428:
1424:
1420:
1413:
1406:
1404:
1402:
1400:
1396:
1391:
1387:
1383:
1382:
1374:
1371:
1365:
1360:
1357:
1353:
1350:
1347:
1343:
1339:
1336:
1335:
1331:
1326:
1322:
1318:
1314:
1310:
1309:leximin order
1306:
1302:
1298:
1294:
1291:
1290:Group Lottery
1287:
1286:
1285:
1278:
1273:
1269:
1266:
1261:
1260:
1257:
1253:
1250:
1249:
1248:
1246:
1238:
1233:
1229:
1225:
1221:
1217:
1213:
1209:
1205:
1202:
1197:
1193:
1192:
1190:
1174:
1171:
1168:
1143:
1139:
1134:
1130:
1127:
1124:
1113:
1109:
1105:
1101:
1097:
1076:
1073:
1070:
1066:
1061:
1057:
1054:
1051:
1040:
1036:
1035:maximin-share
1032:
1028:
1024:
1020:
1019:
1013:
1012:
1011:
1009:
1002:
998:
993:
992:approximately
989:
985:
982:
978:
977:
960:
957:
954:
951:
937:
933:
929:
925:
924:
923:
921:
918:
911:
890:
886:
882:
871:
868:
865:
857:
849:
841:
818:
814:
809:
805:
794:
791:
788:
780:
772:
765:items, where
764:
760:
756:
752:
749:
745:
742:
738:
734:
730:
727:
726:Kneser graphs
723:
719:
718:
713:
709:
704:
700:
699:
673:
670:
667:
662:
659:
656:
653:
641:
629:
626:
623:
618:
615:
612:
609:
589:
585:
581:
577:
576:
575:
573:
566:
562:
558:
555:
551:
547:
546:
545:
543:
541:
540:maximin-share
535:
533:
525:
520:
516:
512:
508:
504:
500:
496:
492:
488:
484:
481:
477:
473:
469:
465:
461:
457:
453:
449:
445:
441:
437:
436:
435:
433:
429:
425:
421:
417:
413:
409:
405:
401:
398:), there are
397:
394:(also called
393:
392:
383:
381:
379:
370:
366:
363:
360:
356:
352:
349:
346:
342:
338:
335:
332:
328:
324:
320:
316:
312:
308:
305:
301:
298:
297:
296:
294:
290:
286:
278:
276:
273:
269:
266:
264:
259:
252:
248:
245:
241:
237:
236:
235:
232:
228:
222:
218:
215:
211:
207:
203:
202:
201:
199:
195:
193:
189:
188:envy-freeness
185:
177:
171:
170:
169:
162:
158:
154:
150:
146:
143:
140:
137:
136:
135:
132:
128:
127:fair division
124:
120:
110:
107:
99:
96:December 2021
88:
85:
81:
78:
74:
71:
67:
64:
60:
57: –
56:
52:
51:Find sources:
45:
41:
35:
34:
29:This article
27:
23:
18:
17:
1957:
1953:
1943:
1898:
1862:
1856:
1821:
1817:
1811:
1776:
1772:
1766:
1731:
1727:
1721:
1686:
1682:
1676:
1651:
1647:
1634:
1599:
1595:
1589:
1564:
1560:
1547:
1515:(1): 79–83.
1512:
1508:
1464:
1460:
1422:
1418:
1380:
1373:
1345:
1341:
1296:
1289:
1282:
1271:
1267:
1255:
1251:
1242:
1231:
1223:
1219:
1215:
1211:
1207:
1188:
1111:
1107:
1095:
1038:
1030:
1026:
1022:
1015:
1007:
1006:
1000:
991:
980:
927:
919:
916:
915:
839:
762:
758:
754:
747:
740:
736:
732:
715:
711:
587:
583:
579:
571:
570:
564:
560:
553:
549:
537:
536:
529:
518:
514:
510:
506:
502:
498:
494:
490:
486:
479:
475:
471:
467:
463:
462:agents with
459:
455:
451:
447:
443:
439:
427:
423:
419:
415:
411:
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403:
399:
395:
389:
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364:
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340:
336:
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326:
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318:
314:
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292:
288:
282:
270:
267:
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230:
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220:
213:
209:
205:
197:
196:
181:
166:
152:
130:
122:
118:
117:
102:
93:
83:
76:
69:
62:
50:
38:Please help
33:verification
30:
1779:: 100–108.
1734:: 179–195.
1654:: 110–123.
1467:: 321–336.
1234:allocation.
497:agents and
1989:Categories
1908:2205.10942
1831:1709.02564
1824:: 103167.
1786:1706.08219
1741:2105.01609
1696:1707.04769
1609:1706.09869
1522:2001.03327
1474:1811.06684
1390:1050345365
1384:(Thesis).
1366:References
1342:individual
981:two groups
712:two groups
580:two groups
550:two groups
485:For every
438:For every
263:referendum
173:time-slot.
66:newspapers
1976:250288879
1935:248986158
1848:203034477
1758:233714947
1713:216283014
1668:220546580
1602:: 40–47.
1539:210157034
1491:0899-8256
1352:Club good
1346:coalition
1325:anonymous
1321:envy-free
1311:). It is
1128:−
1074:−
1055:−
958:
946:Ω
932:envy-free
926:When all
875:Ω
872:≥
866:≥
798:Ω
795:≥
789:≥
470:-1 cuts (
355:connected
148:disputes.
1803:47514346
1307:and the
1270:(called
1254:(called
1114:MMS) is
741:balanced
737:balanced
542:fairness
123:families
1626:3720438
1581:1602396
1439:1602396
1196:NP-hard
1016:binary
703:NP-hard
372:groups.
80:scholar
2000:Voting
1974:
1933:
1923:
1877:
1846:
1801:
1756:
1711:
1666:
1624:
1579:
1537:
1489:
1437:
1388:
1194:It is
1161:. For
701:It is
521:>2.
388:In an
153:single
131:groups
82:
75:
68:
61:
53:
1972:S2CID
1931:S2CID
1903:arXiv
1844:S2CID
1826:arXiv
1799:S2CID
1781:arXiv
1754:S2CID
1736:arXiv
1709:S2CID
1691:arXiv
1664:S2CID
1644:(PDF)
1622:S2CID
1604:arXiv
1577:S2CID
1557:(PDF)
1535:S2CID
1517:arXiv
1469:arXiv
1435:S2CID
1415:(PDF)
87:JSTOR
73:books
1921:ISBN
1875:ISBN
1487:ISSN
1386:OCLC
1323:and
753:For
746:For
731:For
509:-1)(
489:and
442:and
321:log
186:and
121:(or
59:news
1962:doi
1913:doi
1867:doi
1836:doi
1822:277
1791:doi
1746:doi
1732:930
1701:doi
1656:doi
1652:841
1614:doi
1569:doi
1527:doi
1513:128
1479:doi
1465:141
1427:doi
1355:to.
1010::
955:log
327:n k
42:by
1991::
1970:.
1958:36
1956:.
1952:.
1929:.
1919:.
1911:.
1889:^
1873:.
1842:.
1834:.
1820:.
1797:.
1789:.
1777:89
1775:.
1752:.
1744:.
1730:.
1707:.
1699:.
1687:34
1685:.
1662:.
1650:.
1646:.
1620:.
1612:.
1600:92
1598:.
1575:.
1565:53
1563:.
1559:.
1533:.
1525:.
1511:.
1499:^
1485:.
1477:.
1463:.
1459:.
1447:^
1433:.
1423:53
1421:.
1417:.
1398:^
1319:,
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574::
544::
1978:.
1964::
1937:.
1915::
1905::
1883:.
1869::
1850:.
1838::
1828::
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1783::
1760:.
1748::
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1670:.
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1628:.
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1571::
1541:.
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1481::
1471::
1441:.
1429::
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