6,258 research outputs found
Fair Allocation based on Diminishing Differences
Ranking alternatives is a natural way for humans to explain their
preferences. It is being used in many settings, such as school choice, course
allocations and residency matches. In some cases, several `items' are given to
each participant. Without having any information on the underlying cardinal
utilities, arguing about fairness of allocation requires extending the ordinal
item ranking to ordinal bundle ranking. The most commonly used such extension
is stochastic dominance (SD), where a bundle X is preferred over a bundle Y if
its score is better according to all additive score functions. SD is a very
conservative extension, by which few allocations are necessarily fair while
many allocations are possibly fair. We propose to make a natural assumption on
the underlying cardinal utilities of the players, namely that the difference
between two items at the top is larger than the difference between two items at
the bottom. This assumption implies a preference extension which we call
diminishing differences (DD), where X is preferred over Y if its score is
better according to all additive score functions satisfying the DD assumption.
We give a full characterization of allocations that are
necessarily-proportional or possibly-proportional according to this assumption.
Based on this characterization, we present a polynomial-time algorithm for
finding a necessarily-DD-proportional allocation if it exists. Using
simulations, we show that with high probability, a necessarily-proportional
allocation does not exist but a necessarily-DD-proportional allocation exists,
and moreover, that allocation is proportional according to the underlying
cardinal utilities. We also consider chore allocation under the analogous
condition --- increasing-differences.Comment: Revised version, based on very helpful suggestions of JAIR referees.
Gaps in some proofs were filled, more experiments were done, and mor
Paradoxes of Fair Division
Two or more players are required to divide up a set of indivisible items that they can rank from best to worst. They may, as well, be able to indicate preferences over subsets, or packages, of items. The main criteria used to assess the fairness of a division are efficiency (Pareto-optimality) and envy-freeness. Other criteria are also suggested, including a Rawlsian criterion that the worst-off player be made as well off as possible and a scoring procedure, based on the Borda count, that helps to render allocations as equal as possible. Eight paradoxes, all of which involve unexpected conflicts among the criteria, are described and classified into three categories, reflecting (1) incompatibilities between efficiency and envy-freeness, (2) the failure of a unique efficient and envy-free division to satisfy other criteria, and (3) the desirability, on occasion, of dividing up items unequally. While troublesome, the paradoxes also indicate opportunities for achieving fair division, which will depend on the fairness criteria one deems important and the trade-offs one considers acceptable.FAIR DIVISION; ALLOCATION OF INDIVISIBLE ITEMS; ENVY-FREENESS; PARETO- OPTIMALITY; RAWLSIAN JUSTICE; BORDA COUNT.
A Generalization of the AL method for Fair Allocation of Indivisible Objects
We consider the assignment problem in which agents express ordinal
preferences over objects and the objects are allocated to the agents based
on the preferences. In a recent paper, Brams, Kilgour, and Klamler (2014)
presented the AL method to compute an envy-free assignment for two agents. The
AL method crucially depends on the assumption that agents have strict
preferences over objects. We generalize the AL method to the case where agents
may express indifferences and prove the axiomatic properties satisfied by the
algorithm. As a result of the generalization, we also get a speedup on
previous algorithms to check whether a complete envy-free assignment exists or
not. Finally, we show that unless P=NP, there can be no polynomial-time
extension of GAL to the case of arbitrary number of agents
Fair assignment of indivisible objects under ordinal preferences
We consider the discrete assignment problem in which agents express ordinal
preferences over objects and these objects are allocated to the agents in a
fair manner. We use the stochastic dominance relation between fractional or
randomized allocations to systematically define varying notions of
proportionality and envy-freeness for discrete assignments. The computational
complexity of checking whether a fair assignment exists is studied for these
fairness notions. We also characterize the conditions under which a fair
assignment is guaranteed to exist. For a number of fairness concepts,
polynomial-time algorithms are presented to check whether a fair assignment
exists. Our algorithmic results also extend to the case of unequal entitlements
of agents. Our NP-hardness result, which holds for several variants of
envy-freeness, answers an open question posed by Bouveret, Endriss, and Lang
(ECAI 2010). We also propose fairness concepts that always suggest a non-empty
set of assignments with meaningful fairness properties. Among these concepts,
optimal proportionality and optimal weak proportionality appear to be desirable
fairness concepts.Comment: extended version of a paper presented at AAMAS 201
Approximate Maximin Shares for Groups of Agents
We investigate the problem of fairly allocating indivisible goods among
interested agents using the concept of maximin share. Procaccia and Wang showed
that while an allocation that gives every agent at least her maximin share does
not necessarily exist, one that gives every agent at least of her share
always does. In this paper, we consider the more general setting where we
allocate the goods to groups of agents. The agents in each group share the same
set of goods even though they may have conflicting preferences. For two groups,
we characterize the cardinality of the groups for which a constant factor
approximation of the maximin share is possible regardless of the number of
goods. We also show settings where an approximation is possible or impossible
when there are several groups.Comment: To appear in the 10th International Symposium on Algorithmic Game
Theory (SAGT), 201
Chore division on a graph
The paper considers fair allocation of indivisible nondisposable items that
generate disutility (chores). We assume that these items are placed in the
vertices of a graph and each agent's share has to form a connected subgraph of
this graph. Although a similar model has been investigated before for goods, we
show that the goods and chores settings are inherently different. In
particular, it is impossible to derive the solution of the chores instance from
the solution of its naturally associated fair division instance. We consider
three common fair division solution concepts, namely proportionality,
envy-freeness and equitability, and two individual disutility aggregation
functions: additive and maximum based. We show that deciding the existence of a
fair allocation is hard even if the underlying graph is a path or a star. We
also present some efficiently solvable special cases for these graph
topologies
Computing an Approximately Optimal Agreeable Set of Items
We study the problem of finding a small subset of items that is
\emph{agreeable} to all agents, meaning that all agents value the subset at
least as much as its complement. Previous work has shown worst-case bounds,
over all instances with a given number of agents and items, on the number of
items that may need to be included in such a subset. Our goal in this paper is
to efficiently compute an agreeable subset whose size approximates the size of
the smallest agreeable subset for a given instance. We consider three
well-known models for representing the preferences of the agents: ordinal
preferences on single items, the value oracle model, and additive utilities. In
each of these models, we establish virtually tight bounds on the approximation
ratio that can be obtained by algorithms running in polynomial time.Comment: A preliminary version appeared in Proceedings of the 26th
International Joint Conference on Artificial Intelligence (IJCAI), 201
Fair allocation of indivisible goods among two agents
One must allocate a finite set of indivisible goods among two agents without monetary compensation. We impose Pareto-efficiency, anonymity, a weak notion of no-envy, a welfare lower bound based on each agent’s ranking of the sets of goods, and a monotonicity property relative to changes in agents’ preferences. We prove that there is a rule satisfying these axioms. If there are three goods, it is the only rule, with one of its subcorrespondences, satisfying each fairness axiom and not discriminating between goods. Further, we confirm the clear gap between these economies and those with more than two agents.indivisible goods, no monetary compensation, no-envy, lower bound, preference-monotonicity
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