Global Coalition: Interdisciplinary Insights

Updated 12 July 2026

Global Coalition is a polysemous concept defined by whole-system criteria across diverse research domains such as game theory and graph theory.
It encompasses varying formalizations, from coalition strategies in safety games to weighted agreement structures, reflecting practical applications and optimization challenges.
Recent studies emphasize dynamic coalition models that integrate uncertainty, internal rivalry, and distributed learning to enhance global stability.

“Global coalition” is a polysemous technical term used across game theory, graph theory, multi-agent systems, governance, and coalition operations. In some settings it denotes a coalition strategy that must work for every possible number of agents; in others it denotes a system-wide coalition structure over all agents or agreements, the grand coalition containing all participants, or a graph-theoretic pair of sets whose union satisfies a global domination property. The unifying theme is that coalition quality is evaluated against a whole-system criterion rather than a purely local interaction rule (Bertrand et al., 2020, Kulkarni et al., 22 Feb 2025, Besharati et al., 18 Sep 2025).

1. Core meanings across research domains

Several distinct formalizations are currently used.

Domain	Formal object	Global criterion
Parameterized concurrent games	$\stratsys:\Hist \to \Sigma^\omega$	Safety for every agent count
Diplomacy coalition detection	$C=(N,E,Agmt,wt)$	Weighted multilateral agreement structure
Coalition structure generation	Partition $CS$ of all agents	Maximize total social welfare
Graph theory	$gc$ -partition of $V(G)$	Union must dominate $G$ and $\overline G$

In parameterized concurrent games, a coalition is not fixed to a known cardinality. A coalition strategy maps each history to an infinite word, whose $n$ -th letter is the action of agent $n$ , and it is winning when every play compatible with every possible number of agents remains in the safe set (Bertrand et al., 2020).

In natural-language negotiation settings such as Diplomacy, a coalition structure is modeled not as a partition but as an undirected multigraph with parallel edges,

$C=(N,E,Agmt,wt),$

where each edge corresponds to an agreement and the weight records how likely that agreement is to be honored. This formalism is expressly designed for overlapping, private, and bilateral agreements that jointly induce a multilateral structure (Kulkarni et al., 22 Feb 2025).

In coalition structure generation, the relevant global object is a partition of all agents. For induced subgraph games, the target is

$C=(N,E,Agmt,wt)$ 0

so “global” refers to optimization over the full decomposition of the population, not merely the value of a single coalition (Venkatesh et al., 2022).

In graph theory, two disjoint subsets form a global coalition if neither is a global dominating set by itself but their union is one. A global coalition partition requires every part of the vertex partition to have such a partner, and the associated invariant is the global coalition number $C=(N,E,Agmt,wt)$ 1 (Besharati et al., 18 Sep 2025).

2. Arbitrary-large coalition strategies and global safety

A particularly stringent meaning of global coalition arises in parameterized concurrent games with an arbitrary number of agents. The arena is

$C=(N,E,Agmt,wt)$ 2

where $C=(N,E,Agmt,wt)$ 3 is finite, $C=(N,E,Agmt,wt)$ 4 is a finite action alphabet, and $C=(N,E,Agmt,wt)$ 5 is a regular language describing the joint moves that permit transition from $C=(N,E,Agmt,wt)$ 6 to $C=(N,E,Agmt,wt)$ 7. If there are $C=(N,E,Agmt,wt)$ 8 agents, then a word in $C=(N,E,Agmt,wt)$ 9 encodes one joint move, one symbol per agent. Completeness is central because the game must never block for any positive agent count (Bertrand et al., 2020).

The coalition-strategy formulation replaces an infinite tuple of individual strategies by a single map

$CS$ 0

For a safety game $CS$ 1, the coalition is winning exactly when

$CS$ 2

that is, when every play consistent with the strategy remains forever in safe vertices for every possible number of agents. The nontriviality comes from uniformity: the environment determines the hidden agent count, while the coalition strategy must be correct for all counts simultaneously (Bertrand et al., 2020).

The principal decidability theorem states that the safety coalition problem can be solved in exponential space and is PSPACE-hard. Positive instances admit synthesis of a winning coalition strategy in exponential space, and the resulting strategy may require exponential memory in the worst case. The upper bound is obtained by reducing the original graph arena to a finite tree unfolding whose branches stop when a vertex repeats or an unsafe vertex is reached; the height is at most $CS$ 3. A key technical device is the $CS$ 4 function, which collapses loops in histories to a virtual history with no repeated vertex. Winning on the original arena is equivalent to winning on the unfolding tree (Bertrand et al., 2020).

The tree problem is then encoded by a deterministic safety automaton over $CS$ 5, where $CS$ 6 is the number of internal tree nodes. Acceptance is specified branch-by-branch by a Boolean condition ensuring that whenever the edge languages along a maximal path are satisfied for some agent count, the leaf reached is safe. Although the automaton is doubly exponential in the arena size, it is not built explicitly; instead, the procedure guesses a safe prefix and a safe lasso and checks the acceptance formula, yielding a nondeterministic exponential-space algorithm and hence an EXPSPACE procedure via Savitch’s theorem (Bertrand et al., 2020).

The lower bound is established by reduction from QBF-SAT. The construction uses prime numbers, sets of multiples, and regular languages such as $CS$ 7 and $CS$ 8 so that the coalition can infer quantified-variable truth values from the hidden agent count. The result is specific to safety objectives; the paper explicitly notes that the tree-unfolding argument does not directly generalize to reachability or more general $CS$ 9-regular objectives (Bertrand et al., 2020).

3. Welfare, stability, and the grand coalition

A second major line of work studies global coalitions as full-population partitions or as the grand coalition. In induced subgraph games, a coalition $gc$ 0 is feasible when it induces a connected subgraph, and its value is

$gc$ 1

GCS-Q approaches the global optimization problem top-down: it starts from the grand coalition and recursively solves an optimal bipartition subproblem, accepting a split only when it increases coalition-structure value. The abstract states that, for an $gc$ 2-agent induced subgraph game, the method solves the optimal split problem $gc$ 3 times using quantum annealing, explores $gc$ 4 partitions at each step, has runtime in the order of $gc$ 5, and achieves an expected worst-case approximation ratio of $gc$ 6 on standard benchmark datasets (Venkatesh et al., 2022).

Global optimization and global stability are distinct issues. In sharing-based coalition formation, a rule $gc$ 7 induces agents’ coalition rankings through the allocations they receive. The paper “Solidarity to achieve stability” proves that solidarity is equivalent to endowment monotonicity plus consistency, and also characterizes exactly the induced coalition formation problems that are non-circular. Since non-circularity implies a non-empty core, solidarity is sufficient to guarantee stable coalition structures for every induced problem. The result applies across bargaining, rationing, surplus-sharing, and ranking problems, and shows that stability can be derived from the structure of coalition-wide redistribution rather than from an exogenous stability postulate (Alcalde-Unzu et al., 2023).

The grand coalition is especially prominent in communication networks. In cooperative wireless networks, the user set is $gc$ 8, and the grand coalition is $gc$ 9 itself. Stability depends on both the communication model and whether utility is transferable. When receivers cooperate by jointly decoding in a transferable-utility setting, the game is superadditive and cohesive, the grand coalition is sum-rate optimal, and the core is non-empty. Under linear multiuser detectors, the conclusion becomes detector-dependent: with MMSE the grand coalition is always stable and sum-rate maximizing, whereas with the decorrelator it is stable only in the high-SNR regime. For transmitter cooperation, stability depends on channel gains and jamming strengths, and the grand coalition may fail to be stable even when it is sum-rate optimal (0804.3421).

The contrast between a single inclusive coalition and multiple coalitions also appears in public-goods governance. In the model of coalition-structured governance, a single coalition without enforcement and with uninformed self-interested players collapses to the stable state $V(G)$ 0, with no coalition and no contributions. By contrast, multiple overlapping coalitions can let uninformed players recognize marginal gains of cooperation, improving cooperation across diverse levels of excludability, congestability, and return-to-contribution functions. This does not imply that a single global coalition is always inferior, but it does show that global welfare and global participation may be sustained by polycentric rather than monolithic structures (Vasconcelos et al., 2019).

4. Dynamic global coalitions under uncertainty and internal rivalry

Recent work increasingly treats global coalitions as dynamic objects whose membership, value, and stability evolve with state variables, shocks, and internal conflict. In climate-club modeling, coalition membership is state-contingent in a continuous-time dynamic game with heterogeneous regions. Temperatures evolve via cumulative emissions, tipping risk is governed by a temperature-dependent hazard,

$V(G)$ 1

and coalition stability is defined by internal and external stability inequalities. The main result is that coalitions tend to shrink over time as temperatures rise because free-riding incentives eventually dominate; the threat of tipping reduces stable coalition size relative to the no-tipping case; after a tipping event, coalitions temporarily expand and then shrink again; and technology-sharing yields larger stable coalitions and greater collective benefits than sanctions in heterogeneous numerical simulations (Zhu et al., 19 Jun 2025).

A different dynamic perspective is offered by the Compound Coalition-Attrition Game. There, coalitions are mutually exclusive and collectively exhaustive subsets of the player set $V(G)$ 2, but coalition power is endogenous: $V(G)$ 3 External rivalry between coalitions and internal competition among coalition members are modeled simultaneously. A player’s expected payoff combines the coalition’s chance of winning externally with the player’s internal share of the prize. The paper states that no pure-strategy Nash equilibrium exists in the intra-coalition attrition game and characterizes a unique mixed-strategy Nash equilibrium in the abstract. Its central substantive claim is a two-way feedback: external competition intensifies internal conflict, while internal discord weakens external performance (Gordji et al., 4 May 2026).

Multiscale models make this feedback explicit. In the exit-join framework, the coalition structure $V(G)$ 4 evolves on a slow strategic timescale, while each coalition runs a fast DeGroot consensus process

$V(G)$ 5

until it approaches a coalition consensus $V(G)$ 6. Coalition value is then generated endogenously by

$V(G)$ 7

Strategic movement depends on Aumann-Dreze payoffs, switching frictions, and acceptance rules such as Pareto non-decrease for incumbents. The paper proves a fixed-point characterization of joint tactical-strategic equilibrium and derives conditions for tactical unanimity, strategic unanimity, segregation, polarization, and cognitive barriers. Its numerical experiments identify an instability-consensus paradox: low or negative switching barriers can prevent convergence of the coalition partition while still creating enough temporal mixing to produce global tactical consensus (Zhu, 19 Jun 2026).

Taken together, these models replace the static notion of a global coalition with a state-dependent object. A plausible implication is that “global” increasingly refers not only to cardinality or coverage, but also to the coupling between coalition structure and system dynamics.

5. Detection, learning, and operational infrastructures

In operational multi-agent settings, a global coalition may need to be inferred, learned, or administered rather than solved for analytically. In Diplomacy, the central difficulty is that coalition-relevant agreements are private, ambiguous, bilateral, and potentially overlapping. The proposed method first extracts agreements discussed in dialogue using a parsing-based filter plus CICERO’s 2.7-billion-parameter intent model, then scores each agreement with a hypergame-theoretic rationalizability metric. For player $V(G)$ 8, the score is

$V(G)$ 9

and the observer’s score is

$G$ 0

Empirically, the hybrid agreement detector achieves F1 $G$ 1, versus $G$ 2 for the intent-distribution classifier and $G$ 3 for GPT-4o alone. On hand-labeled agreements, honored deals obtain top-1 and top-5 MRRs of $G$ 4 and $G$ 5, while violated deals obtain $G$ 6 and $G$ 7; analogous separations appear on the hybrid-labeled set. The broader conclusion is that the relevant global object is a weighted dynamic coalition multigraph, not a static alliance partition (Kulkarni et al., 22 Feb 2025).

Coalition operations also motivate distributed learning architectures. In federated learning for coalition operations, partners may collaborate either by data sharing or by model sharing. Data and models are filtered by policies keyed to trust, source identity, format, labels, QoI, and VoI. The paper defines quality of information as $G$ 8 and value of information as $G$ 9. It also discusses performance improvement in terms of effective clean data size and identifies major unresolved issues, including architecture mismatch, heterogeneous label spaces, incomplete knowledge, and federation beyond learning into inference and action phases. The emphasis is that coalition-wide AI is a policy-governed collaboration pattern rather than a single federated algorithm (Verma et al., 2019).

Administrative and security infrastructures for coalitions are treated in work on Federated Coalition Networks. The proposed framework uses a three-tier hierarchy—Coalition Layer, National Layer, and Tactical Layer—and focuses on infrastructure management, especially policy enforcement and PKI management. The proof of concept on Avalanche Testnet implements functions for policy creation, retrieval, modification, authorization, and denial, as well as public-key publication, update, retrieval, revocation, and authorization. The reported evaluation is functional rather than performance-heavy: authorized access succeeds, unauthorized access is blocked, transaction records are securely encrypted, and blockchain validation supports availability and integrity; no detailed latency, throughput, gas-cost, or scalability benchmarks are reported (González et al., 12 Mar 2025).

These strands show that, in deployed systems, global coalition research has shifted from equilibrium existence alone to the full stack of extraction, valuation, learning, trust management, and secure orchestration.

6. Graphs, alliances, and structural conditions

In graph-theoretic and network-structural research, global coalition is defined directly in terms of graph properties. For a graph $\overline G$ 0, a set $\overline G$ 1 is a global dominating set if it dominates both $\overline G$ 2 and its complement $\overline G$ 3. Two disjoint sets form a global coalition when neither is globally dominating but their union is. A $\overline G$ 4-partition is a partition of $\overline G$ 5 in which every part has such a partner, and the maximum partition size is the global coalition number $\overline G$ 6. The paper proves that every graph admits a $\overline G$ 7-partition, establishes $\overline G$ 8, and derives a general upper bound on the number of global coalitions in which a partition class $\overline G$ 9 can participate: $n$ 0 It also proves that if $n$ 1, then $n$ 2, linking the new invariant to the classical coalition number. Exact values are obtained for paths, cycles, and extensive families of unicyclic graphs, and comparisons are made with perfect coalition partitions (Besharati et al., 18 Sep 2025).

A different structural approach models countries as spins $n$ 3, with bilateral propensities $n$ 4 and alliance-induced external fields. The effective propensity becomes

$n$ 5

and coalition stability is governed by sign consistency on cycles: instability is associated with a closed circle whose product of effective bonds is negative, while stability in the one-factor alliance case requires

$n$ 6

for every cycle $n$ 7. The model shows how global alliances can stabilize an unstable network of historical relations by reweighting local interactions, while multi-factor superposition can either stabilize or destabilize depending on the resulting cycle structure (Vinogradova et al., 2013).

These formalisms place “global” at two different levels. In graph domination, the adjective refers to simultaneous control in $n$ 8 and $n$ 9. In alliance-spin models, it refers to external fields that act across the full system and alter local couplings. This suggests that a global coalition can be characterized either by a global property of a set union or by a global reparameterization of the interaction network.

7. Conceptual synthesis and limitations

Across the literatures surveyed here, at least four recurring motifs are visible. First, global coalitions are often defined by universal quantification: every agent, every possible count, every vertex, or every relevant agreement must be covered. Second, the global coalition is frequently not identical to the grand coalition; Diplomacy models use weighted multigraphs rather than partitions, and public-goods models can favor multiple overlapping coalitions over a single inclusive one (Kulkarni et al., 22 Feb 2025, Vasconcelos et al., 2019). Third, coalition value is increasingly endogenous, generated by consensus, strategic interaction, communication, or technology transfer rather than treated as fixed data (Zhu, 19 Jun 2026, Zhu et al., 19 Jun 2025). Fourth, stability is highly model-dependent: it can follow from solidarity in sharing rules, fail for the grand coalition under unfavorable wireless channel conditions, or require satisfaction of cycle conditions in alliance networks (Alcalde-Unzu et al., 2023, 0804.3421, Vinogradova et al., 2013).

The limitations are equally domain-specific. The synthesis result for arbitrary-large coalitions is presently proved only for safety objectives (Bertrand et al., 2020). The blockchain-based FCN framework provides feasibility evidence but not a quantitative performance characterization (González et al., 12 Mar 2025). Coalition detection from language remains constrained by ambiguity, private information, and subjective beliefs (Kulkarni et al., 22 Feb 2025). Federated coalition learning still leaves architecture mismatch and applicability-aware model fusion as open issues (Verma et al., 2019). In dynamic coalition models, the long-run persistence of broad cooperation is repeatedly challenged by free-riding, switching frictions, or internal rivalry (Zhu et al., 19 Jun 2025, Gordji et al., 4 May 2026).

For contemporary research, “global coalition” therefore denotes not one object but a family of whole-system coalition concepts. Their common concern is the same: how coordinated subsets of agents can be defined, detected, optimized, stabilized, or synthesized when the relevant criterion is global rather than local.