24 research outputs found
Fair Division of a Graph
We consider fair allocation of indivisible items under an additional
constraint: there is an undirected graph describing the relationship between
the items, and each agent's share must form a connected subgraph of this graph.
This framework captures, e.g., fair allocation of land plots, where the graph
describes the accessibility relation among the plots. We focus on agents that
have additive utilities for the items, and consider several common fair
division solution concepts, such as proportionality, envy-freeness and maximin
share guarantee. While finding good allocations according to these solution
concepts is computationally hard in general, we design efficient algorithms for
special cases where the underlying graph has simple structure, and/or the
number of agents -or, less restrictively, the number of agent types- is small.
In particular, despite non-existence results in the general case, we prove that
for acyclic graphs a maximin share allocation always exists and can be found
efficiently.Comment: 9 pages, long version of accepted IJCAI-17 pape
Fairly Allocating Contiguous Blocks of Indivisible Items
In this paper, we study the classic problem of fairly allocating indivisible
items with the extra feature that the items lie on a line. Our goal is to find
a fair allocation that is contiguous, meaning that the bundle of each agent
forms a contiguous block on the line. While allocations satisfying the
classical fairness notions of proportionality, envy-freeness, and equitability
are not guaranteed to exist even without the contiguity requirement, we show
the existence of contiguous allocations satisfying approximate versions of
these notions that do not degrade as the number of agents or items increases.
We also study the efficiency loss of contiguous allocations due to fairness
constraints.Comment: Appears in the 10th International Symposium on Algorithmic Game
Theory (SAGT), 201
Computing Socially-Efficient Cake Divisions
We consider a setting in which a single divisible good ("cake") needs to be
divided between n players, each with a possibly different valuation function
over pieces of the cake. For this setting, we address the problem of finding
divisions that maximize the social welfare, focusing on divisions where each
player needs to get one contiguous piece of the cake. We show that for both the
utilitarian and the egalitarian social welfare functions it is NP-hard to find
the optimal division. For the utilitarian welfare, we provide a constant factor
approximation algorithm, and prove that no FPTAS is possible unless P=NP. For
egalitarian welfare, we prove that it is NP-hard to approximate the optimum to
any factor smaller than 2. For the case where the number of players is small,
we provide an FPT (fixed parameter tractable) FPTAS for both the utilitarian
and the egalitarian welfare objectives
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
Redividing the Cake
A heterogeneous resource, such as a land-estate, is already divided among
several agents in an unfair way. It should be re-divided among the agents in a
way that balances fairness with ownership rights. We present re-division
protocols that attain various trade-off points between fairness and ownership
rights, in various settings differing in the geometric constraints on the
allotments: (a) no geometric constraints; (b) connectivity --- the cake is a
one-dimensional interval and each piece must be a contiguous interval; (c)
rectangularity --- the cake is a two-dimensional rectangle or rectilinear
polygon and the pieces should be rectangles; (d) convexity --- the cake is a
two-dimensional convex polygon and the pieces should be convex.
Our re-division protocols have implications on another problem: the
price-of-fairness --- the loss of social welfare caused by fairness
requirements. Each protocol implies an upper bound on the price-of-fairness
with the respective geometric constraints.Comment: Extended IJCAI 2018 version. Previous name: "How to Re-Divide a Cake
Fairly
Truthful Facility Assignment with Resource Augmentation: An Exact Analysis of Serial Dictatorship
We study the truthful facility assignment problem, where a set of agents with
private most-preferred points on a metric space are assigned to facilities that
lie on the metric space, under capacity constraints on the facilities. The goal
is to produce such an assignment that minimizes the social cost, i.e., the
total distance between the most-preferred points of the agents and their
corresponding facilities in the assignment, under the constraint of
truthfulness, which ensures that agents do not misreport their most-preferred
points.
We propose a resource augmentation framework, where a truthful mechanism is
evaluated by its worst-case performance on an instance with enhanced facility
capacities against the optimal mechanism on the same instance with the original
capacities. We study a very well-known mechanism, Serial Dictatorship, and
provide an exact analysis of its performance. Although Serial Dictatorship is a
purely combinatorial mechanism, our analysis uses linear programming; a linear
program expresses its greedy nature as well as the structure of the input, and
finds the input instance that enforces the mechanism have its worst-case
performance. Bounding the objective of the linear program using duality
arguments allows us to compute tight bounds on the approximation ratio. Among
other results, we prove that Serial Dictatorship has approximation ratio
when the capacities are multiplied by any integer . Our
results suggest that even a limited augmentation of the resources can have
wondrous effects on the performance of the mechanism and in particular, the
approximation ratio goes to 1 as the augmentation factor becomes large. We
complement our results with bounds on the approximation ratio of Random Serial
Dictatorship, the randomized version of Serial Dictatorship, when there is no
resource augmentation