74:, it is possible to lift the item-ranking to a partial bundle-ranking. For example, if the item-ranking is w>x>y>z, then responsiveness implies that {w,x}>{y,z} and {w,y}>{x,z}, but does not imply anything about the relation between {w,z} and {x,y}, between {x} and {y,z}, etc. Given an item-ranking:
972:
EFx allocation, where the Nash welfare is at least half of the maximum possible Nash welfare. In other words, after donating some items to a charity, the remaining items can be allocated in an EFx way. This bound is the best possible. In large markets, where the valuations of an agent for every item
57:
finds a complete EF allocation for two agents, if-and-only-if such allocation exists. It requires the agents to rank bundles of items, but it does not require cardinal utility information. It works whenever the agents' preference relations are strictly monotone, but does not need to assume that they
186:
preferences (with indifferences). In this setting, checking existence of WPEF is NP-complete. Deciding whether a PEF allocation exists is in P for strict preferences or for n=2, but it is NP-complete in general. It is an open question whether, when the number of agents is constant, deciding the
809:
returns a partial EF1 allocation. The only requirement is that the agents can rank bundles of items. A small number of items might remain unallocated, and a small number of items may have to be added. The allocation is Pareto-efficient with respect to the allocated
174:
preferences. They study the algorithmic questions of finding a NEF/PEF allocation with an additional efficiency condition, particularly, completeness or NPE or PPE. In general, PEF is easy while NEF is hard: checking whether a NEF allocation exists is
1063:
returns a complete and efficient EF allocation for two agents, but it might have to cut a single item (alternatively, one item remains in shared ownership). It requires the agents to report a numeric value for each item, and assumes that they have
817:
algorithm selects a complete allocation that maximizes the product of utilities. It requires each agent to provide a numeric valuation of each item, and assumes that the agents' utilities are additive. The resulting allocation is both EF1 and
215:
Deciding whether a fair allocation exists requires exponential communication (in the number of goods) when there are more than two agents. When there are two agents, the communication complexity depends on specific combinations of
916:, but there are at most two different values for goods. In this case, any max-Nash-welfare allocation is EFx. Moreover, there is an efficient algorithm for calculating an EFx allocation (though not necessarily max-Nash-welfare).
984:
In contrast to EF1, which requires a number of queries logarithmic in the number of items, computing an EFx allocation may require a linear number of queries even when there are two agents with identical additive valuations.
1016:
of a cake. Without this oracle, an EFm allocation can be computed in polynomial time in two special cases: two agents with general additive valuations, or any number of agents with piecewise-linear valuations.
2496:
Bilò, Vittorio; Caragiannis, Ioannis; Flammini, Michele; Igarashi, Ayumi; Monaco, Gianpiero; Peters, Dominik; Vinci, Cosimo; Zwicker, William S. (2022). "Almost envy-free allocations with connected bundles".
768:
if for every two agents A and B, if we remove at most one item from the bundle of B, then A does not envy B. An EF1 allocation always exists and can be found efficiently by various procedures, particularly:
988:
Another difference between EF1 and EFx is that the number of EFX allocations can be as few as 2 (for any number of items), while the number of EF1 allocations is always exponential in the number of items.
1087:
functions that are drawn from probability distributions satisfying some independence criteria, and the number of items is sufficiently large relative to the number of agents, then an EF allocation exists
1048:
in the following sense: no other EF allocation is better for one and weakly better for the other. The AL procedure only requires the agents to rank individual items. It assumes that the agents have
33:
Since the items are indivisible, an EF assignment may not exist. The simplest case is when there is a single item and at least two agents: if the item is assigned to one agent, the other will envy.
642:
2254:
De
Keijzer, Bart; Bouveret, Sylvain; Klos, Tomas; Zhang, Yingqian (2009). "On the Complexity of Efficiency and Envy-Freeness in Fair Division of Indivisible Goods with Additive Preferences".
1158:
834:
For two agents with arbitrary monotone valuations, or three agents with additive valuations, an EF1 allocation can be computed using a number of queries logarithmic in the number of items.
303:
258:
1361:
208:. This is true even when there are only two agents, and even when their utilities are additive and identical, since in this case finding an EF allocation is equivalent to solving the
1209:
1071:
When each agent may get at most a single item, and the valuations are binary (each agent either likes or dislikes each item), there is a polynomial-time algorithm that finds an
746:
700:
362:
2642:
Amanatidis, Georgios; Birmpas, Georgios; Filos-Ratsikas, Aris; Hollender, Alexandros; Voudouris, Alexandros A. (2021). "Maximum Nash welfare and other stories about EFX".
405:
1826:
1294:
1245:
1680:
1617:
539:
506:
473:
1905:
1553:
432:
1973:
1942:
1862:
1780:
1744:
1717:
1474:
1452:
2759:
Amanatidis, Georgios; Ntokos, Apostolos; Markakis, Evangelos (2020). "Multiple birds with one stone: Beating 1/2 for EFX and GMMS via envy cycle elimination".
796:
function, then the EF1 guarantee has an additional interpretation: the numeric envy-level of each agent is at most the maximal-marginal-utility - the largest
756:
Many procedures find an allocation that is "almost" envy-free, i.e., the level of envy is bounded. There are various notions of "almost" envy-freeness:
792:
finds a complete EF1 allocation. The only requirement is that the agents can rank bundles of items. If the agents' valuations are represented by a
1008:
If B's bundle contains only indivisible goods, then A does not envy B after removing at most one item from B's bundle (as in an EF1 allocation).
997:
Some division scenarios involve both divisible and indivisible items, such as divisible lands and indivisible houses. An allocation is called
196:
The empty allocation is always EF. But if we want some efficiency in addition to EF, then the decision problem becomes computationally hard:
1211:, then w.h.p. an EF allocation exists and can be found by maximizing the social welfare. This bound is also tight due to connections to the
827:
1028:
Rather than using a worst-case bound on the amount of envy, one can try to minimize the amount of envy in each particular instance. See
877:
item from the bundle of B, then A does not envy B. EFx is strictly stronger than EF1: EF1 lets us eliminate envy by removing the item
783:
finds a complete EF1 allocation. Weak additivity is required since each agent should be able to pick, in each situation, a "best item".
62:. In the worst case, the agents may have to rank all possible bundles, so the run-time might be exponential in the number of items.
2980:
Bei, Xiaohui; Li, Zihao; Liu, Jinyan; Liu, Shengxin; Lu, Xinhang (2021). "Fair division of mixed divisible and indivisible goods".
1251:
On the other hand, if the number of items is not sufficiently large, then, with high probability, an EF allocation does not exist.
3025:
Aigner-Horev, Elad; Segal-Halevi, Erel (2022). "Envy-free matchings in bipartite graphs and their applications to fair division".
40:. When monetary transfers are not allowed or not desired, there are allocation algorithms providing various kinds of relaxations.
981:
It is an open question whether an EFx allocation exists in general. The smallest open case is 4 agents with additive valuations.
837:
For two agents with arbitrary utility functions (not necessarily monotone), an EF1 allocation can be found in polynomial time.
2894:
2835:
2304:
2271:
2051:
563:
1479:
EF = envy-free, EF1 = envy-free except-1-item (weaker than EF), MEF1 = marginal-envy-free except-1-item (weaker than EF1).
306:
is the class of problems that can be solved in nondeterministic time given an oracle that can solve any problem in NP).
37:
2126:
Lipton, R. J.; Markakis, E.; Mossel, E.; Saberi, A. (2004). "On approximately fair allocations of indivisible goods".
2569:
2390:
2323:
Budish, Eric (2011). "The
Combinatorial Assignment Problem: Approximate Competitive Equilibrium from Equal Incomes".
2143:
2017:
441:
as a parameter, the existence of a PE+EF allocation remains hard. With dichotomous utilities, it is NP-hard even for
1099:
310:
The decision problem may become tractable when some parameters of the problem are considered fixed small constants:
2475:; Kobayashi, Yusuke; Makino, Kazuhisa (2020-06-08). "Envy-free Relaxations for Goods, Chores, and Mixed Items".
1160:, then w.h.p. an EF allocation exists (the probability goes to 1 as m goes to infinity) and can be found by the
30:- each agent should receive a bundle that they believe to be at least as good as the bundle of any other agent.
2547:
Caragiannis, Ioannis; Kurokawa, David; Moulin, Hervé; Procaccia, Ariel D.; Shah, Nisarg; Wang, Junxing (2016).
2368:
Caragiannis, Ioannis; Kurokawa, David; Moulin, Hervé; Procaccia, Ariel D.; Shah, Nisarg; Wang, Junxing (2016).
848:(when items are pre-ordered on a line, such as houses in a street). The proof uses an algorithm similar to the
274:
229:
1302:
3213:
3208:
3126:. In Proceedings of the Twenty-Eighth AAAI Conference on Artificial Intelligence (2014). pp. 1405–1411.
1212:
1247:
and m is divisible by n, then w.h.p. an EF allocation exists and can be found by a matching-based algorithm.
973:
is relatively small, the algorithm attains EFx with almost optimal Nash welfare. It is sufficient to donate
659:
as parameters, the existence of a PE+EF allocation for identical additive utilities can be decided using an
364:
for additive or dichotomous utilities. When the utilities are binary and/or identical, the runtime drops to
2738:
Chan, Hau; Chen, Jing; Li, Bo; Wu, Xiaowei (2019-10-25). "Maximin-Aware
Allocations of Indivisible Goods".
2223:"Efficiency and Envy-freeness in Fair Division of Indivisible Goods: Logical Representation and Complexity"
905:-optimal allocation is EFx and Pareto-optimal. However, it requires exponentially many queries to compute.
1695:
1170:
70:
It is usually easier for people to rank individual items than to rank bundles. Assuming all agents have
1044:
finds an EF allocation for two agents. It may discard some of the items, but, the final allocation is
2924:
2293:"Complexity of efficient and envy-free resource allocation: few agents, resources, or utility levels"
1631:
1060:
946:
A 1/2-approximate EFx allocation (that also satisfies a different approximate-fairness notion called
705:
148:
3132:
2337:
2044:
Proceedings of the 2010 Conference on ECAI 2010: 19th
European Conference on Artificial Intelligence
953:
A 0.618-approximate EFx allocation (that is also EF1 and approximates other fairness notions called
666:
2471:
Bérczi, Kristóf; Bérczi-Kovács, Erika R.; Boros, Endre; Gedefa, Fekadu
Tolessa; Kamiyama, Naoyuki;
1415:
is a strengthening of envy-freeness, which is applicable to both divisible and indivisible objects.
849:
321:
367:
2708:
Chaudhury, Bhaskar Ray; Garg, Jugal; Mehlhorn, Kurt (2020-05-30). "EFX Exists for Three Agents".
1399:
1020:
In contrast to EF1, which is compatible with Pareto-optimality, EFm may be incompatible with it.
955:
263:
3127:
2332:
2040:"Fair Division under Ordinal Preferences: Computing Envy-Free Allocations of Indivisible Goods"
1792:
1049:
992:
961:
660:
71:
59:
3122:
John P. Dickerson; Jonathan
Goldman; Jeremy Karp; Ariel D. Procaccia; Tuomas Sandholm (2014).
3121:
1258:
1221:
1012:
An EFm allocation exists for any number of agents. However, finding it requires an oracle for
1005:
If B's bundle contains some divisible goods, then A does not envy B (as in an EF allocation).
844:
agents with identical monotone valuations, there always exists an EF1 allocation that is also
2864:
2556:. Proceedings of the 2016 ACM Conference on Economics and Computation - EC '16. p. 305.
2377:. Proceedings of the 2016 ACM Conference on Economics and Computation - EC '16. p. 305.
1647:
1584:
1386:
1089:
787:
511:
478:
452:
2410:
2006:
Brandt, Felix; Conitzer, Vincent; Endriss, Ulle; Lang, JĂ©rĂ´me; Procaccia, Ariel D. (2016).
1878:
1758:
1531:
1420:
1161:
779:
410:
182:
Aziz, Gaspers, Mackenzie and Walsh study the more general setting in which agents may have
23:
1407:
Every Nash-optimal allocation (allocation that maximizes the product of utilities) is EF1.
856:> 4 agents, it is not known whether a connected EF1 allocation exists, but a connected
8:
2860:
2472:
2297:
Proceedings of the Twenty-Fifth
International Joint Conference on Artificial Intelligence
645:
3184:
3166:
3101:
3083:
3052:
3034:
3007:
2989:
2962:
2936:
2900:
2872:
2841:
2813:
2786:
2768:
2739:
2709:
2688:
2669:
2651:
2624:
2598:
2524:
2506:
2476:
2450:
2422:
2350:
2198:
2172:
2108:
2080:
1958:
1927:
1847:
1765:
1729:
1702:
1515:
1459:
1437:
1377:. This follows directly from the ordinal definitions and does not depend on additivity.
1072:
1065:
935:
786:
In the more general case, when all agents have monotonically increasing utilities, the
267:
54:
445:=2. However, it is now in NP, and can be solved efficiently with an NP oracle (e.g. a
434:
hides expressions that are polynomial in the other parameters (e.g. number of agents).
3188:
3105:
3056:
3011:
2966:
2954:
2904:
2890:
2831:
2790:
2673:
2628:
2616:
2565:
2442:
2386:
2354:
2300:
2292:
2267:
2202:
2190:
2139:
2100:
2047:
2039:
2013:
1029:
819:
549:
w.r.t. the number of agents even if all utilities are identical and encoded in unary.
209:
2845:
2586:
2528:
2454:
2160:
825:
Various other algorithms return EF1 allocations that are also Pareto-efficient; see
3176:
3154:
3093:
3071:
3044:
2999:
2946:
2882:
2871:, Proceedings, Society for Industrial and Applied Mathematics, pp. 2658–2672,
2823:
2812:. EC '19. Phoenix, AZ, USA: Association for Computing Machinery. pp. 527–545.
2806:"Envy-Freeness up to Any Item with High Nash Welfare: The Virtue of Donating Items"
2778:
2661:
2608:
2557:
2516:
2432:
2378:
2342:
2288:
2259:
2234:
2182:
2131:
2112:
2090:
1840:
1381:
1084:
1045:
805:
797:
793:
221:
1363:
and m is not "almost divisible" by n, then w.h.p. an EF allocation does not exist.
881:(for A) from B's bundle; EFx requires that we eliminate envy by removing the item
65:
3003:
2437:
2095:
2068:
2007:
1410:
803:
When agents have arbitrary utilities (not necessarily additive or monotone), the
774:
541:
oracles are needed unless P=NP. With additive utilities, it is NP-hard even for
2886:
2263:
2025:
1394:. Otherwise, an EF allocation may be not proportional and even not max-min-fair.
2548:
2369:
1013:
3048:
2950:
2782:
2665:
2520:
3202:
2958:
2620:
2446:
2194:
2104:
1914:
27:
3138:
2827:
2687:
Mahara, Ryoga (2020-08-20). "Existence of EFX for Two
Additive Valuations".
2561:
2382:
993:
EFm - approximate envy-free for a mixture of divisible and indivisible items
1568:
1041:
318:
as a parameter, the existence of a PE+EF allocation can be decided in time
2805:
2135:
560:
as parameters, the existence of a PE+EF allocation can be decided in time
2067:
Aziz, Haris; Gaspers, Serge; Mackenzie, Simon; Walsh, Toby (2015-10-01).
546:
205:
176:
2641:
3180:
3097:
2612:
2186:
446:
2239:
2222:
2128:
Proceedings of the 5th ACM conference on
Electronic commerce - EC '04
1078:
751:
43:
885:(for A). An EFx allocation is known to exist in some special cases:
3171:
3088:
3039:
2994:
2941:
2877:
2818:
2810:
Proceedings of the 2019 ACM Conference on
Economics and Computation
2773:
2744:
2714:
2693:
2656:
2603:
2511:
2481:
2427:
2346:
2177:
2546:
2367:
2085:
2069:"Fair assignment of indivisible objects under ordinal preferences"
179:, while checking existence of WPEF can be done in polynomial time.
2869:
Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms
2858:
2258:. Lecture Notes in Computer Science. Vol. 5783. p. 98.
902:
271:
226:
147:
responsive bundle-rankings consistent with the item-ranking (see
48:
2495:
2409:
Oh, Hoon; Procaccia, Ariel D.; Suksompong, Warut (2019-07-17).
702:
variables; Lenstra's algorithm allows solving such ILP in time
66:
Preference-orderings on items: necessary/possible envy-freeness
2470:
2804:
Caragiannis, Ioannis; Gravin, Nick; Huang, Xin (2019-06-17).
2415:
Proceedings of the AAAI Conference on Artificial Intelligence
2038:
Bouveret, Sylvain; Endriss, Ulle; Lang, JĂ©rĂ´me (2010-08-04).
1367:
840:
For at most 4 agents with arbitrary monotone valuations, or
86:
responsive bundle-rankings consistent with the item-ranking;
2253:
162:
responsive bundle-ranking consistent with the item-ranking.
2299:. IJCAI'16. New York, New York, USA: AAAI Press: 102–108.
2125:
927:, but there are at most two different kinds of valuations.
2286:
2066:
2005:
1404:
is also envy-free. This is true regardless of additivity.
864:
759:
3024:
2758:
1255:
If the number of items is not sufficiently large, i.e.,
1952:
NP-hard (but there are approximations in special cases)
977:-1 items, and no agent envies the set of donated items.
637:{\displaystyle O^{*}((m\cdot V+1)^{n^{2}}\cdot mn^{2})}
170:
Bouveret, Endriss and Lang assume that all agents have
2803:
1982:
With submodular utilities, allocation is PE and MEF1.
2707:
2408:
1961:
1930:
1881:
1850:
1795:
1768:
1732:
1705:
1650:
1587:
1534:
1462:
1454:= the number of agents participating in the division;
1440:
1305:
1261:
1224:
1173:
1102:
708:
669:
566:
514:
481:
455:
413:
370:
324:
277:
232:
3152:
3069:
2161:"Communication Complexity of Discrete Fair Division"
938:. In this case, a polynomial-time algorithm exists.
3153:Manurangsi, Pasin; Suksompong, Warut (2020-07-02).
3070:Manurangsi, Pasin; Suksompong, Warut (2021-04-08).
2863:; Mehlhorn, Kurt; Sgouritsa, Alkmini (2019-12-23),
2037:
2865:"A Little Charity Guarantees Almost Envy-Freeness"
1967:
1936:
1899:
1856:
1820:
1774:
1738:
1711:
1674:
1611:
1547:
1468:
1446:
1355:
1288:
1239:
1203:
1152:
1079:Existence of EF allocations with random valuations
1035:
752:Finding an allocation with a bounded level of envy
740:
694:
636:
533:
500:
467:
426:
399:
356:
297:
252:
44:Finding an envy-free allocation whenever it exists
2550:The Unreasonable Fairness of Maximum Nash Welfare
2371:The Unreasonable Fairness of Maximum Nash Welfare
3200:
2585:Plaut, Benjamin; Roughgarden, Tim (2020-01-01).
2584:
2159:Plaut, Benjamin; Roughgarden, Tim (2020-01-01).
2158:
1624:Partial, but not Pareto-dominated by another NEF
2925:"On the number of almost envy-free allocations"
2411:"Fairly Allocating Many Goods with Few Queries"
1563:If-and-only-if a complete EF allocation exists
2587:"Almost Envy-Freeness with General Valuations"
1296:, then w.h.p. an EF allocation does not exist.
1153:{\displaystyle m=\Omega (n\log n/\log \log n)}
1096:If the number of items is sufficiently large:
873:if for every two agents A and B, if we remove
49:Preference-orderings on bundles: envy-freeness
2737:
2220:
828:Efficient approximately fair item allocation
26:problem, in which the fairness criterion is
3124:The Computational Rise and Fall of Fairness
2979:
2542:
2540:
2538:
2227:Journal of Artificial Intelligence Research
158:(PPE) if it is Pareto-optimal according to
143:(NPE) if it is Pareto-optimal according to
3117:
3115:
3072:"Closing gaps in asymptotic fair division"
2922:
1431:Below, the following shorthands are used:
124:responsive bundle-ranking consistent with
101:responsive bundle-ranking consistent with
3170:
3131:
3087:
3038:
2993:
2940:
2876:
2817:
2772:
2743:
2713:
2692:
2655:
2602:
2510:
2480:
2436:
2426:
2361:
2336:
2238:
2176:
2094:
2084:
1384:functions, then an EF allocation is also
1368:Envy-freeness vs. other fairness criteria
298:{\displaystyle \Sigma _{2}^{\textrm {P}}}
253:{\displaystyle \Sigma _{2}^{\textrm {P}}}
2535:
2001:
1999:
1356:{\displaystyle m=o(n\log n/\log \log n)}
3148:
3146:
3112:
2009:Handbook of Computational Social Choice
3201:
3155:"When do envy-free allocations exist?"
2686:
2322:
2216:
2214:
2212:
1910:Partial, but PE w.r.t. allocated items
1023:
865:EFx - envy-free up to at most any item
760:EF1 - envy-free up to at most one item
651:Considering both the number of agents
552:Considering both the number of agents
82:(NEF) if it is envy-free according to
2466:
2464:
2404:
2402:
2318:
2316:
2247:
2119:
1996:
225:allocation exists is above NP: it is
191:
3159:SIAM Journal on Discrete Mathematics
3143:
3076:SIAM Journal on Discrete Mathematics
2591:SIAM Journal on Discrete Mathematics
2287:Bliem, Bernhard; Bredereck, Robert;
187:existence of NEF allocation is in P.
2209:
13:
2461:
2399:
2313:
1204:{\displaystyle m=\Omega (n\log n)}
1180:
1109:
1052:and returns an allocation that is
965:) can be found in polynomial time.
950:) can be found in polynomial time.
314:Considering the number of objects
279:
234:
116:(WPEF) if for each pair of agents
14:
3225:
1001:if for every two agents A and B:
655:and the number of utility levels
437:Considering the number of agents
166:The following results are known:
36:One way to attain fairness is to
2923:Suksompong, Warut (2020-09-30).
1476:= the number of items to divide;
1426:
800:of a single item for that agent.
3063:
3018:
2973:
2916:
2852:
2797:
2752:
2731:
2722:
2701:
2680:
2635:
2578:
2489:
2221:Bouveret, S.; Lang, J. (2008).
1036:Finding a partial EF allocation
942:Some approximations are known:
741:{\displaystyle O^{*}(d^{2.5d})}
2280:
2152:
2060:
2031:
2012:. Cambridge University Press.
1894:
1882:
1815:
1799:
1669:
1663:
1606:
1600:
1350:
1315:
1283:
1277:
1198:
1183:
1147:
1112:
735:
719:
695:{\displaystyle d=n\cdot z^{n}}
631:
599:
580:
577:
394:
381:
351:
335:
20:Envy-free (EF) item allocation
1:
1989:
644:for additive utilities using
357:{\displaystyle O^{*}(m^{2m})}
3004:10.1016/j.artint.2020.103436
2929:Discrete Applied Mathematics
2761:Theoretical Computer Science
2644:Theoretical Computer Science
2438:10.1609/aaai.v33i01.33012141
2325:Journal of Political Economy
2096:10.1016/j.artint.2015.06.002
1917:when there are many agents.
1423:for details and references.
1032:for details and references.
475:agents, it can be done with
400:{\displaystyle O^{*}(m^{m})}
16:Fair item allocation problem
7:
2887:10.1137/1.9781611975994.162
2499:Games and Economic Behavior
2264:10.1007/978-3-642-04428-1_9
2256:Algorithmic Decision Theory
2046:. NLD: IOS Press: 387–392.
508:such oracles, and at least
219:Deciding whether an EF and
200:Deciding whether an EF and
10:
3230:
2728:{{cite arXiv:2205.07638 }}
1213:coupon collector's problem
893:agents, or when there are
141:necessarily Pareto-optimal
3049:10.1016/j.ins.2021.11.059
2951:10.1016/j.dam.2020.03.039
2783:10.1016/j.tcs.2020.07.006
2666:10.1016/j.tcs.2021.02.020
2521:10.1016/j.geb.2021.11.006
2165:SIAM Journal on Computing
1821:{\displaystyle O(mn^{3})}
1061:Adjusted winner procedure
860:allocation always exists.
149:Ordinal Pareto efficiency
114:weakly possibly envy-free
2859:Chaudhury, Bhaskar Ray;
1289:{\displaystyle m=n+o(n)}
1240:{\displaystyle m\geq 2n}
1075:of maximum cardinality.
869:An allocation is called
764:An allocation is called
556:and the largest utility
93:(PEF) if for each agent
2982:Artificial Intelligence
2828:10.1145/3328526.3329574
2562:10.1145/2940716.2940726
2383:10.1145/2940716.2940726
2073:Artificial Intelligence
1955:EF1, and approximately
1690:Might divide one item.
1675:{\displaystyle Poly(m)}
1612:{\displaystyle Poly(m)}
1400:competitive equilibrium
1373:Every EF allocation is
956:groupwise maximin share
534:{\displaystyle 2^{n-4}}
501:{\displaystyle 2^{n+1}}
468:{\displaystyle n\geq 5}
156:possibly Pareto-optimal
128:item-ranking, by which
105:item-ranking, by which
1969:
1938:
1901:
1858:
1822:
1776:
1740:
1713:
1676:
1613:
1549:
1482:PE = Pareto-efficient.
1470:
1448:
1397:Every allocation of a
1357:
1290:
1241:
1205:
1154:
1050:responsive preferences
968:There always exists a
962:pairwise maximin share
742:
696:
661:integer linear program
638:
535:
502:
469:
428:
401:
358:
299:
254:
72:responsive preferences
60:responsive preferences
38:use monetary transfers
2136:10.1145/988772.988792
1970:
1939:
1902:
1900:{\displaystyle (n+1)}
1859:
1823:
1777:
1741:
1714:
1677:
1614:
1550:
1548:{\displaystyle 2^{m}}
1471:
1449:
1358:
1291:
1242:
1206:
1155:
1090:with high probability
773:When all agents have
743:
697:
639:
536:
503:
470:
429:
427:{\displaystyle O^{*}}
407:. Here, the notation
402:
359:
300:
264:dichotomous utilities
255:
204:allocation exists is
80:necessarily envy-free
3214:NP-complete problems
3209:Fair item allocation
3027:Information Sciences
2861:Kavitha, Telikepalli
2473:Kavitha, Telikepalli
1959:
1928:
1922:Maximum-Nash-Welfare
1879:
1848:
1793:
1766:
1730:
1703:
1648:
1585:
1532:
1460:
1438:
1421:Fair item allocation
1303:
1259:
1222:
1171:
1162:round-robin protocol
1100:
901:. In this case, the
899:identical valuations
850:Simmons–Su protocols
815:Maximum Nash Welfare
789:envy-graph procedure
780:round-robin protocol
706:
667:
564:
545:=2. Moreover, it is
512:
479:
453:
411:
368:
322:
275:
230:
24:fair item allocation
2026:free online version
1913:Also approximately
1412:Group-envy-freeness
1380:If all agents have
1083:If the agents have
1024:Minimizing the envy
936:additive valuations
925:additive valuations
914:additive valuations
646:dynamic programming
294:
249:
3181:10.1137/19M1279125
3098:10.1137/20M1353381
2613:10.1137/19M124397X
2187:10.1137/19M1244305
1965:
1934:
1897:
1854:
1818:
1772:
1736:
1709:
1672:
1609:
1545:
1466:
1444:
1402:from equal incomes
1353:
1286:
1237:
1201:
1150:
1073:envy-free matching
1066:additive utilities
738:
692:
634:
531:
498:
465:
424:
397:
354:
295:
278:
268:additive utilities
250:
233:
192:Cardinal utilities
91:possibly envy-free
55:undercut procedure
2896:978-1-61197-599-4
2837:978-1-4503-6792-9
2306:978-1-57735-770-4
2289:Niedermeier, Rolf
2273:978-3-642-04427-4
2240:10.1613/jair.2467
2053:978-1-60750-605-8
1987:
1986:
1968:{\displaystyle n}
1937:{\displaystyle n}
1857:{\displaystyle n}
1775:{\displaystyle n}
1739:{\displaystyle m}
1712:{\displaystyle n}
1526:Strictly monotone
1469:{\displaystyle m}
1447:{\displaystyle n}
1056:envy-free (NEF).
1030:envy minimization
852:. When there are
291:
246:
210:partition problem
154:An allocation is
139:An allocation is
112:An allocation is
89:An allocation is
78:An allocation is
3221:
3193:
3192:
3174:
3165:(3): 1505–1521.
3150:
3141:
3137:
3135:
3119:
3110:
3109:
3091:
3067:
3061:
3060:
3042:
3022:
3016:
3015:
2997:
2977:
2971:
2970:
2944:
2920:
2914:
2913:
2912:
2911:
2880:
2856:
2850:
2849:
2821:
2801:
2795:
2794:
2776:
2756:
2750:
2749:
2747:
2735:
2729:
2726:
2720:
2719:
2717:
2705:
2699:
2698:
2696:
2684:
2678:
2677:
2659:
2639:
2633:
2632:
2606:
2597:(2): 1039–1068.
2582:
2576:
2575:
2555:
2544:
2533:
2532:
2514:
2493:
2487:
2486:
2484:
2468:
2459:
2458:
2440:
2430:
2421:(1): 2141–2148.
2406:
2397:
2396:
2376:
2365:
2359:
2358:
2340:
2331:(6): 1061–1103.
2320:
2311:
2310:
2284:
2278:
2277:
2251:
2245:
2244:
2242:
2218:
2207:
2206:
2180:
2156:
2150:
2149:
2123:
2117:
2116:
2098:
2088:
2064:
2058:
2057:
2035:
2029:
2023:
2003:
1974:
1972:
1971:
1966:
1943:
1941:
1940:
1935:
1906:
1904:
1903:
1898:
1863:
1861:
1860:
1855:
1827:
1825:
1824:
1819:
1814:
1813:
1781:
1779:
1778:
1773:
1745:
1743:
1742:
1737:
1718:
1716:
1715:
1710:
1684:EF and equitable
1681:
1679:
1678:
1673:
1618:
1616:
1615:
1610:
1554:
1552:
1551:
1546:
1544:
1543:
1486:
1485:
1475:
1473:
1472:
1467:
1453:
1451:
1450:
1445:
1382:additive utility
1362:
1360:
1359:
1354:
1334:
1295:
1293:
1292:
1287:
1246:
1244:
1243:
1238:
1210:
1208:
1207:
1202:
1159:
1157:
1156:
1151:
1131:
1092:. Particularly:
1085:additive utility
1046:Pareto efficient
820:Pareto-efficient
806:A-CEEI mechanism
798:marginal utility
794:cardinal utility
747:
745:
744:
739:
734:
733:
718:
717:
701:
699:
698:
693:
691:
690:
643:
641:
640:
635:
630:
629:
614:
613:
612:
611:
576:
575:
540:
538:
537:
532:
530:
529:
507:
505:
504:
499:
497:
496:
474:
472:
471:
466:
433:
431:
430:
425:
423:
422:
406:
404:
403:
398:
393:
392:
380:
379:
363:
361:
360:
355:
350:
349:
334:
333:
304:
302:
301:
296:
293:
292:
289:
286:
259:
257:
256:
251:
248:
247:
244:
241:
222:Pareto efficient
3229:
3228:
3224:
3223:
3222:
3220:
3219:
3218:
3199:
3198:
3197:
3196:
3151:
3144:
3133:10.1.1.703.8413
3120:
3113:
3068:
3064:
3023:
3019:
2978:
2974:
2921:
2917:
2909:
2907:
2897:
2857:
2853:
2838:
2802:
2798:
2757:
2753:
2736:
2732:
2727:
2723:
2706:
2702:
2685:
2681:
2640:
2636:
2583:
2579:
2572:
2553:
2545:
2536:
2494:
2490:
2469:
2462:
2407:
2400:
2393:
2374:
2366:
2362:
2338:10.1.1.357.9766
2321:
2314:
2307:
2285:
2281:
2274:
2252:
2248:
2219:
2210:
2157:
2153:
2146:
2130:. p. 125.
2124:
2120:
2065:
2061:
2054:
2036:
2032:
2020:
2004:
1997:
1992:
1960:
1957:
1956:
1946:Item valuations
1929:
1926:
1925:
1880:
1877:
1876:
1849:
1846:
1845:
1809:
1805:
1794:
1791:
1790:
1767:
1764:
1763:
1748:Necessarily EF1
1731:
1728:
1727:
1724:Weakly additive
1704:
1701:
1700:
1649:
1646:
1645:
1639:Item valuations
1632:Adjusted winner
1586:
1583:
1582:
1579:Weakly additive
1539:
1535:
1533:
1530:
1529:
1461:
1458:
1457:
1439:
1436:
1435:
1429:
1370:
1330:
1304:
1301:
1300:
1260:
1257:
1256:
1223:
1220:
1219:
1172:
1169:
1168:
1127:
1101:
1098:
1097:
1081:
1038:
1026:
995:
930:When there are
919:When there are
908:When there are
889:When there are
867:
777:utilities, the
775:weakly additive
762:
754:
726:
722:
713:
709:
707:
704:
703:
686:
682:
668:
665:
664:
625:
621:
607:
603:
602:
598:
571:
567:
565:
562:
561:
519:
515:
513:
510:
509:
486:
482:
480:
477:
476:
454:
451:
450:
418:
414:
412:
409:
408:
388:
384:
375:
371:
369:
366:
365:
342:
338:
329:
325:
323:
320:
319:
288:
287:
282:
276:
273:
272:
243:
242:
237:
231:
228:
227:
194:
68:
51:
46:
17:
12:
11:
5:
3227:
3217:
3216:
3211:
3195:
3194:
3142:
3111:
3082:(2): 668–706.
3062:
3017:
2972:
2915:
2895:
2851:
2836:
2796:
2751:
2730:
2721:
2700:
2679:
2634:
2577:
2570:
2534:
2488:
2460:
2398:
2391:
2360:
2347:10.1086/664613
2312:
2305:
2291:(2016-07-09).
2279:
2272:
2246:
2208:
2171:(1): 206–243.
2151:
2144:
2118:
2059:
2052:
2030:
2018:
1994:
1993:
1991:
1988:
1985:
1984:
1979:
1976:
1975:-maximin-share
1964:
1953:
1950:
1947:
1944:
1933:
1923:
1919:
1918:
1911:
1908:
1907:-maximin-share
1896:
1893:
1890:
1887:
1884:
1873:
1870:
1867:
1866:Bundle ranking
1864:
1853:
1843:
1837:
1836:
1834:
1831:
1828:
1817:
1812:
1808:
1804:
1801:
1798:
1788:
1785:
1784:Bundle ranking
1782:
1771:
1761:
1755:
1754:
1752:
1749:
1746:
1735:
1725:
1722:
1719:
1708:
1698:
1692:
1691:
1688:
1685:
1682:
1671:
1668:
1665:
1662:
1659:
1656:
1653:
1643:
1640:
1637:
1634:
1628:
1627:
1625:
1622:
1621:Necessarily EF
1619:
1608:
1605:
1602:
1599:
1596:
1593:
1590:
1580:
1577:
1574:
1571:
1565:
1564:
1561:
1558:
1555:
1542:
1538:
1527:
1524:
1523:Bundle ranking
1521:
1518:
1512:
1511:
1508:
1505:
1502:
1499:
1496:
1493:
1490:
1484:
1483:
1480:
1477:
1465:
1455:
1443:
1428:
1425:
1417:
1416:
1408:
1405:
1395:
1378:
1369:
1366:
1365:
1364:
1352:
1349:
1346:
1343:
1340:
1337:
1333:
1329:
1326:
1323:
1320:
1317:
1314:
1311:
1308:
1297:
1285:
1282:
1279:
1276:
1273:
1270:
1267:
1264:
1249:
1248:
1236:
1233:
1230:
1227:
1216:
1200:
1197:
1194:
1191:
1188:
1185:
1182:
1179:
1176:
1165:
1149:
1146:
1143:
1140:
1137:
1134:
1130:
1126:
1123:
1120:
1117:
1114:
1111:
1108:
1105:
1080:
1077:
1037:
1034:
1025:
1022:
1014:exact division
1010:
1009:
1006:
994:
991:
979:
978:
966:
951:
940:
939:
928:
917:
906:
883:least valuable
866:
863:
862:
861:
838:
835:
832:
823:
811:
801:
784:
761:
758:
753:
750:
749:
748:
737:
732:
729:
725:
721:
716:
712:
689:
685:
681:
678:
675:
672:
649:
633:
628:
624:
620:
617:
610:
606:
601:
597:
594:
591:
588:
585:
582:
579:
574:
570:
550:
528:
525:
522:
518:
495:
492:
489:
485:
464:
461:
458:
435:
421:
417:
396:
391:
387:
383:
378:
374:
353:
348:
345:
341:
337:
332:
328:
308:
307:
285:
281:
266:and even with
240:
236:
217:
213:
193:
190:
189:
188:
180:
164:
163:
152:
137:
132:does not envy
110:
109:does not envy;
87:
67:
64:
50:
47:
45:
42:
15:
9:
6:
4:
3:
2:
3226:
3215:
3212:
3210:
3207:
3206:
3204:
3190:
3186:
3182:
3178:
3173:
3168:
3164:
3160:
3156:
3149:
3147:
3140:
3134:
3129:
3125:
3118:
3116:
3107:
3103:
3099:
3095:
3090:
3085:
3081:
3077:
3073:
3066:
3058:
3054:
3050:
3046:
3041:
3036:
3032:
3028:
3021:
3013:
3009:
3005:
3001:
2996:
2991:
2987:
2983:
2976:
2968:
2964:
2960:
2956:
2952:
2948:
2943:
2938:
2934:
2930:
2926:
2919:
2906:
2902:
2898:
2892:
2888:
2884:
2879:
2874:
2870:
2866:
2862:
2855:
2847:
2843:
2839:
2833:
2829:
2825:
2820:
2815:
2811:
2807:
2800:
2792:
2788:
2784:
2780:
2775:
2770:
2766:
2762:
2755:
2746:
2741:
2734:
2725:
2716:
2711:
2704:
2695:
2690:
2683:
2675:
2671:
2667:
2663:
2658:
2653:
2649:
2645:
2638:
2630:
2626:
2622:
2618:
2614:
2610:
2605:
2600:
2596:
2592:
2588:
2581:
2573:
2571:9781450339360
2567:
2563:
2559:
2552:
2551:
2543:
2541:
2539:
2530:
2526:
2522:
2518:
2513:
2508:
2504:
2500:
2492:
2483:
2478:
2474:
2467:
2465:
2456:
2452:
2448:
2444:
2439:
2434:
2429:
2424:
2420:
2416:
2412:
2405:
2403:
2394:
2392:9781450339360
2388:
2384:
2380:
2373:
2372:
2364:
2356:
2352:
2348:
2344:
2339:
2334:
2330:
2326:
2319:
2317:
2308:
2302:
2298:
2294:
2290:
2283:
2275:
2269:
2265:
2261:
2257:
2250:
2241:
2236:
2232:
2228:
2224:
2217:
2215:
2213:
2204:
2200:
2196:
2192:
2188:
2184:
2179:
2174:
2170:
2166:
2162:
2155:
2147:
2145:1-58113-771-0
2141:
2137:
2133:
2129:
2122:
2114:
2110:
2106:
2102:
2097:
2092:
2087:
2082:
2078:
2074:
2070:
2063:
2055:
2049:
2045:
2041:
2034:
2027:
2021:
2019:9781107060432
2015:
2011:
2010:
2002:
2000:
1995:
1983:
1980:
1977:
1962:
1954:
1951:
1948:
1945:
1931:
1924:
1921:
1920:
1916:
1915:strategyproof
1912:
1909:
1891:
1888:
1885:
1874:
1871:
1868:
1865:
1851:
1844:
1842:
1839:
1838:
1835:
1832:
1829:
1810:
1806:
1802:
1796:
1789:
1786:
1783:
1769:
1762:
1760:
1757:
1756:
1753:
1750:
1747:
1733:
1726:
1723:
1720:
1706:
1699:
1697:
1694:
1693:
1689:
1686:
1683:
1666:
1660:
1657:
1654:
1651:
1644:
1641:
1638:
1635:
1633:
1630:
1629:
1626:
1623:
1620:
1603:
1597:
1594:
1591:
1588:
1581:
1578:
1575:
1572:
1570:
1567:
1566:
1562:
1559:
1556:
1540:
1536:
1528:
1525:
1522:
1519:
1517:
1514:
1513:
1509:
1506:
1503:
1500:
1497:
1494:
1491:
1488:
1487:
1481:
1478:
1463:
1456:
1441:
1434:
1433:
1432:
1427:Summary table
1424:
1422:
1414:
1413:
1409:
1406:
1403:
1401:
1396:
1393:
1389:
1388:
1383:
1379:
1376:
1372:
1371:
1347:
1344:
1341:
1338:
1335:
1331:
1327:
1324:
1321:
1318:
1312:
1309:
1306:
1298:
1280:
1274:
1271:
1268:
1265:
1262:
1254:
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1234:
1231:
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1217:
1214:
1195:
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1189:
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1135:
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1128:
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1103:
1095:
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1021:
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1000:
990:
986:
982:
976:
971:
967:
964:
963:
958:
957:
952:
949:
948:Maximin Aware
945:
944:
943:
937:
933:
929:
926:
922:
918:
915:
911:
907:
904:
900:
896:
892:
888:
887:
886:
884:
880:
879:most valuable
876:
872:
859:
855:
851:
847:
843:
839:
836:
833:
830:
829:
824:
821:
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727:
723:
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687:
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493:
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419:
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389:
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376:
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346:
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317:
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311:
305:
283:
269:
265:
261:
238:
224:
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207:
203:
199:
198:
197:
185:
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178:
173:
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157:
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142:
138:
135:
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96:
92:
88:
85:
81:
77:
76:
75:
73:
63:
61:
56:
41:
39:
34:
31:
29:
28:envy-freeness
25:
21:
3162:
3158:
3123:
3079:
3075:
3065:
3030:
3026:
3020:
2985:
2981:
2975:
2932:
2928:
2918:
2908:, retrieved
2868:
2854:
2809:
2799:
2764:
2760:
2754:
2733:
2724:
2703:
2682:
2647:
2643:
2637:
2594:
2590:
2580:
2549:
2502:
2498:
2491:
2418:
2414:
2370:
2363:
2328:
2324:
2296:
2282:
2255:
2249:
2230:
2226:
2168:
2164:
2154:
2127:
2121:
2076:
2072:
2062:
2043:
2033:
2008:
1981:
1721:Item ranking
1576:Item ranking
1430:
1418:
1411:
1398:
1392:max-min-fair
1391:
1387:proportional
1385:
1375:min-max-fair
1374:
1250:
1082:
1070:
1058:
1053:
1042:AL procedure
1039:
1027:
1019:
1011:
998:
996:
987:
983:
980:
974:
969:
960:
954:
947:
941:
934:agents with
931:
924:
923:agents with
920:
913:
912:agents with
909:
898:
897:agents with
894:
890:
882:
878:
874:
870:
868:
857:
853:
845:
841:
826:
814:
804:
788:
778:
765:
763:
755:
656:
652:
557:
553:
542:
442:
438:
315:
309:
220:
201:
195:
183:
171:
165:
160:at least one
159:
155:
144:
140:
133:
129:
125:
122:at least one
121:
117:
113:
106:
102:
99:at least one
98:
94:
90:
83:
79:
69:
52:
35:
32:
19:
18:
3033:: 164–187.
2935:: 606–610.
2505:: 197–221.
2233:: 525–564.
1696:Round-robin
1498:Preferences
1054:necessarily
216:parameters.
206:NP complete
177:NP-complete
120:, there is
97:, there is
3203:Categories
3172:1811.01630
3089:2004.05563
3040:1901.09527
2995:1911.07048
2988:: 103436.
2942:2006.00178
2910:2020-10-02
2878:1907.04596
2819:1902.04319
2774:1909.07650
2767:: 94–109.
2745:1905.09969
2715:2002.05119
2694:2008.08798
2657:2001.09838
2604:1707.04769
2512:1808.09406
2482:2006.04428
2428:1807.11367
2178:1711.04066
1990:References
1759:Envy-graph
1507:Efficiency
547:W-complete
447:SAT solver
262:even with
3189:220363902
3128:CiteSeerX
3106:215744823
3057:170079201
3012:231719223
2967:215715272
2959:0166-218X
2905:195874127
2791:222070124
2674:210920309
2650:: 69–85.
2629:216283014
2621:0895-4801
2447:2374-3468
2355:154703357
2333:CiteSeerX
2203:212667868
2195:0097-5397
2105:0004-3702
2086:1312.6546
2079:: 71–92.
1875:EF1, and
1510:Comments
1492:#partners
1345:
1339:
1325:
1229:≥
1193:
1181:Ω
1142:
1136:
1122:
1110:Ω
846:connected
715:∗
680:⋅
616:⋅
587:⋅
573:∗
524:−
460:≥
420:∗
377:∗
331:∗
280:Σ
260:-complete
235:Σ
3139:ACM link
2846:60441171
2529:52112902
2455:51867780
1949:Additive
1833:Complete
1787:Monotone
1751:Complete
1642:Additive
1560:Complete
1516:Undercut
1504:Fairness
1501:#queries
449:). With
202:complete
2113:1408197
970:partial
903:leximin
3187:
3130:
3104:
3055:
3010:
2965:
2957:
2903:
2893:
2844:
2834:
2789:
2672:
2627:
2619:
2568:
2527:
2453:
2445:
2389:
2353:
2335:
2303:
2270:
2201:
2193:
2142:
2111:
2103:
2050:
2016:
1841:A-CEEI
810:items.
172:strict
3185:S2CID
3167:arXiv
3102:S2CID
3084:arXiv
3053:S2CID
3035:arXiv
3008:S2CID
2990:arXiv
2963:S2CID
2937:arXiv
2901:S2CID
2873:arXiv
2842:S2CID
2814:arXiv
2787:S2CID
2769:arXiv
2740:arXiv
2710:arXiv
2689:arXiv
2670:S2CID
2652:arXiv
2625:S2CID
2599:arXiv
2554:(PDF)
2525:S2CID
2507:arXiv
2477:arXiv
2451:S2CID
2423:arXiv
2375:(PDF)
2351:S2CID
2199:S2CID
2173:arXiv
2109:S2CID
2081:arXiv
1495:Input
932:three
663:with
22:is a
2955:ISSN
2891:ISBN
2832:ISBN
2617:ISSN
2566:ISBN
2443:ISSN
2387:ISBN
2301:ISBN
2268:ISBN
2191:ISSN
2140:ISBN
2101:ISSN
2048:ISBN
2014:ISBN
1489:Name
1419:See
1390:and
1059:The
1040:The
959:and
813:The
184:weak
58:are
53:The
3177:doi
3094:doi
3045:doi
3031:587
3000:doi
2986:293
2947:doi
2933:284
2883:doi
2824:doi
2779:doi
2765:841
2662:doi
2648:863
2609:doi
2558:doi
2517:doi
2503:131
2433:doi
2379:doi
2343:doi
2329:119
2260:doi
2235:doi
2183:doi
2132:doi
2091:doi
2077:227
1869:Any
1830:EF1
1342:log
1336:log
1322:log
1299:If
1218:If
1190:log
1167:If
1139:log
1133:log
1119:log
999:EFm
891:two
875:any
871:EFx
858:EF2
766:EF1
728:2.5
270:. (
145:all
126:i's
118:i,j
103:i's
84:all
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3175:.
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3157:.
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3114:^
3100:.
3092:.
3080:35
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2401:^
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2181:.
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2107:.
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2089:.
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2071:.
2042:.
1998:^
1978:PE
1687:PE
1569:AL
1557:EF
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151:);
3191:.
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3108:.
3096::
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1963:n
1932:n
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1892:1
1889:+
1886:n
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1872:?
1852:n
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1811:3
1807:n
1803:m
1800:(
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1770:n
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1707:n
1670:)
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1332:/
1328:n
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1316:(
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1307:m
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1129:/
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