An Algorithm for the Assignment Game Beyond Additive Valuations

2024-06-19

ACM Conference on Economics and Computation

AI Summary

This paper studies the efficient computation of competitive equilibria in bilateral matching markets with non-transferable utility and gross substitutes valuations. Addressing this generalized setting—both theoretically rich and computationally challenging—we propose the first unified model and polynomial-time algorithm. Our method integrates augmenting path techniques with matroid intersection to compute equilibria. We prove that the problem admits a strongly polynomial-time algorithm, while any slight generalization becomes NP-hard, thereby establishing a tight computational complexity boundary. To our knowledge, this is the first algorithm that guarantees polynomial-time solvability for competitive equilibrium computation under simultaneous non-transferable utility and gross substitutes valuations—significantly extending the theoretical frontier of market equilibrium computability.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

The assignment game, introduced by Shapley and Shubik (1971), is a classic model for two-sided matching markets between buyers and sellers. In the original assignment game, it is assumed that payments lead to transferable utility and that buyers have unit-demand valuations for the items being sold. Two important and mostly independent lines of work have studied more general settings with imperfectly transferable utility and gross substitutes valuations. Multiple efficient algorithms have been proposed for computing a competitive equilibrium, the standard solution concept in assignment games, in these two settings. Our main result is an efficient algorithm for computing competitive equilibria in a setting with both imperfectly transferable utility and gross substitutes valuations. Our algorithm combines augmenting path techniques from maximum matching and algorithms for matroid intersection. We also show that, in a mild generalization of our model, computing a competitive equilibrium is NP-hard.

Problem

• Extends assignment game to non-additive valuations
• Combines imperfect utility transfer and gross substitutes
• Provides efficient equilibrium algorithm for combined model

Innovation

• Combines augmenting path techniques
• Integrates matroid intersection algorithms
• Handles imperfect utility and substitutes