36 research outputs found
Cake Cutting Algorithms for Piecewise Constant and Piecewise Uniform Valuations
Cake cutting is one of the most fundamental settings in fair division and
mechanism design without money. In this paper, we consider different levels of
three fundamental goals in cake cutting: fairness, Pareto optimality, and
strategyproofness. In particular, we present robust versions of envy-freeness
and proportionality that are not only stronger than their standard
counter-parts but also have less information requirements. We then focus on
cake cutting with piecewise constant valuations and present three desirable
algorithms: CCEA (Controlled Cake Eating Algorithm), MEA (Market Equilibrium
Algorithm) and CSD (Constrained Serial Dictatorship). CCEA is polynomial-time,
robust envy-free, and non-wasteful. It relies on parametric network flows and
recent generalizations of the probabilistic serial algorithm. For the subdomain
of piecewise uniform valuations, we show that it is also group-strategyproof.
Then, we show that there exists an algorithm (MEA) that is polynomial-time,
envy-free, proportional, and Pareto optimal. MEA is based on computing a
market-based equilibrium via a convex program and relies on the results of
Reijnierse and Potters [24] and Devanur et al. [15]. Moreover, we show that MEA
and CCEA are equivalent to mechanism 1 of Chen et. al. [12] for piecewise
uniform valuations. We then present an algorithm CSD and a way to implement it
via randomization that satisfies strategyproofness in expectation, robust
proportionality, and unanimity for piecewise constant valuations. For the case
of two agents, it is robust envy-free, robust proportional, strategyproof, and
polynomial-time. Many of our results extend to more general settings in cake
cutting that allow for variable claims and initial endowments. We also show a
few impossibility results to complement our algorithms.Comment: 39 page
Counting Houses of Pareto Optimal Matchings in the House Allocation Problem
Let with and be two sets. We assume that every
element has a reference list over all elements from . We call an
injective mapping from to a matching. A blocking coalition of
is a subset of such that there exists a matching that
differs from only on elements of , and every element of
improves in , compared to according to its preference list. If
there exists no blocking coalition, we call the matching an exchange
stable matching (ESM). An element is reachable if there exists an
exchange stable matching using . The set of all reachable elements is
denoted by . We show This is
asymptotically tight. A set is reachable (respectively exactly
reachable) if there exists an exchange stable matching whose image
contains as a subset (respectively equals ). We give bounds for the
number of exactly reachable sets. We find that our results hold in the more
general setting of multi-matchings, when each element of is matched
with elements of instead of just one. Further, we give complexity
results and algorithms for corresponding algorithmic questions. Finally, we
characterize unavoidable elements, i.e., elements of that are used by all
ESM's. This yields efficient algorithms to determine all unavoidable elements.Comment: 24 pages 2 Figures revise
Fair integer programming under dichotomous preferences
One cannot make truly fair decisions using integer linear programs unless one
controls the selection probabilities of the (possibly many) optimal solutions.
For this purpose, we propose a unified framework when binary decision variables
represent agents with dichotomous preferences, who only care about whether they
are selected in the final solution. We develop several general-purpose
algorithms to fairly select optimal solutions, for example, by maximizing the
Nash product or the minimum selection probability, or by using a random
ordering of the agents as a selection criterion (Random Serial Dictatorship).
As such, we embed the black-box procedure of solving an integer linear program
into a framework that is explainable from start to finish. Moreover, we study
the axiomatic properties of the proposed methods by embedding our framework
into the rich literature of cooperative bargaining and probabilistic social
choice. Lastly, we evaluate the proposed methods on a specific application,
namely kidney exchange. We find that while the methods maximizing the Nash
product or the minimum selection probability outperform the other methods on
the evaluated welfare criteria, methods such as Random Serial Dictatorship
perform reasonably well in computation times that are similar to those of
finding a single optimal solution
Serial Dictatorship Mechanism for Project Scheduling with Non-Renewable Resources
This paper considers a resource-constrained project scheduling problem
with self-interested agents. A novel resource allocation model is
presented and studied in a mechanism design setting without money.
The novelties and specialties of our contribution include that the nonrenewable
resources are supplied at different dates, the jobs requiring
the resources are related with precedence relations, and the utilities of
the agents are based on the tardiness values of their jobs. We modify a
classical scheduling algorithm for implementing the Serial Dictatorship
Mechanism, which is then proven to be truthful and Pareto-optimal.
Furthermore, the properties of the social welfare are studied