[Paper Review] Forming Probably Stable Communities with Limited Interactions

Ayumi Igarashi, Jakub Śliwiński+1 more|arXiv (Cornell University)|Nov 12, 2018

Game Theory and Voting Systems10 references4 citations

TL;DR

This paper proposes a statistical framework for learning probably approximately stable (PAC stable) community structures in hedonic games under limited interactions, modeled via an underlying interaction network. It proves that PAC stability is efficiently achievable if and only if the interaction network is a forest, establishing a tight characterization of when structural constraints enable stable coalition formation from data without full preference knowledge.

ABSTRACT

A community needs to be partitioned into disjoint groups; each community member has an underlying preference over the groups that they would want to be a member of. We are interested in finding a stable community structure: one where no subset of members $S$ wants to deviate from the current structure. We model this setting as a hedonic game, where players are connected by an underlying interaction network, and can only consider joining groups that are connected subgraphs of the underlying graph. We analyze the relation between network structure, and one's capability to infer statistically stable (also known as PAC stable) player partitions from data. We show that when the interaction network is a forest, one can efficiently infer PAC stable coalition structures. Furthermore, when the underlying interaction graph is not a forest, efficient PAC stabilizability is no longer achievable. Thus, our results completely characterize when one can leverage the underlying graph structure in order to compute PAC stable outcomes for hedonic games. Finally, given an unknown underlying interaction network, we show that it is NP-hard to decide whether there exists a forest consistent with data samples from the network.

Motivation & Objective

• To address the challenge of forming stable communities when player preferences are unknown but interactions are limited by a network structure.
• To investigate whether statistical learning from interaction data can yield PAC stable coalition structures in hedonic games.
• To characterize the structural conditions—specifically, the role of the interaction graph—under which PAC stable outcomes can be efficiently computed.
• To explore the computational complexity of learning consistent interaction network structures from data samples, especially when the true graph is unknown.

Proposed method

• Models community formation as a hedonic game constrained by a graph where only connected coalitions are feasible.
• Introduces a statistical learning framework based on Probably Approximately Correct (PAC) stability, using samples of coalition preferences to infer stable partitions.
• Proves that when the interaction network is a forest, a PAC stable outcome can be computed using a polynomial number of samples.
• Employs a reduction from (3,B2)-SAT to show that if the interaction graph contains a cycle, no PAC stable outcome can be guaranteed, even with unlimited samples.
• Uses a novel learning-theoretic approach to infer forest-structured interaction networks from data, showing that while learning consistent forests is NP-hard, communication complexity remains low.
• Applies structural graph theory and preference modeling to construct hardness instances that demonstrate the necessity of the forest condition for PAC stabilizability.

Experimental results

Research questions

• RQ1 Under what structural conditions on the interaction network can PAC stable coalition structures be efficiently learned from data in hedonic games?
• RQ2 Is it possible to guarantee PAC stability when player preferences are unknown but interactions are constrained by a graph?
• RQ3 What is the computational complexity of learning a forest-structured interaction network consistent with observed data samples?
• RQ4 Can PAC stable outcomes be achieved when the underlying interaction graph contains cycles, even with access to a large number of samples?
• RQ5 How does the presence of a forest structure in the interaction network enable efficient inference of stable community partitions without full preference elicitation?

Key findings

• PAC stable outcomes can be efficiently computed if and only if the underlying interaction network is a forest, establishing a tight characterization of computable stability.
• Even when the true forest structure is unknown, PAC stabilizability holds, meaning stable outcomes can be learned from data without identifying the exact network.
• If the interaction graph contains a single cycle, it is impossible to guarantee PAC stability, regardless of the number of samples, due to inherent structural limitations.
• Learning a forest structure consistent with observed data samples is NP-hard, even when both connected and disconnected coalitions are observed.
• The communication complexity of learning a consistent forest structure is low, suggesting practical feasibility with strong optimization tools despite theoretical intractability.
• The framework does not require cardinal utility representations of preferences, making it more general and realistic than prior approaches relying on numerical utility models.