Testing Stability Properties in Graphical Hedonic Games

TL;DR

This paper introduces sublinear time property testing algorithms for stability concepts in large graphical hedonic games, enabling efficient verification of stability properties without examining the entire game data.

Contribution

It pioneers the application of property testing to stability verification in large-scale hedonic games represented by graphs with bounded degree, with algorithms having input-size-independent complexity.

Findings

Testing algorithms for various stability notions are developed.

Existence of perfect coalition structures can be efficiently tested.

All testers operate with one-sided error and constant time complexity.

Abstract

In hedonic games, players form coalitions based on individual preferences over the group of players they could belong to. Several concepts to describe the stability of coalition structures in a game have been proposed and analysed in the literature. However, prior research focuses on algorithms with time complexity that is at least linear in the input size. In the light of very large games that arise from, e.g., social networks and advertising, we initiate the study of sublinear time property testing algorithms for existence and verification problems under several notions of coalition stability in a model of hedonic games represented by graphs with bounded degree. In graph property testing, one shall decide whether a given input has a property (e.g., a game admits a stable coalition structure) or is far from it, i.e., one has to modify at least an $\epsilon$-fraction of the input (e.g., the game's preferences) to make it have the property. In particular, we consider verification of perfection, individual rationality, Nash stability, (contractual) individual stability, and core stability. While there is always a Nash-stable coalition structure (which also implies individually stable coalitions), we show that the existence of a perfect coalition structure is not tautological but can be tested. All our testers have one-sided error and time complexity that is independent of the input size.