Optimizing over Serial Dictatorships

TL;DR

This paper investigates the effectiveness of serial dictatorship mechanisms in combinatorial optimization, providing bounds, algorithms, and truthfulness considerations for approximating social welfare.

Contribution

It introduces query-efficient algorithms for serial dictatorship in combinatorial problems, achieving tight bounds and 2-approximation in specific domains, and discusses truthfulness and solution quality measures.

Findings

Polynomial query algorithms achieve tight bounds on social welfare.

Simple algorithms yield 2-approximations for matchings, arborescences, and Boolean satisfiability.

Some algorithms can be implemented truthfully with VCG-like payments.

Abstract

Motivated by the success of the serial dictatorship mechanism in social choice settings, we explore its usefulness in tackling various combinatorial optimization problems. We do so by considering an abstract model, in which a set of agents are asked to act in a particular ordering, called the action sequence. Each agent acts in a way that gives her the maximum possible value, given the actions of the agents who preceded her in the action sequence. Our goal is to compute action sequences that yield approximately optimal total value to the agents (a.k.a., social welfare). We assume query access to the value $v_i(S)$ that the agent i gets when she acts after the agents in the ordered set $S$. We establish tight bounds on the social welfare that can be achieved using polynomially many queries. Even though these bounds show a marginally sublinear approximation of optimal social welfare in general, excellent approximations can be obtained when the valuations stem from an underlying combinatorial domain. Indicatively, when the valuations are defined using bipartite matchings, arborescences in directed graphs, and satisfiability of Boolean expressions, simple query-efficient algorithms yield $2$-approximations. We discuss issues related to truthfulness and show how some of our algorithms can be implemented truthfully using VCG-like payments. Finally, we introduce and study the price of serial dictatorship, a notion that provides an optimistic measure of the quality of combinatorial optimization solutions generated by action sequences.