Overlapping Coalition Formation Game

• Overlapping coalition formation (OCF) games are cooperative models in which agents allocate divisible resources simultaneously across several coalitions while respecting resource endowment limits.
• They extend classical coalition theory by incorporating structured stability notions through arbitration rules—conservative, refined, and optimistic—that guide partial deviation and payoff redistribution.
• OCF frameworks find applications in smartphone sensing, interference management, and UAV logistics, offering practical insights into decentralized resource allocation and efficient coalition stability.

Overlapping coalition formation (OCF) games are cooperative-game models in which agents distribute divisible resources among several coalitions simultaneously, so the outcome is an overlapping coalition structure rather than a partition of the player set. In the canonical formulation, a coalition is a resource vector, an outcome consists of a feasible list of such partial coalitions together with a payoff imputation, and the framework is intended for settings in which one agent can contribute to multiple tasks at once (Chalkiadakis et al., 2014). In communication and multiagent systems, OCF has been used to model decentralized cooperation under power, bandwidth, interference, secrecy, or service constraints, with the survey literature explicitly presenting it as a framework for emerging communication networks (Wang et al., 2015).

1. Formal model and semantic primitives

In the foundational OCF model, the player set is $N=\{1,\dots,n\}$, and each agent $i$ has a divisible resource endowment, written either as $R_i$ or $w_i$ depending on the formulation. A partial coalition is a vector $r=(r_1,\dots,r_n)\in[0,1]^n$ or $\mathbf c=(c^1,\dots,c^n)$ with $0\le r_i\le 1$ or $0\le c^i\le w_i$, and $\mathrm{supp}(r)$ or $\mathrm{supp}(\mathbf c)$ denotes the participating agents. The characteristic function assigns a value to each partial coalition, e.g. $i$0 or $i$1, with the empty coalition worth $i$2 (Chalkiadakis et al., 2014).

An overlapping coalition structure is a finite list of partial coalitions, such as $i$3 or $i$4, subject to a feasibility condition requiring that each agent’s total allocated resource across all coalitions not exceed its endowment. In the generic weighted formulation this is

$i$5

In application-specific models the same idea appears as task- or channel-specific constraints. For smartphone sensing, for example, feasibility is written as

$i$6

so a user can join multiple task-coalitions only within its available wireless feedback rate (Di et al., 2016).

An outcome couples the coalition structure with a payoff allocation. In transferable-utility OCF, each coalition payoff vector $i$7 or $i$8 must satisfy efficiency, meaning that the members’ shares sum to the coalition value, and zero-support, meaning that nonmembers receive zero. Agent $i$9’s total payoff is then the sum over all coalitions it participates in. The superadditive cover $R_i$0 plays a central role: it is the maximum total value a resource bundle can generate after being redistributed among any subcoalitions. This object converts OCF from a purely structural description into a welfare benchmark, because it captures the best value obtainable by arbitrary overlapping decomposition (Zick et al., 2014).

A common misconception is to identify OCF with “multiple memberships” alone. The formal literature is more restrictive. Overlap is meaningful because agents split explicit resources across coalitions, coalition values depend on these allocations, and feasibility is enforced at the level of total contributed weight, power, bandwidth, or similar budgets. Without those ingredients, the model loses the partial-coalition semantics that distinguish OCF from ordinary coalition overlap in informal network analysis (Chalkiadakis et al., 2014).

2. Stability, deviations, and the geometry of the core

Deviation in OCF is more intricate than in classical coalition games because deviators may withdraw only part of their resource from some coalitions while remaining involved in others. The foundational framework therefore introduces arbitration functions to specify how non-deviators react to partial withdrawal. For a deviating set $R_i$1, the key quantity is the best value $R_i$2 can obtain under a given arbitration rule $R_i$3, denoted $R_i$4. An outcome is $R_i$5-stable iff

$R_i$6

where $R_i$7 is the total current payoff of $R_i$8 (Zick et al., 2014).

Three arbitration rules recur throughout the literature. Under the conservative rule, deviators receive nothing from coalitions they partially leave. Under the refined rule, they keep their old payoff only from untouched coalitions. Under the optimistic rule, non-deviators keep their original shares and the residual value can flow to deviators. These rules induce the $R_i$9-core, $w_i$0-core, and $w_i$1-core, respectively, with the inclusion relation

$w_i$2

The inclusions can be strict (Chalkiadakis et al., 2014).

For the conservative core, the theory is especially sharp. Under monotonicity, boundedness, continuity, and $w_i$3-finiteness, an outcome lies in the $w_i$4-core iff for every coalition $w_i$5,

$w_i$6

This extends the standard core inequalities to overlapping coalitions. A further consequence is that every $w_i$7-core outcome maximizes social welfare, i.e. attains $w_i$8 (Chalkiadakis et al., 2014).

Two additional structural notions generalize classical cooperative-game theory. Balancedness gives a Bondareva–Shapley-type characterization of when a fixed overlapping coalition structure can be supported by a core imputation. Convexity is generalized so that, under continuity, boundedness, and $w_i$9-finiteness, convex OCF games have non-empty $r=(r_1,\dots,r_n)\in[0,1]^n$0-core (Chalkiadakis et al., 2014). These results show that overlap does not merely complicate coalition stability; it also admits a systematic extension of the duality-based and marginal-contribution ideas familiar from non-overlapping cooperative games.

Application papers often replace the classical core with domain-specific stability notions. In collaborative smartphone sensing, the relevant concept is $r=(r_1,\dots,r_n)\in[0,1]^n$1-stability: no user can feasibly transfer, quit, or join in a way that strictly improves its own utility without decreasing the payoffs of affected coalition-mates (Di et al., 2016). In MEC-assisted blockchain, the target is an individually stable overlapping coalition structure under five atomic moves: merge-A, merge-B, split-A, split-B, and leave (Ye et al., 8 Aug 2025). In dynamic AAV logistics, the coalition process is formulated as an exact potential game and convergence is to a Nash-stable equilibrium (Zhou et al., 26 May 2026). This suggests that OCF is best viewed as a family of overlapping-cooperation models whose stability notion is chosen to match the deviation semantics of the application.

3. Computational frontier and tractable subclasses

The computational theory of OCF is shaped by a clear tractability boundary. In the unrestricted case, deciding whether the $r=(r_1,\dots,r_n)\in[0,1]^n$2-core is nonempty is NP- or coNP-hard already for very small games, including instances with only two agents and small integer weights. The tractability study identifies three main sources of hardness: large agent weights $r=(r_1,\dots,r_n)\in[0,1]^n$3, unbounded coalition sizes, and complex interaction structure, formalized through high treewidth (Zick et al., 2014).

Positive results arise when one or more of these sources is restricted. If the game admits only 2-agent coalitions and the interaction graph is a tree, then the optimal social welfare $r=(r_1,\dots,r_n)\in[0,1]^n$4 can be computed by dynamic programming in time $r=(r_1,\dots,r_n)\in[0,1]^n$5, where $r=(r_1,\dots,r_n)\in[0,1]^n$6 bounds the relevant weights. Under the same assumptions and a local arbitration function such as $r=(r_1,\dots,r_n)\in[0,1]^n$7, $r=(r_1,\dots,r_n)\in[0,1]^n$8, or $r=(r_1,\dots,r_n)\in[0,1]^n$9, deviation values and core-membership checks are polynomial-time computable. These results extend from trees to interaction graphs of treewidth $\mathbf c=(c^1,\dots,c^n)$0, with running time $\mathbf c=(c^1,\dots,c^n)$1 (Zick et al., 2014).

A particularly important tractable class is the class of Linear Bottleneck Games (LBGs). In an LBG, each task $\mathbf c=(c^1,\dots,c^n)$2 requires exactly a set of agents $\mathbf c=(c^1,\dots,c^n)$3 and yields payoff

$\mathbf c=(c^1,\dots,c^n)$4

whenever the coalition support is exactly $\mathbf c=(c^1,\dots,c^n)$5, and $\mathbf c=(c^1,\dots,c^n)$6 otherwise. The optimal coalition structure is obtained from a linear program, and the dual variables directly define a payoff division. The resulting outcome is an imputation and lies in the optimistic core. Hence every LBG has a nonempty optimistic core, and a core outcome can be found in polynomial time (Zick et al., 2014).

The survey treatment of wireless OCF adds a complementary algorithmic perspective. Under the optimistic core and a bound $\mathbf c=(c^1,\dots,c^n)$7 on the size of a deviating set, two subclasses are singled out. In $\mathbf c=(c^1,\dots,c^n)$8-coalition OCF games, each player can participate in at most $\mathbf c=(c^1,\dots,c^n)$9 coalitions, so deviation checking reduces to computing a superadditive cover by dynamic programming. In $0\le r_i\le 1$0-task OCF games, coalitions correspond to a fixed set of tasks, and deviations are reallocations among predetermined coalitions, which makes deviation checks much simpler (Wang et al., 2015). The broader implication is not that OCF is intrinsically intractable, but that tractability depends sharply on how overlap, interaction structure, and deviation freedom are parametrized.

4. Distributed coalition formation and equilibrium computation

Much of the applied OCF literature studies distributed formation rather than direct core computation. A recurring template is local-improvement dynamics over a finite space of overlapping structures, with convergence ensured by a monotonic potential-like quantity such as user utility, total utility, or social welfare.

In smartphone sensing, the distributed OCF algorithm has two phases. Phase 1 allocates subcarriers either randomly or by a priority order based on descending $0\le r_i\le 1$1. Phase 2 initializes each user by ranking tasks according to $0\le r_i\le 1$2, then iteratively applies feasible transfer, quit, and join operations until a $0\le r_i\le 1$3-stable overlapping coalition structure is reached. Theorem 2 establishes finite convergence because the number of distinct OCSs is finite, every permitted move strictly raises some user’s utility, and no other user’s utility decreases. The worst-case per-iteration complexity is $0\le r_i\le 1$4, and simulation reports convergence in 20–80 iterations even for $0\le r_i\le 1$5 up to 100 (Di et al., 2016).

In hierarchical HetNet spectrum sharing, OCF is embedded in a Stackelberg game. At the lower tier, unlicensed users choose sub-band memberships and powers; at the upper tier, the macro-cell operator updates prices. For fixed prices and payoff-division factors, each unlicensed user solves a constrained utility maximization problem whose KKT solution is a water-filling-style power allocation,

$0\le r_i\le 1$6

The paper states that the OCF core is non-empty, gives a distributed sub-band allocation scheme, and proves existence, uniqueness, and optimality of the Stackelberg equilibrium (Yuan et al., 2015).

In MEC-assisted blockchain, coalition formation is part of a two-stage Stackelberg architecture. Given the edge-computing price, mobile users first update overlapping memberships through one of five atomic moves, and then coalitions solve an edge resource competition game with a closed-form best response and a unique integer Nash equilibrium after rounding. The coalition-formation phase converges in finitely many steps to an individually stable structure because every accepted move improves one user’s utility without harming others and the number of overlapping coalition structures is finite (Ye et al., 8 Aug 2025).

The survey viewpoint under the optimistic core gives a unifying explanation for why such dynamics terminate. Any $0\le r_i\le 1$7-profitable deviation strictly increases total social welfare $0\le r_i\le 1$8. Since $0\le r_i\le 1$9 is bounded above, iterative $0\le c^i\le w_i$0-profitable moves terminate in finitely many steps at an $0\le c^i\le w_i$1-stable outcome (Wang et al., 2015). In practice, many domain-specific algorithms operationalize this principle through admissible local moves and explicit feasibility checks.

5. Communication, sensing, and security applications

The engineering literature uses OCF when cooperative groups are task-specific, resource-bounded, and naturally overlapping. The same mathematical idea has been instantiated with transferable utility, nontransferable utility, and mixed hierarchical models.

Domain	OCF representation	Reported outcome
Smartphone sensing (Di et al., 2016)	One coalition per sensing task; users join multiple tasks under wireless feedback-rate constraints	20–30 % higher platform utility and sensing performance than the non-cooperative benchmark; within 5–10 % of the centralized upper bound
Small-cell interference management (Zhang et al., 2014)	SBSs split OFDMA subcarriers among multiple coalitions	Up to 32% gain over noncooperative and $0\le c^i\le w_i$2 over disjoint coalition formation at $0\le c^i\le w_i$3
Distributed cooperative sensing (Wang et al., 2014)	Each SU may belong to multiple “report-from” coalitions	Up to 25 % reduction in total error probability, up to 20 % reduction in missed detection probability, overhead up to 80 %, and total report number up to 10 %
Two-tier UAV secrecy (Xu et al., 2020)	A UT may relay for several source UTs across different time slots	$0\le c^i\le w_i$4 improvement over FGS, $0\le c^i\le w_i$5 over DCS, and more than 50% over AS

In collaborative smartphone sensing, OCF addresses two limitations of non-cooperative incentives identified in the paper: time-varying wireless channels and omission of channel-resource constraints. Coalition value is tied to task performance $0\le c^i\le w_i$6, which increases linearly with total contribution until a threshold $0\le c^i\le w_i$7 and then saturates at $0\le c^i\le w_i$8. Coalition payoffs are divided proportionally to individual contributions, and user utility subtracts the per-task feedback cost $0\le c^i\le w_i$9 (Di et al., 2016).

In small-cell interference management, the coalition value depends on the entire overlapping structure because one coalition’s transmissions create co-tier interference for others. The paper explicitly characterizes this as negative externalities. The game is non-superadditive, and stability is defined through single resource-unit reallocations that must improve both the moving SBS’s payoff and the global value without hurting the receiving coalition (Zhang et al., 2014).

In distributed cooperative sensing for cognitive radio, the coalition value is a concave function of coalition size under both the $\mathrm{supp}(r)$0 and $\mathrm{supp}(r)$1 criteria. This concavity is operationally important: the paper argues that distributing small amounts of reporting power and bandwidth across several moderate-size coalitions can dominate committing all resources to one coalition, because marginal gains diminish with coalition size (Wang et al., 2014). This is a direct structural rationale for overlap rather than a purely empirical observation.

In two-tier UAV networks, the OCF stage is explicitly nontransferable-utility. A coalition is the relay set helping a source UT in its time slot, and a UT’s utility is the positive part of cooperative secrecy gain minus broadcast-phase secrecy loss, with infeasible coalitions assigned $\mathrm{supp}(r)$2. The resulting distributed Quit/Join/Switch algorithm converges in finite rounds with probability 1 to a stable overlapping structure (Xu et al., 2020). Together, these examples show that OCF is not tied to a single utility model; it functions as a structural language for resource splitting across overlapping cooperative groups.

6. Learning-based and dynamic extensions

Later work extends OCF beyond settings where coalition values are known and the environment is static. One line of research assumes that coalitional values are induced by latent relational structure. “Overlapping Coalition Formation via Probabilistic Topic Modeling” introduces Relational Rules, which extend MC-nets to overlapping settings by allowing rule contributions to scale with the fractional investment $\mathrm{supp}(r)$3 of participating agents. Because the rules are unknown to the agents, coalition outcomes are encoded as documents and online Latent Dirichlet Allocation is used to learn profitable collaboration patterns. In the reported experiments with $\mathrm{supp}(r)$4, the best OVERPRO configuration achieves more than twice the welfare of the best baseline, while maintaining practical per-step runtime (Mamakos et al., 2018).

A second line studies dynamic OCF under stochastic task arrivals. In heterogeneous AAV logistics, the OCF process is coupled to a generalized logistics cost

$\mathrm{supp}(r)$5

where each task cost combines payload deficiency, service timeliness, and operational cost. The paper proves that the coalition formation process is an exact potential game by showing that a unilateral cost change equals the change in a global potential $\mathrm{supp}(r)$6, which yields the finite-improvement property and convergence to a Nash-stable equilibrium. A transformer-based Soft Actor-Critic policy then replaces heuristic coalition updates. In a scenario with 32 AAVs and 80 tasks, the algorithm achieves a 39.76% cost reduction compared with the heuristic OCF baseline, and indoor flight experiments are reported to validate practicality (Zhou et al., 26 May 2026).

These extensions shift OCF from static equilibrium analysis toward adaptive decision-making. The survey literature already identified multi-dimensional resources, temporal dynamics, incentive compatibility, and richer arbitration functions as open challenges (Wang et al., 2015). The more recent learning-based papers suggest a concrete methodological response: preserve the overlapping-coalition semantics, but learn either the coalition values, the profitable move structure, or both from repeated interaction and environmental feedback.