51 research outputs found
Deterministic, Strategyproof, and Fair Cake Cutting
We study the classic cake cutting problem from a mechanism design
perspective, in particular focusing on deterministic mechanisms that are
strategyproof and fair. We begin by looking at mechanisms that are non-wasteful
and primarily show that for even the restricted class of piecewise constant
valuations there exists no direct-revelation mechanism that is strategyproof
and even approximately proportional. Subsequently, we remove the non-wasteful
constraint and show another impossibility result stating that there is no
strategyproof and approximately proportional direct-revelation mechanism that
outputs contiguous allocations, again, for even the restricted class of
piecewise constant valuations. In addition to the above results, we also
present some negative results when considering an approximate notion of
strategyproofness, show a connection between direct-revelation mechanisms and
mechanisms in the Robertson-Webb model when agents have piecewise constant
valuations, and finally also present a (minor) modification to the well-known
Even-Paz algorithm that has better incentive-compatible properties for the
cases when there are two or three agents.Comment: A shorter version of this paper will appear at IJCAI 201
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
The Incentive Guarantees Behind Nash Welfare in Divisible Resources Allocation
We study the problem of allocating divisible resources among agents,
hopefully in a fair and efficient manner. With the presence of strategic
agents, additional incentive guarantees are also necessary, and the problem of
designing fair and efficient mechanisms becomes much less tractable. While the
maximum Nash welfare (MNW) mechanism has been proven to be prominent by
providing desirable fairness and efficiency guarantees as well as other
intuitive properties, no incentive property is known for it.
We show a surprising result that, when agents have piecewise constant value
density functions, the incentive ratio of the MNW mechanism is for cake
cutting, where the incentive ratio of a mechanism is defined as the ratio
between the largest possible utility that an agent can gain by manipulation and
his utility in honest behavior. Remarkably, this result holds even without the
free disposal assumption, which is hard to get rid of in the design of truthful
cake cutting mechanisms. We also show that the MNW mechanism is group
strategyproof when agents have piecewise uniform value density functions.
Moreover, we show that, for cake cutting, the Partial Allocation (PA)
mechanism proposed by Cole et al., which is truthful and -MNW for
homogeneous divisible items, has an incentive ratio between
and when randomization is allowed, can be turned to be truthful in expectation.
Given two alternatives for a trade-off between incentive ratio and Nash welfare
provided by the MNW and PA mechanisms, we establish an interpolation between
them for both cake cutting and homogeneous divisible items.
Finally, we study the existence of fair mechanisms with a low incentive ratio
in the connected pieces setting. We show that any envy-free cake cutting
mechanism with the connected pieces constraint has an incentive ratio of at
least
Chore Cutting: Envy and Truth
We study the fair division of divisible bad resources with strategic agents
who can manipulate their private information to get a better allocation. Within
certain constraints, we are particularly interested in whether truthful
envy-free mechanisms exist over piecewise-constant valuations. We demonstrate
that no deterministic truthful envy-free mechanism can exist in the
connected-piece scenario, and the same impossibility result occurs for hungry
agents. We also show that no deterministic, truthful dictatorship mechanism can
satisfy the envy-free criterion, and the same result remains true for
non-wasteful constraints rather than dictatorship. We further address several
related problems and directions.Comment: arXiv admin note: text overlap with arXiv:2104.07387 by other author
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On the Trade-offs between Modeling Power and Algorithmic Complexity
Mathematical modeling is a central component of operations research. Most of the academic research in our field focuses on developing algorithmic tools for solving various mathematical problems arising from our models. However, our procedure for selecting the best model to use in any particular application is ad hoc. This dissertation seeks to rigorously quantify the trade-offs between various design criteria in model construction through a series of case studies. The hope is that a better understanding of the pros and cons of different models (for the same application) can guide and improve the model selection process.
In this dissertation, we focus on two broad types of trade-offs. The first type arises naturally in mechanism or market design, a discipline that focuses on developing optimization models for complex multi-agent systems. Such systems may require satisfying multiple objectives that are potentially in conflict with one another. Hence, finding a solution that simultaneously satisfies several design requirements is challenging. The second type addresses the dynamics between model complexity and computational tractability in the context of approximation algorithms for some discrete optimization problems. The need to study this type of trade-offs is motivated by certain industry problems where the goal is to obtain the best solution within a reasonable time frame. Hence, being able to quantify and compare the degree of sub-optimality of the solution obtained under different models is helpful. Chapters 2-5 of the dissertation focus on trade-offs of the first type and Chapters 6-7 the second type
Multi-type Resource Allocation with Partial Preferences
We propose multi-type probabilistic serial (MPS) and multi-type random
priority (MRP) as extensions of the well known PS and RP mechanisms to the
multi-type resource allocation problem (MTRA) with partial preferences. In our
setting, there are multiple types of divisible items, and a group of agents who
have partial order preferences over bundles consisting of one item of each
type. We show that for the unrestricted domain of partial order preferences, no
mechanism satisfies both sd-efficiency and sd-envy-freeness. Notwithstanding
this impossibility result, our main message is positive: When agents'
preferences are represented by acyclic CP-nets, MPS satisfies sd-efficiency,
sd-envy-freeness, ordinal fairness, and upper invariance, while MRP satisfies
ex-post-efficiency, sd-strategy-proofness, and upper invariance, recovering the
properties of PS and RP
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