Stability Under Valuation Updates in Coalition Formation

TL;DR

This paper investigates the complexity of maintaining stable coalition structures in hedonic games after valuation updates, providing both NP-completeness results and efficient algorithms for certain stability concepts.

Contribution

It offers a comprehensive complexity analysis and introduces polynomial-time algorithms for specific stability notions under restricted valuation conditions.

Findings

NP-completeness for finding nearby stable structures in general cases

Polynomial algorithms for contractual stability under symmetric valuations

Bounded average distance over update sequences with the proposed algorithms

Abstract

Coalition formation studies how to partition a set of agents into disjoint coalitions under consideration of their preferences. We study the classical objective of stability in a variant of additively separable hedonic games where agents can change their valuations. Our objective is to find a stable partition after each change. To minimize the reconfiguration cost, we search for nearby stable coalition structures. Our focus is on stability concepts based on single-agent deviations. We present a detailed picture of the complexity of finding nearby stable coalition structures in additively separable hedonic games, for both symmetric and non-symmetric valuations. Our results show that the problem is NP-complete for Nash stability, individual stability, contractual Nash stability, and contractual individual stability. We complement these results by presenting polynomial-time algorithms for contractual Nash stability and contractual individual stability under restricted symmetric valuations. Finally, we show that these algorithms guarantee a bounded average distance over long sequences of updates.