Hedonic Coalition Formation

Updated 19 December 2025

Hedonic coalition formation is a decentralized process where agents join groups based solely on their internal preferences about group membership.
Stability notions like Nash, individual, and core stability provide critical benchmarks, with complexities from NP-hardness to Σp2-completeness shaping the modeling approach.
The topic underpins applications in distributed resource allocation, autonomous coordination, and dynamic coalition protocols, supported by simulation and empirical analysis.

Hedonic coalition formation describes the decentralized process by which a set of agents partition themselves into coalitions based solely on their internal preferences regarding group composition. In hedonic settings, each agent evaluates possible coalitions entirely by the identities of the other coalition members, without regard to external rewards, underlying structure, or how agents outside its coalition are grouped. Research has mapped extensive connections between preference modeling, solution concepts such as Nash stability and core stability, algorithmic tractability, and complexity results. Hedonic coalition formation underpins distributed resource allocation, autonomous agent coordination, social segregation models, and foundational bargaining theory.

1. Formal Models and Preference Structures

The central primitives are:

Coalitions and partitions: Given $N$ agents, any nonempty subset $S\subseteq N$ is a coalition. A coalition structure (partition) $\pi = \{S_1, ..., S_k\}$ is a set of disjoint coalitions whose union is $N$ (Woeginger, 2012).
Hedonic preference relations: Each agent $i$ has a preference $\succeq_i$ (complete, transitive) over all coalitions containing $i$ , i.e., over $\mathcal{N}_i = \{S\subseteq N: i\in S\}$ .
Utility representations: Models include
- Additively separable: $u_i(S) = \sum_{j\in S\setminus\{i\}} v_i(j)$ (Brandt et al., 2022).
- Fractional: $u_i(S) = \frac{1}{|S|} \sum_{j\in S} v_i(j)$ (Aziz et al., 2017).
- Common ranking property (CRP): All agents rank coalitions by the same function $U(S)$ (Caskurlu et al., 2022).
- Graph-restricted games: Coalitions must be connected subsets of $G=(N,E)$ (Igarashi et al., 2016).
- Diversity games: Preferences determined by proportions of agent types (e.g., homophily, heterophily) (Bredereck et al., 2019).
- Boolean/dichotomous: Each agent identifies satisfactory and unsatisfactory coalitions via propositional formulas (Aziz et al., 2015).
- History-based/trust-augmented: Utility adjusted by trust from past dynamics (Ghaffarizadeh et al., 2013).

Preference extensions such as best-player (B-hedonic) and worst-player (W-hedonic) games rank coalitions by the best or worst member (other than $i$ ), with robust implications for stability existence and computational complexity (Aziz et al., 2011).

2. Solution Concepts and Stability Notions

Stability notions form a structural hierarchy:

Nash Stability (NS): No agent can profitably unilaterally move to another coalition (possibly singleton) (Woeginger, 2012).
Individual Stability (IS): No agent can profitably move if the destination coalition is weakly accepting (no member strictly worse off) (Brandt et al., 2022).
Contractual IS/CNS: Strengthened IS where the agent's old coalition must also weakly accept her departure (Rey et al., 2022).
Core Stability (CR): No subset can regroup so all its members strictly prefer the new coalition (Woeginger, 2012).
Strict Core: Blocks weak deviations (at least one strictly prefers, all others weakly prefer) (Suksompong, 2018).
Strong Nash Stability (SNS): No nonempty group can coordinate moves so every member strictly benefits (Aziz et al., 2012).
Pareto Optimality (PO): No partition universally weakly improves and strictly improves for at least one agent (Caskurlu et al., 2022).
Popularity: Partition not losing a majority vote to any other partition (Condorcet-winner in partition space) (Bullinger et al., 8 Nov 2024).

A summary of major implications:

Stability Concept	Description	Complexity Status
Nash Stability (NS)	No profitable unilateral deviation	NP-hard (general), P in restricted B/W games
Individual Stability	No profitable, mutually-accepted unilateral deviation	NP-hard (general), P in B-hedonic, anonymous
Core Stability	No group can block by strictly preferring regrouping	$\Sigma^p_2$ -complete (additive/fractional)
Strict Core	No group weakly blocks with at least one strict preference	NP-hard (fractional, additive), P in trees
Popularity	No partition loses majority vote against any other	$\Sigma^p_2$ -complete

Existence results depend on preference restrictions. Top-responsiveness and mutuality yield strictly strong Nash stable partitions efficiently (Aziz et al., 2012). In subset-neutral and neutrally anonymous settings, Nash and individual stability are always achievable, often in polynomial time (Suksompong, 2018).

3. Algorithmic and Complexity Landscape

Algorithmic tractability is tightly coupled with the succinctness and structure of preference representations:

Polynomial-time algorithms: Exist for core/individual stability in graph-restricted games on acyclic graphs (Igarashi et al., 2016), for IS in B-hedonic games (Aziz et al., 2011), and for CRP when coalition size is bounded (max-weight matching) (Caskurlu et al., 2022).
NP/combinatorial hardness: Verifying NS, IS, CR, and strict core existence is NP-hard or $\Sigma^p_2$ -complete for additively separable, fractional, and anonymous representations, even with restricted preferences (Peters et al., 2015, Aziz et al., 2017, Woeginger, 2012).
PLS-completeness: For symmetric additively separable games and Nash/in-neighbor stability in star-shaped graphs (Igarashi et al., 2016).
Random hedonic games: For large random populations, Nash-stable partitions rarely exist, but individual and contractual Nash-stable partitions exist with high probability and can be computed in polynomial time via greedy clustering algorithms (Bullinger et al., 3 Jun 2024).

In diversity games, core stable partitions need not exist and deciding non-emptiness of the core is NP-complete, but IS partitions exist efficiently under single-peakedness (Bredereck et al., 2019). In weighted modified fractional models, strong Nash equilibria may fail to exist, but unrestricted core partitions are always attainable and efficiently computable (Monaco et al., 2018).

4. Distributed and Dynamic Coalition Formation

Distributed protocols and dynamic processes define state-of-the-art models:

Wireless agent allocation: Agents and tasks iteratively self-organize into Nash-stable coalitions via distributed switch rules balancing throughput and delay, with history-based exclusion to prevent cycling (Saad et al., 2010). Simulation shows adaptation to dynamic environments (task arrival/removal, mobility), and significant performance gain over non-coalitional baselines.
Dynamic utilities: Agents' utilities update (“resent” or “appreciation”) after coalition changes, yielding potential functions that enforce convergence to strong stability under mild conditions (Boehmer et al., 2022).
History/trust-based formation: Agents score coalitions by combining expected direct payoff and trust valuation from coalition history. Departure penalties, honesty, and risk-aversion modulate coalition switching dynamics (Ghaffarizadeh et al., 2013).

5. Compact Representations and Preference Extensions

Succinct representations span:

Ordinal rankings: B- and W-hedonic (rank-based extensions), friends-enemies-neutral ballots (FEN), Boolean logic formulas for dichotomous games, and anonymous (size-only) preferences (Rey et al., 2022, Aziz et al., 2015, Bredereck et al., 2019).
Coalition nets and subset-additive models: Provide universal expressivity (subset-additive), subset-neutral models guarantee Nash/individual stability; common ranking property admits Pareto, core, and individual stability simultaneously (Suksompong, 2018, Caskurlu et al., 2022).
Distance-based models: Use directed Hausdorff–Kendall-tau distance to compare ordinal preferences to coalition composition, achieving metric-like axiomatic robustness and efficient stability verification for bounded-degree networks (Rey et al., 2022).

Preference structure and locality (e.g., connected coalitions in graphs (Igarashi et al., 2016)) crucially determine tractability and existence of stable structures.

6. Empirical Methods, Simulation, and Practical Mechanisms

Simulation tools for hedonic coalition formation allow:

Modeling and visualizing agents, weighted utilities, and preferences (Miles, 2017).
Exploring the impact of dynamic rules: best-response, merge-and-split, serial dictatorship.
Evaluating coalition stability for all major concepts (Core, NS, IS, CIS, PO).
Empirical findings: for small to moderate $n$ (e.g., $n \leq 20$ ), core and individual stability checks and best-response dynamics converge rapidly; qualitative analysis of coalition formation time, agent payoff, and lifetime under various dynamic settings (Saad et al., 2010, Ghaffarizadeh et al., 2013, Miles, 2017).

7. Open Research Problems and Directions

Several fundamental questions remain open:

Precise complexity of strong core existence (strict core and group deviations) in additive, fractional, and Boolean models (Woeginger, 2012, Aziz et al., 2017, Caskurlu et al., 2022).
Tight algorithmic bounds for Nash and Individual stability in W-hedonic games and diversity games with arbitrary (non-single-peaked) preferences (Aziz et al., 2011, Bredereck et al., 2019, Brandt et al., 2022).
Extension of neutrality/model restrictions to large-scale and infinite populations, externalities, and networked settings (Suksompong, 2018).
Characterizing practical stable coalition formation in dynamic, trust-based, or history-dependent strategic environments (Boehmer et al., 2022, Ghaffarizadeh et al., 2013).
Approximate and parameterized algorithms for stable outcome finding (especially with APX-hardness and inapproximability in CRP models for general coalition sizes) (Caskurlu et al., 2022).

Hedonic coalition formation remains an essential research area for understanding decentralized group structure emergence, with direct implications in multi-agent systems, networked resource allocation, team formation, and political/economic bargaining.