Achieving rental harmony with a secretive roommate

Frick, Florian; Houston-Edwards, Kelsey; Meunier, Frédéric

doi:10.48550/arxiv.1702.07325

Cited by 3 publications

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“…We extend these results and prove that, considering the utility functions of n − 1 agents, 3 we can propose a rent division among the n rooms such that for every possible room selection of the secretive agent there exists an allocation (of the remaining n − 1 rooms among the n − 1 agents) which ensures overall envy freeness (Theorem 2). Hence, analogous to the results of [AFP + 18] and [FHEM17] (which considered ordinal preferences), we show that fair rent division with a secretive agent can be achieved in the cardinal setting as well. 4 Efficient algorithms for fair rent division are known for the quasilinear utility model [Ara95,GMPZ17,Kli00].…”

Section: Fair Rent Divisionsupporting

confidence: 83%

“…The work of Frick et al [FHEM17] shows that an envy-free rent division can also be achieved, albeit inefficiently, in the absence of full information. Rent division is another well-studied problem in the fairdivision literature, and it entails allocating n rooms among n agents and splitting the rent such that envy freeness is achieved; see, e.g., [Su99], [Sve83], [ADG91], [Ara95], [Kli00] and [GMPZ17].…”

Section: Introductionmentioning

confidence: 99%

“…Rent division is another well-studied problem in the fairdivision literature, and it entails allocating n rooms among n agents and splitting the rent such that envy freeness is achieved; see, e.g., [Su99], [Sve83], [ADG91], [Ara95], [Kli00] and [GMPZ17]. Considering agents who have ordinal preferences over the rooms (for every possible division of the rent), Frick et al [FHEM17] proved that the preferences of n − 1 agents suffice to find a rent division such that secretive agent can select an arbitrary room (at its price) and still an overall envy-free allocation of the remaining n − 1 rooms (at their respective prices and among the n − 1 agents) would exist.…”

Section: Introductionmentioning

confidence: 99%

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Fair Division with a Secretive Agent

Arunachaleswaran

Barman

Rathi

2019

AAAI

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We study classic fair-division problems in a partial information setting. This paper respectively addresses fair division of rent, cake, and indivisible goods among agents with cardinal preferences. We will show that, for all of these settings and under appropriate valuations, a fair (or an approximately fair) division among n agents can be efficiently computed using only the valuations of n − 1 agents. The nth (secretive) agent can make an arbitrary selection after the division has been proposed and, irrespective of her choice, the computed division will admit an overall fair allocation.For the rent-division setting we prove that the (well-behaved) utilities of n−1 agents suffice to find a rent division among n rooms such that, for every possible room selection of the secretive agent, there exists an allocation (of the remaining n − 1 rooms among the n − 1 agents) which ensures overall envy freeness (fairness). We complement this existential result by developing a polynomial-time algorithm that finds such a fair rent division under quasilinear utilities.In this partial information setting, we also develop efficient algorithms to compute allocations that are envy-free up to one good (EF1) and ε-approximate envy free. These two notions of fairness are applicable in the context of indivisible goods and divisible goods (cake cutting), respectively. This work also addresses fairness in terms of proportionality and maximin shares. Our key result here is an efficient algorithm that, even with a secretive agent, finds a 1/19-approximate maximin fair allocation (of indivisible goods) under submodular valuations of the non-secretive agents.One of the main technical contributions of this paper is the development of novel connections between different fair-division paradigms, e.g., we use our existential results for envy-free rent-division to develop an efficient EF1 algorithm. * Indian Institute of Science. eshwarram.arunachaleswaran@gmail.com † Indian Institute of Science. barman@iisc.ac.in ‡ Indian Institute of Science. nidhirathi@iisc.ac.in 1 http://www.spliddit.org/ 2 http://www.nyu.edu/projects/adjustedwinner/ 1 agent) and, then, the second agent can select her most preferred piece. This protocol leads to an envy-free cake division, i.e., in the resulting allocation no agent has a strictly stronger preference for the other agent's piece. Envy freeness is a standard notion of fairness [Fol67, Var74, Str80] and the divide-andchoose method shows a fair division with respect to this notion can be found even if the second agent is secretive and just selects a piece after we partition the cake.Asada et al. [AFP + 18] showed that this result extends to higher number of agents who are endowed with ordinal preferences: there exists a division of the cake into n parts, which depends on the (subjective) preferences of only n − 1 agents, such that the nth (secretive) agent can select an arbitrary piece and still we would be able to allocate the remaining n − 1 pieces in an envy-free manner. In other words, independent of the cho...

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Section: Fair Rent Divisionsupporting

confidence: 83%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Fair Division with a Secretive Agent

Arunachaleswaran

Barman

Rathi

2019

AAAI

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“…The same idea was used by Su [18] to prove a colorful Sperner's lemma. For related Sperner-type results for multiple Sperner labelings see Babson [3], Bapat [4], and Frick, Houston-Edwards, and Meunier [7].…”

Section: A Colorful Komiya Theoremmentioning

confidence: 98%

Colorful Coverings of Polytopes and Piercing Numbers of Colorful d-Intervals

Frick

Zerbib

2019

Combinatorica

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We prove a common strengthening of Bárány's colorful Carathéodory theorem and the KKMS theorem. In fact, our main result is a colorful polytopal KKMS theorem, which extends a colorful KKMS theorem due to Shih and Lee [Math. Ann. 296 (1993), no. 1, 35-61] as well as a polytopal KKMS theorem due to Komiya [Econ. Theory 4 (1994), no. 3, 463-466]. The (seemingly unrelated) colorful Carathéodory theorem is a special case as well. We apply our theorem to establish an upper bound on the piercing number of colorful d-interval hypergraphs, extending earlier results of Tardos [Combinatorica 15 (1995), no. 1, 123-134] and Kaiser [Discrete Comput.

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“…al. [6] show that an envy-free rent division can be achieved among n people, even if the preferences of only n − 1 housemates are known.…”

Section: Introductionmentioning

confidence: 99%

Fair division with multiple pieces

Nyman

Zerbib

2020

Discrete Applied Mathematics

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Given a set of p players we consider problems concerning envy-free allocation of collections of k pieces from a given set of goods or chores. We show that if p ≤ n and each player can choose k pieces out of n pieces of a cake, then there exist a division of the cake and an allocation of the pieces where at least p 2(k 2 −k+1) players get their desired k pieces each. We further show that if p ≤ k(n − 1) + 1 and each player can choose k pieces, one from each of k cakes that are divided into n pieces each, then there exist a division of the cakes and allocation of the pieces where at least p 2k(k−1) players get their desired k pieces. Finally we prove that if p ≥ k(n− 1)+ 1 and each player can choose one shift in each of k days that are partitioned into n shifts each, then, given that the salaries of the players are fixed, there exist n(1 + ln k) players covering all the shifts, and moreover, if k = 2 then n players suffice. Our proofs combine topological methods and theorems of Füredi, Lovász and Gallai from hypergraph theory.

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Achieving rental harmony with a secretive roommate

Cited by 3 publications

References 0 publications

Fair Division with a Secretive Agent

Fair Division with a Secretive Agent

Colorful Coverings of Polytopes and Piercing Numbers of Colorful d-Intervals

Fair division with multiple pieces

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