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Discrete fair division of resources is a fundamental problem in various multi-agent settings, where the goal is to partition a set M of m indivisible goods among n agents in a fair and efficient manner

# Fair and Efficient Allocations under Subadditive Valuations

AAAI, pp.5269-5276, (2021)

EI

We study the problem of allocating a set of indivisible goods among agents with subadditive valuations in a fair and efficient manner. Envy-Freeness up to any good (EFX) is the most compelling notion of fairness in the context of indivisible goods. Although the existence of EFX is not known beyond the simple case of two agents with suba...更多

• Discrete fair division of resources is a fundamental problem in various multi-agent settings, where the goal is to partition a set M of m indivisible goods among n agents in a fair and efficient manner.
• The authors design a polynomial-time algorithm that outputs an allocation that satisfies either of the two approximations of EFX as well as achieves an O(n) approximation to the Nash welfare.
• EFX allocation can be computed in polynomial time when agents have subadditive valuations.

• Discrete fair division of resources is a fundamental problem in various multi-agent settings, where the goal is to partition a set M of m indivisible goods among n agents in a fair and efficient manner
• As mentioned earlier in Section 1.1 the singleton sets allocated to the agents are the barriers to proving our desired approximation for any (α, c)-EFX allocation
• We showed that the allocation Z computed by Algorithm 1 is an (α, c)-EFX allocation and Mp(Z) ≥

• There is a polynomial time algorithm to find an (1 − ε)-EFX allocation with bounded charity for general valuations for any ε > 0 [CKMS19]2.
• In case of additive valuations[3], an allocation that maximizes Nash welfare is both EF1 and Pareto optimal4 [CKM+16].
• It is not intuitive that the allocation that maximizes welfare will be fair.[5] the authors manage to give a polynomial time algorithm that achieves a good approximation to the p-mean welfare while still retaining all the fairness properties.
• In the same paper they show that when agents have identical valuations, there is an algorithm that provides an O(1) factor approximation to the p-mean welfare.
• The authors show that the authors can determine an (α, c)-EFX allocation with an O(n) approximation on the p-mean welfare.
• As mentioned earlier in Section 1.1 the singleton sets allocated to the agents are the barriers to proving the desired approximation for any (α, c)-EFX allocation.
• The authors prove a lower bound on vi(Zi) in terms of the initial allocation Y and the set of low valuable goods for agent i, i.e., M \ Hi. 10.
• Algorithm 1 Determining an (α, c)-EFX allocation with O(n) approximation on optimum p-mean.

• Let X∗ be the allocation with the highest p-mean value and let gi∗ be agent i’s most valuable good in Xi∗.
• Section 3.2, with the only difference that since p is positive and the authors compute a Maximum weight matching in the bipartite graph G = ([n] ∪ M, [n] × M ) where the weight of an edge from p agent i to good g, wig = n · vi(g) + vi(M \ Hi) and the authors will have lower bounds on R(Z) and lower bounds on Mp(Z).
• The authors get an O(n) approximation algorithm for asymmetric Nash welfare when agents have submodular valuations (improving the current best bound of O(n · log n) by Garg et al [GKK20]).

• Discrete fair division of resources is a fundamental problem in various multi-agent settings, where the goal is to partition a set M of m indivisible goods among n agents in a fair and efficient manner.
• The authors design a polynomial-time algorithm that outputs an allocation that satisfies either of the two approximations of EFX as well as achieves an O(n) approximation to the Nash welfare.
• EFX allocation can be computed in polynomial time when agents have subadditive valuations.
• There is a polynomial time algorithm to find an (1 − ε)-EFX allocation with bounded charity for general valuations for any ε > 0 [CKMS19]2.
• In case of additive valuations[3], an allocation that maximizes Nash welfare is both EF1 and Pareto optimal4 [CKM+16].
• It is not intuitive that the allocation that maximizes welfare will be fair.[5] the authors manage to give a polynomial time algorithm that achieves a good approximation to the p-mean welfare while still retaining all the fairness properties.
• In the same paper they show that when agents have identical valuations, there is an algorithm that provides an O(1) factor approximation to the p-mean welfare.
• The authors show that the authors can determine an (α, c)-EFX allocation with an O(n) approximation on the p-mean welfare.
• As mentioned earlier in Section 1.1 the singleton sets allocated to the agents are the barriers to proving the desired approximation for any (α, c)-EFX allocation.
• The authors prove a lower bound on vi(Zi) in terms of the initial allocation Y and the set of low valuable goods for agent i, i.e., M \ Hi. 10.
• Algorithm 1 Determining an (α, c)-EFX allocation with O(n) approximation on optimum p-mean.
• Let X∗ be the allocation with the highest p-mean value and let gi∗ be agent i’s most valuable good in Xi∗.
• Section 3.2, with the only difference that since p is positive and the authors compute a Maximum weight matching in the bipartite graph G = ([n] ∪ M, [n] × M ) where the weight of an edge from p agent i to good g, wig = n · vi(g) + vi(M \ Hi) and the authors will have lower bounds on R(Z) and lower bounds on Mp(Z).
• The authors get an O(n) approximation algorithm for asymmetric Nash welfare when agents have submodular valuations (improving the current best bound of O(n · log n) by Garg et al [GKK20]).

• Fair division has been extensively studied for more than seventy years since the seminal work of Steinhaus [Ste48]. A complete survey of all different settings and the fairness and efficiency notions used is well beyond the scope of this paper. We limit our attention to the discrete setting (as mentioned in Section 1) and the two most universal notions of fairness, namely envy-freeness and proportionality[9]. Both of these properties can be guaranteed in case of di- the rest of the

In any division there n−1 agents who do not get

1 n of on the set of n goods visible goods. For indivisible goods, there are trivial instances (mentioned in Section 1) where neither of these notions can be achieved by any allocation. However there has been extensive studies on relaxations of envy-freeness like EF1 [BCKO17, BBMN18, LMMS04, CKM+16] and EFX [CKMS20, CGH19, CKM+16, PR18] and relaxations of proportionality like maximin shares (MMS) [Bud[11], BL16, AMNS17, BK17, KPW18, GHS+18, GMT19, GT19] and proportionality up to one good (PROP1) [CFS17, BK19, GM19].

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