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Exit-and-Join Dynamics and Equilibrium in Continuum Cooperative Games

Quanyan Zhu

Year: 2026Venue: arXiv preprintArea: cs.GTType: PreprintEmbeddings: 99

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Summary

This paper develops a continuum theory for exit-and-join coalition dynamics in nonatomic cooperative games. It extends the Aumann-Shapley and Aumann-Drèze values to define payoff densities that drive agent mobility between coalitions. By deriving deterministic mean-field dynamics from decentralized switching rules, the authors show that payoff-difference switching recovers replicator dynamics. The paper characterizes equilibrium through the absence of profitable deviations, proving its equivalence to Wardrop equilibrium and stationarity of mass dynamics. It establishes global convergence via a Lyapunov function, connects the framework to variational inequalities, and extends the model to incorporate switching costs and endogenous acceptance constraints, unifying cooperative value allocation, noncooperative mobility, and evolutionary game theory for large-scale multi-agent systems.

Entities (11)

New York University Tandon School of Engineering · organization · 99%Quanyan Zhu · person · 99%Aumann-Shapley Value · concept · 98%Exit-and-Join Dynamics · process · 97%Aumann-Drèze Value · concept · 95%Continuum Cooperative Games · concept · 95%Wardrop Equilibrium · concept · 94%Replicator dynamics · process · 93%Mean-Field Dynamics · process · 92%Variational Inequalities · mathematical-tool · 90%Lyapunov Function · mathematical-tool · 88%

Relation Signals (8)

Quanyan Zhu → affiliatedwith → New York University Tandon School of Engineering

confidence 99% · Department of Electrical and Computer Engineering New York University Tandon School of Engineering

Quanyan Zhu → authored → Exit-and-Join Dynamics and Equilibrium in Continuum Cooperative Games

confidence 99% · Quanyan Zhu Department of Electrical and Computer Engineering New York University Tandon School of Engineering

Exit-and-Join Equilibrium → equivalentto → Wardrop Equilibrium

confidence 96% · We further show that the equilibrium is equivalent to a Wardrop equilibrium of an induced nonatomic population game and admits a variational inequality formulation.

Aumann-Shapley Value → extends → Classical Shapley Value

confidence 95% · The key insight of Aumann and Shapley is that, in a nonatomic setting, discrete insertions can be replaced by a continuous participation process.

Exit-and-Join Dynamics → recovers → Replicator dynamics

confidence 94% · show that payoff-difference switching recovers replicator dynamics as a special case.

Aumann-Drèze Value → extends → Aumann-Shapley Value

confidence 92% · We define the continuum Aumann-Drèze value by restricting the game to each coalition and applying the Aumann-Shapley construction inside the restricted nonatomic measure space.

Exit-and-Join Equilibrium → characterizedby → Variational Inequalities

confidence 90% · admits a variational inequality formulation. The framework is extended to incorporate switching costs and endogenous coalition acceptance rules, leading to constrained equilibria characterized by quasi-variational inequalities.

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Abstract

Abstract:This paper develops a continuum theory of exit-and-join coalition dynamics in nonatomic cooperative games. We extend the Aumann-Shapley value and the Aumann-Drèze value to coalition structures in which each coalition is treated as a restricted nonatomic game, yielding a marginal-contribution-based payoff density that governs incentives for agents to remain in, exit, or join coalitions. We derive deterministic mean-field dynamics from decentralized switching rules and show that payoff-difference switching recovers replicator dynamics as a special case. We characterize exit-and-join equilibrium by the absence of profitable positive-mass deviations and prove its equivalence with stationarity of the induced mass dynamics under incentive-compatible and strictly payoff-responsive switching rates. For mass-based cooperative games, we construct a Lyapunov function and establish global convergence under strict concavity. We further show that the equilibrium is equivalent to a Wardrop equilibrium of an induced nonatomic population game and admits a variational inequality formulation. The framework is extended to incorporate switching costs and endogenous coalition acceptance rules, leading to constrained equilibria characterized by quasi-variational inequalities. The proposed theory unifies cooperative value allocation, noncooperative coalition mobility, mean-field dynamics, evolutionary game theory, and population games within a common framework for analyzing coalition formation and adaptation in large-scale multi-agent systems.

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EXIT-AND-JOIN DYNAMICS AND EQUILIBRIUM IN CONTINUUM COOPERATIVE GAMES Quanyan Zhu Department of Electrical and Computer Engineering New York University Tandon School of Engineering Brooklyn, NY, USA quanyan.zhu@nyu.edu ABSTRACT This paper develops a continuum theory of exit-and-join coalition dynamics in nonatomic cooper- ative games. We first extend the Aumann-Shapley value and the Aumann-Drèze value to coalition structures in which each coalition is a nonatomic restricted game. The resulting payoff density gives a marginal-contribution-based incentive for agents to remain in, exit, or join coalitions. We then derive deterministic mean-field dynamics from decentralized switching rules and show that payoff- difference switching includes replicator dynamics as a special case. The associated exit-and-join equilibrium is characterized by the absence of profitable positive-mass deviations and is equivalent to stationarity of the induced mass dynamics under incentive-compatible and strictly payoff-responsive switching rates. For mass-based cooperative games, we construct a Lyapunov function and obtain global convergence under strict concavity. We further connect the equilibrium concept to Wardrop equilibria and variational inequalities, and we extend the model to switching costs and endogenous acceptance constraints. The framework links cooperative value allocation, noncooperative mobility, and evolutionary selection in large-population coalition systems. Keywords nonatomic cooperative games·Aumann-Shapley value·exit-and-join dynamics·mean-field limits· Wardrop equilibrium· evolutionary selection 1 Introduction Cooperative game theory studies how groups of agents create value and how that value should be allocated among participants. The classical Shapley value [1, 2] provides a canonical allocation rule for finite-player games by averaging marginal contributions over all orders of entry. In large populations, however, individual agents are negligible. A single agent has zero measure and cannot create a discrete jump in coalition value. The Aumann-Shapley value replaces finite permutations by infinitesimal marginal contributions along continuous participation paths [3, 4]. Coalition structures add a second layer to this problem. In many economic, organizational, and multi-agent systems, agents are not arranged in one grand coalition but are distributed across several coalitions. The Aumann-Drèze value [5, 6] addresses this issue in finite games by applying the Shapley principle within each coalition of a fixed coalition structure. The first objective of this paper is to develop the corresponding construction for nonatomic cooperative games. Each coalition is treated as a restricted nonatomic game, and the payoff assigned to an agent is its Aumann-Shapley marginal contribution within that coalition. The second objective is dynamic. Coalition structures are rarely fixed in applications: workers move across firms, agents reallocate across tasks, users migrate across platforms, and alliances form or dissolve in response to changing incentives. We therefore introduce exit-and-join dynamics in which agents compare the payoff density in their current coalition with the payoff densities available elsewhere. A positive-mass flow from one coalition to another occurs only when the destination offers a strict payoff improvement, possibly subject to switching costs or acceptance constraints. Since the player space is nonatomic, the primitive individual switches aggregate into deterministic mean-field dynamics for coalition masses. arXiv:2606.28824v1 [cs.GT] 27 Jun 2026 Exit-and-Join Dynamics in Continuum Cooperative Games The resulting model connects three viewpoints. First, it is cooperative: the payoff field is generated by a transferable- utility cooperative game and by continuum Aumann-Drèze values. Second, it is noncooperative: agents move unilaterally whenever another coalition offers a better payoff. Third, it is evolutionary: mass grows in coalitions with higher payoff and shrinks in coalitions with lower payoff, yielding replicator-type dynamics under standard payoff-difference switching rules. Main contributions. The paper makes the following contributions. (1) We formulate nonatomic cooperative games, participation paths, marginal contribution densities, and the Aumann-Shapley value in a form suited for coalition structures. (2)We define the continuum Aumann-Drèze value by restricting the game to each coalition and applying the Aumann-Shapley construction inside the restricted nonatomic measure space. (3) We derive exit-and-join mass dynamics from decentralized switching intensities and show that payoff-difference switching recovers the replicator equation. (4)We characterize exit-and-join equilibrium and prove its equivalence with stationarity of the dynamics under incentive-compatible and strictly payoff-responsive switching rates. (5)For mass-based games, we construct a Lyapunov function and establish global convergence under strict concavity. (6)We show that the equilibrium condition coincides with Wardrop equilibrium in the induced nonatomic population game, and we give the corresponding variational inequality formulation. (7) We extend the framework to switching costs and endogenous acceptance rules, which produce state-dependent feasible deviation cones and a quasi-variational inequality structure. (8)We interpret exit-and-join dynamics as an evolutionary selection process over cooperative coalition structures. Related work. The paper builds on the classical Shapley value [1] and its nonatomic extension by Aumann and Shapley [3, 4]. Coalition structures and coalitional values trace back to Aumann and Drèze [5, 6]. Dynamic cooperative games and coalition formation have been studied in several directions, including dynamic cooperative games [7, 8], coalition formation processes [9,10], and computational or learning-based approaches [11–13]. The population-dynamic part of the paper is related to evolutionary game theory and population games [14–19]. The Wardrop-equilibrium component is also close to traffic and security models that study adversarial perturbations and non-equilibrium learning in congestion networks [20, 21]. The mean-field derivation is aligned with classical convergence ideas for Markov processes [22]. Organization. Section 2 develops the continuum Aumann-Shapley and Aumann-Drèze values. Section 3 derives the exit-and-join dynamics and the equilibrium-stationarity connection. Section 5 relates the model to Wardrop equilibrium and variational inequalities. Section 6 introduces switching costs and acceptance constraints. Section 8 discusses the relationship between cooperative value creation and noncooperative mobility. Section 9 gives an evolutionary interpretation, and Section 10 concludes the paper. 2 Continuum Extension of the Shapley and Aumann-Drèze Values 2.1 Nonatomic Cooperative Games and the Aumann-Shapley Value Let(I,I,μ)be a nonatomic probability space, whose elements index a continuum of agents. A nonatomic transferable- utility cooperative game is a set functionv : I → Rwithv(∅) = 0, which assigns a real value to each measurable coalition. Coalitions are identified with measurable subsets of I , and all equalities are understood up to μ-null sets. Because the player space is nonatomic, individual agents have zero measure and cannot affect coalition values on their own. As a result, discrete constructions based on permutations of players, central to the classical Shapley value, are no longer meaningful. Instead, solution concepts must be formulated in terms of infinitesimal marginal contributions of population mass. Throughout this section, we assume that the gamevsatisfies sufficient regularity conditions to ensure the existence of well-defined marginal contribution densities. In finite cooperative games, the Shapley value assigns payoffs by averaging a player’s marginal contribution over all permutations of the player set. Each permutation represents a possible order of coalition formation, and the Shapley value measures the expected incremental value created when a player joins the coalition formed by its predecessors. In a continuum of players, however, individual agents are nonatomic and have zero measure. No agent can be meaningfully inserted at a specific position in an ordering, and the permutation-based definition of the Shapley value 2 Exit-and-Join Dynamics in Continuum Cooperative Games does not apply. Nevertheless, the underlying Shapley principle remains meaningful: payoffs should reflect average marginal contributions to coalition value. The key insight of Aumann and Shapley is that, in a nonatomic setting, discrete insertions can be replaced by a continuous participation process. Rather than considering permutations of players, one considers a path along which the measure of participating agents grows continuously from zero to full participation. Marginal contributions are then defined infinitesimally along this path. 2.2 From Participation Paths to the Aumann-Shapley Value In finite cooperative games, the Shapley value assigns to each player the average of its marginal contributions over all permutations of the player set. Each permutation represents a possible order of coalition formation, and a player’s payoff reflects the incremental value created when it joins the coalition formed by its predecessors. In a nonatomic game this logic no longer applies. Individual agents have zero measure,μ(i) = 0for alli∈ I, and therefore cannot generate discrete jumps in coalition value. There is no meaningful notion of inserting a single agent at a specific position in an ordering. The permutation-based construction must be replaced by a continuous analogue. The key idea of Aumann and Shapley is to reinterpret coalition formation as a continuous participation process. Instead of averaging over permutations, we average marginal productivity over participation levels. Definition 2.1. Let(I,I,μ)be a nonatomic probability space withμ(I) = 1. A nonatomic transferable–utility cooperative game [4] is a set functionv :I → Rwithv(∅) = 0, where coalitions are measurable subsets ofI. Two coalitions that differ on a μ–null set are identified. The nonatomic assumption means that value cannot change because of the addition of a single agent. Coalition growth must therefore be modeled at the level of population mass. Definition 2.2. Let S ∈I. A participation path for S is a measurable familyS λ λ∈[0,1] such that S 0 = ∅, S 1 = S, S λ ⊂ S λ ′ whenever λ < λ ′ , and μ(S λ ) = λμ(S). Example 2.3. LetI = [0, 1]equipped with Lebesgue measure and letS ⊂ [0, 1]be a measurable set withμ(S) = α. Define, forλ∈ [0, 1],S λ =i∈ S : i≤ λα. ThenS 0 = ∅andS 1 = S. Moreover, ifλ < λ ′ , thenS λ ⊂ S λ ′ , and μ(S λ ) = λμ(S) = λα. HenceS λ λ∈[0,1] is a participation path for S. A participation path represents the gradual growth of a coalition from empty (λ = 0) to full size (λ = 1). The parameter λ measures the aggregate level of participation. This replaces the discrete insertion of players in the finite case. Along such a path, coalition value evolves continuously. Definition 2.4. Let(I,I,μ)be a nonatomic probability space, letv :I → Rbe a cooperative game, and letS λ λ∈[0,1] be a participation path for a measurable coalition S. Assume that the mapλ 7−→ v(S λ )is absolutely continuous on[0, 1]. Then its derivative exists for almost every λ∈ [0, 1], and the functionλ7−→ d dλ v(S λ )is called the pathwise marginal value of coalition growth at participation level λ. The derivative above measures the instantaneous rate at which value is created when coalition size increases infinitesi- mally. It plays the role of the discrete marginal contribution v(S∪i)− v(S) in finite games. Example 2.5. Let (I,I,μ) be a nonatomic probability space with μ(I) = 1 and suppose the game is mass-based: v(S) = F (μ(S)), F ∈ C 1 ([0, 1]). LetS λ be any participation path for S. Since μ(S λ ) = λμ(S), we have v(S λ ) = F (λμ(S)). By the chain rule, d dλ v(S λ ) = F ′ (λμ(S))μ(S). Thus the pathwise marginal value at levelλequals the marginal productivity of coalition size evaluated at the current coalition mass. To attribute this aggregate marginal value to individual agents, we require that it admits an integral decomposition. 3 Exit-and-Join Dynamics in Continuum Cooperative Games Definition 2.6. The game v admits a marginal contribution density if there exists a measurable function m : I × [0, 1]→ R such that for every measurable coalition S and every participation pathS λ , d dλ v(S λ ) = Z S m(i,λ)dμ(i) for a.e. λ. The functionm(i,λ)represents the marginal productivity of agentiwhen the aggregate participation level isλ. Thus, λcaptures coalition scale;m(i,λ)captures individual productivity at that scale. This is the continuum analogue of assigning discrete marginal increments to individual players. Example 2.7. Continuing the mass–based example, we now identify a marginal contribution density. Since d dλ v(S λ ) = F ′ (λμ(S))μ(S), and since μ(S λ ) = λμ(S), we may rewrite this as d dλ v(S λ ) = F ′ (μ(S λ ))μ(S). Because μ(S) = R S 1dμ(i), we obtain d dλ v(S λ ) = Z S F ′ (μ(S λ ))dμ(i). Thus the function m(i,λ) = F ′ (μ(S λ )) is a marginal contribution density. In particular, the density does not depend on the identity ofibut only on the current coalition mass. Hence the game admits a marginal contribution density, and it is uniform across agents. In the finite Shapley value, payoffs are obtained by averaging marginal contributions over all permutations. In the continuum, “averaging over permutations” becomes “averaging over participation levels.” Definition 2.8. Given a marginal contribution densitym(i,λ), the average marginal contribution of agentiacross all participation levels is defined as Z 1 0 m(i,λ)dλ. This averaging step preserves the core Shapley principle: payoffs reflect average marginal productivity across coalition scales. Definition 2.9. Let(I,I,μ)be a nonatomic probability space withμ(I) = 1, and letv :I → Rbe a cooperative game admitting a marginal contribution density m : I × [0, 1]→ R in the sense that for every measurable coalition S and every participation pathS λ , d dλ v(S λ ) = Z S m(i,λ)dμ(i) for a.e. λ∈ [0, 1]. The Aumann-Shapley value of v is the function φ AS (v)∈ L 1 (I,μ) defined μ–almost everywhere by φ AS i (v) = Z 1 0 m(i,λ)dλ. Theorem 2.10. Suppose v admits a marginal contribution density m. Then the Aumann-Shapley value satisfies Z I φ AS i (v)dμ(i) = v(I). 4 Exit-and-Join Dynamics in Continuum Cooperative Games Proof. LetI λ λ∈[0,1] be a participation path for the grand coalition I . Since μ(I λ ) = λ, we have v(I 1 )− v(I 0 ) = Z 1 0 d dλ v(I λ )dλ. By the defining property of the marginal contribution density, d dλ v(I λ ) = Z I m(i,λ)dμ(i). Hence, v(I) = Z 1 0 Z I m(i,λ)dμ(i)dλ. By Fubini’s theorem, v(I) = Z I Z 1 0 m(i,λ)dλ dμ(i) = Z I φ AS i (v)dμ(i). Proposition 2.11. Suppose that v(S) = F (μ(S)), F ∈ C 1 ([0, 1]). Then the game admits a marginal contribution density given by m(i,λ) = F ′ (λ) for all i∈ I, and consequently φ AS i (v) = Z 1 0 F ′ (λ)dλ = F (1)− F (0), μ–a.e. i. Proof. LetS λ be a participation path for S. Since μ(S λ ) = λμ(S), v(S λ ) = F (λμ(S)). Differentiating, d dλ v(S λ ) = F ′ (λμ(S))μ(S). For the grand coalition I , where μ(I) = 1, this reduces to d dλ v(I λ ) = F ′ (λ). Because F ′ (λ) = Z I F ′ (λ)dμ(i), the function m(i,λ) = F ′ (λ) is a marginal contribution density. The Aumann-Shapley value therefore satisfies φ AS i (v) = Z 1 0 F ′ (λ)dλ = F (1)− F (0). 2.3 Axiomatic Characterization As in the finite-player case, the Aumann-Shapley value is uniquely characterized by natural axioms extending the classical Shapley principle to nonatomic cooperative games. LetVdenote the class of cooperative games on(I,I,μ)admitting a marginal contribution density. An allocation rule is a mapping Φ :V → L 1 (I,μ) that assigns to each game v a payoff density Φ(v). Theorem 2.12. The Aumann-Shapley value is the unique allocation ruleΦ :V → L 1 (I,μ), defined up toμ-null sets, satisfying the following properties: 5 Exit-and-Join Dynamics in Continuum Cooperative Games (1) Efficiency. Z I Φ i (v)dμ(i) = v(I). (2) Symmetry. If two agents i,j ∈ I satisfy m(i,λ) = m(j,λ) for a.e. λ∈ [0, 1], then Φ i (v) = Φ j (v) for μ-a.e. i,j. (3) Linearity. For all v,w ∈V and α,β ∈ R, Φ(αv + βw) = αΦ(v) + βΦ(w). (4) Marginality. If two games v,w ∈V admit marginal contribution densities m v and m w satisfying m v (i,λ) = m w (i,λ) for μ-a.e. i and a.e. λ, then Φ(v) = Φ(w) in L 1 (I,μ). Under these axioms, Φ(v) = φ AS (v). The axioms mirror the classical Shapley axioms but are formulated in terms of marginal contribution densities rather than discrete marginal increments. Efficiency ensures that the grand coalition value is fully distributed. Symmetry guarantees equal treatment of agents with identical marginal productivity functions. Linearity preserves additivity across games. Marginality ensures that payoffs depend only on infinitesimal marginal contributions and not on absolute value levels. Remark 2.13. If the cooperative gamevis convex, the Aumann-Shapley value coincides with the competitive (Walrasian) payoff allocation in the associated exchange or production economy. In this case, there exists a price functionalpsuch that each agent’s payoff equals its marginal productivity evaluated at equilibrium prices. The grand coalition outcome can therefore be decentralized through price-taking behavior. This establishes a structural equivalence between the Shapley principle, marginal productivity pricing, and general equilibrium theory in large economies. Example 2.14. Let(I,I,μ)be a nonatomic probability space withμ(I) = 1. Suppose each agent supplies one unit of a homogeneous input and total output is generated according to F (x) = x 2 , x∈ [0, 1]. Define the cooperative game v(S) = F (μ(S)) = μ(S) 2 . Since F is convex, the game is convex. The marginal productivity at participation level x is F ′ (x) = 2x. Hence the Aumann-Shapley value assigns φ AS i (v) = Z 1 0 2xdx = 1 for μ-a.e. i. Efficiency holds: Z I φ AS i (v)dμ(i) = 1 = v(I). Thus each agent receives the same payoff density, reflecting both symmetry and the price-taking nature of large competitive economies. 6 Exit-and-Join Dynamics in Continuum Cooperative Games 2.4 Non-Mass-Based Cooperative Games and Kernel Interactions The mass–based specification of Proposition 2.11 is a special case in which coalition value depends only on its measure. We now extend Definitions 2.2, 2.4, and 2.6 to cooperative games whose value depends on the composition of agents. 2.4.1 Finite-Dimensional Functional Games Let(I,I,μ)be a nonatomic probability space withμ(I) = 1. Letψ 1 ,...,ψ k ∈ L 1 (I,μ)be measurable characteristics (types). For every measurable coalitionS ∈ I, defineT r (S) := R S ψ r (i)dμ(i), r = 1,...,k.LetF : R k → Rbe continuously differentiable. Define the cooperative game v(S) = F T 1 (S),...,T k (S) . This specification allows coalition value to depend on composition rather than only on mass. Ifψ 1 ≡ 1andk = 1, the model reduces to the mass–based case. Proposition 2.15. Letvbe defined as above and letS λ λ∈[0,1] be a participation path forSas in Definition 2.2. Assume that λ7→ S λ is absolutely continuous and that d dλ μ S λ (A) = μ(A∩ S) for all measurable A, i.e., coalition growth is proportional. Define T r (λ) := Z S λ ψ r (i)dμ(i), r = 1,...,k. Then v admits a marginal contribution density m : I × [0, 1]→ R given by m(i,λ) = k X r=1 ∂F ∂x r T 1 (λ),...,T k (λ) ψ r (i). Proof. Write T (λ) = T 1 (λ),...,T k (λ) ∈ R k . Then v(S λ ) = F T (λ) . Since F ∈ C 1 (R k ), the chain rule yields d dλ v(S λ ) =∇F T (λ) · T ′ (λ) = k X r=1 ∂F ∂x r T (λ) d dλ T r (λ). By absolute continuity of the participation path, d dλ T r (λ) = Z S ψ r (i)dμ(i). Hence d dλ v(S λ ) = Z S k X r=1 ∂F ∂x r T (λ) ψ r (i) ! dμ(i). Comparing with Definition 2.6 establishes the marginal contribution density. Corollary 2.16. Under the above assumptions, the Aumann–Shapley value exists and satisfies φ AS i (v) = Z 1 0 m(i,λ)dλ = k X r=1 ψ r (i) Z 1 0 ∂ r F T (λ) dλ. Thus the Aumann–Shapley value is a linear combination of the agent’s characteristicsψ r (i), with coefficients determined by averaged marginal productivity along the participation path. Remark 2.17. Unless allψ r are constantμ–almost everywhere, the Aumann–Shapley value is heterogeneous across agents. Equality of payoffs arises only under symmetry of the marginal contribution density (cf. Theorem 2.12). 7 Exit-and-Join Dynamics in Continuum Cooperative Games Example 2.18. Let(I,I,μ)be a nonatomic probability space withμ(I) = 1. Suppose each agent has two measurable characteristics ψ 1 (i) = θ(i) and ψ 2 (i) = 1, where θ ∈ L 1 (I,μ) represents productivity. For any coalitionS, defineT 1 (S) = R S θ(i)dμ(i)andT 2 (S) = μ(S). LetF (x 1 ,x 2 ) = x 1 +βx 1 x 2 withβ > 0. Then v(S) = T 1 (S) + βT 1 (S)T 2 (S) = Z S θ(i)dμ(i) + β Z S θ(i)dμ(i) μ(S). Coalition value therefore depends on total productivity and its interaction with coalition size. The model reduces to a mass–based game only if θ is constant almost everywhere. LetS λ be a proportional participation path. ThenT 1 (λ) = R S λ θ(i)dμ(i)andT 2 (λ) = λμ(S). Since ∂F ∂x 1 = 1 +βx 2 and ∂F ∂x 2 = βx 1 , Proposition 2.15 gives m(i,λ) = (1 + βT 2 (λ))θ(i) + βT 1 (λ). Substituting T 2 (λ) = λμ(S), m(i,λ) = θ(i) + βλμ(S)θ(i) + βT 1 (λ). Integrating over λ, φ AS i (v) = Z 1 0 m(i,λ)dλ = θ(i) + βθ(i) Z 1 0 λμ(S)dλ + β Z 1 0 T 1 (λ)dλ. Since R 1 0 λdλ = 1 2 , we obtain φ AS i (v) = θ(i) + β 2 μ(S)θ(i) + β Z 1 0 T 1 (λ)dλ. The Aumann–Shapley value decomposes into a direct productivity termθ(i), an interaction amplification term β 2 μ(S)θ(i), and a common surplus termβ R 1 0 T 1 (λ)dλ independent ofi. Unlessθis constant almost everywhere, φ AS i (v)̸= φ AS j (v), so the allocation is heterogeneous. 2.4.2 Infinite-Dimensional Functional Formulation The finite-dimensional construction extends naturally to fully general measure-dependent cooperative games. We first illustrate the idea with a kernel interaction example and then present the general formulation. Example 2.19. Let(I,I,μ)be a nonatomic probability space withμ(I) = 1. LetK : I × I → Rbe a symmetric measurable kernel with K ∈ L 1 (I × I). Define the cooperative game v(S) = Z S×S K(i,j)dμ(i)dμ(j). This specification captures pairwise complementarities among agents. Coalition value depends on the entire distribution of agents in S, not merely on its mass. LetS λ λ∈[0,1] be a proportional participation path for S. Then v(S λ ) = Z S λ ×S λ K(i,j)dμ(i)dμ(j). Differentiating with respect to λ and using symmetry of K yields d dλ v(S λ ) = 2 Z S λ Z S λ K(i,j)dμ(j) d dμ S λ dλ (i). Under proportional growth, d dλ μ S λ (A) = μ(A∩ S), so d dλ v(S λ ) = Z S 2 Z S λ K(i,j)dμ(j)dμ(i). 8 Exit-and-Join Dynamics in Continuum Cooperative Games Comparing with Definition 2.6, the marginal contribution density is m(i,λ) = 2 Z S λ K(i,j)dμ(j). By Definition 2.9, φ AS i (v) = Z 1 0 2 Z S λ K(i,j)dμ(j)dλ. Thus each agent’s payoff equals its average interaction intensity with the coalition along the participation path. If K(i,j) depends on i, then generally φ AS i (v)̸= φ AS j (v). The mass–based model arises as the special caseK(i,j)≡ c, for whichv(S) = cμ(S) 2 and the marginal contribution density is uniform across agents. We now abstract this structure. Definition 2.20. LetM(I)denote the space of finite signed measures onI. Elements ofM(I)describe distributions of agents and allow coalition value to depend on the entire population profile, not merely on its total mass. For each coalitionS ∈ I, define the restricted measureμ S (A) = μ(A∩ S), A ∈ I.Thusμ S encodes the full distribution of agents insideS. LetV : M(I) → Rbe Gateaux differentiable. We define the cooperative game v(S) =V(μ S ). Hence coalition value depends on the measure μ S , that is, on the distribution of agents in S. Gateaux differentiability means that for every measureμand every signed measure perturbationν, the directional derivative DV(μ)(ν) := lim ε→0 V(μ + εν)−V(μ) ε exists and is linear in ν. By the Riesz representation principle for measures, there exists a measurable function δV δμ (μ) : I → R such that DV(μ)(ν) = Z I δV δμ (μ)(i)dν(i). The function δV δμ (μ)(i)represents the marginal productivity of an infinitesimal increase of mass at agentiwhen the coalition measure is μ. Proposition 2.21. LetS λ be a proportional participation path, so that d dλ μ S λ (A) = μ(A∩ S) . IfVis Gateaux differentiable at μ S λ , then the game admits a marginal contribution density given by m(i,λ) = δV δμ μ S λ (i). Proof. Along the participation path, v(S λ ) =V(μ S λ ). By the chain rule in Banach spaces, d dλ v(S λ ) = DV(μ S λ ) d dλ μ S λ . Using the representation of the Gateaux derivative, d dλ v(S λ ) = Z I δV δμ μ S λ (i)d d dλ μ S λ (i). Under proportional growth, d dλ μ S λ (A) = μ(A∩ S), so the derivative simplifies to d dλ v(S λ ) = Z S δV δμ μ S λ (i)dμ(i). 9 Exit-and-Join Dynamics in Continuum Cooperative Games Comparing with Definition 2.6, the marginal contribution density is m(i,λ) = δV δμ μ S λ (i). Corollary 2.22. Under the above assumptions, the Aumann–Shapley value equals φ AS i (v) = Z 1 0 δV δμ μ S λ (i)dλ. Thus the Aumann–Shapley allocation equals the average marginal productivity of agentialong the participation path. Heterogeneity arises whenever the functional derivative depends on i. 2.5 Coalition Structures in the Continuum At any given time, the population of agents is assumed to be fully organized into a finite number of coalitions. Coalitions do not overlap, and no agent remains unaffiliated. Formally, a coalition structure is a measurable partition of the entire player space. Definition 2.23. Let (I,I,μ) be a nonatomic probability space. A coalition structure is a finite measurable partition T =T 1 ,...,T m , m <∞, of I such that: μ(T i ∩ T j ) = 0 for i̸= j, and μ   I \ m [ j=1 T j   = 0. The partition condition ensures that every agent belongs to exactly one coalition, up toμ-null sets. Since the player space is nonatomic, coalition structures that differ only on null sets are identified. Thus coalition membership is defined almost everywhere. At a fixed time, the coalition structure is treated as exogenously given, reflecting organizational, institutional, or technological constraints that limit feasible coalitions. Dynamic reconfiguration occurs through exit–and–join moves, but at every instant the coalitions jointly exhaust the population. Fix a coalition T j ∈ T . The restriction of the player space to T j is the measure space (T j ,I T j ,μ T j ), where I T j =A⊂ T j : A = S∩ T j for some S ∈I, and μ T j (A) = μ(A), A∈I T j . Since(I,I,μ)is nonatomic, each restricted measure space(T j ,I T j ,μ T j )is also nonatomic. Thus each coalition inherits the same measure-theoretic structure as the entire population. Let v :I → R be a cooperative game. The restricted game on coalition T j is the function v |T j :I T j → R defined by v |T j (A) = v(A), A∈I T j . Equivalently, for any S ∈I, v |T j (S∩ T j ) = v(S∩ T j ). The restricted game captures the value generated by subcoalitions formed entirely withinT j , abstracting from inter- actions with agents outside the coalition. In other words, each coalition in a structureTinduces its own nonatomic cooperative game, to which solution concepts such as the Aumann-Shapley value may be applied independently. This restriction principle is fundamental for extending the Aumann-Drèze construction to the continuum: the coalition structure is treated as fixed, the original game is restricted to each coalition, and payoffs are computed separately within each restricted measure space. 10 Exit-and-Join Dynamics in Continuum Cooperative Games Example 2.24. LetI = [0, 1]equipped with Lebesgue measure and fixα ∈ (0, 1). DefineT 1 = [0,α]and T 2 = (α, 1]. ThenT =T 1 ,T 2 is a coalition structure. Indeed, bothT 1 andT 2 are measurable,μ(T 1 ∩ T 2 ) = 0, and μ(I \ (T 1 ∪ T 2 )) = 0. The coalition masses are μ(T 1 ) = α and μ(T 2 ) = 1− α. The restricted measure spaces are(T 1 ,I T 1 ,μ T 1 )and(T 2 ,I T 2 ,μ T 2 ), each of which remains nonatomic since the original space(I,I,μ)is nonatomic. If the cooperative game is mass-based, that is,v(S) = F (μ(S)), then the restricted games satisfy v |T 1 (A) = F (μ(A)) for A⊂ T 1 , and similarly v |T 2 (A) = F (μ(A)) for A⊂ T 2 . Thus each coalition behaves as an independent nonatomic cooperative game with total massαand1− α, respectively. Remark 2.25. A coalition structure can be embedded into the finite–dimensional functional game framework by introducing indicator characteristics ψ j (i) = 1 T j (i), j = 1,...,m. For any measurable coalition S ⊂ I , T j (S) = Z S ψ j (i)dμ(i) = μ(S∩ T j ). Thus a cooperative game of the form v(S) = F T 1 (S),...,T m (S) = F μ(S∩ T 1 ),...,μ(S∩ T m ) depends only on the mass ofSinside each block of the partition. In this sense, coalition membership functions as a finite type system, and the coalition structure is a special case of the finite–dimensional functional game. Under this representation, agents within the same blockT j are symmetric, and heterogeneity arises only across blocks. The mass–based model corresponds to the degenerate case m = 1. 2.6 Continuum Aumann-Drèze Value We now extend the Aumann-Drèze construction (see [5]) to a continuum of players. The logic mirrors the finite case: the coalition structure is treated as fixed, the original game is restricted to each coalition, and the Shapley principle (in its continuum form) is applied within each restricted game. Let(I,I,μ)be a nonatomic player space withμ(I) = 1, letv : I → Rbe a transferable–utility cooperative game, and letT =T 1 ,...,T m be a coalition structure. For each coalitionT j ∈ T, denote by(T j ,I T j ,μ T j )the restricted measure space, whereI T j = S ∩ T j : S ∈ I andμ T j (A) = μ(A) forA ∈ I T j , and define the restricted game v |T j :I T j → R by v |T j (A) = v(A). Definition 2.26. Assume that for every coalitionT j ∈ Tthe restricted gamev |T j admits an Aumann-Shapley value on (T j ,I T j ,μ T j ). The continuum Aumann-Drèze value is the payoff density Ω AD (v;T ) = Ω AD i (v;T ) i∈I ∈ L 1 (I,μ) defined μ-almost everywhere by Ω AD i (v;T ) = φ AS i v |T j ,for i∈ T j , where φ AS denotes the Aumann-Shapley value computed within the restricted game v |T j . Thus each agent’s payoff is determined by its infinitesimal marginal contribution within its own coalition, averaged over participation levels. Each coalition is treated as an independent nonatomic cooperative game, and payoffs are computed separately within each coalition. Remark 2.27. Since(I,I,μ)is nonatomic,Ω AD (v;T )is defined only up toμ-null sets. All payoff identities and efficiency statements are therefore understood to hold μ-almost everywhere. Remark 2.28. If the player space consists of finitely many atoms with equal mass, the above construction reduces to the classical Aumann-Drèze value. The continuum Aumann-Drèze value may therefore be interpreted as the large-population limit of the finite exit-and-join payoff rule. Proposition 2.29. For every coalition T j ∈ T , Z T j Ω AD i (v;T )dμ(i) = v(T j ). 11 Exit-and-Join Dynamics in Continuum Cooperative Games Proof.By definition,Ω AD i (v;T )coincidesμ-almost everywhere onT j withφ AS i (v |T j ). Efficiency of the Aumann- Shapley value on the restricted measure space implies Z T j φ AS i (v |T j )dμ(i) = v |T j (T j ) = v(T j ), which yields the claim. Remark 2.30. As in the finite Aumann-Drèze construction, the coalition structure is treated as exogenously fixed. Payoffs within each coalition depend only on the restricted game on that coalition; cross-coalition externalities are not internalized. Example 2.31. Let(I,I,μ)be a nonatomic probability space withμ(I) = 1and consider the cooperative game v(S) = μ(S) 2 forS ∈ I. LetT = T 1 ,T 2 be a coalition structure withμ(T 1 ) = αandμ(T 2 ) = 1− αfor some α∈ (0, 1). Denotem 1 = μ(T 1 ) = αandm 2 = μ(T 2 ) = 1− α. Forj ∈1, 2the restricted game onT j is given by v |T j (A) = μ(A) 2 forA∈I T j . In particular the value of the grand coalition withinT j isv |T j (T j ) = v(T j ) = m 2 j . Fix j ∈1, 2 and choose any participation pathA λ λ∈[0,1] for T j . By definition, μ(A λ ) = λm j . Along this path, v |T j (A λ ) = μ(A λ ) 2 = (λm j ) 2 . Differentiating with respect to λ yields d dλ v |T j (A λ ) = 2(λm j )m j = 2λm 2 j . Equivalently, if we reparametrize by coalition massx = μ(A λ ) = λm j , then the marginal productivity at level x∈ [0,m j ]is2x. The Aumann-Shapley value averages marginal productivity over participation levels. Using the mass parameterization on [0,m j ] gives φ AS i (v |T j ) = 1 m j Z m j 0 2xdx = 1 m j x 2 m j 0 = m j , i∈ T j . By definition of Ω AD , Ω AD i (v;T ) = m j for i∈ T j , so Ω AD i (v;T ) = α,i∈ T 1 , 1− α, i∈ T 2 . We verify efficiency within each coalition: Z T 1 Ω AD i (v;T )dμ(i) = αμ(T 1 ) = α 2 = v(T 1 ), Z T 2 Ω AD i (v;T )dμ(i) = (1− α)μ(T 2 ) = (1− α) 2 = v(T 2 ). Theorem 2.32. Let T =T 1 ,...,T m be a coalition structure on (I,I,μ). Define membership functions ψ j (i) = 1 T j (i), j = 1,...,m, and consider a cooperative game of finite-dimensional functional form v(S) = F T 1 (S),...,T m (S) , T j (S) = μ(S∩ T j ), where F : R m → R is continuously differentiable. Assume that F exhibits no cross-coalition externalities in the sense that, for every j, F (x 1 ,...,x m ) = F j (x j ) whenever x k = 0 for k ̸= j. Then the continuum Aumann–Drèze value satisfies Ω AD i (v;T ) = Z 1 0 ∂F j ∂x λμ(T j ) dλ, i∈ T j , and coincides μ-almost everywhere with the blockwise Aumann–Shapley value computed within each coalition T j . 12 Exit-and-Join Dynamics in Continuum Cooperative Games Proof.Under the membership representation,T j (S) = R S 1 T j (i)dμ(i) = μ(S∩ T j ).Fix a coalitionT j and consider the restricted game v |T j . For any A⊂ T j , T k (A) = 0 for k ̸= j, T j (A) = μ(A). Hence, by the no-externality assumption, v |T j (A) = F (0,...,μ(A),..., 0) = F j (μ(A)). Thus the restricted game is mass-based on T j . LetA λ be a participation path for T j . Then μ(A λ ) = λμ(T j ) and v |T j (A λ ) = F j (λμ(T j )). Differentiating, d dλ v |T j (A λ ) = F ′ j (λμ(T j ))μ(T j ). By the definition of the Aumann–Shapley value within T j , φ AS i (v |T j ) = Z 1 0 F ′ j (λμ(T j ))dλ, i∈ T j . By Definition 2.6, Ω AD i (v;T ) = φ AS i (v |T j ), i∈ T j , which proves the claim. 3 Exit-and-Join Dynamics in the Continuum This section defines exit-and-join dynamics for a continuum of players in a measure-theoretically rigorous manner. Because the player space is nonatomic, individual agents have zero measure and cannot be assigned trajectories. Accordingly, coalition reconfiguration is described as an evolution of population measures, induced by payoff differences under the continuum Aumann-Drèze value. 3.1 Primitives, State Space, and Coalition Payoffs We begin by specifying the primitives of the model and the induced state representation. Let(I,I,μ)be a nonatomic probability space, so thatμ(I) = 1andμ(i) = 0for alli∈ I. Letv :I → Rbe a transferable–utility cooperative game defined on measurable coalitions. We assume throughout thatvsatisfies the regularity conditions required for the existence of the Aumann–Shapley value on measurable subsets and, consequently, for the continuum Aumann–Drèze value on measurable partitions. Fix m <∞. A coalition structure is a finite measurable partition T =T 1 ,...,T m ⊂I satisfying μ(T i ∩ T j ) = 0 for i̸= j, μ I \ m [ i=1 T i ! = 0. Coalition structures that differ only on μ–null sets are identified. At time t≥ 0, the system is described by a coalition structure T (t). The associated population state is x(t) = (x 1 (t),...,x m (t))∈ R m + , x i (t) := μ(T i (t)). Since P m i=1 x i (t) = 1, the state lies in the simplex ∆ m = ( x∈ R m + : m X i=1 x i = 1 ) . Remark 3.1. The mappingT 7→ xis many–to–one: distinct coalition structures may induce the same population state. However, coalition payoffs and exit–and–join dynamics will depend only on the induced state x∈ ∆ m . 13 Exit-and-Join Dynamics in Continuum Cooperative Games For each coalition structure T =T 1 ,...,T m , let Ω AD (v;T ) = Ω AD k (v;T ) k∈I denote the continuum Aumann–Drèze payoff density. Assumption 3.2. For every coalitionT i ∈ T, the restricted gamev |T i is symmetric in the sense that its Aumann–Shapley value assigns the same payoff density to μ–almost every agent in T i . Lemma 3.3. Under Assumption 3.2, for each coalition T i there exists a constant r i (T )∈ R such that Ω AD k (v;T ) = r i (T ) for μ–almost every k ∈ T i . Proof. By coalition symmetry,Ω AD k (v;T )is constantμ–almost everywhere onT i . The constant is uniquely defined up to μ–null sets and is denoted by r i (T ). Let x∈ ∆ m be induced by a coalition structure T . Define the coalition payoff function ρ i : ∆ m → R, ρ i (x) := r i (T ), i = 1,...,m. The corresponding payoff vector field is ρ(x) = (ρ 1 (x),...,ρ m (x)). Remark 3.4. Although the definitionρ i (x) := r i (T )is expressed in terms of the coalition structureTinducingx, the dependence onTis only through the coalition massμ(T i ) = x i . Under Assumption 3.2, the Aumann–Shapley value within a coalition depends on the restricted game solely via the coalition measure. Hence thex–dependence ofρ i (x)is implicit through x i , and no additional structural features of the partition affect the payoff level. Proposition 3.5. Under Assumption 3.2, the payoff vectorρ(x)depends on the coalition structure only through the induced population state x∈ ∆ m . Proof. Suppose coalition structuresTandT ′ induce the same statex ∈ ∆ m . Thenμ(T i ) = μ(T ′ i )for alli. Under coalition symmetry, the Aumann–Shapley value within each coalition depends only on the restricted game and the coalition mass. Hencer i (T ) = r i (T ′ )for alli, soρ i (x)is well defined independently of the specific partition realizing x. 3.2 Admissible Switching Incentives Because the player space(I,I,μ)is nonatomic, each individual agent has zero measure and therefore cannot affect the population state on its own. Incentives must therefore be understood at the level of individual switching decisions, which aggregate into movements of population mass. Letx∈ ∆ m be the population state induced by a coalition structureT =T 1 ,...,T m . Under Assumption 3.2, the continuum Aumann–Drèze value assigns a payoff density that is constantμ–almost everywhere on each coalition. We denote byρ i (x)the payoff level assigned to coalitionT i . Thus, forμ–almost every agentk ∈ T i , its payoff equals ρ i (x). Definition 3.6. Coalition j offers an admissible switching opportunity to agents in coalition i at state x if ρ j (x) > ρ i (x). Admissibility is evaluated at the current population statex. It captures the existence of a strict payoff advantage for agents currently in coalition i to switch voluntarily to coalition j. Lemma 3.7. Ifρ j (x) > ρ i (x), then for every measurable subsetS ⊆ T i withμ(S) > 0, almost every agent inS strictly prefers switching to coalition T j over remaining in T i . Proof.Under coalition symmetry, almost every agent inT i receives payoffρ i (x). Hence almost every agent inS ⊆ T i receives payoff ρ i (x). If such an agent were to switch to coalitionT j while the population state remainsx, its payoff would equalρ j (x). Since ρ j (x) > ρ i (x), switching strictly increases payoff for almost every agent in S. Because μ(S) > 0, there exists a set of positive measure whose members strictly prefer to switch. Remark 3.8. In the nonatomic setting, strict payoff inequality between coalitions generates decentralized switching incentives. No central planner reallocates agents. Rather, whenρ j (x) > ρ i (x), almost every agent in coalitioni individually prefers coalitionj. Aggregate movements of population mass therefore arise endogenously from individual incentives. 14 Exit-and-Join Dynamics in Continuum Cooperative Games 3.3 Finite-Population Approximation and Mean-Field Limit The continuum dynamics should not be interpreted as assigning an independent Poisson clock to each individual point of the nonatomic space. A cleaner construction is to approximate the continuum by finite populations and then pass to a law-of-large-numbers limit. This yields the deterministic coalition-mass equation as the limit of density-dependent Markov chains. For each ordered pair (i,j) with i̸= j, let λ ij : ∆ m → R + be the per-agent switching intensity from coalitionito coalitionj. The intensity may depend on the current population state. Assumption 3.9. For eachi̸= j, the functionλ ij is Lipschitz continuous on∆ m . Moreover, there is a constantΛ <∞ such that 0≤ λ ij (x)≤ Λ, x∈ ∆ m , i̸= j. The boundedness condition is automatic if theλ ij are continuous on the compact simplex. We state it explicitly because it is the estimate used in the martingale bound below. For N ∈ N, consider N agents distributed among the m coalitions. Let X N (t) = X N 1 (t),...,X N m (t) ∈ ∆ N m ,∆ N m :=x∈ ∆ m : Nx i ∈ N, whereX N i (t)is the fraction of agents in coalitioniat timet. Conditional onX N (t) = x, a transition from coalitioni to coalition j changes the empirical state by x7−→ x + 1 N (e j − e i ), i̸= j, and occurs at aggregate rate q N ij (x) = Nx i λ ij (x).(3.1) Thus each of the Nx i agents currently in coalition i switches to coalition j with intensity λ ij (x). The generatorL N of X N (·) acts on bounded test functions f : ∆ N m → R by L N f (x) = X i̸=j Nx i λ ij (x) f x + 1 N (e j − e i ) − f (x) .(3.2) Define b : ∆ m → R m by b i (x) = X j̸=i λ ji (x)x j − X j̸=i λ ij (x)x i , i = 1,...,m.(3.3) This is the net inflow into coalitioniminus the outflow from coalitioni. Under Assumption 3.9, the mapbis Lipschitz on ∆ m and bounded. Theorem 3.10. Assume Assumption 3.9. If X N (0)→ x 0 ∈ ∆ m in probability, then for every T <∞, sup 0≤t≤T ∥X N (t)− x(t)∥ P −→ N→∞ 0, where x(·) is the unique solution of ̇x(t) = b(x(t)), x(0) = x 0 .(3.4) Equivalently, for each i = 1,...,m, ̇x i (t) = X j̸=i λ ji (x(t))x j (t)− X j̸=i λ ij (x(t))x i (t). Proof. Sincebis Lipschitz on the compact simplex, it admits a Lipschitz extension to a neighborhood of∆ m . Picard– Lindelof therefore gives a unique local solution, and boundedness ofbgives global existence. The solution remains in ∆ m : indeed, P i b i (x) = 0, and if x i = 0 then b i (x) = P j̸=i λ ji (x)x j ≥ 0. For each coordinate i, Dynkin’s formula applied to the coordinate map f i (x) = x i gives the martingale M N i (t) := X N i (t)− X N i (0)− Z t 0 b i (X N (s))ds.(3.5) 15 Exit-and-Join Dynamics in Continuum Cooperative Games Indeed, using (3.2) with f i (x) = x i gives L N f i (x) = X j̸=i x j λ ji (x)− X j̸=i x i λ ij (x) = b i (x). Each jump changes a single coordinate by at most 1/N . Since the total jump rate is bounded by X i̸=j Nx i λ ij (x)≤ N (m− 1)Λ, the predictable quadratic variation satisfies, for a constant C independent of N , ⟨M N i ⟩(t)≤ Ct N . Doob’s inequality therefore gives, for each T <∞, E sup 0≤t≤T |M N i (t)| 2 ≤ C T N . Thus sup 0≤t≤T ∥M N (t)∥→ 0 in probability. Let e N (t) = X N (t)− x(t). Combining (3.5) with (3.4) yields e N (t) = X N (0)− x 0 + Z t 0 b(X N (s))− b(x(s)) ds + M N (t). If L is a Lipschitz constant for b, then sup 0≤s≤t ∥e N (s)∥≤∥X N (0)− x 0 ∥ + L Z t 0 sup 0≤r≤s ∥e N (r)∥ds + sup 0≤s≤t ∥M N (s)∥. Grönwall’s inequality gives sup 0≤t≤T ∥e N (t)∥≤ e LT ∥X N (0)− x 0 ∥ + sup 0≤t≤T ∥M N (t)∥ . The right-hand side converges to zero in probability, proving the uniform-on-compact convergence. Remark 3.11. The deterministic ODE(3.4)is the continuum model. It is not obtained by counting events in an uncountable population. Rather, it is the hydrodynamic limit of finite empirical coalition distributions. The nonatomic model records only the limiting coalition masses. Assumption 3.12. The switching intensities are incentive compatible if, for every x∈ ∆ m and i̸= j, ρ j (x)≤ ρ i (x)=⇒ λ ij (x) = 0. They are strictly payoff-responsive if, in addition, ρ j (x) > ρ i (x)=⇒ λ ij (x) > 0. Under incentive compatibility, the vector field generates no flow from a coalition to another coalition with weakly lower payoff. Under strict payoff responsiveness, every strictly profitable destination generates a positive per-agent switching rate. 3.4 Invariance and Incentive-Compatible Dynamics Proposition 3.13. Let x(t) evolve according to the exit–and–join dynamics ̇x i (t) = X j̸=i λ ji (x(t))x j (t)− X j̸=i λ ij (x(t))x i (t), i = 1,...,m. Then the simplex ∆ m = n x∈ R m + : m X i=1 x i = 1 o is forward invariant. 16 Exit-and-Join Dynamics in Continuum Cooperative Games Proof. Summing the right–hand side over i yields m X i=1 ̇x i (t) = m X i=1 X j̸=i λ ji (x)x j − m X i=1 X j̸=i λ ij (x)x i . Reindexing the first double sum shows that each termλ ij (x)x i appears exactly once with positive sign and once with negative sign. Hence m X i=1 ̇x i (t) = 0. Therefore P i x i (t) remains constant along trajectories, and if x(0)∈ ∆ m then P i x i (t) = 1 for all t≥ 0. To verify nonnegativity, supposex i (t 0 ) = 0for somet 0 . Then all outflow terms from coalitionivanish since they are proportional tox i (t 0 ). The inflow terms are nonnegative. Thus ̇x i (t 0 ) ≥ 0. Therefore trajectories cannot cross the boundary of R m + , and ∆ m is forward invariant. 3.4.1 Replicator Dynamics as a Special Case Proposition 3.14. Suppose the switching intensities satisfy the payoff–difference rule λ ij (x) = κx j [ρ j (x)− ρ i (x)] + , κ > 0, where [a] + = maxa, 0. Then the exit–and–join dynamics reduce to the replicator equation ̇x i = κx i ρ i (x)− ̄ρ(x) , ̄ρ(x) = m X j=1 x j ρ j (x). Proof. Substituting the rate specification into the mass–balance system yields ̇x i = κx i X j̸=i x j [ρ i − ρ j ] + − κx i X j̸=i x j [ρ j − ρ i ] + . Using the identity [a− b] + − [b− a] + = a− b for all a,b∈ R, we obtain x i X j̸=i x j [ρ i − ρ j ] + − x i X j̸=i x j [ρ j − ρ i ] + = x i X j̸=i x j (ρ i − ρ j ). Since P j̸=i x j = 1− x i , we have X j̸=i x j (ρ i − ρ j ) = ρ i − m X j=1 x j ρ j = ρ i − ̄ρ. Combining the expressions gives ̇x i = κx i (ρ i − ̄ρ), which is precisely the replicator equation. 3.4.2 General Incentive-Compatible Switching Rules The payoff difference rule leading to replicator dynamics is not unique. More generally, the one-way incentive condition λ ij (x) > 0 ⇒ ρ j (x) > ρ i (x), i̸= j, admits a broad class of admissible switching specifications. Proposition 3.15. Let Φ : R→ R + be a locally Lipschitz function satisfying Φ(z) = 0 for z ≤ 0,Φ(z) > 0 for z > 0. Define the switching intensities by λ ij (x) = x j Φ ρ j (x)− ρ i (x) , i̸= j. Then the induced exit–and–join dynamics are incentive compatible. 17 Exit-and-Join Dynamics in Continuum Cooperative Games Proof.Ifρ j (x) ≤ ρ i (x), thenρ j (x)− ρ i (x) ≤ 0, soΦ(ρ j − ρ i ) = 0and henceλ ij (x) = 0. Ifρ j (x) > ρ i (x), then Φ(ρ j − ρ i ) > 0, so switching from coalitionito coalitionjoccurs with positive intensity whenever the destination coalition has positive mass,x j > 0. Ifx j = 0, the rule is still incentive compatible, but the empty destination is not accessible under this pairwise-imitation specification. Under this specification, the mass–balance system becomes ̇x i = x i X j̸=i x j h Φ ρ i (x)− ρ j (x) − Φ ρ j (x)− ρ i (x) i , i = 1,...,m. 3.4.3 Switching Rules and Induced Population Dynamics If Φ(z) = κz + with κ > 0 and z + = maxz, 0, then we obtain ̇x i = κx i X j̸=i x j (ρ i − ρ j ) = κx i ρ i (x)− ̄ρ(x) , where ̄ρ(x) = m X j=1 x j ρ j (x). Hence the classical replicator equation arises as the linear payoff–difference case. Let Φ(z) = κz α + , α > 0. Then the induced dynamics become ̇x i = κx i X j̸=i x j h (ρ i − ρ j ) α + − (ρ j − ρ i ) α + i . Whenα = 1, this reduces to the replicator equation. Forα̸= 1, the adjustment intensity depends nonlinearly on payoff differences. Large payoff gaps may generate disproportionately strong flows whenα > 1, whileα < 1dampens large differences. The caseα≥ 1is locally Lipschitz and fits the regularity assumption above; the sublinear case0 < α < 1 is continuous but not Lipschitz at zero payoff gaps and requires separate well-posedness arguments. Let Φ(z) = κ1 z>ε , ε > 0. Then ̇x i = κx i X j̸=i x j h 1 ρ i −ρ j >ε − 1 ρ j −ρ i >ε i . Here the vector field is piecewise constant in payoff space. Population mass moves only when payoff advantages exceed the thresholdε, creating regions of local inertia in which small payoff differences generate no adjustment. This discontinuous rule is an illustrative adjustment protocol; it falls outside Assumption 3.9 unless it is smoothed or interpreted as a differential inclusion. Let Φ(z) = κ e βz − 1 β 1 z>0 , β > 0. The induced dynamics are ̇x i = κx i X j̸=i x j e β(ρ i −ρ j ) − 1 β 1 ρ i >ρ j − e β(ρ j −ρ i ) − 1 β 1 ρ j >ρ i . Using the expansion e βz − 1 β = z + O(β), we see that asβ → 0the dynamics converge to the replicator equation. For largeβ, switching becomes highly sensitive to large payoff gaps, generating steep directional flows. All the above dynamics share two fundamental properties: (i) the growth rate ofx i is proportional to its current mass, and (i) total mass is conserved along trajectories. Consequently, they admit a common multiplicative representation on the simplex. This motivates the following abstract formulation. 18 Exit-and-Join Dynamics in Continuum Cooperative Games Definition 3.16. Let∆ m =x∈ R m + : P m i=1 x i = 1. A population dynamic on∆ m is called aG-dynamic [15, 17] if it admits the multiplicative form ̇x i = x i G i (x), i = 1,...,m, where G : ∆ m → R m is a measurable vector field satisfying the balance condition m X i=1 x i G i (x) = 0for all x∈ ∆ m . Proposition 3.17. EveryG-dynamic leaves the simplex∆ m forward invariant. Moreover, ifx ∗ ∈ ∆ m is a rest point, then x ∗ i > 0=⇒ G i (x ∗ ) = 0. In the replicator case G i (x) = κ ρ i (x)− ̄ρ(x) , rest points satisfy payoff equalization across all coalitions with positive mass: x ∗ i > 0, x ∗ j > 0=⇒ ρ i (x ∗ ) = ρ j (x ∗ ). Proof. Summing the differential equation yields d dt m X i=1 x i (t) = m X i=1 x i (t)G i (x(t)) = 0, so P i x i (t) is constant and equals one if it does initially. Nonnegativity follows from the multiplicative structure, since x i = 0 implies ̇x i = 0. Ifx ∗ is a rest point, then ̇x i = 0for alli. Hencex ∗ i G i (x ∗ ) = 0for alli, which impliesG i (x ∗ ) = 0wheneverx ∗ i > 0. For the replicator field,G i (x ∗ ) = 0on the active support impliesρ i (x ∗ ) = ̄ρ(x ∗ )for every activei, which gives payoff equalization across active coalitions. Remark 3.18. Replicator dynamics correspond to the special case G i (x) = κ ρ i (x)− ̄ρ(x) , but many other incentive-compatible switching rules generateG-dynamics. Thus replicator dynamics are one represen- tative of a broader class of mean-field exit-and-join population systems. 4 Exit-and-Join Equilibrium and Stationarity 4.1 Equilibrium and Stationarity We now formalize the equilibrium concept induced by the exit-and-join dynamics and establish its equivalence with stationarity of the mean-field system. Definition 4.1. A population statex ⋆ ∈ ∆ m is an exit-and-join equilibrium if no coalition with positive mass admits a strictly profitable deviation. Equivalently, ρ j (x ⋆ )≤ ρ i (x ⋆ )for all i with x ⋆ i > 0 and all j. Thus, at equilibrium, every populated coalition offers the same maximal payoff level, and no coalition, populated or empty, offers a strictly higher payoff. Theorem 4.2. A state x ⋆ ∈ ∆ m is an exit-and-join equilibrium if and only if there exists ρ ⋆ ∈ R such that ρ i (x ⋆ ) = ρ ⋆ for all i with x ⋆ i > 0, and ρ j (x ⋆ )≤ ρ ⋆ for all j with x ⋆ j = 0. Proof.Supposex ⋆ is an equilibrium. For anyi,kwithx ⋆ i > 0andx ⋆ k > 0, we have bothρ k (x ⋆ ) ≤ ρ i (x ⋆ )and ρ i (x ⋆ )≤ ρ k (x ⋆ ), hence equality. Denote the common value by ρ ⋆ . If somejwithx ⋆ j = 0satisfiedρ j (x ⋆ ) > ρ ⋆ , agents in any populated coalition would have a profitable deviation, contradicting equilibrium. Thus ρ j (x ⋆ )≤ ρ ⋆ . Conversely, if these conditions hold, no agent in a populated coalition can strictly improve by moving, hencex ⋆ is an equilibrium. 19 Exit-and-Join Dynamics in Continuum Cooperative Games The equilibrium concept admits an exact dynamical characterization. Theorem 4.3. Assume the switching intensities are incentive compatible and strictly payoff-responsive in the sense of Assumption 3.12. For x ⋆ ∈ ∆ m , the following are equivalent: (1) x ⋆ is an exit-and-join equilibrium; (2) no admissible positive–measure deviation exists at x ⋆ ; (3) x ⋆ is a stationary point of the exit-and-join dynamics, ̇x(t) = 0 whenever x(t) = x ⋆ ; (4) all active incentive-compatible transition fluxes vanish at x ⋆ , x ⋆ i λ ij (x ⋆ ) = 0 for all i,j. Proof. Suppose first thatx ⋆ is an equilibrium. Then for everyiwithx ⋆ i > 0and everyj,ρ j (x ⋆ ) ≤ ρ i (x ⋆ ). By incentive compatibility, λ ij (x ⋆ ) = 0. If x ⋆ i = 0, then x ⋆ i λ ij (x ⋆ ) = 0 anyway. Next, if all active transition fluxes vanish, all inflow and outflow terms in the mass dynamics vanish, hence ̇x(t) = 0. To prove the converse implication from stationarity, argue by contrapositive. Supposex ⋆ is not an exit-and-join equilibrium. Then there exist an active coalitioniwithx ⋆ i > 0and a coalitionjsuch thatρ j (x ⋆ ) > ρ i (x ⋆ ). Letkbe an active coalition with minimal payoff among active coalitions: ρ k (x ⋆ ) = min ℓ:x ⋆ ℓ >0 ρ ℓ (x ⋆ ). Thenρ j (x ⋆ ) > ρ k (x ⋆ ). By strict payoff-responsiveness,λ kj (x ⋆ ) > 0, so there is positive active outflow from coalition kto coalitionj. No active coalition has payoff strictly belowρ k (x ⋆ ), and incentive compatibility rules out inflows tok from coalitions with weakly higher payoff. Therefore every inflow term intokis zero, while at least one outflow term is strictly positive. Hence ̇x k = X ℓ̸=k λ ℓk (x ⋆ )x ⋆ ℓ − X ℓ̸=k λ kℓ (x ⋆ )x ⋆ k < 0, so x ⋆ is not stationary. Thus stationarity implies that no admissible positive-measure deviation exists. The implication from the absence of admissible positive-measure deviations to equilibrium is immediate from the definition of equilibrium. Remark 4.4. Strict payoff-responsiveness is essential for the converse direction. For pairwise imitation specifications such asλ ij (x) = κx j [ρ j (x)− ρ i (x)] + , an empty destination coalition hasx j = 0and therefore cannot be entered. Such dynamics may have stationary boundary states with an unused higher-payoff coalition. On the relative interior of a fixed support, or after adding an exploration term that permits entry into empty coalitions, the strict-responsiveness condition is restored. Remark 4.5. An exit-and-join equilibrium equalizes payoff levels across all coalitions with positive mass. Coalition sizes need not be equal, and empty coalitions may coexist with populated ones. 4.2 Mass–Based Cooperative Games and Lyapunov Structure We now specialize the exit-and-join framework to a class of cooperative games in which coalition value depends only on coalition mass. In this setting, the induced population dynamics admit a natural global Lyapunov function derived directly from the primitive game. Assumption 4.6. There exists a function F ∈ C 1 ([0, 1]) with F (0) = 0 such that the cooperative game satisfies v(S) = F (μ(S)), S ∈I. Under this assumption, coalition value depends only on its measure and not on its composition. LetT = T 1 ,...,T m be a coalition structure and define the population statex ∈ ∆ m byx i = μ(T i ). Since the restricted gamev |T i is again mass–based, its total value equalsF (x i ). Applying the continuum Aumann–Shapley formula to v |T i yields the payoff density assigned to almost every agent in T i : ρ i (x) = 1 x i Z x i 0 F ′ (s)ds, x i > 0. 20 Exit-and-Join Dynamics in Continuum Cooperative Games By continuity of F ′ , the limit ρ i (0) := lim x i ↓0 ρ i (x i ) = F ′ (0) exists, so ρ i extends continuously to [0, 1]. Thus, under the mass–based structure, the payoff vector ρ(x) depends only on coalition masses. Define V : ∆ m → R by V (x) = m X i=1 Z x i 0 ρ i (s)ds.(4.1) Since eachρ i (·)is continuous on[0, 1],Vis continuously differentiable on∆ m . Moreover, by the fundamental theorem of calculus, ∂V (x) ∂x i = ρ i (x), i = 1,...,m. Hence ∇V (x) = ρ(x). The functionVaggregates the marginal payoff levels across coalitions and therefore represents total cooperative surplus consistent with the Aumann–Drèze allocation. The exit-and-join dynamics will be shown to ascendV, aligning individual payoff improvements with global surplus maximization. 4.3 Lyapunov Structure and Global Convergence We now establish a complete Lyapunov analysis of the exit-and-join dynamics in the class of mass-based cooperative games and derive global convergence results. Throughout this subsection, Assumption 4.6 remains in force. The continuum Aumann-Drèze payoff level of a coalition of mass x i > 0 is ρ i (x) = 1 x i Z x i 0 F ′ (s)ds, and extends continuously to x i = 0 with ρ i (0) = F ′ (0). Define V (x) = m X i=1 Z x i 0 ρ i (s)ds, x∈ ∆ m .(4.2) Lemma 4.7. The function V belongs to C 1 (∆ m ) and satisfies ∇V (x) = ρ(x). Proof. Since F ∈ C 1 ([0, 1]), the function ρ i (x i ) = 1 x i Z x i 0 F ′ (s)ds is continuous on [0, 1]. By the fundamental theorem of calculus, ∂V (x) ∂x i = ρ i (x i ), which establishes differentiability and yields∇V (x) = ρ(x). Theorem 4.8. Assume Assumption 4.6 and suppose the switching intensities are incentive compatible. Then the function Vdefined in(4.2)is a Lyapunov function for the exit-and-join dynamics. For every absolutely continuous solutionx(t), d dt V (x(t))≥ 0. If the switching intensities are also strictly payoff-responsive, then equality holds at a statex(t)if and only ifx(t)is an exit-and-join equilibrium. 21 Exit-and-Join Dynamics in Continuum Cooperative Games Proof. Let x(t) satisfy the exit-and-join dynamics. By the chain rule and Lemma 4.7, d dt V (x(t)) =∇V (x(t)) ⊤ ̇x(t) = m X i=1 ρ i (x(t)) ̇x i (t). Substituting the mass dynamics ̇x i = X j̸=i λ ji (x)x j − X j̸=i λ ij (x)x i and rearranging terms yields d dt V (x) = X i̸=j λ ij (x)x i ρ j (x)− ρ i (x) . By incentive compatibility, λ ij (x) > 0 ⇒ ρ j (x) > ρ i (x). Since x i ≥ 0, each term in the sum is nonnegative, hence dV (x(t))/dt≥ 0. Moreover, under strict payoff-responsiveness, d dt V (x(t)) = 0 if and only if λ ij (x(t))x i (t) = 0 for all i,j. This condition is equivalent to ρ j (x(t))≤ ρ i (x(t)) for all i with x i (t) > 0, which is precisely the exit-and-join equilibrium condition. 4.4 Global Convergence Assumption 4.9. The function F is strictly concave on [0, 1]. Under strict concavity of F , coalition marginal productivity is strictly decreasing in coalition mass. Theorem 4.10. Suppose Assumptions 4.6 and 4.9 hold. Suppose also that the switching intensities satisfy Assumption 3.9 and are incentive compatible and strictly payoff-responsive. Then the exit-and-join dynamics admit a unique equilibrium x ⋆ ∈ ∆ m , and for every initial condition x(0)∈ ∆ m , lim t→∞ x(t) = x ⋆ . Proof. Strict concavity of F implies that F ′ is strictly decreasing on [0, 1]. Since ρ i (x i ) = 1 x i Z x i 0 F ′ (s)ds, the function x i 7→ ρ i (x i ) is strictly decreasing on (0, 1]. Consequently, the mapping x i 7−→ Z x i 0 ρ i (s)ds is strictly concave on[0, 1]. BecauseVis the sum of these coordinate functions and∆ m is convex, it follows thatVis strictly concave on ∆ m . Since∆ m is compact and convex andVis continuous,Vattains a maximizerx ⋆ ∈ ∆ m . Strict concavity guarantees that this maximizer is unique. The first-order optimality condition for maximizing the differentiable concave function V over ∆ m is ρ j (x ⋆ )≤ ρ i (x ⋆ )for all i with x ⋆ i > 0 and all j, which is exactly the exit-and-join equilibrium condition. Hence x ⋆ is the unique equilibrium. For any absolutely continuous solution x(t) of the dynamics, Theorem 4.8 implies d dt V (x(t))≥ 0, 22 Exit-and-Join Dynamics in Continuum Cooperative Games with strict inequality wheneverx(t)is not an equilibrium. ThusV (x(t))is nondecreasing along trajectories and strictly increasing outside x ⋆ . Since V is continuous on the compact set ∆ m , it is bounded above, and therefore lim t→∞ V (x(t)) exists. The setx∈ ∆ m : ̇ V (x) = 0coincides with the set of equilibria. Because the equilibrium is unique, this invariant set reduces tox ⋆ . By LaSalle’s invariance principle, every trajectory converges to x ⋆ , completing the proof. Remark 4.11. IfFis concave but not strictly concave, thenVis concave but may admit multiple maximizers. In this case, every trajectory converges to the compact set of equilibria, but convergence need not be to a unique point. Remark 4.12. The Lyapunov functionVrepresents aggregate cooperative surplus. The exit-and-join dynamics imple- ment a decentralized, incentive-compatible ascent of total surplus. Strict concavity ensures that surplus maximization selects a unique coalition size distribution, so individual rational adjustments lead globally to the socially efficient allocation. 5 Population Game Formulation and Wardrop Equivalence This section establishes that exit–and–join equilibria coincide with Wardrop equilibria of an induced nonatomic population game. The cooperative structure determines the payoff vector fieldρ(x), while the equilibrium concept itself is entirely noncooperative. This link places the model alongside Wardrop formulations of traffic assignment and recent security and learning variants of congestion games [20, 21, 23]. 5.1 The Induced Population Game Let (I,I,μ) be a nonatomic probability space with μ(I) = 1. Fix m <∞ and define the simplex ∆ m = ( x∈ R m + : m X i=1 x i = 1 ) . From Section 3, the continuum Aumann–Drèze construction induces a payoff vector field ρ : ∆ m → R m , ρ(x) = (ρ 1 (x),...,ρ m (x)), where ρ i (x) denotes the payoff density assigned to coalition i when the population state is x. Definition 5.1. The induced nonatomic population game is the tripleG = (I,A,u)defined as follows. The player set is I. The strategy set isA =1,...,m. A (pure) strategy profile is a measurable mapσ : I → A. The induced population state is given byx i (σ) = μ(k ∈ I : σ(k) = i)fori = 1,...,m, so thatx(σ) = (x 1 (σ),...,x m (σ))∈ ∆ m . If the population state is x∈ ∆ m , the payoff to a player choosing strategy i∈ A is u i (x) = ρ i (x). Thus payoffs depend only on aggregate coalition masses. 5.2 Wardrop Equilibrium We now introduce the appropriate equilibrium notion for nonatomic populations. Definition 5.2. A state x ⋆ ∈ ∆ m is a Wardrop equilibrium of G if x ⋆ i > 0 =⇒ u i (x ⋆ ) = max j=1,...,m u j (x ⋆ ). Equivalently, u j (x ⋆ )≤ u i (x ⋆ ) for all i with x ⋆ i > 0. Since u i (x) = ρ i (x), the Wardrop condition can be written as ρ j (x ⋆ )≤ ρ i (x ⋆ ) ∀i with x ⋆ i > 0. 23 Exit-and-Join Dynamics in Continuum Cooperative Games 5.3 Equivalence with Exit–and–Join Equilibrium Recall from Definition 4.1 that x ⋆ ∈ ∆ m is an exit–and–join equilibrium if ρ j (x ⋆ )≤ ρ i (x ⋆ ) ∀i with x ⋆ i > 0. Theorem 5.3. Letρ : ∆ m → R m be the payoff vector induced by the continuum Aumann–Drèze value. Then for x ⋆ ∈ ∆ m the following are equivalent: (1) x ⋆ is an exit–and–join equilibrium; (2) x ⋆ is a Wardrop equilibrium of the population game G. Proof. By Definition 5.1, the payoff to strategyiisu i (x) = ρ i (x). Hence the Wardrop condition in Definition 5.2 is identical to the exit–and–join equilibrium condition. 5.4 Variational Inequality Characterization Wardrop equilibria admit an equivalent variational inequality (VI) formulation. Proposition 5.4. A state x ⋆ ∈ ∆ m is a Wardrop equilibrium if and only if ⟨ρ(x ⋆ ),x− x ⋆ ⟩≤ 0 ∀x∈ ∆ m . Proof. Suppose first that x ⋆ is a Wardrop equilibrium. Let x∈ ∆ m . Partition the index set into P =i : x ⋆ i > 0, Z =i : x ⋆ i = 0. For i∈ P , ρ i (x ⋆ ) = max j ρ j (x ⋆ ). For i∈ Z, ρ i (x ⋆ )≤ max j ρ j (x ⋆ ). Hence m X i=1 ρ i (x ⋆ )x i ≤ max j ρ j (x ⋆ ) m X i=1 x i = max j ρ j (x ⋆ ). Similarly, m X i=1 ρ i (x ⋆ )x ⋆ i = max j ρ j (x ⋆ ). Subtracting yields ⟨ρ(x ⋆ ),x− x ⋆ ⟩≤ 0. Conversely, suppose the variational inequality holds. If there existediwithx ⋆ i > 0andjwithρ j (x ⋆ ) > ρ i (x ⋆ ), consider the direction x = x ⋆ + ε(e j − e i ) for sufficiently small ε > 0. Then ⟨ρ(x ⋆ ),x− x ⋆ ⟩ = ε(ρ j (x ⋆ )− ρ i (x ⋆ )) > 0, contradicting the VI condition. Thus the Wardrop condition must hold. 5.5 Mass–Based Games and Potential Structure Under Assumption 4.6, v(S) = F (μ(S)), the payoff field satisfies ∇V (x) = ρ(x), where V (x) = m X i=1 Z x i 0 ρ i (s)ds. Proposition 5.5. In the mass–based case, the induced population game is a potential game with potentialV. Wardrop equilibria coincide with maximizers of V on ∆ m . Proof.Since∇V (x) = ρ(x), the variational inequality in Proposition 5.4 is equivalent to the first-order optimality condition for maximizing V over ∆ m . 24 Exit-and-Join Dynamics in Continuum Cooperative Games 6 Switching Costs and Endogenous Acceptance Rules In this section we extend the exit–and–join framework by incorporating (i) switching costs and (i) endogenous acceptance rules. Under this extension, coalition transitions become bilateral: a deviation from coalitionito coalitionj must be individually profitable and acceptable to the incumbent members of coalition j. Throughout this section, assume that the payoff field ρ : ∆ m → R m is continuously differentiable. 6.1 Switching Costs For each ordered pair(i,j)withi ̸= j, letc ij : ∆ m → R + denote the switching cost incurred by an agent moving from coalition i to coalition j. Definition 6.1. For x∈ ∆ m , define ∆ ij (x) = ρ j (x)− ρ i (x)− c ij (x). A deviation from i to j satisfies individual rationality at x if ∆ ij (x) > 0. 6.2 Derivation of the Acceptance Rule Because the continuum Aumann–Drèze value assigns identical payoff density to all members of a coalition, every incumbent member of coalition j receives payoff ρ j (x). Consider a statex∈ ∆ m and an infinitesimal massε > 0moving from coalitionito coalitionj. The perturbed state is x ε = x + ε(e j − e i ), where e k denotes the k-th canonical basis vector. Definition 6.2. Coalition j accepts the entrant at state x if and only if ρ j (x ε )≥ ρ j (x) for all sufficiently small ε > 0. Since ρ is continuously differentiable, the directional derivative of ρ j in direction d = e j − e i is Dρ j (x)[d] =∇ρ j (x)· (e j − e i ) = ∂ρ j ∂x j (x)− ∂ρ j ∂x i (x). To first order, ρ j (x ε ) = ρ j (x) + εDρ j (x)[e j − e i ] + o(ε). Hence coalition j weakly benefits from entry if and only if Dρ j (x)[e j − e i ]≥ 0. Proposition 6.3. Under continuous differentiability ofρ, entry from coalitioniinto coalitionjat statexis admissible if and only if Dρ j (x)[e j − e i ]≥ 0. Thus acceptance depends on the first-order effect of the proposed transfer on the payoff of incumbent members of the destination coalition. In the important own-mass case, where ρ j (x) depends only on x j , this condition reduces to ∂ρ j ∂x j (x)≥ 0. 6.3 Admissible Deviations and Switching Dynamics For compactness, define the acceptance margin A ij (x) := Dρ j (x)[e j − e i ]. Definition 6.4. A deviation from coalition i to coalition j is admissible at state x∈ ∆ m if x i > 0 and both ∆ ij (x) > 0 and A ij (x)≥ 0. 25 Exit-and-Join Dynamics in Continuum Cooperative Games Thus switching requires both individual profitability and coalitional consent. Let Φ : R→ R + satisfy Φ(z) = 0 for z ≤ 0 and Φ(z) > 0 for z > 0. Define the admissible switching intensity λ ij (x) = Φ ∆ ij (x) 1 A ij (x)≥0 . The induced mass dynamics are ̇x i = X j̸=i λ ji (x)x j − X j̸=i λ ij (x)x i , i = 1,...,m. Proposition 6.5. These dynamics constitute a G-dynamic in the sense of Definition 3.16. Proof. Summing over i yields m X i=1 ̇x i = X i̸=j λ ji (x)x j − X i̸=j λ ij (x)x i = 0, so total mass is preserved. The multiplicative structure follows by factoring x i . 6.4 Constrained Equilibrium under Switching Costs and Acceptance The introduction of switching costs and endogenous acceptance rules alters the equilibrium concept in a fundamental way. Without these constraints, equilibrium coincides with a Wardrop equilibrium and admits a classical variational inequality characterization over the simplex∆ m . With bilateral admissibility, equilibrium becomes constrained by state-dependent feasibility conditions, and the set of admissible deviations depends endogenously on the state. Recall that the net gain from switching from coalition i to coalition j is ∆ ij (x) = ρ j (x)− ρ i (x)− c ij (x). A deviation from i to j is admissible at x∈ ∆ m if x i > 0, ∆ ij (x) > 0 and A ij (x)≥ 0. The first condition enforces individual rationality. The second condition ensures that coalitionjweakly benefits from the proposed transfer at the margin. Definition 6.6. For x∈ ∆ m , define D(x) =e j − e i : x i > 0, ∆ ij (x) > 0 and A ij (x)≥ 0. The admissible deviation cone is T (x) = cone D(x) . Unlike the unconstrained case,T (x) is generally a strict subset of the tangent cone of ∆ m . Definition 6.7. A state x ⋆ ∈ ∆ m is a constrained equilibrium if ⟨ρ(x ⋆ ),d⟩≤ 0 for all d∈D(x ⋆ ). Equivalently, ⟨ρ(x ⋆ ),x− x ⋆ ⟩≤ 0 ∀x− x ⋆ ∈T (x ⋆ ). Thus no admissible bilateral deviation yields a first-order payoff improvement. Proposition 6.8. A statex ⋆ ∈ ∆ m is a constrained equilibrium if and only if for everyiwithx ⋆ i > 0and everyj ̸= i, either ρ j (x ⋆ )− c ij (x ⋆ )≤ ρ i (x ⋆ ), or A ij (x ⋆ ) < 0. Proof. If both inequalities fail, then ∆ ij (x ⋆ ) > 0 and A ij (x ⋆ )≥ 0, so d = e j − e i ∈D(x ⋆ ) and ⟨ρ(x ⋆ ),d⟩ = ρ j (x ⋆ )− ρ i (x ⋆ ) > 0, contradicting constrained equilibrium. The converse follows directly from the definition ofD(x ⋆ ). 26 Exit-and-Join Dynamics in Continuum Cooperative Games 6.4.1 Relation to Wardrop Equilibrium Let T ∆ m (x) denote the tangent cone of the simplex ∆ m at x, that is, T ∆ m (x) = ( d∈ R m : m X i=1 d i = 0, d i ≥ 0 whenever x i = 0 ) . Proposition 6.9. Suppose that (1) c ij (x)≡ 0 for all i,j, (2) A ij (x)≥ 0 for all i̸= j and all x∈ ∆ m . Then constrained equilibrium coincides with Wardrop equilibrium. Proof. If c ij ≡ 0 and acceptance is automatic, then a deviation i→ j is admissible whenever ρ j (x) > ρ i (x). Hence D(x) =e j − e i : ρ j (x) > ρ i (x). Thus a constrained equilibrium has no elementary transfer from a populated coalition to a coalition with strictly higher payoff. Equivalently, ρ j (x ⋆ )≤ ρ i (x ⋆ ) ∀i with x ⋆ i > 0, which is precisely the Wardrop equilibrium condition. Thus Wardrop equilibrium is a special case of constrained equilibrium under full feasibility of deviations. 6.4.2 Modified Variational Inequality In the absence of switching costs and acceptance constraints, equilibrium is characterized by the variational inequality ⟨ρ(x ⋆ ),x− x ⋆ ⟩≤ 0 ∀x∈ ∆ m , or equivalently, ⟨ρ(x ⋆ ),d⟩≤ 0 ∀d∈ T ∆ m (x ⋆ ), where T ∆ m (x ⋆ ) denotes the tangent cone of the simplex. This is the classical Wardrop condition. Under switching costs and acceptance rules, the feasible comparison set is restricted to the admissible deviation cone T (x ⋆ )⊆ T ∆ m (x ⋆ ). Accordingly, equilibrium satisfies the state-dependent variational inequality ⟨ρ(x ⋆ ),d⟩≤ 0 ∀d∈T (x ⋆ ). SinceT (x ⋆ )⊆ T ∆ m (x ⋆ )in general, the constrained equilibrium condition is weaker than the Wardrop condition and depends endogenously onx ⋆ . The problem is therefore a state-dependent variational inequality, i.e., a quasi-variational inequality (QVI). The classical variational inequality is equivalent to the normal cone inclusion −ρ(x ⋆ )∈ N ∆ m (x ⋆ ), where N ∆ m (x ⋆ ) is the normal cone of ∆ m =x∈ R m + : m X i=1 x i = 1. Explicitly, N ∆ m (x ⋆ ) =λ1− μ : λ∈ R, μ i ≥ 0, μ i x ⋆ i = 0. Hence Wardrop equilibrium is equivalent to the existence of λ∈ R such that ρ i (x ⋆ ) = λ if x ⋆ i > 0, ρ i (x ⋆ )≤ λ if x ⋆ i = 0. The scalar λ is the Lagrange multiplier associated with the mass constraint P i x i = 1. Under switching costs and acceptance rules, the dual inclusion becomes−ρ(x ⋆ )∈ N A(x ⋆ ) (x ⋆ ),whereA(x ⋆ )denotes the locally admissible feasible set generated byT (x ⋆ ). BecauseA(x ⋆ )depends on the state, the associated normal cone is state-dependent, and the equilibrium condition is a quasi-variational inequality. SinceT (x ⋆ ) ⊆ T ∆ m (x ⋆ ), we haveN ∆ m (x ⋆ ) ⊆ N A(x ⋆ ) (x ⋆ ),so constrained equilibrium allows payoff differentials that are ruled out under Wardrop equilibrium. 27 Exit-and-Join Dynamics in Continuum Cooperative Games 6.4.3 Structural differences from Wardrop equilibrium. The constrained equilibrium differs from the classical Wardrop equilibrium in three essential respects. Switching costs introduce wedges between payoffs at equilibrium. In particular, it may occur thatρ j (x ⋆ ) > ρ i (x ⋆ )for someiwith x ⋆ i > 0, provided that ρ j (x ⋆ )− ρ i (x ⋆ )≤ c ij (x ⋆ ). Thus payoff equalization across active coalitions is no longer necessary. Equilibrium permits bounded payoff differen- tials that are sustained by switching frictions. Acceptance rules generate endogenous entry restrictions. If A ij (x ⋆ ) < 0, coalitionjrejects entry from coalitioniat the margin. Consequently, coalitions may stabilize at interior sizes, even whenρ j (x ⋆ )exceeds the payoff of other coalitions. Equilibrium coalition sizes are therefore determined not only by payoff levels, but also by the marginal effect of transfers on coalition payoff. The admissible deviation coneT (x ⋆ ) depends explicitly on the equilibrium state. As a result, equilibrium is characterized by a state-dependent constrained variational inequality. Unlike the Wardrop case—where the feasible deviation set is the tangent cone of∆ m — the feasible directions here are determined endogenously by switching costs and marginal acceptance conditions. 6.4.4 Special Case: Concave Mass-Based Games Consider the mass-based case in whichv(S) = F (μ(S)),and assumeF : R + → Ris twice continuously differentiable. Recall that the induced payoff density satisfies ρ j (x j ) = 1 x j Z x j 0 F ′ (s)ds, x j > 0. Proposition 6.10. IfFis strictly concave, i.e.,F ′ < 0, then for every interior statex∈ ∆ m and every destination coalition j with x j > 0, A ij (x) < 0 for every i̸= j. Proof. Define R(x j ) = 1 x j R x j 0 F ′ (s)ds. Differentiating with respect to x j , dR dx j = x j F ′ (x j )− R x j 0 F ′ (s)ds x 2 j . By strict concavity ofF,F ′ is strictly decreasing, so R x j 0 F ′ (s)ds > x j F ′ (x j ). Hence the numerator is negative, and therefore ∂ρ j ∂x j (x) = dR dx j < 0.Since the mass-based payoffρ j depends only onx j , we haveA ij (x) = Dρ j (x)[e j −e i ] = dR/dx j < 0. By Proposition 6.10, interior coalitions strictly reject entry. Thus for any interior state, A ij (x) < 0 whenever x j > 0. Consequently, no admissible expansion of an interior coalition is possible. Theorem 6.11. Consider the mass-based casev(S) = F (μ(S)). AssumeFis strictly concave andc ij ≡ 0. Adopt the convention that an empty coalition has no incumbents and therefore accepts first entry automatically. Then: (1) For every interior state x∈ ∆ m and every j with x j > 0, A ij (x) < 0 for all i̸= j. Hence every active coalition strictly rejects marginal entry. (2)Ifxis an interior state, then no admissible deviation exists. Thus every interior state is a constrained equilibrium under the acceptance rule, even though it need not be a Wardrop equilibrium. (3) At a boundary statex ⋆ , admissible deviations can only target empty coalitions. Consequently,x ⋆ is a constrained equilibrium if and only if ρ j (x ⋆ )≤ ρ i (x ⋆ ) for every i with x ⋆ i > 0 and every j with x ⋆ j = 0. 28 Exit-and-Join Dynamics in Continuum Cooperative Games (4)Therefore acceptance-constrained equilibrium is generally weaker than Wardrop equilibrium and need not coincide with the maximizers of the potentialVunless additional acceptance or feasibility assumptions restore all payoff-improving directions. Proof. In the mass-based case v(S) = F (μ(S)), the induced payoff density is ρ j (x j ) = 1 x j Z x j 0 F ′ (s)ds, x j > 0. Define the potential V (x) = m X i=1 Z x i 0 ρ i (s)ds. A direct differentiation shows that∇V (x) = ρ(x). Assume F is strictly concave, so F ′ < 0 and F ′ is strictly decreasing. Differentiating ρ j yields ∂ρ j ∂x j (x) = x j F ′ (x j )− R x j 0 F ′ (s)ds x 2 j . Since F ′ is strictly decreasing, Z x j 0 F ′ (s)ds > x j F ′ (x j ), which implies ∂ρ j ∂x j (x) < 0 whenever x j > 0. Because ρ j depends only on x j , this is exactly A ij (x) < 0 for every i̸= j, proving statement (1). Ifxis interior, every destination coalition has positive mass. By statement (1), every possible destination rejects marginal entry. HenceD(x) = ∅, proving statement (2). At a boundary state, any admissible deviationi → jmust havex ⋆ i > 0and, by statement (1), cannot havex ⋆ j > 0. Thus it can only target an empty coalition. Since switching costs are zero and empty coalitions accept first entry by convention, such a deviation is admissible exactly whenρ j (x ⋆ ) > ρ i (x ⋆ ). Absence of admissible deviations is therefore equivalent to the inequality in statement (3). Wardrop equilibrium requires the same inequality against all coalitions, not only empty destinations. The acceptance rule removes all transfers into active coalitions from the feasible deviation set, so the constrained condition is generally weaker than Wardrop equilibrium and weaker than the first-order optimality condition for maximizingVover the whole simplex. This proves statement (4). 7 Numerical Studies in the Large-Population Regime This section illustrates the exit-and-join mechanism in a large population. The examples are not intended as calibration exercises. They are designed to make visible the qualitative mechanisms established above: payoff-responsive mass movement, convergence of the finite population process to the deterministic mean-field dynamics, and the persistence of payoff gaps when switching frictions constrain mobility. We consider four coalitions with payoff densities ρ i (x) = θ i − β i x i , i = 1,..., 4, whereθ = (1.25, 1.10, 0.92, 0.80)andβ = (1.80, 1.20, 0.90, 0.70). This specification captures diminishing returns within each coalition and is the gradient of the concave potentialV (x) = P i θ i x i − 1 2 P i β i x 2 i . The initial population state isx(0) = (0.62, 0.24, 0.10, 0.04), so the first coalition begins large but has low payoff because of congestion. Unless otherwise stated, switching uses the payoff-difference intensity λ ij (x) = κx j [ρ j (x)− ρ i (x)] + , κ = 3. The resulting mean-field dynamics are the replicator-type exit-and-join dynamics derived in Section 3. For the chosen parameters, the unique interior equilibrium and common active payoff are x ⋆ ≈ (0.302, 0.328, 0.237, 0.133), ρ ⋆ ≈ 0.707. 29 Exit-and-Join Dynamics in Continuum Cooperative Games 051015 0 0.2 0.4 0.6 t x i ( t ) C 1 C 2 C 3 C 4 (a) Coalition masses. 051015 0.2 0.4 0.6 0.8 t ρ i ( x ( t )) common payoff (b) Payoff equalization. 051015 0.8 0.85 0.9 t V ( x ( t )) (c) Surplus ascent. Figure 1: Mean-field exit-and-join dynamics. Mass leaves the initially congested coalition and reallocates toward coalitions with higher payoff density. The active payoff densities equalize at equilibrium, while the potential increases monotonically along the trajectory. 02468 0 2 4 6 ·10 −2 t ∥ x N ( t ) − x ( t ) ∥ 2 N = 200 N = 1000 N = 5000 (a) Mean path error. 10 2.5 10 3 10 3.5 10 −1.5 N mean sup error simulation N −1/2 (b) Error scaling. Figure 2: Finite-population approximation. The stochastic exit-and-join process concentrates around the deterministic trajectory asNgrows, and the mean sup-norm path error is close to theN −1/2 fluctuation scale. Each curve averages 24 simulated paths. Figure 1 shows the deterministic large-population dynamics. The first coalition initially has the largest mass but the lowest payoff density, so it loses agents. The other coalitions attract mass until the payoff densities are equalized. The potential rises throughout the adjustment, illustrating the Lyapunov structure in Theorem 4.8. To connect the deterministic equation with a finite but large population, letn N i (t)be the number of agents in coalitioni and letx N i (t) = n N i (t)/N. Over a short time interval, an agent in coalitioniswitches to coalitionjwith probability approximatelyλ ij (x N (t))∆t. This finite population Markov process has drift equal to the mean-field vector field, and its fluctuations vanish at the usual orderN −1/2 , consistent with classical law-of-large-numbers approximations for density-dependent population processes [16, 22]. Figure 2 displays this large-population approximation. ForN = 200, stochastic switching creates visible deviations from the ODE path. AtN = 1000andN = 5000, the paths concentrate much more tightly around the deterministic trajectory. The log-log comparison shows that the finite-population error decreases at approximately the N −1/2 scale, which is the expected order for aggregate fluctuations in a large population. The final experiment introduces a symmetric switching costc = 0.055. The switching intensity becomesλ ij (x) = κx j [ρ j (x)− ρ i (x)− c] + . The cost does not change the payoff field, but it changes which payoff differences are actionable. Hence the dynamics need not eliminate all raw payoff differences; they eliminate only net profitable deviations. Figure 3 illustrates the constrained equilibrium logic. In the costless case, payoff densities are equalized. With switching costs, the dynamics stop earlier: the raw payoff range is not eliminated, yet the maximum net gain from any admissible switch approaches zero. Numerically, the constrained steady state therefore matches the theory in Section 6. Frictions can sustain payoff differences that would be unstable under unrestricted Wardrop or exit-and-join equilibrium. 8 Noncooperative Cooperation vs. Cooperative Noncooperation This work reveals a structural duality between cooperation and noncooperation. Cooperative structures form the substrate: players organize into coalitions to accomplish tasks that generate mutual gains. Coalition formation creates 30 Exit-and-Join Dynamics in Continuum Cooperative Games 051015 0 0.2 0.4 0.6 solid: no cost dashed: cost t x i ( t ) (a) Mass trajectories. 051015 0 0.2 0.4 0.6 t payoff gap range, no cost range, cost max net gain cost (b) Payoff gaps. Figure 3: Switching costs and constrained stability. Without costs, payoff differences vanish. With switching cost c = 0.055, the raw payoff range remains positive, but the maximum net profitable deviation converges to zero. The terminal state is stable because remaining payoff gains are too small to justify switching. potential surplus and admits Pareto improvements. In this sense, cooperation defines the space of feasible collective value. Yet this cooperative substrate is governed by a noncooperative layer. Players retain unilateral strategic autonomy: if a more advantageous coalition becomes available, they may exit and join another group. Thus cooperation is sustained only insofar as it remains stable against individual deviation. Noncooperation defines the stability constraints imposed on cooperative arrangements. The duality arises because cooperation defines potential surplus, whereas noncooperation defines stability under deviation. An outcome may be efficient but unstable; conversely, a stable equilibrium may be inefficient. The tension between efficiency and stability is the core of the cooperation–noncooperation duality. There is therefore an inherent interplay between the two layers. Cooperation induces noncooperation: precisely because agents seek better cooperative outcomes, they must retain the option to deviate unilaterally. At the same time, noncooperative mobility can improve cooperation: unilateral deviations reconfigure coalition structures, sometimes increasing overall efficiency or enabling superior collective performance. Improvements may be local, but they can also propagate globally through structural reorganization. The relationship is thus dualistic rather than contradictory. Cooperation and noncooperation are intertwined mechanisms of the same system. One operates at the structural level (coalition value creation), the other at the strategic level (individual incentive compatibility). Their interaction admits a natural dual perspective: the cooperative problem specifies feasible collective structures, while the noncooperative problem governs admissible deviations within that feasible set. Historically, this tension has appeared in debates over collective action and governance. Hardin’s tragedy of the commons and Olson’s logic of collective action show how individually rational behavior can undermine shared resources and public goods [24, 25]. A pure cooperation view, by contrast, assumes that common interest suffices to sustain collective action. Ostrom’s common-pool resource framework asked a question closely aligned with the present framework: how can individually rational agents sustain cooperation without centralized enforcement [26]? Her empirical studies demonstrated that cooperation and noncooperation coexist within structured rule systems. Agents remain strategic and capable of deviation, yet institutions reshape the incentive landscape so that cooperative behavior becomes self- enforcing. In our terminology, institutions restrict the admissible deviation set and alter the effective payoff functional. They mediate the cooperation–noncooperation duality by aligning collective efficiency with individual stability. Our framework differs in emphasis but parallels her insight. Rather than assuming externally imposed rules, we show that decentralized exit–and–join dynamics can endogenously reconcile cooperative value creation with noncooperative stability. Cooperation need not be imposed from above; it can emerge from below through structured strategic mobility. Classical coordination games such as the Battle of the Sexes illustrate this duality. The objective is cooperative alignment, yet equilibrium selection proceeds through noncooperative reasoning. As emphasized by Adam Smith, decentralized self-interest can, under appropriate structure, generate cooperative order [27]. Cooperation defines what is desirable. Noncooperation determines what is sustainable. The equilibrium of a multi-agent system lies at their intersection. 31 Exit-and-Join Dynamics in Continuum Cooperative Games 9 Exit-and-Join and Evolution This section interprets the preceding model as an evolutionary system on coalition structures. The interpretation does not introduce a new equilibrium concept. Rather, it shows that the same payoff density that allocates cooperative value also acts as a selection index for coalition growth, in the sense of evolutionary population dynamics [15–19, 28–33]. 9.1 Marginal Contribution Densities and Fitness In the continuum cooperative model, a coalition is represented by a restricted measureμ S , and its value is generated by a functional V on measures. The local object that determines payoff is the functional derivative δV δμ (μ S )(i), which gives the marginal contribution density of an infinitesimal mass at player locationi. The Aumann–Shapley value averages this marginal contribution along a participation path [3, 4]: φ AS i (v) = Z 1 0 δV δμ (μ S λ )(i)dλ. Thus value allocation is determined by accumulated marginal productivity along a coalition-growth path. In evolutionary population models, the analogous local object is theG-function: the per-capita growth rate of a rare type inserted into a resident population [14, 15]. Written in measure terms, both objects are directional marginal values. The cooperative model uses the marginal value to allocate surplus; the evolutionary model uses it to determine growth. The correspondence is direct: coalition measures play the role of population distributions, the cooperative value functional plays the role of a fitness landscape, and the marginal contribution density plays the role of aG-function. This explains why the payoff density ρ i (x) in the exit-and-join model can be interpreted as a coalition fitness index. 9.2 Selection Dynamics Exit-and-join dynamics turn payoff comparisons into population movement. If agents leave lower-payoff coalitions for higher-payoff coalitions, then coalition mass is reallocated in the direction of higher marginal productivity. Under the pairwise payoff-difference rule λ ij (x) = κx j [ρ j (x)− ρ i (x)] + , the mean-field equation reduces to the replicator form familiar from evolutionary game theory [15, 16, 30–32], ̇x i = κx i ρ i (x)− ̄ρ(x) , ̄ρ(x) = X j x j ρ j (x). Coalitions with payoff above the population average expand, while those below the average contract. The replicator equation is only one specification. Other incentive-compatible switching rules preserve the same principle: mass moves toward higher payoff, with speed determined by the rate rule [16, 31, 32]. 9.3 Micro Incentives and Macro Evolution The evolutionary behavior is generated by agents, not by coalitions. Coalitions do not choose to grow or shrink. Agents compare their current payoff with feasible alternatives and move when a profitable destination is admissible. Aggregating these decentralized choices yields the deterministic mass dynamics derived in Section 3, paralleling classical law-of-large-numbers limits for Markov population processes [22]. This produces a two-level structure. Within each coalition, agents cooperate to generate transferable surplus. Across coalitions, the resulting payoff densities determine which coalitions attract mass. Cooperation therefore supplies the value functional, while mobility creates selection among cooperative arrangements, connecting coalition formation dynamics with population adjustment dynamics [9, 10, 16, 29, 31]. In this sense, Darwinian selection is an emergent property of exit-and-join incentives. It does not require a planner or an organization-level optimization rule. It requires only that agents can move in response to relative payoff differences. 32 Exit-and-Join Dynamics in Continuum Cooperative Games 9.4 Equilibrium and Constrained Stability The evolutionary interpretation also clarifies stability. In the unconstrained model, an exit-and-join equilibrium is a state in which no positive-mass group of agents can improve by moving to another coalition. Under incentive-compatible and strictly payoff-responsive switching rates, this condition coincides with stationarity of the mean-field dynamics. Equivalently, no active transition flux remains in a payoff-improving direction. With switching costs or acceptance constraints, stability becomes conditional on the feasible deviation set. Payoff differences may persist because a profitable move is too costly or because the destination coalition rejects marginal entry. The relevant notion is therefore constrained evolutionary stability: no admissible positive-mass deviation can invade the current coalition structure. This interpretation is close to the invasion-resistance logic of evolutionary stability and to viability and mutational viewpoints in constrained dynamical systems [28–30, 34, 35]. 10 Conclusion This paper developed a continuum model of coalition formation in nonatomic cooperative games. It extended the Aumann–Shapley and Aumann–Drèze values to finite coalition structures by treating each coalition as a restricted nonatomic game and assigning payoff densities through infinitesimal marginal contributions [3–6]. The paper then derived exit-and-join dynamics from decentralized switching rules. A finite-population approximation yields a deterministic mean-field ODE for coalition masses, and payoff-difference switching recovers replicator dynamics as a special case. The associated exit-and-join equilibrium rules out profitable positive-mass deviations and, under incentive-compatible and strictly payoff-responsive switching rates, coincides with stationarity of the mass dynamics [15–19, 22]. For mass-based cooperative games, the dynamics admit a Lyapunov function derived from aggregate cooperative surplus. Under strict concavity and the stated regularity and responsiveness assumptions, trajectories converge globally to the unique equilibrium. The same equilibrium condition is equivalent to Wardrop equilibrium in the induced nonatomic population game and admits a variational inequality formulation [20, 21, 23, 36]. Switching costs and acceptance rules restrict feasible deviations and lead to constrained equilibria. In that setting, payoff differences can persist because mobility is no longer unrestricted. 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