Coalition-Structure Core (CS-Core)

• Coalition-Structure Core (CS-Core) is a solution concept in cooperative game theory that extends the classical core by allowing arbitrary coalition structures.
• It ensures stability by enforcing coalitional rationality and within-coalition efficiency, preventing subsets of agents from deviating for better outcomes.
• The CS-Core has broad applications in transferable-utility games, hedonic coalition formation, network flows, and distributed dynamics in multiagent systems.

The coalition-structure core (CS-Core) is a central solution concept in cooperative game theory underpinning coalition formation, stability, and allocation in multiagent systems. The CS-Core extends the classical core by accommodating arbitrary coalition structures (partitions) of players, ensuring that no subset of agents has an incentive to deviate to another coalition under the value they could jointly achieve. Its mathematical formulation, existence conditions, algorithmic properties, and applications span transferable-utility (TU) games, hedonic coalition formation, and combinatorial structures such as network flows.

1. Formal Definitions and Core Characterizations

Let $N$ denote a finite set of players. In a transferable-utility (TU) coalitional game $(N, v)$, the characteristic function $v : 2^N \rightarrow \mathbb{R}$ assigns worth to every coalition. A coalition structure $\rho = \{S_1, \ldots, S_k\}$ is a partition of $N$ into pairwise-disjoint nonempty subsets whose union is $N$. An allocation $x \in \mathbb{R}^n$ assigns payoff $x_i$ to agent $i$, with $x(S) = \sum_{i \in S} x_i$.

The CS-Core, $\Pi(v)$, consists of all pairs $(x, \rho)$ such that:

1. Coalitional rationality: $x(S) \geq v(S)$ for all $S \subseteq N$,
2. Internal consistency (within-coalition efficiency): $x(S) = v(S)$ for all $S \in \rho$,

ensuring that no coalition $C \subseteq N$ can improve upon the allocation by forming outside $\rho$ ($x(C) < v(C)$ is not permitted). Every $(x, \rho)$ in the CS-Core satisfies $$x(N) = K_v = \max_{\tau \in \mathcal{P}(N)} \sum_{S \in \tau} v(S)$$, i.e., total welfare is maximized over all partitions (Hamed et al., 2023). In the setting of hedonic games, the CS-Core is realized as the set of all core-stable partitions—a partition is core-stable if no subset of players all strictly prefer to form a new coalition over their current assignment (Woeginger, 2012).

2. Existence, Characterizations, and Relationships

The non-emptiness of the CS-Core is not guaranteed in general cooperative or hedonic games. In TU coalitional games, emptiness of $\Pi(v)$ means that for any feasible partition and allocation, some coalition consistently has incentive to block (Hamed et al., 2023). Explicit examples illustrate that the CS-Core may be empty even for small instances; e.g., a 3-player hedonic game with cyclical preferences (each player prefers to pair, but every partition is blocked) (Woeginger, 2012).

However, specific sufficient conditions for non-emptiness have been established in literature:

• Common Ranking Property: If a shared ranking function $f:\ 2^N\to\mathbb{R}$ exists so that each player's preference is consistent with $f$, the CS-Core is non-empty.
• Friend-Oriented or Enemy-Oriented Preferences: Structured preferences where each agent classifies others as friends/enemies guarantee a non-empty CS-Core via component analysis in the friendship graph.
• Max-Preferences Without Ties: When players rank coalitions by their most preferred member's utility, a top-trading-cycle argument ensures core-stable partitions (Woeginger, 2012).

In path cooperative games (PC-games) and flow-based cooperative games, the CS-Core is always non-empty and coincides with the cut-polytope of the associated flow-game (Fang et al., 2015). Here, the flow-game core is the convex hull of minimum $s$-$t$ cuts in the network.

3. Linear Programming, Polyhedral Structure, and Algorithmics

The CS-Core for flow-based TU games such as edge-path cooperative games (EPC) is fully characterized via linear programming formulations. Specifically, the core polyhedron is specified by:

$\begin{align*} \sum_{e \in E} x_e & = f^* \ \sum_{e \in P} x_e & \geq 1 \quad \text{for every %%%%26%%%%-%%%%27%%%% path %%%%28%%%%}\ x_e & \geq 0 \quad \forall e \in E \end{align*}$

where $f^*$ is the value of the maximum flow (number of disjoint $s$-$t$ paths). This describes the convex hull of indicator vectors of minimum $s$-$t$ cuts (Fang et al., 2015).

Efficient algorithms:

• Membership testing: To check if $x$ is in the CS-Core, verify $\sum_e x_e = f^*$ and check that the minimum weighted $s$-$t$ cut (computed on the current $x$) is at least 1. Both operations are polynomial via max-flow/min-cut algorithms.
• Vertex computation: Obtain an extreme point of the CS-Core by finding a minimum $s$-$t$ cut and assigning unit value to included edges (or vertices).

Similar efficiency holds for the least-core and nucleolus of such games: computation reduces to single or iterative max-flow/min-cut computations (Fang et al., 2015).

For general TU games, verification (“Given partition $\rho$ and $x$, is $(x, \rho)$ in the CS-Core?”) is in NP; specifically, in polynomial time for succinct encodings (additive, anonymous, friend/enemy-oriented), but existence is $\Sigma_2^p$-complete under additive preferences and NP-complete under several other encodings (Woeginger, 2012).

In hedonic coalition formation, certain special cases are polynomial-time tractable, such as Gale–Shapley for bipartite matchings, Irving’s algorithm for strict preferences, and strongly connected components for friend-oriented preferences (Woeginger, 2012).

4. Distributed Dynamics and Decentralized Convergence

Recent distributed learning dynamics allow self-interested agents to negotiate toward absorption into a CS-Core solution (Hamed et al., 2023). Each agent maintains:

• An aspiration level $a_i(t)$ (estimate of final payoff),
• A current coalition $C_i(t)$.

The dynamics proceed in discrete random-activation rounds, where activated agents propose new coalitions based on current aspirations, perform a feasibility check with characteristic function $v(\cdot)$, and adjust aspirations and coalition membership according to success or failure of coalition formation. Communications are local: exchanging aspirations within candidate coalitions and broadcasting coalition dissolutions/formations.

Key convergence guarantees:

• If the CS-Core is non-empty and discretization aligns with some solution, the Markov chain over joint aspirations and coalition memberships almost surely reaches an absorbing CS-Core allocation.
• The protocol requires $O(n^2)$ messages per round in the worst case, with per-agent computation being trivial (single coalition value and aspiration sum).
• No monotonicity or superadditivity assumptions are needed apart from non-emptiness of the CS-Core.
• The convergence time is bounded in expectation relative to the discretization and starting state ($O((D+|\rho^*|)/p)$, with $D$ the grid-based $\ell_1$ distance and $p$ a constant lower bound on desired proposal occurrence).
• If the CS-Core is empty, the system never absorbs—a signal of non-stability (Hamed et al., 2023).

A worked example in multiagent task allocation demonstrates eventual convergence to the unique CS-Core payoff/partition, confirming the protocol’s correctness.

5. Special Cases: Hedonic Coalition Formation

Hedonic games focus on coalition formation where player utilities depend only on the identity of coalition members, not on division of monetary payoffs. The CS-Core here is the set of all core-stable partitions—those unblocked by any coalition whose members all strictly prefer to group together. This setting highlights several critical complexity phenomena:

• The CS-Core can be empty even for small player sets and, in general, is difficult to test for existence (NP-complete or higher in succinct encodings).
• When specific structure exists (friend/enemy orientation, common ranking, max-preferences without ties), efficient algorithms are available for CS-Core partition computation.
• Dynamic processes for coalition improvement (such as iteratively blocking by pairs) may cycle, indicating that naive dynamic stability is elusive (Woeginger, 2012).

Notable open problems include the precise complexity of CS-Core existence under naïve encoding, polynomial-time verifiability for friend-oriented preferences, and combinatorial variants such as partitioning into largest cliques or more complex group structures.

6. Generalizations, Further Solution Concepts, and Extensions

In path cooperative games, the CS-Core, least-core, and nucleolus are tightly linked. The least-core is the rescaling of the cut-polytope by $1/f^*$ (the reciprocal of the max-flow value), enabling efficient polynomial-time computation. The nucleolus coincides with the flow-game nucleolus up to scaling, also solvable in polynomial time through standard algorithms (Fang et al., 2015).

Undirected networks are handled by reductions to the directed case using antiparallel arc replacements and preserving solution concepts. This suggests wide applicability to various network types.

In distributed and hedonic models, the CS-Core provides a unifying solution concept encompassing Pareto optimality, group rationality, and coalition stability. Algorithmic and complexity-theoretic directions continue to be an active area of research, especially in the exploration of new preference structures and succinct encodings.

7. Illustrative Table: Existence and Complexity of the CS-Core in Selected Models

Game Model	CS-Core Existence	Verification Complexity
TU Cooperative (Flow)	Always non-empty for PC-games; cut-polytope equivalence	Polynomial (via max-flow/min-cut)
Hedonic (General)	May be empty	In NP; often polynomial in special cases
Additive Hedonic	May be empty, $\Sigma_2^p$-complete to decide	Polynomial
Friend-Oriented Hedonic	Always non-empty	Open (conjectured polynomial)

This table aggregates model-specific existence and computational properties based on structural and preference assumptions (Fang et al., 2015, Woeginger, 2012, Hamed et al., 2023).

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The coalition-structure core (CS-Core) thus forms a rigorous, broadly applicable stability concept in coalition formation and cooperative allocation, extending across transferable-utility optimization, hedonic grouping, and distributed dynamics. Its interplay with LP-duality, combinatorial optimization, and algorithmic game theory continues to underwrite both foundational results and ongoing open questions.