On Fair and Efficient Allocations of Indivisible Public Goods

TL;DR

This paper investigates fair and efficient allocation methods for indivisible public goods under constraints, establishing connections with private goods and public decision models, and analyzing computational complexity and algorithms.

Contribution

It introduces polynomial-time reductions linking public goods to private goods and decision models, and analyzes fairness, efficiency, and computational complexity in this setting.

Findings

MNW allocations guarantee Prop1, RRS approximation, and Pareto efficiency.

Finding MNW or leximin allocations is NP-hard even with few agents or binary valuations.

Algorithms for exact and approximate solutions are developed for specific cases.

Abstract

We study fair allocation of indivisible public goods subject to cardinality (budget) constraints. In this model, we have n agents and m available public goods, and we want to select $k \leq m$ goods in a fair and efficient manner. We first establish fundamental connections between the models of private goods, public goods, and public decision making by presenting polynomial-time reductions for the popular solution concepts of maximum Nash welfare (MNW) and leximin. These mechanisms are known to provide remarkable fairness and efficiency guarantees in private goods and public decision making settings. We show that they retain these desirable properties even in the public goods case. We prove that MNW allocations provide fairness guarantees of Proportionality up to one good (Prop1), $1/n$ approximation to Round Robin Share (RRS), and the efficiency guarantee of Pareto Optimality (PO). Further, we show that the problems of finding MNW or leximin-optimal allocations are NP-hard, even in the case of constantly many agents, or binary valuations. This is in sharp contrast to the private goods setting that admits polynomial-time algorithms under binary valuations. We also design pseudo-polynomial time algorithms for computing an exact MNW or leximin-optimal allocation for the cases of (i) constantly many agents, and (ii) constantly many goods with additive valuations. We also present an O(n)-factor approximation algorithm for MNW which also satisfies RRS, Prop1, and 1/2-Prop.