[Paper Review] $\Sigma_2^p$-complete Problems on Hedonic Games

Dominik Peters|arXiv (Cornell University)|Sep 8, 2015

Game Theory and Voting Systems3 citations

TL;DR

This paper establishes the Σ₂^p-completeness of deciding core- and strict-core-stable partitions in hedonic games with additively separable preferences, even under symmetric valuations and bounded interaction. It further proves Σ₂^p-completeness for non-emptiness of the strict core in dichotomous preference games, and shows TRUE-∃∀-3DNF remains Σ₂^p-hard with bounded variable occurrences.

ABSTRACT

Hedonic games provide a general model of coalition formation, in which a set of agents is partitioned into coalitions, with each agent having preferences over which other players are in her coalition. We prove that with additively separable preferences, it is $\Sigma_2^p$-complete to decide whether a core- or strict-core-stable partition exists, extending a result of Woeginger (2013). Our result holds even if valuations are symmetric and non-zero only for a constant number of other agents. We also establish $\Sigma_2^p$-completeness of deciding non-emptiness of the strict core for hedonic games with dichotomous preferences. Of more general interest, we prove that TRUE-$\exists\forall$-3DNF remains $\Sigma_2^p$-hard even if no variable occurs more than three times.

Motivation & Objective

• To determine the computational complexity of deciding core- and strict-core-stable partitions in hedonic games with additively separable preferences.
• To extend prior work by Woeginger (2013) by proving Σ₂^p-completeness under symmetric valuations and limited interaction.
• To investigate the complexity of non-emptiness of the strict core in hedonic games with dichotomous preferences.
• To examine the complexity of TRUE-∃∀-3DNF when variable occurrences are bounded.
• To establish tight complexity bounds for stable coalition formation under various preference models.

Proposed method

• Reduction from known Σ₂^p-complete problems to show hardness for core stability in additively separable hedonic games.
• Construction of symmetric, sparse valuation functions where only a constant number of agents have non-zero valuations to prove hardness under structural constraints.
• Use of a novel reduction framework to show Σ₂^p-completeness for non-emptiness of the strict core in dichotomous preference games.
• Proof that TRUE-∃∀-3DNF remains Σ₂^p-hard even when each variable appears at most three times.
• Leveraging properties of quantified Boolean formulas to establish lower bounds on computational complexity.
• Formal proof of completeness via both hardness and membership in Σ₂^p for all studied problems.

Experimental results

Research questions

• RQ1 Is deciding the existence of a core-stable partition in additively separable hedonic games Σ₂^p-complete?
• RQ2 Does the complexity remain Σ₂^p-complete when valuations are symmetric and non-zero for only a constant number of agents?
• RQ3 Is the non-emptiness of the strict core in dichotomous preference hedonic games Σ₂^p-complete?
• RQ4 Is TRUE-∃∀-3DNF Σ₂^p-hard when each variable appears at most three times?
• RQ5 Can the complexity of stable coalition formation be tightly bounded under structural constraints on preferences?

Key findings

• Deciding core-stable partitions in additively separable hedonic games is Σ₂^p-complete, even with symmetric valuations and bounded interaction.
• The same Σ₂^p-completeness result holds for strict-core-stable partitions under the same constraints.
• Non-emptiness of the strict core in hedonic games with dichotomous preferences is Σ₂^p-complete.
• TRUE-∃∀-3DNF remains Σ₂^p-hard even when each variable occurs at most three times.
• The results establish tight complexity bounds for stable coalition formation under various preference models.
• The findings extend and strengthen prior work by Woeginger (2013) by proving completeness under stronger structural constraints.