arXiv subjects

Barton E. Lee

Publications and source records attributed to Barton E. Lee.

10 recordsLinked to original sources

arXiv2026

Proportionally Representative Clustering

In recent years, there has been a surge in effort to formalize notions of fairness in machine learning. We focus on centroid clustering--one of the fundamental tasks in unsupervised machine learning. We propose a new axiom ``proportionally representative fairness'' (PRF) that is designed for clustering problems where the selection of centroids reflects the distribution of data points and how tightly they are clustered together. Our fairness concept is not satisfied by existing fair clustering algorithms. We design efficient algorithms to achieve PRF both for unconstrained and discrete clustering problems. Our algorithm for the unconstrained setting is also the first known polynomial-time approximation algorithm for the well-studied Proportional Fairness (PF) axiom. Our algorithm for the discrete setting also matches the best known approximation factor for PF.

cs.LG↗

arXiv2023

Strategyproof and Proportionally Fair Facility Location

We focus on a simple, one-dimensional collective decision problem (often referred to as the facility location problem) and explore issues of strategyproofness and proportionality-based fairness. We introduce and analyze a hierarchy of proportionality-based fairness axioms of varying strength: Individual Fair Share (IFS), Unanimous Fair Share (UFS), Proportionality (as in Freeman et al, 2021), and Proportional Fairness (PF). For each axiom, we characterize the family of mechanisms that satisfy the axiom and strategyproofness. We show that imposing strategyproofness renders many of the axioms to be equivalent: the family of mechanisms that satisfy proportionality, unanimity, and strategyproofness is equivalent to the family of mechanisms that satisfy UFS and strategyproofness, which, in turn, is equivalent to the family of mechanisms that satisfy PF and strategyproofness. Furthermore, there is a unique such mechanism: the Uniform Phantom mechanism, which is studied in Freeman et al. (2021). We also characterize the outcomes of the Uniform Phantom mechanism as the unique (pure) equilibrium outcome for any mechanism that satisfies continuity, strict monotonicity, and UFS. Finally, we analyze the approximation guarantees, in terms of optimal social welfare and minimum total cost, obtained by mechanisms that are strategyproof and satisfy each proportionality-based fairness axiom. We show that the Uniform Phantom mechanism provides the best approximation of the optimal social welfare (and also minimum total cost) among all mechanisms that satisfy UFS.

cs.GT↗

arXiv2021

A characterization of proportionally representative committees

A well-known axiom for proportional representation is Proportionality of Solid Coalitions (PSC). We characterize committees satisfying PSC as possible outcomes of the Minimal Demand rule, which generalizes an approach pioneered by Michael Dummett.

cs.GT↗

arXiv2020

Proportionally Representative Participatory Budgeting with Ordinal Preferences

Participatory budgeting (PB) is a democratic paradigm whereby voters decide on a set of projects to fund with a limited budget. We consider PB in a setting where voters report ordinal preferences over projects and have (possibly) asymmetric weights. We propose proportional representation axioms and clarify how they fit into other preference aggregation settings, such as multi-winner voting and approval-based multi-winner voting. As a result of our study, we also discover a new solution concept for approval-based multi-winner voting, which we call Inclusion PSC (IPSC). IPSC is stronger than proportional justified representation (PJR), incomparable to extended justified representation (EJR), and yet compatible with EJR. The well-studied Proportional Approval Voting (PAV) rule produces a committee that satisfies both EJR and IPSC; however, both these axioms can also be satisfied by an algorithm that runs in polynomial-time.

cs.GT↗

arXiv2019

Facility Location Problem with Capacity Constraints: Algorithmic and Mechanism Design Perspectives

We consider the facility location problem in the one-dimensional setting where each facility can serve a limited number of agents from the algorithmic and mechanism design perspectives. From the algorithmic perspective, we prove that the corresponding optimization problem, where the goal is to locate facilities to minimize either the total cost to all agents or the maximum cost of any agent is NP-hard. However, we show that the problem is fixed-parameter tractable, and the optimal solution can be computed in polynomial time whenever the number of facilities is bounded, or when all facilities have identical capacities. We then consider the problem from a mechanism design perspective where the agents are strategic and need not reveal their true locations. We show that several natural mechanisms studied in the uncapacitated setting either lose strategyproofness or a bound on the solution quality for the total or maximum cost objective. We then propose new mechanisms that are strategyproof and achieve approximation guarantees that almost match the lower bounds.

cs.GT↗

arXiv2019

The Capacity Constrained Facility Location problem

We initiate the study of the capacity constrained facility location problem from a mechanism design perspective. The capacity constrained setting leads to a new strategic environment where a facility serves a subset of the population, which is endogenously determined by the ex-post Nash equilibrium of an induced subgame and is not directly controlled by the mechanism designer. Our focus is on mechanisms that are ex-post dominant-strategy incentive compatible (DIC) at the reporting stage. We provide a complete characterization of DIC mechanisms via the family of Generalized Median Mechanisms (GMMs). In general, the social welfare optimal mechanism is not DIC. Adopting the worst-case approximation measure, we attain tight lower bounds on the approximation ratio of any DIC mechanism. The well-known median mechanism is shown to be optimal among the family of DIC mechanisms for certain capacity ranges. Surprisingly, the framework we introduce provides a new characterization for the family of GMMs, and is responsive to gaps in the current social choice literature highlighted by Border and Jordan (1983) and Barbar{à}, Mass{ó} and Serizawa (1998).

cs.GT↗

arXiv2018

Representing the Insincere: Strategically Robust Proportional Representation

Proportional representation (PR) is a fundamental principle of many democracies world-wide which employ PR-based voting rules to elect their representatives. The normative properties of these voting rules however, are often only understood in the context of sincere voting. In this paper we consider PR in the presence of strategic voters. We construct a voting rule such that for every preference profile there exists at least one costly voting equilibrium satisfying PR with respect to voters' private and unrevealed preferences - such a voting rule is said to be strategically robust. In contrast, a commonly applied voting rule is shown not be strategically robust. Furthermore, we prove a limit on `how strategically robust' a PR-based voting rule can be; we show that there is no PR-based voting rule which ensures that every equilibrium satisfies PR. Collectively, our results highlight the possibility and limit of achieving PR in the presence of strategic voters and a positive role for mechanisms, such as pre-election polls, which coordinate voter behaviour towards equilibria which satisfy PR.

cs.GT↗

arXiv2017

Sub-committee Approval Voting and Generalised Justified Representation Axioms

Social choice is replete with various settings including single-winner voting, multi-winner voting, probabilistic voting, multiple referenda, and public decision making. We study a general model of social choice called Sub-Committee Voting (SCV) that simultaneously generalizes these settings. We then focus on sub-committee voting with approvals and propose extensions of the justified representation axioms that have been considered for proportional representation in approval-based committee voting. We study the properties and relations of these axioms. For each of the axioms, we analyse whether a representative committee exists and also examine the complexity of computing and verifying such a committee.

cs.GT↗

arXiv2017

How long is a piece of string? An exploration of multi-winner approval voting and ballot-length restrictions

Multi-winner approval elections are seen in a variety of settings ranging from academic societies and associations to public elections. In such elections, it is often the case that ballot-length restrictions are enforced; that is, where voters have a limit on the number of candidates which they can vote for. Despite this common feature, there does not seem to be any theoretical justification for ballot-length restrictions (Laslier and Van der Straeten, 2016). This work endogenously derives the set of voter best-response ballot lengths under complete information and with general assumptions on voter utilities and voting rules. These results provide justification for some ballot-length restrictions observed in practice, however when considering equilibrium outcomes our analysis shows that this justification is no longer valid. Equilibrium analysis is considered for voters with lazy and truth-bias second-order tendencies and the equilibrium solution concept is pure-Nash equilibria. The key insights show that ballot-length restrictions or institutional features which make voting costly may lead to instability in election outcomes when voters have diverse preferences, via the non-existence of equilibria. On the other hand, when equilibria do exist they satisfy desirable properties which are not guaranteed by equilibria attained under costless voting and in the absence of ballot-length restrictions. In summary our results highlight a stark trade-off between stable and desirable election outcomes.

cs.GT↗

arXiv2017

Consensus and Voting on Large Graphs: An Application of Graph Limit Theory

Building on recent work by Medvedev (2014) we establish new connections between a basic consensus model, called the voting model, and the theory of graph limits. We show that in the voting model if consensus is attained in the continuum limit then solutions to the finite model will eventually be close to a constant function, and a class of graph limits which guarantee consensus is identified. It is also proven that the dynamics in the continuum limit can be decomposed as a direct sum of dynamics on the connected components, using Janson's definition of connectivity for graph limits. This implies that without loss of generality it may be assumed that the continuum voting model occurs on a connected graph limit.

math.DS↗