Fractional Hedonic Games

The work we present in this paper initiated the formal study of fractional hedonic games, coalition formation games in which the utility of a player is the average value he ascribes to the members of his coalition. Among other settings, this covers situations in which players only distinguish between friends and non-friends and desire to be in a coalition in which the fraction of friends is maximal. Fractional hedonic games thus not only constitute a natural class of succinctly representable coalition formation games, but also provide an interesting framework for network clustering. We propose a number of conditions under which the core of fractional hedonic games is non-empty and provide algorithms for computing a core stable outcome. By contrast, we show that the core may be empty in other cases, and that it is computationally hard in general to decide non-emptiness of the core.

💡 Research Summary

This paper initiates the formal study of fractional hedonic games (FHGs), a subclass of hedonic coalition formation games in which each player’s utility from a coalition is the average of the values he assigns to the members of that coalition. The authors adopt a compact representation: each player i has a valuation function v_i : N → ℝ, and the utility of i in a coalition S (with i ∈ S) is u_i(S) = (∑_{j∈S} v_i(j))/|S|. By setting v_i(i)=0 the model avoids self‑valuation. When valuations are binary (0/1) and symmetric, the game can be visualized as an undirected simple graph where edges indicate mutual friendship. This “simple symmetric” case captures many natural scenarios—friend‑of‑friend clustering, political party formation, or the classic “Bakers and Millers” market where two types of agents prefer to be surrounded by the opposite type.

The paper’s central focus is the core, the set of partitions (coalition structures) that admit no blocking coalition: a coalition C blocks a partition π if every member of C would obtain a strictly higher utility in C than in his current coalition under π. The authors explore three fundamental questions: (1) Does the core always exist? (2) How hard is it to decide whether a given FHG has a non‑empty core? (3) If the core is non‑empty, can we compute a core‑stable partition efficiently?

Key contributions

1. Core may be empty even for simple symmetric games.
   The authors construct a concrete example with 40 players whose underlying graph is simple and symmetric, yet no partition lies in the core. This contrasts sharply with additively separable hedonic games, where the grand coalition is always core‑stable in the unweighted undirected case.

2. Complexity of core existence.
   Deciding whether a simple symmetric FHG has a non‑empty core is shown to be Σ₂^P‑complete. The proof reduces from a quantified Boolean formula with one alternation, establishing that the problem sits at the second level of the polynomial hierarchy. Consequently, finding a core‑stable partition is NP‑hard. Moreover, verifying that a given partition belongs to the core is coNP‑complete, because one must rule out the existence of any blocking coalition.

3. Structural graph classes guaranteeing a non‑empty core.
   The authors identify several families of graphs for which the core is always non‑empty and can be found in polynomial time:

   • Graphs of maximum degree two (paths and cycles).
   • Forests (acyclic graphs).
   • Complete multipartite graphs (including complete bipartite graphs).
   • Bipartite graphs that admit a perfect matching.
   • Graphs with girth at least five (no short cycles).
     For each class they present explicit algorithms (often as simple as grouping connected components or matching pairs) that output a core‑stable partition.

4. Bakers and Millers (and its generalizations).
   In the “Bakers and Millers” setting, agents belong to one of k types; each type prefers coalitions where the proportion of agents of other types is maximal. This can be modeled as a complete k‑partite graph with edges only between different types. The paper proves that the strict core is always non‑empty and, remarkably, the grand coalition is always core‑stable. They further characterize all strict‑core partitions and give a polynomial‑time algorithm that computes the unique finest partition in the strict core.

5. Connections to community detection.
   Because a simple symmetric FHG’s utility is exactly the fraction of friends inside a coalition, a core‑stable partition corresponds to a clustering where no group of agents can improve their average internal friendship by moving together. This provides a game‑theoretic perspective on graph clustering, complementing existing methods based on modularity or spectral techniques.

6. Related work and extensions.
   The paper situates FHGs relative to additively separable hedonic games, social distance games, and segregation models. It notes that while average‑utility (fractional) and total‑utility (additive) models share similar definitions, their stability properties diverge dramatically. Subsequent work (cited in the paper) has explored welfare maximization, Nash stability, price of anarchy, and empirical heuristics for large instances, confirming the relevance of the model.

Implications and future directions
The results demonstrate that a modest modeling choice—averaging rather than summing valuations—creates a rich landscape where core existence hinges on graph topology. For many practical networks (social, economic, political) the underlying graph often falls into one of the tractable classes identified, suggesting that core‑stable outcomes can be computed efficiently in realistic settings. Conversely, the Σ₂^P‑completeness result warns that for arbitrary networks, guaranteeing core stability may be computationally infeasible, motivating the study of approximation, heuristics, or alternative solution concepts (e.g., Nash stability, Pareto optimality).

Overall, the paper establishes fractional hedonic games as a natural, mathematically intriguing, and practically useful framework for coalition formation, opening numerous avenues for algorithmic game theory, network science, and economic modeling.