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Welfare-Optimal Serial Dictatorships have Polynomial Query Complexity

Ioannis Caragiannis, Kurt Mehlhorn, Nidhi Rathi

TL;DR

This work addresses finding welfare-optimal serial dictatorships under limited query access to agent-item weights in a complete bipartite graph. It introduces a proxy-graph approach that overestimates true weights and progressively learns edge information through a sequence of queries, guided by a maximum-weight PO matching computed on the proxy graph. The main result is an algorithm that computes a welfare-optimal action sequence using only $n^5$ queries (with $n^4$ in the distinct-weight-per-agent case), thereby resolving the primary open problem from CR23. The approach combines a potential-function analysis with a learning-on-the-fly strategy and offers a pathway to truthful implementations via VCG payments. This advances the understanding of query complexity in welfare optimization problems solved via serial dictatorship and opens avenues for generalization to other combinatorial problems.

Abstract

Serial dictatorship is a simple mechanism for coordinating agents in solving combinatorial optimization problems according to their preferences. The most representative such problem is one-sided matching, in which a set of n agents have values for a set of n items, and the objective is to compute a matching of the agents to the items of maximum total value (a.k.a., social welfare). Following the recent framework of Caragiannis and Rathi [10], we consider a model in which the agent-item values are not available upfront but become known by querying agent sequences. In particular, when the agents are asked to act in a sequence, they respond by picking their favorite item that has not been picked by agents who acted before and reveal their value for it. Can we compute an agent sequence that induces a social welfare-optimal matching? We answer this question affirmatively and present an algorithm that uses polynomial number (n^5) of queries. This solves the main open problem stated by Caragiannis and Rathi [CR23]. Our analysis uses a potential function argument that measures progress towards learning the underlying edge-weight information. Furthermore, the algorithm has a truthful implementation by adapting the paradigm of VCG payments.

Welfare-Optimal Serial Dictatorships have Polynomial Query Complexity

TL;DR

This work addresses finding welfare-optimal serial dictatorships under limited query access to agent-item weights in a complete bipartite graph. It introduces a proxy-graph approach that overestimates true weights and progressively learns edge information through a sequence of queries, guided by a maximum-weight PO matching computed on the proxy graph. The main result is an algorithm that computes a welfare-optimal action sequence using only queries (with in the distinct-weight-per-agent case), thereby resolving the primary open problem from CR23. The approach combines a potential-function analysis with a learning-on-the-fly strategy and offers a pathway to truthful implementations via VCG payments. This advances the understanding of query complexity in welfare optimization problems solved via serial dictatorship and opens avenues for generalization to other combinatorial problems.

Abstract

Serial dictatorship is a simple mechanism for coordinating agents in solving combinatorial optimization problems according to their preferences. The most representative such problem is one-sided matching, in which a set of n agents have values for a set of n items, and the objective is to compute a matching of the agents to the items of maximum total value (a.k.a., social welfare). Following the recent framework of Caragiannis and Rathi [10], we consider a model in which the agent-item values are not available upfront but become known by querying agent sequences. In particular, when the agents are asked to act in a sequence, they respond by picking their favorite item that has not been picked by agents who acted before and reveal their value for it. Can we compute an agent sequence that induces a social welfare-optimal matching? We answer this question affirmatively and present an algorithm that uses polynomial number (n^5) of queries. This solves the main open problem stated by Caragiannis and Rathi [CR23]. Our analysis uses a potential function argument that measures progress towards learning the underlying edge-weight information. Furthermore, the algorithm has a truthful implementation by adapting the paradigm of VCG payments.
Paper Structure (9 sections, 11 theorems, 5 equations, 1 algorithm)

This paper contains 9 sections, 11 theorems, 5 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Results and Techniques:
  3. Further Related Work:
  4. Preliminaries
  5. Intuition
  6. Finding an Optimal Action Sequence
  7. Overview of our algorithm:
  8. Computing maximum-weight PO matchings (${\sf MWPO}$):
  9. Conclusion and Open Problems

Key Result

Theorem 4.1

CR23 Any welfare-optimal action sequence induces a maximum-weight matching in a complete bipartite weighted graph.

Theorems & Definitions (18)

  • Theorem 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Lemma 4.4
  • proof
  • Corollary 4.5
  • proof
  • Lemma 4.6
  • proof
  • Lemma 4.7
  • ...and 8 more