7 research outputs found
Existence of EFX for Two Additive Valuations
Fair division of indivisible items is a well-studied topic in Economics and
Computer Science.The objective is to allocate items to agents in a fair manner,
where each agent has a valuation for each subset of items. Envy-freeness is one
of the most widely studied notions of fairness. Since complete envy-free
allocations do not always exist when items are indivisible, several relaxations
have been considered. Among them, possibly the most compelling one is
envy-freeness up to any item (EFX), where no agent envies another agent after
the removal of any single item from the other agent's bundle. However, despite
significant efforts by many researchers for several years, it is known that a
complete EFX allocation always exists only in limited cases. In this paper, we
show that a complete EFX allocation always exists when each agent is of one of
two given types, where agents of the same type have identical additive
valuations. This is the first such existence result for non-identical
valuations when there are any number of agents and items and no limit on the
number of distinct values an agent can have for individual items. We give a
constructive proof, in which we iteratively obtain a Pareto dominating
(partial) EFX allocation from an existing partial EFX allocation.Comment: 14 pages, 2 figure
EFX配分の存在に関する非加法的評価関数への拡張
京都大学新制・課程博士博士(理学)甲第24394号理博第4893号新制||理||1699(附属図書館)京都大学大学院理学研究科数学・数理解析専攻(主査)准教授 小林 佑輔, 教授 牧野 和久, 教授 長谷川 真人学位規則第4条第1項該当Doctor of ScienceKyoto UniversityDGA
Extension of Additive Valuations to General Valuations on the Existence of EFX
Envy-freeness is one of the most widely studied notions in fair division. Since envy-free allocations do not always exist when items are indivisible, several relaxations have been considered. Among them, possibly the most compelling concept is envy-freeness up to any item (EFX). We study the existence of EFX allocations for general valuations. The existence of EFX allocations is a major open problem. For general valuations, it is known that an EFX allocation always exists (i) when n = 2 or (ii) when all agents have identical valuations, where n is the number of agents. it is also known that an EFX allocation always exists when one can leave at most n-1 items unallocated.
We develop new techniques and extend some results of additive valuations to general valuations on the existence of EFX allocations. We show that an EFX allocation always exists (i) when all agents have one of two general valuations or (ii) when the number of items is at most n+3. We also show that an EFX allocation always exists when one can leave at most n-2 items unallocated. In addition to the positive results, we construct an instance with n = 3 in which an existing approach does not work as it is
Proportional Allocation of Indivisible Goods up to the Least Valued Good on Average
We study the problem of fairly allocating a set of indivisible goods to multiple agents and focus on the proportionality, which is one of the classical fairness notions. Since proportional allocations do not always exist when goods are indivisible, approximate notions of proportionality have been considered in the previous work. Among them, proportionality up to the maximin good (PROPm) has been the best approximate notion of proportionality that can be achieved for all instances. In this paper, we introduce the notion of proportionality up to the least valued good on average (PROPavg), which is a stronger notion than PROPm, and show that a PROPavg allocation always exists. Our results establish PROPavg as a notable non-trivial fairness notion that can be achieved for all instances. Our proof is constructive, and based on a new technique that generalizes the cut-and-choose protocol
Reconfiguration of the Union of Arborescences
An arborescence in a digraph is an acyclic arc subset in which every vertex
execpt a root has exactly one incoming arc. In this paper, we reveal the
reconfigurability of the union of arborescences for fixed in the
following sense: for any pair of arc subsets that can be partitioned into
arborescences, one can be transformed into the other by exchanging arcs one by
one so that every intermediate arc subset can also be partitioned into
arborescences. This generalizes the result by Ito et al. (2023), who showed the
case with . Since the union of arborescences can be represented as a
common matroid basis of two matroids, our result gives a new non-trivial
example of matroid pairs for which two common bases are always reconfigurable
to each other
Proportional Allocation of Indivisible Goods up to the Least Valued Good on Average
We study the problem of fairly allocating a set of indivisible goods to
multiple agents and focus on the proportionality, which is one of the classical
fairness notions. Since proportional allocations do not always exist when goods
are indivisible, approximate concepts of proportionality have been considered
in the previous work. Among them, proportionality up to the maximin good
(PROPm) has been the best approximate notion of proportionality that can be
achieved for all instances. In this paper, we introduce the notion of
proportionality up to the least valued good on average (PROPavg), which is a
stronger notion than PROPm, and show that a PROPavg allocation always exists.
Our results establish PROPavg as a notable non-trivial fairness notion that can
be achieved for all instances. Our proof is constructive, and based on a new
technique that generalizes the cut-and-choose protocol.Comment: 17 pages, 3 figures, 2 table