Core stability in hedonic coalition formation

In many economic, social and political situations individuals carry out activities in groups (coalitions) rather than alone and on their own. Examples range from households and sport clubs to research networks, political parties and trade unions. The underlying game theoretic framework is known as coalition formation. This survey discusses the notion of core stability in hedonic coalition formation (where each player’s happiness only depends on the other members of his coalition but not on how the remaining players outside his coalition are grouped). We present the central concepts and algorithmic approaches in the area, provide many examples, and pose a number of open problems.

💡 Research Summary

The surveyed paper provides a comprehensive overview of core stability in hedonic coalition formation games, a class of cooperative games where each player’s utility depends solely on the members of his own coalition and not on how the remaining players are grouped. The authors begin by formalizing the model: a finite set of agents N, and for each agent i a preference (or utility) function u_i defined on subsets of N that contain i. A coalition structure (or partition) Π = {C_1,…,C_k} assigns every agent to exactly one coalition, and the utility each agent receives is u_i(C) where C is the coalition containing i.

Core stability is defined in the usual way: a coalition T ⊆ N blocks a partition Π if every member of T strictly prefers being together in T to staying in their current coalition under Π. A partition that admits no blocking coalition is called a core (or core‑stable) partition. The core thus captures the idea that no group of agents can jointly deviate to improve all of their outcomes, providing a strong notion of collective stability.

The paper then classifies the most studied hedonic preference models and discusses how these affect the existence and computation of the core.

Friendship (or affinity) models – each player i has a set of “friends” F_i and derives higher utility from coalitions containing more friends. The authors recount the classic result that a core always exists in this setting; a simple construction groups each connected component of the friendship graph into a coalition, guaranteeing that no subset can improve by breaking away.
Antagonism models – agents have “enemies” and experience disutility when paired with them. Here core existence is not guaranteed; the paper presents counter‑examples where any partition can be blocked by a coalition of mutually non‑antagonistic agents.
Absolute ranking models – each player ranks all possible coalitions containing him. This is the most general hedonic setting. The authors note that deciding whether a core exists is NP‑complete, and for certain restricted rankings (e.g., those that can be represented by a tree) the problem drops to polynomial time.
Single‑peaked and single‑tree preferences – utilities decline as the coalition moves away from a most‑preferred “peak” or from a designated subtree. In these structured environments, the core is guaranteed to exist and can be found by greedy or dynamic‑programming algorithms that exploit the underlying order.
Weighted‑graph (or additive) models – each edge (i,j) carries a weight w_{ij} reflecting mutual affinity; the utility of a coalition is the sum of internal edge weights (possibly plus a baseline). The authors explain that a maximum‑weight matching yields a core‑stable partition when coalitions are restricted to size two, and extensions exist for larger bounded sizes.

Having established the landscape of existence results, the survey turns to algorithmic issues. For general hedonic games, the core‑existence decision problem is Σ₂^P‑complete, reflecting the need to quantify over both partitions and potential blocking coalitions. Nevertheless, several algorithmic paradigms are highlighted:

Core‑adjustment (or improvement) algorithms start from an arbitrary partition, repeatedly locate a blocking coalition (often via an optimization subroutine that maximizes the collective gain of a deviation), and merge its members into a new coalition. The process terminates when no blocker can be found. While this method is guaranteed to converge in models where the core is known to exist, its worst‑case runtime can be exponential.
Verification algorithms reduce the search for a blocker to a combinatorial optimization problem such as “Maximum Blocking Gain”. In many structured models this subproblem is tractable (e.g., maximum weight clique in a friendship graph), enabling polynomial‑time core verification.
Dynamic programming for single‑peaked/tree preferences exploits the fact that the set of feasible coalitions can be arranged along a line or tree. By solving sub‑instances for each subtree and combining solutions, a core‑stable partition can be constructed in O(n·2^k) time, where k is the treewidth.
Greedy matching for weighted‑graph models computes a maximum‑weight matching (or a maximum‑weight b‑matching for larger coalition sizes) and then forms coalitions around the matched edges. The resulting partition is shown to be core‑stable because any deviating group would have to contain a higher‑weight edge, contradicting maximality.

The paper enriches the theoretical discussion with a variety of illustrative examples: household formation based on familial ties, research collaboration networks where co‑authorship preferences are modeled as edge weights, and political party alliances driven by ideological proximity. In each case, the authors demonstrate how the appropriate hedonic model captures the salient features of the domain and how core‑stable outcomes correspond to intuitively “stable” groupings.

Finally, the survey outlines a set of open problems that delineate promising directions for future research. Among these are:

Quantitative analysis of core size and structure – determining bounds on the number of coalitions, the distribution of coalition sizes, and the minimal/maximal cardinality of core partitions.
Dynamic hedonic games – studying how the core evolves when agents enter or leave the system, or when preferences change over time, and designing incremental algorithms that update a core‑stable partition efficiently.
Multi‑criteria hedonic preferences – extending the model to incorporate several dimensions of utility (e.g., monetary payoff, social status, task satisfaction) and investigating appropriate aggregation methods that preserve core existence.
Incorporating side payments or bargaining – exploring “core‑with‑transfer” concepts where agents can compensate each other to prevent blocking, and analyzing the computational complexity of finding such compensated core outcomes.
Hybrid preference models – combining affinity, antagonism, and absolute rankings into a single framework, and determining under what conditions a core is guaranteed or efficiently computable.

In conclusion, the survey emphasizes that while core stability offers a powerful and conceptually clean notion of group stability in hedonic settings, its existence and tractability are highly sensitive to the underlying preference structure. For well‑structured models (friendship, single‑peaked, bounded‑size additive), polynomial‑time algorithms and strong existence guarantees are available. In the general case, the problem remains computationally intractable, motivating ongoing research at the intersection of game theory, algorithm design, and social choice theory. The open questions identified suggest a vibrant research agenda aimed at bridging theoretical insights with practical applications in economics, political science, and networked societies.