Impartial Selection Under Combinatorial Constraints

TL;DR

This paper studies impartial selection mechanisms under combinatorial constraints, proposing new algorithms with provable approximation guarantees for selecting agents based on weighted nominations while maintaining strategic resistance.

Contribution

It introduces impartial mechanisms for weighted nominations under various combinatorial constraints, including independence systems, matroids, and knapsack constraints, with specific approximation ratios.

Findings

Mechanism extends to general independence systems with 1/4-approximation.

Achieves 1/3 and 1/2 approximation ratios for knapsack and matroid constraints.

Provides an impartial, 1/3-approximate mechanism for graphic matroids with multiple nominations.

Abstract

Impartial selection problems are concerned with the selection of one or more agents from a set based on mutual nominations from within the set. To avoid strategic nominations of the agents, the axiom of impartiality requires that the selection of each agent is independent of the nominations cast by that agent. This paper initiates the study of impartial selection problems where the nominations are weighted and the set of agents that can be selected is restricted by a combinatorial constraint. We call a selection mechanism $\alpha$-optimal if, for every instance, the ratio between the total sum of weighted nominations of the selected set and that of the best feasible set of agents is at least $\alpha$. We show that a natural extension of a mechanism studied for the selection of a single agent remains impartial and $\frac{1}{4}$-optimal for general independence systems, and we generalize upper bounds from the selection of multiple agents by parameterizing them by the girth of the independence system. We then focus on independence systems defined by knapsack and matroid constraints, giving impartial mechanisms that exploit a greedy order of the agents and achieve approximation ratios of $\frac{1}{3}$ and $\frac{1}{2}$, respectively, when agents cast a single nomination. For graphic matroids, we further devise an impartial and $\frac{1}{3}$-optimal mechanism for an arbitrary number of unweighted nominations.