810:(fPO). They prove that, if the agents' valuations are non-degenerate, the number of fPO allocations is polynomial in the number of objects (for a fixed number of agents). Therefore, it is possible to enumerate all of them in polynomial time, and find an allocation that is fair and fPO with the smallest number of sharings. In contrast, of the valuations are degenerate, the problem becomes NP-hard. They present empirical evidence that, in realistic cases, there often exists an allocation with substantially fewer sharings than the worst-case bound.
1119:, except that in multiwinner voting the number of elected candidates is usually much smaller than the number of voters, while in public goods allocation the number of chosen goods is usually much larger than the number of agents. An example is a public library that has to decide which books to purchase, respecting the preferences of the readers; the number of books is usually much larger than the number of readers.
280:: each partner reports a value for each bundle of size at most 2. The value of a bundle is calculated by summing the values for the individual items in the bundle and adding the values of pairs in the bundle. Typically, when there are substitute items, the values of pairs will be negative, and when there are complementary items, the values of pairs will be positive. This idea can be generalized to
2581:
935:(envy-freeness for mixed items), which generalizes both envy-freeness for divisible items and EF1 for indivisible items. They prove that an EFM allocation always exists for any number of agents with additive valuations. They present efficient algorithms to compute EFM allocations for two agents with general additive valuations, and for
1233:
corresponds to the special case in which each issue corresponds to an item, each decision option corresponds to giving that item to a particular agent, and the agents' utilities are zero for all options in which the item is given to someone else. In this case, proportionality means that the utility of each agent is at least 1/
725:: a simple protocol where the agents take turns in selecting items, based on some pre-specified sequence of turns. The goal is to design the picking-sequence in a way that maximizes the expected value of a social welfare function (e.g. egalitarian or utilitarian) under some probabilistic assumptions on the agents' valuations.
1508:
2283:
2786:
577:
competitive equilibrium is reached when the supply matches the demand. The fairness argument is straightforward: prices and budgets are the same for everyone. CEEI implies EF regardless of additivity. When the agents' preferences are additive and strict (each bundle has a different value), CEEI implies
965:
for EFM. They show that, in general, no truthful EFM algorithm exists, even if there is only one indivisible good and one divisible good and only two agents. But, when agents have binary valuations on indivisible goods and identical valuations on a single divisible good, an EFM and truthful mechanism
543:
Every agent weakly prefers his own bundle to any other bundle. Every envy-free allocation of all items is mFS-fair; this follows directly from the ordinal definitions and does not depend on additivity. If the valuations are additive, then an EF allocation is also proportional and MMS-fair. Otherwise,
412:
if all agents receive a bundle that they weakly prefer over their mFS. mFS-fairness can be described as the result of the following negotiation process. A certain allocation is suggested. Each agent can object to it by demanding that a different allocation be made by another agent, letting him choose
1255:
Often, the same items are allocated repeatedly. For example, recurring house chores. If the number of repetitions is a multiple of the number of agents, then it is possible to find in polynomial time a sequence of allocations that is envy-free and complete, and to find in exponential time a sequence
985:
and max Nash welfare). They assume that all agents have binary valuations. It is known that, if only divisible goods or only indivisible goods exist, the problem is polytime solvable. They show that, with mixed goods, the problem is NP-hard even when all indivisible goods are identical. In contrast,
750:
Several works assume that all objects can be divided if needed (e.g. by shared ownership or time-sharing), but sharing is costly or undesirable. Therefore, it is desired to find a fair allocation with the smallest possible number of shared objects, or of sharings. There are tight upper bounds on the
407:
The min-max-fair-share of an agent is the minimal utility that she can hope to get from an allocation if all the other agents have the same preferences as her, when she always receives the best share. It is also the minimal utility that an agent can get for sure in the allocation game “Someone cuts,
2116:
969:
Nishimura and Sumita study the properties of the maximum Nash welfare allocation (MNW) for mixed goods. They prove that, when all agents' valuations are binary and linear for each good, an MNW allocation satisfies a property stronger than EFM, which they call "envy-freeness up to any good for mixed
1080:
In this variant, bundles are given not to individual agents but to groups of agents. Common use-cases are: dividing inheritance among families, or dividing facilities among departments in a university. All agents in the same group consume the same bundle, though they may value it differently. The
958:(objects with negative utilities) and a divisible cake (with positive utility). They present an algorithm for finding an EFM allocation in two special cases: when each agent has the same preference ranking over the set of chores, and when the number of items is at most the number of agents plus 1.
993:
Li, Li, Liu and Wu study a setting in which each agent may have a different "indivisibility ratio" (= proportion of items that are indivisible). Each agent is guaranteed an allocation that is EF/PROP up to a fraction of an item, where the fraction depends on the agent's indivisibility ratio. The
989:
Bei, Liu and Lu study a more general setting, in which the same object can be divisible for some agents and indivisible for others. They show that the best possible approximation for MMS is 2/3, even for two agents; and present algorithms attaining this bound for 2 or 3 agents. For any number of
255:
In the ordinal approach, additivity allows us to infer some rankings between bundles. For example, if a person prefers w to x to y to z, then he necessarily prefers {w,x} to {w,y} or to {x,y}, and {w,y} to {x}. This inference is only partial, e.g., we cannot know whether the agent prefers {w} to
1054:
In this variant, different agents are entitled to different fractions of the resource. A common use-case is dividing cabinet ministries among parties in the coalition. It is common to assume that each party should receive ministries according to the number of seats it has in the parliament. The
576:
This criterion is based on the following argument: the allocation process should be considered as a search for an equilibrium between the supply (the set of objects, each one having a public price) and the demand (the agents’ desires, each agent having the same budget for buying the objects). A
1232:
In this variant, several agents have to accept decisions on several issues. A common use-case is a family that has to decide what activity to do each day (here each issue is a day). Each agent assigns different utilities to the various options in each issue. The classic item allocation setting
272:
have been developed as a compromise between the full expressiveness of combinatorial preferences to the simplicity of additive preferences. They provide a succinct representation to some natural classes of utility functions that are more general than additive utilities (but not as general as
260:
The additivity implies that each partner can always choose a "preferable item" from the set of items on the table, and this choice is independent of the other items that the partner may have. This property is used by some fair assignment algorithms that will be described next.
970:
goods". Their results hold not only for max Nash welfare, but also for a general fairness notion based on minimizing a symmetric strictly convex function. For general additive valuations, they prove that an MNW allocation satisfies an EF approximation that is weaker than EFM.
103:
A naive way to determine the preferences is asking each partner to supply a numeric value for each possible bundle. For example, if the items to divide are a car and a bicycle, a partner may value the car as 800, the bicycle as 200, and the bundle {car, bicycle} as 900 (see
2251:
817:
is not fixed, even for non-degenerate valuations, it is NP-hard to decide whether there exists an fPO envy-free allocation with 0 sharings. They also demonstrate an alterate approach to enumerating distinct consumption graph for allocations with a small number of
48:
The indivisibility of the items implies that a fair division may not be possible. As an extreme example, if there is only a single item (e.g. a house), it must be given to a single partner, but this is not fair to the other partners. This is in contrast to the
554:: for each two agents A and B, if we remove from the bundle of B the item most valuable for A, then A does not envy B (in other words, the "envy level" of A in B is at most the value of a single item). Under monotonicity, an EF1 allocation always exists.
1663:
1292:
361:
is a criterion that should hold for each individual partner, as long as the partner truthfully reports his preferences. Five such criteria are presented below. They are ordered from the weakest to the strongest (assuming the valuations are additive):
950:
fairness. They propose an algorithm that computes an alpha-approximate MMS allocation for any number of agents, where alpha is a constant between 1/2 and 1, which is monotonically increasing with the value of the divisible goods relative to the MMS
1106:
In this variant, each item provides utility not only to a single agent but to all agents. Different agents may attribute different utilities to the same item. The group has to choose a subset of items satisfying some constraints, for example:
861:
necessary for consensus halving when agents' utilities are drawn from probabilistic distributions. For agents with non-additive monotone utilities, consensus halving is PPAD-hard, but there are polynomial-time algorithms for a fixed number of
417:
partitions, there is a bundle that he strongly prefers over his current bundle. An allocation is mFS-fair iff no agent objects to it, i.e., for every agent there exists a partition in which all bundles are weakly worse than his current share.
2576:{\displaystyle \sum _{i=1}^{n}\sum _{A\in {\mathcal {A}}}\left(p_{d^{*}}(A)\cdot u_{i}(A)\right)=\sum _{A\in {\mathcal {A}}}p_{d^{*}}(A)\sum _{i=1}^{n}u_{i}(A)\leq \max _{A\in {\mathcal {A}}\colon p_{d^{*}}(A)>0}\sum _{i=1}^{n}u_{i}(A)}
1256:
that is proportional and Pareto-optimal. But, an envy-free and Pareto-optimal sequence may not exist. With two agents, if the number of repetitions is even, it is always possible to find a sequence that is envy-free and Pareto-optimal.
609:
if it attains the maximum possible egalitarian welfare, i.e., it maximizes the utility of the poorest agent. Since there can be several different allocations maximizing the smallest utility, egalitarian optimality is often refined to
2640:
30:
rather than continuous. The items have to be divided among several partners who potentially value them differently, and each item has to be given as a whole to a single person. This situation arises in various real-life scenarios:
1977:
741:
Traditional papers on fair allocation either assume that all items are divisible, or that all items are indivisible. Some recent papers study settings in which the distinction between divisible and indivisible is more fuzzy.
1904:
1213:. This idea can be formalized to show a general reduction from private-goods allocation to public-goods allocation that retains the maximum Nash welfare allocation, as well as a similar reduction that retains the
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formula, and may assign a value for each formula. For example, a partner may say: "For (x or (y and z)), my value is 5". This means that the agent has a value of 5 for any of the bundles: x, xy, xz, yz,
201:
different bundles, i.e., say which bundle is the best, which is the second-best, and so on. This may be easier than calculating exact numbers, but it is still difficult if the number of items is large.
2788:
In contrast to the utilitarian rule, here, the stochastic setting allows society to achieve higher value — as an example, consider the case where are two identical agents and only one item that worth
220:
the preferences on items to preferences on bundles. Then, the agents report their valuations/rankings on individual items, and the algorithm calculates for them their valuations/rankings on bundles.
2129:
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exists. When agents have binary valuations over both divisible and indivisible goods, an EFM and truthful mechanism exists when there are only two agents, or when there is a single divisible good.
3384:
Nguyen, Nhan-Tam; Nguyen, Trung Thanh; Roos, Magnus; Rothe, Jörg (2013). "Computational complexity and approximability of social welfare optimization in multiagent resource allocation".
1001:
in both indivisible and mixed item allocation. They provide bounds for the price of EF1, EFx, EFM and EFxM. They provide tight bounds for two agents and asymptotically tight bounds for
3339:
Nguyen, Trung Thanh; Roos, Magnus; Rothe, Jörg (2013). "A survey of approximability and inapproximability results for social welfare optimization in multiagent resource allocation".
2631:
1968:
1503:{\displaystyle {\mathcal {A}}=\{(A^{1},\dots ,A^{n})\mid \forall i\in \colon A^{i}\subseteq ,\quad \forall i\neq j\in \colon A^{i}\cap A^{j}=\emptyset ,\quad \cup _{i=1}^{n}A^{i}=\}}
614:: from the subset of allocations maximizing the smallest utility, it selects those allocations that maximize the second-smallest utility, then the third-smallest utility, and so on.
1543:
2850:
1129:
The number of items should be as small as possible, subject to that all agents must agree that the chosen set is better than the non-chosen set. This variant is known as the
719:
Nash-optimal allocation: and prove hardness of calculating utilitarian-optimal and Nash-optimal allocations. present an approximation procedure for Nash-optimal allocations.
1535:
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of his "dictatorship utility", i.e., the utility he could get by picking the best option in each issue. Proportionality might be unattainable, but PROP1 is attainable by
696:
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1801:
199:
160:
2781:{\displaystyle d^{*}={\underset {d\in {\mathcal {D}}}{\operatorname {argmax} }}\min _{i=1,\ldots ,n}\sum _{A\in {\mathcal {A}}}\left(p_{d}(A)\cdot u_{i}(A)\right)}
252:
utility function). Once the agent reports a value for each individual item, it is easy to calculate the value of each bundle by summing up the values of its items.
1037:
More generally: does there always exist an EFM allocation when both divisible items and indivisible items may be positive for some agents and negative for others?
489:
454:
133:
2111:{\displaystyle d^{*}={\underset {d\in {\mathcal {D}}}{\operatorname {argmax} }}\sum _{i=1}^{n}\sum _{A\in {\mathcal {A}}}\left(p_{d}(A)\cdot u_{i}(A)\right)}
1282:
agents. Formally, in the deterministic setting, a solution describes a feasible allocation of the items to the agents — a partition of the set of items into
1070:
4370:
Conitzer, Vincent; Freeman, Rupert; Shah, Nisarg (2017). "Fair public decision making". In
Daskalakis, Constantinos; Babaioff, Moshe; Moulin, Hervé (eds.).
3538:. In Bureš, Tomáš; Dondi, Riccardo; Gamper, Johann; Guerrini, Giovanna; Jurdziński, Tomasz; Pahl, Claus; Sikora, Florian; Wong, Prudence W.H. (eds.).
53:
problem, where the dividend is divisible and a fair division always exists. In some cases, the indivisibility problem can be mitigated by introducing
1809:
372:
The maximin-share (also called: max-min-fair-share guarantee) of an agent is the most preferred bundle he could guarantee himself as divider in
3751:
3489:
Goldberg, Paul W.; Hollender, Alexandros; Igarashi, Ayumi; Manurangsi, Pasin; Suksompong, Warut (2022). "Consensus
Halving for Sets of Items".
294:: for each partner, there is a graph that represents the dependencies between different items. In the cardinal approach, a common tool is the
212:
In the ordinal approach, each partner should report a ranking over the items, i.e., say which item is the best, which is the second-best, etc.
707:
1224:(which implies both Pareto-efficiency and proportionality), maximum Nash welfare, leximin optimality and proportionality up to one item.
3182:
Bouveret, Sylvain; Lemaître, Michel (2015). "Characterizing conflicts in fair division of indivisible goods using a scale of criteria".
3979:
Babaioff, Moshe; Nisan, Noam; Talgam-Cohen, Inbal (2017-03-23). "Competitive
Equilibrium with Indivisible Goods and Generic Budgets".
1727:
1673:
802:
This raises the question of whether it is possible to attain fair allocations with fewer sharings than the worst-case upper bound:
4413:
Igarashi, Ayumi; Lackner, Martin; Nardi, Oliviero; Novaro, Arianna (2023-04-04). "Repeated Fair
Allocation of Indivisible Items".
162:
possible bundles. For example, if there are 16 items then each partner will have to present their preferences using 65536 numbers.
564:
valuable for A, then A does not envy B. EFx is strictly stronger than EF1. It is not known whether EFx allocations always exist.
2925:- a fair division problem in which each agent should get exactly one object, with neither monetary transfers nor randomization.
2246:{\displaystyle A^{*}={\underset {A\in {\mathcal {A}}\colon p_{d^{*}}(A)>0}{\operatorname {argmax} }}\sum _{i=1}^{n}u_{i}(A)}
105:
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4298:
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Kawase and Sumita present an algorithm that, given an algorithm for finding a deterministic allocation that approximates the
873:. They study the run-time complexity of deciding the existence of a fair allocation with s sharings or shared objects, where
2118:
Kawase and Sumita show that maximization of the utilitarian welfare in the stochastic setting can always be achieved with a
1537:. That is, the set of all stochastic allocations (i.e., all feasible solutions to the problem) can be described as follows:
4192:
Babaioff, Moshe; Ezra, Tomer; Feige, Uriel (2021-11-15). "Fair-Share
Allocations for Agents with Arbitrary Entitlements".
3265:
Heinen, Tobias; Nguyen, Nhan-Tam; Rothe, Jörg (2015). "Fairness and Rank-Weighted
Utilitarianism in Resource Allocation".
3047:
2941:- a general measure of the trade-off between fairness and efficiency, with some results about the item assignment setting.
1152:
Allocation of private goods can be seen as a special case of allocating public goods: given a private-goods problem with
939:
agents with piecewise linear valuations over the divisible goods. They also present an efficient algorithm that finds an
38:
Several lecturers want to divide the courses given in their faculty. Each lecturer can teach one or more whole courses.
35:
Several heirs want to divide the inherited property, which contains e.g. a house, a car, a piano and several paintings.
3323:
3025:
521:
Every allocation which gives the first and second items to Bob and Carl and the third item to Alice is proportional.
3217:
Budish, E. (2011). "The
Combinatorial Assignment Problem: Approximate Competitive Equilibrium from Equal Incomes".
865:
Bismuth, Makarov, Shapira and Segal-Halevi study fair allocation with identical valuations, which is equivalent to
729:
4372:
Proceedings of the 2017 ACM Conference on
Economics and Computation, EC '17, Cambridge, MA, USA, June 26-30, 2017
4277:
41st IARCS Annual
Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)
3542:. Lecture Notes in Computer Science. Vol. 12607. Cham: Springer International Publishing. pp. 421–430.
633:
An advantage of global optimization criteria over individual criteria is that welfare-maximizing allocations are
3012:
Sylvain
Bouveret and Yann Chevaleyre and Nicolas Maudet, "Fair Allocation of Indivisible Goods". Chapter 12 in:
3301:
Caragiannis, Ioannis; Kurokawa, David; Moulin, Hervé; Procaccia, Ariel D.; Shah, Nisarg; Wang, Junxing (2016).
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various fairness notions have to be adapted accordingly. Several classes of fairness notions were considered:
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2599:
1936:
1658:{\displaystyle {\mathcal {D}}=\{d\mid p_{d}\colon {\mathcal {A}}\to ,\sum _{A\in {\mathcal {A}}}p_{d}(A)=1\}}
866:
2901:, respectively. Many techniques used for these problems are useful in the case of fair item allocation, too.
2592:
says that society should choose the solution that maximize the utility of the poorest. That is, to choose a
990:
agents, they present a 1/2-MMS approximation. They also show that EFM is incompatible with non-wastefulness.
2949:
2913:- a fair division problem where indivisible items and a fixed total cost have to be divided simultaneously.
1123:
807:
760:
655:
41:
3596:
1009:
Liu, Lu, Suzuki and Walsh survey some recent results on mixed items, and identify several open questions:
491:. Hence, every proportional allocation is MMS-fair. Both inclusions are strict, even when every agent has
1238:
982:
870:
772:
713:
728:
Competitive equilibrium: various algorithms for finding a CE allocation are described in the article on
4213:
Segal-Halevi, Erel; Suksompong, Warut (2019-12-01). "Democratic fair allocation of indivisible goods".
4008:. Aamas '18. International Foundation for Autonomous Agents and Multiagent Systems. pp. 1267–1275.
2820:
1096:
4077:
3074:. Proceedings of the 2010 conference on ECAI 2010: 19th European Conference on Artificial Intelligence
1929:
says that society should choose the solution that maximize the sum of utilities. That is, to choose a
629:
if it maximizes the product of utilities. Nash-optimal allocations have some nice fairness properties.
3093:
Brams, Steven J.; Edelman, Paul H.; Fishburn, Peter C. (2003). "Fair Division of Indivisible Items".
2904:
2812:
764:
701:
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Garg, Jugal; Kulkarni, Pooja; Murhekar, Aniket (2021). Bojańczy, Miko\laj; Chekuri, Chandra (eds.).
3353:
3231:
3142:
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subsets (one for each agent). The set of all deterministic allocations can be described as follows:
1081:
classic item allocation setting corresponds to the special case in which all groups are singletons.
2922:
1268:
3535:
2944:
1214:
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Various algorithms for fair item allocation are surveyed in pages on specific fairness criteria:
594:
569:
339:: many languages for representing combinatorial preferences have been studied in the context of
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3226:
3137:
2916:
1918:
that are suggested for deterministic setting can also be considered in the stochastic setting:
806:
Sandomirskiy and Segal-Halevi study sharing minimization in allocations that are both fair and
3310:. Proceedings of the 2016 ACM Conference on Economics and Computation - EC '16. p. 305.
3071:
Fair Division Under Ordinal Preferences: Computing Envy-Free Allocations of Indivisible Goods
2919:- a fair division problem without money, in which fairness is attained through randomization.
2870:, finds a stochastic allocation that approximates the egalitarian welfare to the same factor
668:
457:
384:
340:
3438:
2817:
and also, that it is NP-hard to approximate the eqalitarian welfare to a factor better than
1122:
The total cost of all items must not exceed a fixed budget. This variant is often known as
3014:
Brandt, Felix; Conitzer, Vincent; Endriss, Ulle; Lang, Jérôme; Procaccia, Ariel D. (2016).
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2256:
2119:
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1141:
177:
138:
3720:
Proceedings of the Thirty-Second International Joint Conference on Artificial Intelligence
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532:
The above implications do not hold when the agents' valuations are not sub/superadditive.
205:
The second problem is often handled by working with individual items rather than bundles:
8:
4134:
3886:
3421:
Envy-ratio and average-nash social welfare optimization in multiagent resource allocation
3106:
2890:
1137:
977:
allocation of mixed goods, where the utility vector should minimize a symmetric strictly-
822:
783:
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number of shared objects / sharings required for various kinds of fair allocations among
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Does there always exist an EFM allocation when there are indivisible chores and a cake?
962:
821:
Goldberg, Hollender, Igarashi, Manurangsi and Suksompong study sharing minimization in
768:
228:
To make the item-assignment problem simpler, it is common to assume that all items are
209:
In the cardinal approach, each partner should report a numeric valuation for each item;
118:
4272:
2970:
829:, there is a polynomial-time algorithm for computing a consensus halving with at most
16:
Fair division problem in which the items to divide are discrete rather than continuous
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Bismuth, Samuel; Makarov, Vladislav; Segal-Halevi, Erel; Shapira, Dana (2023-11-08),
3551:
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1014:
998:
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329:
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50:
4333:. EC '18. New York, NY, USA: Association for Computing Machinery. pp. 575–592.
4283:. Dagstuhl, Germany: Schloss Dagstuhl – Leibniz-Zentrum für Informatik: 22:1–22:19.
4001:
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There are two functions related to each agent, a utility function associated with a
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3147:
3128:
Brams, S. J. (2005). "Efficient Fair Division: Help the Worst off or Avoid Envy?".
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1926:
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1021:
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social welfare is the product of the utilities of the agents. An assignment called
492:
298:(Generalized Additive Independence). In the ordinal approach, a common tool is the
245:
233:
171:
61:, or by discarding some of the items. But such solutions are not always available.
4289:
4019:
Chakraborty, Mithun; Igarashi, Ayumi; Suksompong, Warut; Zick, Yair (2021-08-16).
3405:
2907:- including some case-studies and lab experiments related to fair item assignment.
605:
social welfare is minimum utility of a single agent. An item assignment is called
112:
It may be difficult for a person to calculate exact numeric values to the bundles.
4236:
4103:
3861:
Li, Zihao; Liu, Shengxin; Lu, Xinhang; Tao, Biaoshuai; Tao, Yichen (2024-01-02),
3622:
3547:
3274:
3015:
1513:
In the stochastic setting, a solution is a probability distribution over the set
1027:
Are there bounded or even finite algorithms for computing EFM allocations in the
978:
661:
Minimax-share item allocation: The problem of calculating the mFS of an agent is
167:
4076:
Chakraborty, Mithun; Schmidt-Kraepelin, Ulrike; Suksompong, Warut (2021-12-01).
3791:
Fair Allocation with Binary Valuations for Mixed Divisible and Indivisible Goods
3767:
Envy-freeness and maximum Nash welfare for mixed divisible and indivisible goods
3033:
2806:
Kawase and Sumita prove that finding a stochastic allocation that maximizes the
3716:"Truthful Fair Mechanisms for Allocating Mixed Divisible and Indivisible Goods"
3663:
3302:
2910:
2792:. It is easy to see that in the deterministic setting the egalitarian value is
3912:
3727:
3397:
3362:
3195:
2893:- two well-studied optimization problems that can be seen as special cases of
1088:(fairness in the eyes of all agents in each group), so it is often relaxed to
798:−1) sharings/shared objects are always sufficient, and may be necessary.
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3472:
3151:
2994:
1040:
Is there a truthful EFM algorithm for agents with binary additive valuations?
947:
928:
858:
779:−1 sharings/shared objects are always sufficient, and may be necessary;
662:
649:
536:
399:
if every agent receives a bundle worth at least his proportional-fair-share.
366:
23:
4389:
4338:
3315:
3069:
994:
results are tight up to a constant for EF and asymptotically tight for PROP.
174:. In the ordinal model, each partner should only express a ranking over the
3512:
3464:
4326:
3715:
1899:{\displaystyle E_{i}(d)=\sum _{A\in {\mathcal {A}}}p_{d}(A)\cdot u_{i}(A)}
560:: For each two agents A and B, if we remove from the bundle of B the item
4133:
Chakraborty, Mithun; Segal-Halevi, Erel; Suksompong, Warut (2022-06-28).
3687:"On Approximate Envy-Freeness for Indivisible Chores and Mixed Resources"
1724:
and an expected utility function associated with a stochastic allocation
456:. Hence, every mFS-fair allocation is proportional. For every agent with
3839:
Allocating Mixed Goods with Customized Fairness and Indivisibility Ratio
881:−1. They prove that, for sharings, the problem is NP-hard for any
343:. Some of these languages can be adapted to the item assignment setting.
3956:
3885:
Liu, Shengxin; Lu, Xinhang; Suzuki, Mashbat; Walsh, Toby (2024-03-24).
3067:
1092:(fairness in the eyes of e.g. at least half the agents in each group).
4075:
3934:
Brams, Steven J.; Kaplan, Todd R. (2004). "Dividing the Indivisible".
3595:
Bei, Xiaohui; Li, Zihao; Liu, Shengxin; Lu, Xinhang (5 January 2021).
923:
Bei, Li, Liu, Liu and Lu study a mixture of indivisible and divisible
853:
sharings and an instance that requires 0 sharings. Probabilistically,
380:
if every agent receives a bundle that he weakly prefers over his MMS.
3636:
Bei, Xiaohui; Liu, Shengxin; Lu, Xinhang; Wang, Hongao (2021-06-30).
395:
of his utility from the entire set of items. An allocation is called
4046:
3714:
Li, Zihao; Liu, Shengxin; Lu, Xinhang; Tao, Biaoshuai (2023-08-19).
568:
4419:
4380:
4331:
Proceedings of the 2018 ACM Conference on Economics and Computation
4227:
4198:
4151:
4094:
4037:
3985:
3903:
3871:
3847:
3823:
3799:
3775:
3654:
3613:
3581:
3503:
3488:
3455:
3240:
4020:
3300:
3269:. Lecture Notes in Computer Science. Vol. 9346. p. 521.
1267:
is a type of fair item allocation in which a solution describes a
986:
if all divisible goods are identical, a polytime algorithm exists.
927:(objects with positive utiliies). They define an approximation to
4132:
4018:
3570:
849:, it is NP-hard to distinguish between an instance that requires
665:. The problem of deciding whether an mFS allocation exists is in
4445:"On the Max-Min Fair Stochastic Allocation of Indivisible Goods"
3789:
Kawase, Yasushi; Nishimura, Koichi; Sumita, Hanna (2023-11-08),
510:
Carl values the items as 3,2,1. For him, MMS=1, PFS=2 and mFS=3.
413:
first. Hence, an agent would object to an allocation only if in
4273:"On Fair and Efficient Allocations of Indivisible Public Goods"
813:
Misra and Sethia complement their result by proving that, when
518:
Every allocation which gives an item to each agent is MMS-fair.
507:
Bob values the items as 3,2,1. For him, MMS=1, PFS=2 and mFS=3.
108:
for more examples). There are two problems with this approach:
4078:"Picking sequences and monotonicity in weighted fair division"
1769:{\displaystyle E_{i}\colon {\mathcal {D}}\to \mathbb {R} _{+}}
1715:{\displaystyle u_{i}\colon {\mathcal {A}}\to \mathbb {R} _{+}}
1259:
264:
4449:
Proceedings of the AAAI Conference on Artificial Intelligence
4279:. Leibniz International Proceedings in Informatics (LIPIcs).
4139:
Proceedings of the AAAI Conference on Artificial Intelligence
3891:
Proceedings of the AAAI Conference on Artificial Intelligence
3638:"Maximin fairness with mixed divisible and indivisible goods"
917:
4325:
Fain, Brandon; Munagala, Kamesh; Shah, Nisarg (2018-06-11).
3418:
954:
Bhaskar, Sricharan and Vaish study a mixture of indivisible
4135:"Weighted Fairness Notions for Indivisible Items Revisited"
544:
an EF allocation may be not proportional and even not MMS.
4412:
3837:
Li, Bo; Li, Zihao; Liu, Shengxin; Wu, Zekai (2024-04-28),
1115:
items can be selected. This variant is closely related to
946:
Bei, Liu, Lu and Wang study the same setting, focusing on
698:, but its exact computational complexity is still unknown.
3978:
3691:
DROPS-IDN/V2/Document/10.4230/LIPIcs.APPROX/RANDOM.2021.1
3013:
1220:
Common solution concepts for public goods allocation are
845:
cuts. But sharing minimization is NP-hard: for any fixed
115:
The number of possible bundles can be huge: if there are
3597:"Fair division of mixed divisible and indivisible goods"
3436:
504:
Alice values the items as 2,2,2. For her, MMS=PFS=mFS=2.
4021:"Weighted Envy-freeness in Indivisible Item Allocation"
3813:
Bei, Xiaohui; Liu, Shengxin; Lu, Xinhang (2023-10-02),
3685:
Bhaskar, Umang; Sricharan, A. R.; Vaish, Rohit (2021).
3045:
376:
against adversarial opponents. An allocation is called
82:
Based on the preferences and the fairness criterion, a
4212:
2831:
3788:
3693:. Schloss Dagstuhl – Leibniz-Zentrum für Informatik.
3684:
3383:
2823:
2643:
2602:
2286:
2259:
2132:
1980:
1939:
1812:
1782:
1730:
1676:
1546:
1519:
1295:
1071:
Proportional cake-cutting with different entitlements
671:
469:
434:
180:
141:
121:
4270:
3540:
SOFSEM 2021: Theory and Practice of Computer Science
3437:
Sandomirskiy, Fedor; Segal-Halevi, Erel (May 2022).
3092:
3068:
Sylvain Bouveret; Ulle Endriss; Jérôme Lang (2010).
736:
64:
An item assignment problem has several ingredients:
4369:
3863:
A Complete Landscape for the Price of Envy-Freeness
3722:. IJCAI '23. Macao, P.R.China. pp. 2808–2816.
3046:Barberà, S.; Bossert, W.; Pattanaik, P. K. (2004).
2122:. The reason is that the utilitarian value of the
2844:
2780:
2625:
2575:
2272:
2245:
2110:
1962:
1898:
1795:
1768:
1714:
1657:
1529:
1502:
1059:Notions based on weighted competitive equilibrium;
690:
483:
448:
193:
154:
127:
4191:
3341:Annals of Mathematics and Artificial Intelligence
3304:The Unreasonable Fairness of Maximum Nash Welfare
3264:
1205:essentially represents the decision to give item
745:
90:These ingredients are explained in detail below.
4482:
4324:
4002:"Competitive Equilibrium For almost All Incomes"
3884:
3536:"Fair Division is Hard Even for Amicable Agents"
3181:
2969:Demko, Stephen; Hill, Theodore P. (1988-10-01).
2680:
2479:
495:. This is illustrated in the following example:
383:
352:
86:should be executed to calculate a fair division.
3765:Nishimura, Koichi; Sumita, Hanna (2023-08-13),
3764:
3338:
2971:"Equitable distribution of indivisible objects"
2810:is NP-hard even when agents' utilities are all
1084:With groups, it may be impossible to guarantee
1005:agents, for both scaled and unscaled utilities.
584:
3439:"Efficient Fair Division with Minimal Sharing"
1020:Are there efficient algorithms for maximizing
913:is not fixed, the problem is strongly NP-hard.
877:is smaller than the worst-case upper bound of
328:: each partner describes some bundles using a
302:(Conditional Preferences) and its extensions:
216:Under suitable assumptions, it is possible to
4327:"Fair Allocation of Indivisible Public Goods"
4025:ACM Transactions on Economics and Computation
1101:
391:The proportional-fair-share of an agent is 1/
273:combinatorial utilities). Some examples are:
4442:
3999:
3635:
2931:- a fair division problem in which seats in
1652:
1557:
1497:
1306:
708:Efficient approximately fair item allocation
244:In the cardinal approach, each agent has an
98:
3933:
3533:
3296:
3294:
1271:over the set of deterministic allocations.
1265:Stochastic allocations of indivisible goods
1260:Stochastic allocations of indivisible goods
402:
270:Compact preference representation languages
265:Compact preference representation languages
3860:
3815:Fair Division with Subjective Divisibility
3750:: CS1 maint: location missing publisher (
3713:
3594:
3008:
3006:
3004:
1044:
918:Mixture of divisible and indivisible goods
4460:
4418:
4379:
4288:
4226:
4197:
4160:
4150:
4093:
4036:
3984:
3955:
3902:
3870:
3846:
3836:
3822:
3798:
3774:
3698:
3653:
3642:Autonomous Agents and Multi-Agent Systems
3612:
3580:
3502:
3454:
3386:Autonomous Agents and Multi-Agent Systems
3352:
3230:
3184:Autonomous Agents and Multi-Agent Systems
3141:
2968:
1756:
1702:
1227:
1049:
515:The possible allocations are as follows:
26:problem in which the items to divide are
3812:
3534:Misra, Neeldhara; Sethia, Aditi (2021).
3291:
2796:, while in the stochastic setting it is
1175:, construct a public-goods problem with
1062:Notions based on weighted envy-freeness;
833:sharings, and for computing a consensus
640:
501:There are three agents and three items:
4443:Kawase, Yasushi; Sumita, Hanna (2020).
3419:Trung Thanh Nguyen; Jörg Rothe (2013).
3377:
3260:
3258:
3210:
3017:Handbook of Computational Social Choice
3001:
2626:{\displaystyle d^{*}\in {\mathcal {D}}}
1963:{\displaystyle d^{*}\in {\mathcal {D}}}
1075:
869:, and also the more general setting of
223:
166:The first problem motivates the use of
4483:
4438:
4436:
4434:
4432:
4430:
3332:
3216:
3177:
3175:
3173:
3171:
3169:
1915:
1909:
1250:
593:evaluates a division based on a given
106:Utility functions on indivisible goods
4320:
4318:
4266:
4264:
4262:
3484:
3482:
3432:
3430:
3412:
3127:
2253:is at least the utilitarian value of
1065:Notions based on weighted fair share;
889:−2; but for shared objects and
535:
3927:
3255:
3086:
3061:
1278:items should be distributed between
347:
4427:
3700:10.4230/LIPIcs.APPROX/RANDOM.2021.1
3166:
3121:
2852:even when all agents have the same
997:Li, Liu, Lu, Tao and Tao study the
973:Kawase, Nishimura and Sumita study
893:≥ 3, the problem is polynomial for
825:. They prove that, for agents with
558:Envy-freeness-except-cheapest (EFx)
68:The partners have to express their
13:
4363:
4315:
4259:
4206:
3573:Number Partitioning with Splitting
3491:Mathematics of Operations Research
3479:
3427:
3107:10.1023/B:THEO.0000024421.85722.0a
2720:
2671:
2618:
2491:
2402:
2321:
2160:
2050:
2008:
1955:
1848:
1746:
1692:
1620:
1581:
1549:
1522:
1447:
1394:
1347:
1298:
408:I choose first”. An allocation is
14:
4502:
4000:Segal-Halevi, Erel (2018-07-09).
2935:should be divided among students.
2845:{\displaystyle 1-{\tfrac {1}{e}}}
737:Between divisible and indivisible
365:
4406:
4185:
4126:
4069:
4012:
3993:
3972:
3936:Journal of Theoretical Politics
3887:"Mixed Fair Division: A Survey"
3878:
3854:
3830:
3806:
3782:
3758:
3707:
3678:
3629:
3588:
3564:
3527:
1453:
1393:
1201:and the other items at 0. Item
901:−2 and NP-hard for any
547:Weaker versions of EF include:
72:for the different item-bundles.
3039:
3020:. Cambridge University Press.
2962:
2770:
2764:
2748:
2742:
2570:
2564:
2522:
2516:
2472:
2466:
2432:
2426:
2378:
2372:
2356:
2350:
2240:
2234:
2191:
2185:
2100:
2094:
2078:
2072:
1893:
1887:
1871:
1865:
1829:
1823:
1751:
1697:
1643:
1637:
1601:
1589:
1586:
1530:{\displaystyle {\mathcal {A}}}
1494:
1488:
1415:
1409:
1387:
1381:
1362:
1356:
1341:
1309:
981:(this is a generalization off
746:Bounding the amount of sharing
359:individual guarantee criterion
93:
1:
4290:10.4230/LIPIcs.FSTTCS.2021.22
2955:
2899:indivisible chores allocation
867:Identical-machines scheduling
808:Fractionally Pareto efficient
650:Maximin-share item allocation
591:global optimization criterion
353:Individual guarantee criteria
322:(a simplification of CI net).
318:(Conditional Importance) and
256:{x,y} or even {w,z} to {x,y}.
75:The group should decide on a
4237:10.1016/j.artint.2019.103167
4104:10.1016/j.artint.2021.103578
3623:10.1016/j.artint.2020.103436
3548:10.1007/978-3-030-67731-2_31
3275:10.1007/978-3-319-23114-3_31
3219:Journal of Political Economy
2987:10.1016/0165-4896(88)90047-9
2975:Mathematical Social Sciences
2950:17-animal inheritance puzzle
2895:indivisible goods allocation
943:-approximate EFM allocation.
656:Proportional item allocation
585:Global optimization criteria
552:Envy-freeness-except-1 (EF1)
42:White elephant gift exchange
7:
4374:. {ACM}. pp. 629–646.
3267:Algorithmic Decision Theory
2880:
1776:which defined according to
1239:Round-robin item allocation
983:Egalitarian item allocation
871:Uniform-machines scheduling
714:Egalitarian item allocation
284:for every positive integer
10:
4507:
3664:10.1007/s10458-021-09517-7
3055:Handbook of utility theory
3048:"Ranking sets of objects."
1243:
1102:Allocation of public goods
1097:Fair division among groups
1094:
1068:
1029:Robertson–Webb query model
1022:Utilitarian social welfare
961:Li, Liu, Lu and Tao study
524:No allocation is mFS-fair.
4006:Proceedings of AAMAS 2018
3913:10.1609/aaai.v38i20.30274
3398:10.1007/s10458-013-9224-2
3363:10.1007/s10472-012-9328-4
3196:10.1007/s10458-015-9287-3
2905:Fair division experiments
1670:deterministic allocation
702:Envy-free item allocation
572:from Equal Incomes (CEEI)
99:Combinatorial preferences
84:fair assignment algorithm
4462:10.1609/AAAI.V34I02.5580
4162:10.1609/aaai.v36i5.20425
3948:10.1177/0951629804041118
3152:10.1177/1043463105058317
2923:House allocation problem
2120:deterministic allocation
1269:probability distribution
837:-division with at most (
403:Min-max fair-share (mFS)
4390:10.1145/3033274.3085125
4339:10.1145/3219166.3219174
4215:Artificial Intelligence
4082:Artificial Intelligence
3728:10.24963/ijcai.2023/313
3601:Artificial Intelligence
3316:10.1145/2940716.2940726
3130:Rationality and Society
2945:Fair subset sum problem
1124:participatory budgeting
1045:Variants and extensions
1013:Is EFM compatible with
691:{\displaystyle NP^{NP}}
595:social welfare function
570:Competitive equilibrium
248:function (also called:
3513:10.1287/moor.2021.1249
3465:10.1287/opre.2022.2279
2917:Fair random assignment
2846:
2782:
2627:
2577:
2553:
2455:
2307:
2274:
2247:
2223:
2112:
2036:
1964:
1900:
1797:
1770:
1716:
1659:
1531:
1504:
1228:Public decision making
1050:Different entitlements
1024:among EFM allocations?
692:
485:
450:
341:combinatorial auctions
282:k-additive preferences
278:2-additive preferences
195:
156:
129:
2847:
2783:
2628:
2578:
2533:
2435:
2287:
2275:
2273:{\displaystyle d^{*}}
2248:
2203:
2113:
2016:
1965:
1901:
1798:
1796:{\displaystyle u_{i}}
1771:
1717:
1660:
1532:
1505:
1136:There may be general
693:
641:Allocation algorithms
486:
458:superadditive utility
451:
421:For every agent with
326:Logic based languages
196:
194:{\displaystyle 2^{m}}
157:
155:{\displaystyle 2^{m}}
135:items then there are
130:
4491:Fair item allocation
2887:Bin covering problem
2821:
2641:
2600:
2284:
2257:
2130:
1978:
1937:
1810:
1780:
1728:
1674:
1544:
1517:
1293:
1217:optimal allocation.
1146:knapsack constraints
1142:matching constraints
1076:Allocation to groups
669:
627:Maximum-Nash-Welfare
467:
432:
224:Additive preferences
178:
139:
119:
20:Fair item allocation
3897:(20): 22641–22649.
3443:Operations Research
3095:Theory and Decision
3034:free online version
2891:Bin packing problem
2864:utilitarian welfare
2808:eqalitarian welfare
2635:egalitarian walfare
2633:that maximizes the
1972:utilitarian walfare
1970:that maximizes the
1474:
1251:Repeated allocation
1185:items, where agent
1160:items, where agent
1138:matroid constraints
1090:democratic fairness
963:truthful mechanisms
823:consensus splitting
784:Consensus splitting
607:egalitarian-optimal
484:{\displaystyle 1/n}
449:{\displaystyle 1/n}
425:, the mFS is worth
423:subadditive utility
238:complementary goods
59:time-based rotation
3607:103436. Elsevier.
2933:university courses
2854:submodular utility
2842:
2840:
2778:
2726:
2706:
2677:
2623:
2573:
2532:
2408:
2327:
2270:
2243:
2201:
2108:
2056:
2014:
1960:
1896:
1854:
1793:
1766:
1712:
1655:
1626:
1527:
1500:
1454:
1246:multi-issue voting
1148:on the chosen set.
1117:multiwinner voting
1086:unanimous fairness
827:additive utilities
688:
612:leximin-optimality
481:
460:, the MMSis worth
446:
191:
152:
125:
77:fairness criterion
4399:978-1-4503-4527-9
4348:978-1-4503-5829-3
4300:978-3-95977-215-0
4031:(3): 18:1–18:39.
3737:978-1-956792-03-4
3557:978-3-030-67731-2
3284:978-3-319-23113-6
2939:Price of fairness
2929:Course allocation
2839:
2707:
2679:
2658:
2586:Egalitarian rule:
2478:
2389:
2308:
2147:
2037:
1995:
1910:Fairness criteria
1835:
1607:
1189:values each item
1015:Pareto-efficiency
999:price of fairness
579:Pareto efficiency
374:divide and choose
348:Fairness criteria
337:Bidding languages
330:first order logic
232:(so they are not
230:independent goods
128:{\displaystyle m}
55:monetary payments
51:fair cake-cutting
22:is a kind of the
4498:
4475:
4474:
4464:
4455:(2): 2070–2078.
4440:
4425:
4424:
4422:
4410:
4404:
4403:
4383:
4367:
4361:
4360:
4322:
4313:
4312:
4292:
4268:
4257:
4256:
4230:
4210:
4204:
4203:
4201:
4189:
4183:
4182:
4164:
4154:
4145:(5): 4949–4956.
4130:
4124:
4123:
4097:
4073:
4067:
4066:
4040:
4016:
4010:
4009:
3997:
3991:
3990:
3988:
3976:
3970:
3969:
3959:
3931:
3925:
3924:
3906:
3882:
3876:
3875:
3874:
3858:
3852:
3851:
3850:
3834:
3828:
3827:
3826:
3810:
3804:
3803:
3802:
3786:
3780:
3779:
3778:
3762:
3756:
3755:
3749:
3741:
3711:
3705:
3704:
3702:
3682:
3676:
3675:
3657:
3633:
3627:
3626:
3616:
3592:
3586:
3585:
3584:
3568:
3562:
3561:
3531:
3525:
3524:
3506:
3497:(4): 3357–3379.
3486:
3477:
3476:
3458:
3449:(3): 1762–1782.
3434:
3425:
3424:
3416:
3410:
3409:
3381:
3375:
3374:
3356:
3336:
3330:
3329:
3309:
3298:
3289:
3288:
3262:
3253:
3252:
3234:
3225:(6): 1061–1103.
3214:
3208:
3207:
3179:
3164:
3163:
3145:
3125:
3119:
3118:
3090:
3084:
3083:
3081:
3079:
3065:
3059:
3058:
3052:
3043:
3037:
3031:
3010:
2999:
2998:
2966:
2873:
2869:
2851:
2849:
2848:
2843:
2841:
2832:
2799:
2795:
2791:
2787:
2785:
2784:
2779:
2777:
2773:
2763:
2762:
2741:
2740:
2725:
2724:
2723:
2705:
2678:
2676:
2675:
2674:
2653:
2652:
2632:
2630:
2629:
2624:
2622:
2621:
2612:
2611:
2582:
2580:
2579:
2574:
2563:
2562:
2552:
2547:
2531:
2515:
2514:
2513:
2512:
2495:
2494:
2465:
2464:
2454:
2449:
2425:
2424:
2423:
2422:
2407:
2406:
2405:
2385:
2381:
2371:
2370:
2349:
2348:
2347:
2346:
2326:
2325:
2324:
2306:
2301:
2279:
2277:
2276:
2271:
2269:
2268:
2252:
2250:
2249:
2244:
2233:
2232:
2222:
2217:
2202:
2200:
2184:
2183:
2182:
2181:
2164:
2163:
2142:
2141:
2117:
2115:
2114:
2109:
2107:
2103:
2093:
2092:
2071:
2070:
2055:
2054:
2053:
2035:
2030:
2015:
2013:
2012:
2011:
1990:
1989:
1969:
1967:
1966:
1961:
1959:
1958:
1949:
1948:
1923:Utilitarian rule
1905:
1903:
1902:
1897:
1886:
1885:
1864:
1863:
1853:
1852:
1851:
1822:
1821:
1802:
1800:
1799:
1794:
1792:
1791:
1775:
1773:
1772:
1767:
1765:
1764:
1759:
1750:
1749:
1740:
1739:
1723:
1721:
1719:
1718:
1713:
1711:
1710:
1705:
1696:
1695:
1686:
1685:
1664:
1662:
1661:
1656:
1636:
1635:
1625:
1624:
1623:
1585:
1584:
1575:
1574:
1553:
1552:
1536:
1534:
1533:
1528:
1526:
1525:
1509:
1507:
1506:
1501:
1484:
1483:
1473:
1468:
1443:
1442:
1430:
1429:
1377:
1376:
1340:
1339:
1321:
1320:
1302:
1301:
1285:
1281:
1277:
1184:
1131:agreeable subset
723:Picking sequence
697:
695:
694:
689:
687:
686:
635:Pareto efficient
493:additive utility
490:
488:
487:
482:
477:
455:
453:
452:
447:
442:
387:fair-share (PFS)
292:Graphical models
246:additive utility
234:substitute goods
200:
198:
197:
192:
190:
189:
172:cardinal utility
161:
159:
158:
153:
151:
150:
134:
132:
131:
126:
4506:
4505:
4501:
4500:
4499:
4497:
4496:
4495:
4481:
4480:
4479:
4478:
4441:
4428:
4411:
4407:
4400:
4368:
4364:
4349:
4323:
4316:
4301:
4269:
4260:
4211:
4207:
4190:
4186:
4131:
4127:
4074:
4070:
4047:10.1145/3457166
4017:
4013:
3998:
3994:
3977:
3973:
3932:
3928:
3883:
3879:
3859:
3855:
3835:
3831:
3811:
3807:
3787:
3783:
3763:
3759:
3743:
3742:
3738:
3712:
3708:
3683:
3679:
3634:
3630:
3593:
3589:
3569:
3565:
3558:
3532:
3528:
3487:
3480:
3435:
3428:
3417:
3413:
3382:
3378:
3354:10.1.1.671.3497
3337:
3333:
3326:
3307:
3299:
3292:
3285:
3263:
3256:
3232:10.1.1.357.9766
3215:
3211:
3180:
3167:
3143:10.1.1.118.9114
3126:
3122:
3091:
3087:
3077:
3075:
3066:
3062:
3050:
3044:
3040:
3028:
3011:
3002:
2967:
2963:
2958:
2883:
2871:
2867:
2830:
2822:
2819:
2818:
2813:budget-additive
2797:
2793:
2789:
2758:
2754:
2736:
2732:
2731:
2727:
2719:
2718:
2711:
2683:
2670:
2669:
2662:
2657:
2648:
2644:
2642:
2639:
2638:
2617:
2616:
2607:
2603:
2601:
2598:
2597:
2558:
2554:
2548:
2537:
2508:
2504:
2503:
2499:
2490:
2489:
2482:
2460:
2456:
2450:
2439:
2418:
2414:
2413:
2409:
2401:
2400:
2393:
2366:
2362:
2342:
2338:
2337:
2333:
2332:
2328:
2320:
2319:
2312:
2302:
2291:
2285:
2282:
2281:
2264:
2260:
2258:
2255:
2254:
2228:
2224:
2218:
2207:
2177:
2173:
2172:
2168:
2159:
2158:
2151:
2146:
2137:
2133:
2131:
2128:
2127:
2088:
2084:
2066:
2062:
2061:
2057:
2049:
2048:
2041:
2031:
2020:
2007:
2006:
1999:
1994:
1985:
1981:
1979:
1976:
1975:
1954:
1953:
1944:
1940:
1938:
1935:
1934:
1912:
1881:
1877:
1859:
1855:
1847:
1846:
1839:
1817:
1813:
1811:
1808:
1807:
1787:
1783:
1781:
1778:
1777:
1760:
1755:
1754:
1745:
1744:
1735:
1731:
1729:
1726:
1725:
1706:
1701:
1700:
1691:
1690:
1681:
1677:
1675:
1672:
1671:
1669:
1631:
1627:
1619:
1618:
1611:
1580:
1579:
1570:
1566:
1548:
1547:
1545:
1542:
1541:
1521:
1520:
1518:
1515:
1514:
1479:
1475:
1469:
1458:
1438:
1434:
1425:
1421:
1372:
1368:
1335:
1331:
1316:
1312:
1297:
1296:
1294:
1291:
1290:
1283:
1279:
1275:
1262:
1253:
1248:
1230:
1198:
1176:
1173:
1104:
1099:
1078:
1073:
1052:
1047:
979:convex function
920:
909:−3. When
748:
739:
679:
675:
670:
667:
666:
643:
587:
574:
541:
473:
468:
465:
464:
438:
433:
430:
429:
405:
389:
370:
355:
350:
267:
226:
185:
181:
179:
176:
175:
168:ordinal utility
146:
142:
140:
137:
136:
120:
117:
116:
101:
96:
17:
12:
11:
5:
4504:
4494:
4493:
4477:
4476:
4426:
4405:
4398:
4362:
4347:
4314:
4299:
4258:
4205:
4184:
4125:
4068:
4011:
3992:
3971:
3926:
3877:
3853:
3829:
3805:
3781:
3757:
3736:
3706:
3677:
3628:
3587:
3563:
3556:
3526:
3478:
3426:
3411:
3376:
3347:(1–3): 65–90.
3331:
3324:
3290:
3283:
3254:
3241:10.1086/664613
3209:
3165:
3136:(4): 387–421.
3120:
3085:
3060:
3057:. Springer US.
3038:
3026:
3000:
2981:(2): 145–158.
2960:
2959:
2957:
2954:
2953:
2952:
2947:
2942:
2936:
2926:
2920:
2914:
2911:Rental harmony
2908:
2902:
2882:
2879:
2878:
2877:
2876:
2875:
2857:
2838:
2835:
2829:
2826:
2776:
2772:
2769:
2766:
2761:
2757:
2753:
2750:
2747:
2744:
2739:
2735:
2730:
2722:
2717:
2714:
2710:
2704:
2701:
2698:
2695:
2692:
2689:
2686:
2682:
2673:
2668:
2665:
2661:
2656:
2651:
2647:
2620:
2615:
2610:
2606:
2583:
2572:
2569:
2566:
2561:
2557:
2551:
2546:
2543:
2540:
2536:
2530:
2527:
2524:
2521:
2518:
2511:
2507:
2502:
2498:
2493:
2488:
2485:
2481:
2477:
2474:
2471:
2468:
2463:
2459:
2453:
2448:
2445:
2442:
2438:
2434:
2431:
2428:
2421:
2417:
2412:
2404:
2399:
2396:
2392:
2388:
2384:
2380:
2377:
2374:
2369:
2365:
2361:
2358:
2355:
2352:
2345:
2341:
2336:
2331:
2323:
2318:
2315:
2311:
2305:
2300:
2297:
2294:
2290:
2267:
2263:
2242:
2239:
2236:
2231:
2227:
2221:
2216:
2213:
2210:
2206:
2199:
2196:
2193:
2190:
2187:
2180:
2176:
2171:
2167:
2162:
2157:
2154:
2150:
2145:
2140:
2136:
2106:
2102:
2099:
2096:
2091:
2087:
2083:
2080:
2077:
2074:
2069:
2065:
2060:
2052:
2047:
2044:
2040:
2034:
2029:
2026:
2023:
2019:
2010:
2005:
2002:
1998:
1993:
1988:
1984:
1957:
1952:
1947:
1943:
1911:
1908:
1907:
1906:
1895:
1892:
1889:
1884:
1880:
1876:
1873:
1870:
1867:
1862:
1858:
1850:
1845:
1842:
1838:
1834:
1831:
1828:
1825:
1820:
1816:
1790:
1786:
1763:
1758:
1753:
1748:
1743:
1738:
1734:
1709:
1704:
1699:
1694:
1689:
1684:
1680:
1666:
1665:
1654:
1651:
1648:
1645:
1642:
1639:
1634:
1630:
1622:
1617:
1614:
1610:
1606:
1603:
1600:
1597:
1594:
1591:
1588:
1583:
1578:
1573:
1569:
1565:
1562:
1559:
1556:
1551:
1524:
1511:
1510:
1499:
1496:
1493:
1490:
1487:
1482:
1478:
1472:
1467:
1464:
1461:
1457:
1452:
1449:
1446:
1441:
1437:
1433:
1428:
1424:
1420:
1417:
1414:
1411:
1408:
1405:
1402:
1399:
1396:
1392:
1389:
1386:
1383:
1380:
1375:
1371:
1367:
1364:
1361:
1358:
1355:
1352:
1349:
1346:
1343:
1338:
1334:
1330:
1327:
1324:
1319:
1315:
1311:
1308:
1305:
1300:
1261:
1258:
1252:
1249:
1229:
1226:
1222:core stability
1196:
1171:
1150:
1149:
1134:
1127:
1120:
1103:
1100:
1077:
1074:
1067:
1066:
1063:
1060:
1051:
1048:
1046:
1043:
1042:
1041:
1038:
1035:
1032:
1025:
1018:
1007:
1006:
995:
991:
987:
971:
967:
959:
952:
944:
919:
916:
915:
914:
863:
819:
811:
800:
799:
780:
747:
744:
738:
735:
734:
733:
726:
720:
717:
711:
705:
699:
685:
682:
678:
674:
659:
653:
642:
639:
631:
630:
615:
586:
583:
573:
567:
566:
565:
555:
540:
534:
530:
529:
528:
527:
526:
525:
522:
519:
513:
512:
511:
508:
505:
480:
476:
472:
445:
441:
437:
404:
401:
388:
382:
369:
364:
354:
351:
349:
346:
345:
344:
334:
323:
289:
266:
263:
258:
257:
253:
225:
222:
214:
213:
210:
188:
184:
164:
163:
149:
145:
124:
113:
100:
97:
95:
92:
88:
87:
80:
73:
46:
45:
39:
36:
15:
9:
6:
4:
3:
2:
4503:
4492:
4489:
4488:
4486:
4472:
4468:
4463:
4458:
4454:
4450:
4446:
4439:
4437:
4435:
4433:
4431:
4421:
4416:
4409:
4401:
4395:
4391:
4387:
4382:
4377:
4373:
4366:
4358:
4354:
4350:
4344:
4340:
4336:
4332:
4328:
4321:
4319:
4310:
4306:
4302:
4296:
4291:
4286:
4282:
4278:
4274:
4267:
4265:
4263:
4254:
4250:
4246:
4242:
4238:
4234:
4229:
4224:
4220:
4216:
4209:
4200:
4195:
4188:
4180:
4176:
4172:
4168:
4163:
4158:
4153:
4148:
4144:
4140:
4136:
4129:
4121:
4117:
4113:
4109:
4105:
4101:
4096:
4091:
4087:
4083:
4079:
4072:
4064:
4060:
4056:
4052:
4048:
4044:
4039:
4034:
4030:
4026:
4022:
4015:
4007:
4003:
3996:
3987:
3982:
3975:
3967:
3963:
3958:
3953:
3949:
3945:
3941:
3937:
3930:
3922:
3918:
3914:
3910:
3905:
3900:
3896:
3892:
3888:
3881:
3873:
3868:
3864:
3857:
3849:
3844:
3840:
3833:
3825:
3820:
3816:
3809:
3801:
3796:
3792:
3785:
3777:
3772:
3768:
3761:
3753:
3747:
3739:
3733:
3729:
3725:
3721:
3717:
3710:
3701:
3696:
3692:
3688:
3681:
3673:
3669:
3665:
3661:
3656:
3651:
3647:
3643:
3639:
3632:
3624:
3620:
3615:
3610:
3606:
3602:
3598:
3591:
3583:
3578:
3574:
3567:
3559:
3553:
3549:
3545:
3541:
3537:
3530:
3522:
3518:
3514:
3510:
3505:
3500:
3496:
3492:
3485:
3483:
3474:
3470:
3466:
3462:
3457:
3452:
3448:
3444:
3440:
3433:
3431:
3422:
3415:
3407:
3403:
3399:
3395:
3391:
3387:
3380:
3372:
3368:
3364:
3360:
3355:
3350:
3346:
3342:
3335:
3327:
3325:9781450339360
3321:
3317:
3313:
3306:
3305:
3297:
3295:
3286:
3280:
3276:
3272:
3268:
3261:
3259:
3250:
3246:
3242:
3238:
3233:
3228:
3224:
3220:
3213:
3205:
3201:
3197:
3193:
3189:
3185:
3178:
3176:
3174:
3172:
3170:
3161:
3157:
3153:
3149:
3144:
3139:
3135:
3131:
3124:
3116:
3112:
3108:
3104:
3100:
3096:
3089:
3073:
3072:
3064:
3056:
3049:
3042:
3035:
3029:
3027:9781107060432
3023:
3019:
3018:
3009:
3007:
3005:
2996:
2992:
2988:
2984:
2980:
2976:
2972:
2965:
2961:
2951:
2948:
2946:
2943:
2940:
2937:
2934:
2930:
2927:
2924:
2921:
2918:
2915:
2912:
2909:
2906:
2903:
2900:
2896:
2892:
2888:
2885:
2884:
2865:
2861:
2858:
2855:
2836:
2833:
2827:
2824:
2816:
2814:
2809:
2805:
2802:
2801:
2774:
2767:
2759:
2755:
2751:
2745:
2737:
2733:
2728:
2715:
2712:
2708:
2702:
2699:
2696:
2693:
2690:
2687:
2684:
2666:
2663:
2659:
2654:
2649:
2645:
2636:
2613:
2608:
2604:
2595:
2591:
2587:
2584:
2567:
2559:
2555:
2549:
2544:
2541:
2538:
2534:
2528:
2525:
2519:
2509:
2505:
2500:
2496:
2486:
2483:
2475:
2469:
2461:
2457:
2451:
2446:
2443:
2440:
2436:
2429:
2419:
2415:
2410:
2397:
2394:
2390:
2386:
2382:
2375:
2367:
2363:
2359:
2353:
2343:
2339:
2334:
2329:
2316:
2313:
2309:
2303:
2298:
2295:
2292:
2288:
2265:
2261:
2237:
2229:
2225:
2219:
2214:
2211:
2208:
2204:
2197:
2194:
2188:
2178:
2174:
2169:
2165:
2155:
2152:
2148:
2143:
2138:
2134:
2125:
2124:deterministic
2121:
2104:
2097:
2089:
2085:
2081:
2075:
2067:
2063:
2058:
2045:
2042:
2038:
2032:
2027:
2024:
2021:
2017:
2003:
2000:
1996:
1991:
1986:
1982:
1973:
1950:
1945:
1941:
1932:
1928:
1924:
1921:
1920:
1919:
1917:
1916:same criteria
1890:
1882:
1878:
1874:
1868:
1860:
1856:
1843:
1840:
1836:
1832:
1826:
1818:
1814:
1806:
1805:
1804:
1788:
1784:
1761:
1741:
1736:
1732:
1707:
1687:
1682:
1678:
1649:
1646:
1640:
1632:
1628:
1615:
1612:
1608:
1604:
1598:
1595:
1592:
1576:
1571:
1567:
1563:
1560:
1554:
1540:
1539:
1538:
1491:
1485:
1480:
1476:
1470:
1465:
1462:
1459:
1455:
1450:
1444:
1439:
1435:
1431:
1426:
1422:
1418:
1412:
1406:
1403:
1400:
1397:
1390:
1384:
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761:proportional
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623:Nash-optimal
622:
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3423:. AAMAS 13.
2596:allocation
1933:allocation
1156:agents and
773:egalitarian
603:egalitarian
94:Preferences
70:preferences
4420:2304.01644
4381:1611.04034
4228:1709.02564
4221:: 103167.
4199:2103.04304
4152:2112.04166
4095:2104.14347
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3942:(2): 143.
3904:2306.09564
3872:2401.01516
3848:2404.18132
3824:2310.00976
3800:2306.05986
3776:2302.13342
3655:2002.05245
3614:1911.07048
3582:2204.11753
3504:2007.06754
3456:1908.01669
3392:(2): 256.
3190:(2): 259.
3101:(2): 147.
2956:References
2860:Algorithm:
2594:stochastic
1931:stochastic
1244:See also:
1095:See also:
1069:See also:
240:). Then:
4471:214407880
4309:236154847
4253:203034477
4245:0004-3702
4179:244954009
4171:2374-3468
4120:233443832
4112:0004-3702
4063:202719373
4055:2167-8375
3966:154854134
3921:2374-3468
3746:cite book
3672:1573-7454
3648:(2): 34.
3521:246764981
3473:0030-364X
3349:CiteSeerX
3249:154703357
3227:CiteSeerX
3160:154808734
3138:CiteSeerX
3115:153943630
3078:26 August
2995:0165-4896
2856:function.
2828:−
2804:Hardness:
2752:⋅
2716:∈
2709:∑
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2667:∈
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2609:∗
2535:∑
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2476:≤
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2360:⋅
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2046:∈
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2018:∑
2004:∈
1987:∗
1951:∈
1946:∗
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1837:∑
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1354:∈
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1326:…
1209:to agent
841:−1)
818:sharings.
769:equitable
765:envy-free
312:CP theory
4485:Category
3204:16041218
2881:See also
1133:problem.
1111:At most
755:agents:
427:at least
410:mFS-fair
378:MMS-fair
28:discrete
4357:3331859
3371:6864410
1925:: this
1215:leximin
1180:·
975:optimal
951:values.
941:epsilon
931:called
862:agents.
790:parts,
462:at most
320:SCI net
308:UCP net
304:TCP net
296:GAI net
250:modular
44:parties
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2868:α
2660:argmax
2149:argmax
1997:argmax
956:chores
316:CI net
300:CP net
4467:S2CID
4415:arXiv
4376:arXiv
4353:S2CID
4305:S2CID
4249:S2CID
4223:arXiv
4194:arXiv
4175:S2CID
4147:arXiv
4116:S2CID
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4059:S2CID
4033:arXiv
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3962:S2CID
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3867:arXiv
3843:arXiv
3819:arXiv
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3771:arXiv
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3402:S2CID
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3308:(PDF)
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3111:S2CID
3051:(PDF)
2588:this
925:goods
786:into
562:least
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4295:ISBN
4241:ISSN
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4051:ISSN
3917:ISSN
3752:link
3732:ISBN
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3469:ISSN
3320:ISBN
3279:ISBN
3080:2016
3022:ISBN
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2897:and
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782:For
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759:For
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617:The
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333:xyz.
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218:lift
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