Temporal Elections: Welfare, Strategyproofness, and Proportionality

TL;DR

This paper analyzes the computational complexity and strategic properties of sequential decision-making mechanisms aimed at maximizing utilitarian and egalitarian welfare, revealing hardness results and proposing efficient solutions under certain conditions.

Contribution

It provides a comprehensive complexity analysis of welfare maximization in sequential elections, introduces approximation algorithms, and examines strategyproofness and proportionality constraints.

Findings

Maximizing utilitarian welfare is computationally easy.

Maximizing egalitarian welfare is NP-complete, even in restricted cases.

Mechanisms maximizing utilitarian welfare are strategyproof, but egalitarian maximizers face non-obvious manipulability issues.

Abstract

We investigate a model of sequential decision-making where a single alternative is chosen at each round. We focus on two objectives -- utilitarian welfare (Util) and egalitarian welfare (Egal) -- and consider the computational complexity of maximizing these objectives, as well as their compatibility with strategyproofness and proportionality. We observe that maximizing Util is easy, but the corresponding decision problem for Egal is NP-complete even in restricted cases. We complement this hardness result for Egal with parameterized complexity analysis and an approximation algorithm. Additionally, we show that, while a mechanism that outputs an outcome that maximizes Util is strategyproof, all deterministic mechanisms for computing outcomes that maximize Egal fail a very weak variant of strategyproofness, called non-obvious manipulability (NOM). However, we show that when agents have non-empty approval sets at each timestep, choosing an Egal-maximizing outcome while breaking ties lexicographically satisfies NOM. Regarding proportionality, we prove that a proportional (PROP) outcome can be computed efficiently, but finding an outcome that maximizes Util while guaranteeing PROP is NP-hard. We also derive upper and lower bounds on the (strong) price of proportionality with respect to Util and Egal. Some of our results extend to $p$-mean welfare measures other than Egal and Util.