Coalitional Games with Bounded Coalition Sizes

• Coalitional games with bounded coalition sizes are models for strategic cooperation that enforce strict size limits, reflecting physical, organizational, or legal constraints.
• The framework extends classical models by integrating core stability notions across Transferable-Utility, additively-separable hedonic, and network-based games.
• Computational results vary, with polynomial-time solutions in unweighted cases and NP-hardness in weighted settings, highlighting challenges in stable coalition formation.

A coalitional game with bounded coalition sizes models strategic cooperation among a finite set of self-interested agents, subject to explicit restrictions on the maximum size of any coalition. This generalizes classical coalitional games by incorporating a cardinality constraint in the admissible partitions or coalition structures. The analysis of existence, structure, and computation of stable outcomes—typically those in the core, strict core, and related concepts—differs significantly between transferable-utility (TU), additively separable hedonic, and network-based models. Bounded coalition sizes are especially relevant in settings where physical, organizational, or legal limitations preclude formation of arbitrarily large coalitions.

1. Formal Models and Core Stability Notions

Let $N$ denote the finite set of agents and $k \in \mathbb{N}$ the imposed upper bound on coalition size. A $k$-bounded coalition structure is a partition $P = \{S_1, \ldots, S_m\}$ of $N$ such that $|S_j| \le k$ for all $j$. The specifics of agent preferences or the value of each coalition depend on the class of game:

• Transferable-Utility (TU): $(N, v)$ with $v : 2^N \to \mathbb{R}$, $v(\emptyset) = 0$. Allocations $x \in \mathbb{R}^N$ must satisfy efficiency and coalitional rationality over all admissible coalition structures. The coalition-structure core (CS-core) requires that each block $C^j$ receives $x(C^j) = v(C^j)$ and $x(S) \geq v(S)$ for all $S \subseteq N$ (Hamed et al., 2023).
• Additively-Separable Hedonic Games (ASHGs): Each agent $i$ chooses coalitions based on additive valuations $w_{ij}$ of her partners. For bounded $k$, the $k$-CS-core consists of $k$-bounded partitions $P$ such that no subset $S$ with $|S| \leq k$ is “strongly-blocking,” i.e., $u_i(S) > u_i(P(i))$ for all $i \in S$ (Levinger et al., 2023).
• Hedonic Games (Ordinal): Preferences $\succeq_i$ over coalitions containing $i$; a core-stable $k$-bounded partition admits no blocking coalition $S$ with $|S| \leq k$ such that all $i \in S$ prefer $S \succ_i \pi(i)$ (Woeginger, 2012).
• Path Cooperative Games: Coalitions correspond to edge or vertex sets in a network whose value is determined by enabling a commodity flow from $s$ to $t$; the CS-core coincides with the core of the associated flow game (Fang et al., 2015).

The table summarizes constraint forms and blocking concepts:

Game Class	Coalition Value/Preference	Core Stability Condition
TU games	$v : 2^N \to \mathbb{R}$	No $S$ with $\sum_{i\in S} x_i < v(S)$
ASHGs (additive)	$u_i(S) = \sum_{j\in S \setminus\{i\}} w_{ij}$	No $S$ s.t. $\forall i\in S, u_i(S) > u_i(P(i))$
Hedonic (general)	$\succeq_i$ over $\{S \subseteq N : i \in S\}$	No $S$ with $\forall i \in S, S \succ_i \pi(i)$
Path cooperative	Network flow/cut structure	No $S$ s.t. imputation contradicts min-cut

2. Existence Results and Characterizations

The existence of nonempty CS-core partitions under bounded coalition sizes is highly sensitive to the underlying preference or value structure and the parameter $k$.

• Unweighted Additive/Hedonic Games ($k=3$): For symmetric, unweighted preferences ($w_{ij} \in \{0, 1\}$), the $3$-bounded CS-core is always nonempty and can be computed in polynomial time (Levinger et al., 2023). The proof utilizes iterative blocking of cliques of size $\leq 3$.
• Weighted Additive Games: Allowing arbitrary $w_{ij} \ge 0$, the problem of determining core nonemptiness is strongly NP-hard for any $k \geq 3$, and the core may be empty (Levinger et al., 2023). The size bound does not guarantee existence in the weighted setting.
• TU Games: Existence of a CS-core reduces to feasibility of a linear system reflecting the coalition structure and rationality constraints. The maximum welfare $K_v$ over all $k$-bounded partitions is central. Balancedness-type conditions analogous to Bondareva–Shapley determine CS-core nonemptiness, but the problem grows in difficulty with the coalition-size constraint (Hamed et al., 2023).
• Path Cooperative Games: The CS-core is always nonempty, following from min-cut properties and flow-game duality. Existence is guaranteed regardless of coalition size due to the structural properties of flows and cuts (Fang et al., 2015).

A plausible implication is that bounded coalition sizes facilitate existence and computation only in restricted settings (e.g., unweighted, low $k$), but are not a panacea in general hedonic or weighted games.

3. Algorithms and Computational Complexity

The computational landscape is heterogeneous. Key results include:

• ASHGs, $k=3$, Unweighted: A constructive polynomial-time algorithm, based on iterative blocking of strongly-blocking $2$- or $3$-cliques, always yields a core partition. Each step either improves total welfare or permanently eliminates blocking opportunities for certain agents (Levinger et al., 2023).
• ASHGs, Weighted: Existence is NP-hard to decide, and no efficient algorithm exists unless P=NP. The reduction uses Metric 3-Dimensional Stable Roommates (Levinger et al., 2023).
• TU Games: Coalition proposal algorithms using distributed learning dynamics, where agents iteratively propose coalition formations and adjust aspiration levels, exhibit almost-sure convergence to the CS-core (when nonempty), under mild conditions such as value and aspiration discretization (Hamed et al., 2023). The proposal mechanism operates via local feasibility checks and Markovian state transitions.
• Path Cooperative: The CS-core can be characterized and solved as a linear program dual to the maximum flow problem, with computation via the ellipsoid method and shortest-path separation oracles—guaranteeing polynomial-time solution despite exponential constraint sets (Fang et al., 2015).
• General Hedonic Games: Deciding existence of a core stable partition in additive games is $\Sigma^P_2$-complete. Various tractable subclasses are known for friend-/enemy-oriented or matching-based structures, often relying on specialized combinatorial algorithms (Woeginger, 2012).

4. Related Core Concepts Under Size Constraints

Alternative stability notions address some of the limitations of the strict core with bounded coalitions:

• Strict Core (SC): Allows for weakly-blocking coalitions. SC can be empty even in unweighted settings and is NP-hard to check for nonemptiness for $k \geq 3$ (Levinger et al., 2023).
• Contractual Strict Core (CSC): More permissive, forbidding blocks only if agents strictly improve while someone outside loses utility. The CSC is always nonempty and can be found using a greedy merge algorithm in polynomial time (Levinger et al., 2023).

These findings suggest that while the CS-core may be elusive in several settings, weaker notions of stability can be robust and computationally manageable under bounded coalition sizes.

5. Applications and Illustrative Examples

Coalitional games with bounded sizes model a diverse range of scenarios. Notable applications include:

• Multi-Agent Task Allocation: Agents and tasks are represented in a combinatorial setting; coalitions form if they fulfill task requirements and maximize net value. Distributed coalition proposal algorithms achieve near-optimal welfare and are resilient under communication constraints (Hamed et al., 2023).
• Social and Friendship Networks: Coalition formation is impeded by both the structure of underlying relationships (modeled as cliques or graphs) and size restrictions, reflecting real organizational or structural limits (Levinger et al., 2023).
• Network Flow and Path Games: Coalition value is determined by the ability to support flows across a network; the CS-core directly corresponds to minimal edge/vertex cuts, with allocations linked to flow-cut duality (Fang et al., 2015).

Examples from the literature demonstrate that for path cooperative games, coalition structure solutions are min-cut incidences in the network, and that for ASHGs, iterative reblocking based on local gains suffices for small $k$, but not with heterogeneous (weighted) preferences.

6. Open Problems and Future Directions

Several key open questions remain unresolved:

• Maximal Coalition Size $k > 3$, Unweighted: Whether the guarantee of nonempty CS-core partitions extends beyond $k=3$ in unweighted ASHGs is open. Empirical evidence suggests core existence is likely, but no general proof nor tight characterization is available (Levinger et al., 2023).
• Approximation of Maximum Social Welfare: The MaxUtil partition problem under size bounds is computationally hard; improved approximation algorithms are needed (Levinger et al., 2023).
• General Preference Models: Extensions to games with skill-based requirements, non-additive utilities, dynamic coalition structures, or network-constrained communication remain active research directions (Hamed et al., 2023).
• Complexity Boundaries: The exact boundaries of tractable vs intractable cases in general hedonic (especially additive or enemy-oriented) games with bounded coalition sizes are incompletely mapped (Woeginger, 2012).

7. Connections to Classical and Network Games

Coalitional games with bounded coalition sizes bridge traditional cooperative game theory, hedonic coalition formation, network flow problems, and combinatorial optimization. The interplay of structural constraints (such as cardinality bounds) and preference expressiveness leads to sharp distinctions in core existence, algorithmic tractability, and solution structure. These models are foundational for multiagent systems, decentralized coordination, and networked resource-sharing environments.

For comprehensive accounts, see Levinger et al. (Levinger et al., 2023), Mohri et al. (Hamed et al., 2023), and Woeginger (Woeginger, 2012) for hedonic foundations. For networked cooperative structures, see Yang et al. (Fang et al., 2015).