Computing Stable Coalitions: Approximation Algorithms for Reward Sharing

TL;DR

This paper develops approximation algorithms for stable coalition formation with finite, non-identical projects, achieving near-optimal welfare and core stability in complex valuation settings, and connects these results to auction equilibria.

Contribution

It introduces a black-box mechanism reducing core stability computation to welfare maximization, with improved approximations for various valuation classes.

Findings

Achieves one-fourth of optimal social welfare for subadditive valuations.

Provides new algorithms for welfare maximization with anonymous functions.

Establishes a link between coalition stability and auction equilibria.

Abstract

Consider a setting where selfish agents are to be assigned to coalitions or projects from a fixed set P. Each project k is characterized by a valuation function; v_k(S) is the value generated by a set S of agents working on project k. We study the following classic problem in this setting: "how should the agents divide the value that they collectively create?". One traditional approach in cooperative game theory is to study core stability with the implicit assumption that there are infinite copies of one project, and agents can partition themselves into any number of coalitions. In contrast, we consider a model with a finite number of non-identical projects; this makes computing both high-welfare solutions and core payments highly non-trivial. The main contribution of this paper is a black-box mechanism that reduces the problem of computing a near-optimal core stable solution to the purely algorithmic problem of welfare maximization; we apply this to compute an approximately core stable solution that extracts one-fourth of the optimal social welfare for the class of subadditive valuations. We also show much stronger results for several popular sub-classes: anonymous, fractionally subadditive, and submodular valuations, as well as provide new approximation algorithms for welfare maximization with anonymous functions. Finally, we establish a connection between our setting and the well-studied simultaneous auctions with item bidding; we adapt our results to compute approximate pure Nash equilibria for these auctions.