Exit-and-Join Dynamics for Decentralized Coalition Formation

# Exit–and–Join Dynamics for Decentralized Coalition Formation
Source: [https://arxiv.org/html/2606.19683](https://arxiv.org/html/2606.19683)
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 studies coalition formation as a decentralized dynamical process driven by unilateral exit–and–join decisions\. Agents evaluate local moves using the Aumann–Drèze value, so payoffs are computed within the agent’s current coalition rather than through a globally negotiated coalition structure\. The resulting model links cooperative payoff allocation with noncooperative best\-response behavior: a terminal partition is precisely a coalition structure with no admissible, individually profitable exit–and–join deviation\. We establish equilibrium characterizations, identify conditions under which the dynamics admit scalar Lyapunov or exact\-potential representations, and analyze how switching and acceptance costs shape local stability\. Numerical experiments test finite\-time stabilization, cost sensitivity, and a special convex\-game benchmark\.

*K*eywordscoalition formation⋅\\cdotAumann–Drèze value⋅\\cdotexit–and–join dynamics⋅\\cdotpotential games⋅\\cdotLyapunov analysis⋅\\cdotmulti\-agent systems

## 1Introduction

In many economic, social, and socio–technical systems, agents continuously move across organizational boundaries\. Individuals leave existing groups, join new ones, or form new organizations altogether\. These transitions are not orchestrated by a central authority; rather, they arise from a large number of decentralized decisions made by self\-interested agents\. As a result, coalition structures are not designed ex ante, but emerge endogenously from ongoing exit–and–join behavior, as in endogenous and dynamic coalition formation models\[[15](https://arxiv.org/html/2606.19683#bib.bib41),[27](https://arxiv.org/html/2606.19683#bib.bib43),[19](https://arxiv.org/html/2606.19683#bib.bib37),[32](https://arxiv.org/html/2606.19683#bib.bib50)\]\. Despite this reality, much of the existing literature models coalition formation as a top–down process\. Classical split–and–merge algorithms, hierarchical clustering methods, and centralized optimization approaches typically assume global information, coordinated decision making, or a planner capable of evaluating alternative coalition structures at the system level\[[28](https://arxiv.org/html/2606.19683#bib.bib48),[9](https://arxiv.org/html/2606.19683#bib.bib49)\]\. While such models are analytically convenient, they obscure the distributed nature of coalition formation observed in practice\.

This paper adopts a fundamentally different perspective\. We study coalition formation as a decentralized dynamical process driven by local exit–and–join decisions\. Each agent evaluates whether to remain in its current coalition, join another coalition, or form a singleton based solely on local payoff comparisons\. No agent requires knowledge of the global coalition structure, and no centralized coordination or negotiation is assumed\. Coalition structures therefore arise as the outcome of distributed individual decisions rather than as the solution to a centralized design problem\. This local\-decision viewpoint is close in spirit to hedonic coalition formation, where each agent evaluates coalitions through the members of its own group\[[10](https://arxiv.org/html/2606.19683#bib.bib40),[7](https://arxiv.org/html/2606.19683#bib.bib30)\]\. Extensions to settings with limited information, for example through learning or belief updating, fit naturally within this framework and are left for future work\[[8](https://arxiv.org/html/2606.19683#bib.bib9),[14](https://arxiv.org/html/2606.19683#bib.bib10)\]\.

Within this decentralized process, coalitions may dissolve when they no longer generate sufficient value for their members, while new coalitions may emerge as agents split off and reorganize\. The resulting patterns of formation, dissolution, and reconfiguration closely mirror those observed in firms, alliances, online communities, and modular socio–technical systems\. From a game–theoretic standpoint, coalition formation in this setting lies at the intersection of noncooperative and cooperative games\. Agents behave noncooperatively in the sense that they pursue individually rational improvements through unilateral exit–and–join moves\. At the same time, payoffs within each coalition are determined cooperatively through solution concepts such as the Aumann–Drèze or Owen values\. The coalition structure itself is thus an emergent object, shaped by individual incentives interacting with cooperative payoff allocation rules\[[3](https://arxiv.org/html/2606.19683#bib.bib23),[25](https://arxiv.org/html/2606.19683#bib.bib25),[26](https://arxiv.org/html/2606.19683#bib.bib24)\]\.

### 1\.1Relevant Applications

In multi–agent artificial intelligence systems, agents must decide which tasks, projects, or workflows to participate in\. An agent may initially join a collaborative task based on partial or local information and subsequently exit when a higher–value or better–matched task becomes available\. Such decisions are typically made autonomously and without centralized coordination, relying instead on local assessments of expected utility, resource requirements, or compatibility with other agents\. The resulting allocation of agents to tasks therefore emerges from distributed exit–and–join decisions rather than from a global planning mechanism\. In this interpretation, tasks correspond to coalitions, and acceptance reflects task capacity, compatibility, or coordination constraints\[[30](https://arxiv.org/html/2606.19683#bib.bib47),[28](https://arxiv.org/html/2606.19683#bib.bib48)\]\.

Labor markets provide a canonical example of exit–and–join dynamics\. Workers leave existing organizations and join new ones when they expect higher compensation, improved working conditions, or better career prospects\. These decisions are decentralized and often myopic, based on local information such as job offers, peer outcomes, and internal organizational policies\. Firms impose acceptance constraints through hiring decisions, compensation structures, and promotion rules\. Public policies and institutional factors, including hiring frictions, noncompete clauses, and labor regulations, directly affect switching costs and acceptance conditions, thereby shaping patterns of workforce mobility and talent flow\. The resulting organizational structure of the labor market emerges endogenously from individual incentives interacting with these constraints, paralleling coalition\-formation models in which admissible partitions are shaped by local incentives and institutional rules\[[15](https://arxiv.org/html/2606.19683#bib.bib41),[4](https://arxiv.org/html/2606.19683#bib.bib44)\]\.

Subscription markets, such as mobile service providers or digital streaming platforms, exhibit similar decentralized reconfiguration dynamics\. Consumers periodically reassess their memberships and may exit one service to join another based on price, quality, bundled offerings, or network effects\. Service providers influence acceptance indirectly through pricing tiers, contract terms, and service differentiation\. From a modeling perspective, platforms correspond to competing coalitions, while consumers act as agents who repeatedly evaluate whether to remain or switch\. Market structure, including user distribution and churn rates, thus arises from many individual exit–and–join decisions rather than from centralized coordination\.

Across these applications, coalition formation is driven by distributed individual decisions, shaped by local payoff comparisons and acceptance constraints, rather than by global optimization\. Coalitions form, grow, shrink, or disappear as agents respond to changing incentives, costs, and opportunities\. This common structure makes exit–and–join dynamics a unifying modeling framework for organizational formation and reconfiguration in both human and artificial systems\. The paper makes four contributions\. First, it formulates coalition formation as an asynchronous exit–and–join process in which agents use only coalition\-local Aumann–Drèze payoffs\. Second, it characterizes terminal coalition structures as fixed points of an induced noncooperative best\-response problem\. Third, it separates efficiency properties of the underlying cooperative game from dynamic implementability, showing that scalar Lyapunov and exact\-potential representations require explicit incentive\-alignment conditions\. Fourth, it uses numerical experiments to examine finite termination, switching and acceptance frictions, and a special convex benchmark in which the grand coalition is dynamically selected\.

## 2Related Work

Coalition formation has a long history in cooperative and noncooperative game theory\. Classical cooperative analysis studies stability and efficiency of coalitions through solution concepts such as the core and the Shapley value\[[13](https://arxiv.org/html/2606.19683#bib.bib33),[29](https://arxiv.org/html/2606.19683#bib.bib1),[20](https://arxiv.org/html/2606.19683#bib.bib18),[24](https://arxiv.org/html/2606.19683#bib.bib26),[22](https://arxiv.org/html/2606.19683#bib.bib27)\], while endogenous coalition formation models study how partitions arise from strategic behavior rather than being imposed exogenously\. Early and influential treatments include coalition formation through game forms and binding agreements\[[15](https://arxiv.org/html/2606.19683#bib.bib41),[27](https://arxiv.org/html/2606.19683#bib.bib43)\], sequential coalition formation with externalities\[[6](https://arxiv.org/html/2606.19683#bib.bib42)\], dynamic coalition formation processes\[[19](https://arxiv.org/html/2606.19683#bib.bib37)\], split–merge dynamics for Shapley\-fair coalition formation\[[32](https://arxiv.org/html/2606.19683#bib.bib50)\], and dynamic cooperative games\[[12](https://arxiv.org/html/2606.19683#bib.bib38),[5](https://arxiv.org/html/2606.19683#bib.bib39)\]\. The present paper is closest in spirit to this dynamic literature, but it differs by using the Aumann–Drèze value as the payoff rule at every intermediate partition and by focusing on local exit–and–join deviations with acceptance and switching costs\.

A second related line of work studies hedonic coalition formation, where each agent’s preferences depend only on the members of its own coalition\. Hedonic coalitions were introduced by Drèze and Greenberg\[[10](https://arxiv.org/html/2606.19683#bib.bib40)\], and stability notions for hedonic partitions were developed extensively in later work, including Bogomolnaia and Jackson’s analysis of hedonic coalition structures\[[7](https://arxiv.org/html/2606.19683#bib.bib30)\]\. Subsequent work has studied stable partitions and generic coalition\-formation procedures\[[2](https://arxiv.org/html/2606.19683#bib.bib20),[1](https://arxiv.org/html/2606.19683#bib.bib34)\], as well as simple models of core stability under unilateral or group deviations\[[4](https://arxiv.org/html/2606.19683#bib.bib44)\]\. Our model shares the locality of hedonic games because an agent’s payoff is computed from its current coalition\. However, preferences are not primitive: they are induced endogenously by a transferable\-utility game and a cooperative allocation rule, which makes it possible to connect local incentives to surplus and potential functions\.

The allocation rule used here builds directly on cooperative values with coalition structures\. The Aumann–Drèze value allocates worth within each coalition of a fixed partition\[[3](https://arxiv.org/html/2606.19683#bib.bib23)\], whereas the Owen value first treats coalitions as a priori unions in a quotient game and then allocates within each union\[[25](https://arxiv.org/html/2606.19683#bib.bib25),[26](https://arxiv.org/html/2606.19683#bib.bib24)\]\. Related cooperative models also study restrictions induced by networks or communication structures, as in Myerson’s graph\-restricted games\[[23](https://arxiv.org/html/2606.19683#bib.bib46)\], and strategic network formation models study how links and cooperation structures arise from individual incentives\[[18](https://arxiv.org/html/2606.19683#bib.bib19)\]\. These approaches provide alternative ways of embedding organizational constraints into payoff allocation\. The present paper uses the Aumann–Drèze value because its coalition\-local structure is compatible with decentralized exit–and–join updates and with local acceptance tests performed by destination coalitions\.

There is also a substantial computational and multi\-agent literature on coalition formation\. Coalition\-structure generation algorithms typically seek a partition that maximizes total value or provides approximation guarantees under large search spaces\[[28](https://arxiv.org/html/2606.19683#bib.bib48),[9](https://arxiv.org/html/2606.19683#bib.bib49)\]\. Task\-allocation methods for multi\-agent systems use coalition formation to assign agents to tasks under resource and compatibility constraints\[[30](https://arxiv.org/html/2606.19683#bib.bib47)\], coalitional control uses cooperative\-game tools for networked control and coordination\[[11](https://arxiv.org/html/2606.19683#bib.bib11)\], and learning\-based approaches study coalition formation under uncertainty\[[8](https://arxiv.org/html/2606.19683#bib.bib9)\]\. By contrast, the exit–and–join dynamics studied here do not solve a centralized coalition\-structure generation problem\. They describe a decentralized adjustment process in which agents move myopically, destination coalitions apply local acceptance constraints, and convergence is established through Lyapunov or potential arguments rather than through global optimization\.

Finally, the convergence analysis is related to potential games and learning dynamics\. Potential games show how unilateral incentives can be represented by a single scalar function\[[21](https://arxiv.org/html/2606.19683#bib.bib45)\], and cooperative\-game work has connected values, potentials, and consistency conditions\[[16](https://arxiv.org/html/2606.19683#bib.bib8),[17](https://arxiv.org/html/2606.19683#bib.bib21)\]\. Recent work on learning in coalitional games also emphasizes distributed adaptation and convergence\[[14](https://arxiv.org/html/2606.19683#bib.bib10),[31](https://arxiv.org/html/2606.19683#bib.bib35)\]\. The contribution here is to identify when Aumann–Drèze exit–and–join incentives admit exact or ordinal alignment with coalition surplus\. This separation clarifies why cooperative convexity can imply efficiency of the grand coalition without guaranteeing that myopic decentralized dynamics will select it\.

## 3Aumann–Drèze Value

### 3\.1Motivation

LetN=\{1,…,n\}N=\\\{1,\\dots,n\\\}be a finite set of agents and letv:2N→ℝv:2^\{N\}\\to\\mathbb\{R\}be a transferable–utility \(TU\) cooperative game withv​\(∅\)=0v\(\\varnothing\)=0\. The Shapley valueϕ​\(v\)\\phi\(v\)\[[29](https://arxiv.org/html/2606.19683#bib.bib1)\]presumes that the grand coalitionNNforms and that the entire surplusv​\(N\)v\(N\)is distributed among agents according to expected marginal contributions averaged over all permutations of players\. Under this assumption, all potential synergies encoded in the characteristic functionvvare fully realizable, regardless of any intermediate organizational, institutional, or technological constraints\.

In many socio–technical, economic, and organizational systems, however, cooperation is constrained by an existing*coalition structure*T=\{T1,…,Tm\}∈Π​\(N\)T=\\\{T\_\{1\},\\dots,T\_\{m\}\\\}\\in\\Pi\(N\), which specifies the coalitions within which binding agreements, transfers, and coordination are feasible\. Agents belonging to different coalitions inTTcannot form binding subcoalitions or share surplus directly, even if the characteristic functionvvassigns positive value to such cross–coalition cooperation\. Examples include firms with rigid departmental boundaries, political alliances, supply–chain consortia, or modular cyber–physical systems in which coordination across subsystems is costly or infeasible\. The Aumann–Drèze \(AD\) value\[[3](https://arxiv.org/html/2606.19683#bib.bib23)\]is designed precisely for this setting\. It provides a Shapley–consistent allocation rule that respects the constraints imposed by a given coalition structure\. Rather than assuming that the grand coalition forms, the AD value conditions on the coalition structureTTand allocates surplus*within*each coalition\.

Conceptually, the Aumann–Drèze value can be understood as a two–stage construction with a*fixed*first stage\. In the first stage, the coalition structureTTis taken as exogenously given and no bargaining or strategic interaction takes place across coalitions\. In the second stage, each coalitionTjT\_\{j\}independently distributes its worthv​\(Tj\)v\(T\_\{j\}\)among its members according to the Shapley value of the restricted gamev\|Tjv\_\{\|T\_\{j\}\}\. Thus, while marginal contributions are averaged over permutations of players within each coalition, no marginal contributions across distinct coalitions are ever considered\. As a consequence, surplus generated outside a coalition is not shared, and potential cross–coalition synergies encoded invvremain unrealized\. The Aumann–Drèze allocation therefore generally differs from the Shapley value whenever the characteristic function exhibits complementarities across distinct coalitions\.

The Owen value\[[25](https://arxiv.org/html/2606.19683#bib.bib25),[26](https://arxiv.org/html/2606.19683#bib.bib24)\]also admits a two–stage interpretation, but of a fundamentally different nature\. In the first stage, coalitions inTTbargain strategically as aggregate players in a*quotient game*induced by the original characteristic functionvv\. This stage explicitly internalizes cross–coalition externalities at the level of coalitions\. In the second stage, the surplus allocated to each coalition is distributed internally among its members according to the Shapley value\. The Owen value coincides with the Aumann–Drèze value only in special cases, most notably when the game is*additively separable across the coalition structure*TT, so that no strategic externalities arise between distinct coalitions\[[3](https://arxiv.org/html/2606.19683#bib.bib23),[25](https://arxiv.org/html/2606.19683#bib.bib25)\]\. In general, however, the Owen value internalizes cross–coalition interactions at the coalition level, whereas the Aumann–Drèze value does not\.

From a dynamic perspective, the Aumann–Drèze value is particularly well suited to modeling decentralized coalition formation and adaptation\. Because payoffs depend only on local coalition membership, agents can evaluate unilateral exit–and–join deviations without requiring global coordination, coalition–level renegotiation, or computation of a quotient game\. This locality property makes the Aumann–Drèze value a natural foundation for the exit–and–join dynamics studied in this paper, where coalition structures evolve endogenously through myopic, individually rational moves subject to acceptance and switching costs\.

### 3\.2Definition

LetN=\{1,…,n\}N=\\\{1,\\dots,n\\\}be a finite set of players and letv:2N→ℝv:2^\{N\}\\to\\mathbb\{R\}be a transferable–utility cooperative game withv​\(∅\)=0v\(\\varnothing\)=0\. LetΠ​\(N\)\\Pi\(N\)denote the set of all partitions ofNN\. For any coalition structureT=\{T1,…,Tm\}∈Π​\(N\)T=\\\{T\_\{1\},\\dots,T\_\{m\}\\\}\\in\\Pi\(N\)and any playeri∈Ni\\in N, denote byCT​\(i\)∈TC\_\{T\}\(i\)\\in Tthe unique coalition containingii\. GivenTTandii, the restriction ofvvtoCT​\(i\)C\_\{T\}\(i\)is defined by

v\|CT\(i\)​\(S\):=v​\(S\),∀S⊆CT​\(i\)\.v\_\{\|C\_\{T\}\(i\)\}\(S\):=v\(S\),\\qquad\\forall\\,S\\subseteq C\_\{T\}\(i\)\.\(3\.1\)
###### Example 3\.1\(Restriction of the game to a coalition\)\.

LetN=\{1,2,3\}N=\\\{1,2,3\\\}and consider the transferable–utility gamev:2N→ℝv:2^\{N\}\\to\\mathbb\{R\}defined by the coalition values in Table[1](https://arxiv.org/html/2606.19683#S3.T1)\.

Table 1:Characteristic function values of the TU gamevvConsider the coalition structureT=\{\{1,2\},\{3\}\}∈Π​\(N\)T=\\\{\\\{1,2\\\},\\\{3\\\}\\\}\\in\\Pi\(N\)\. Under the partitionTT, the restricted games are obtained by restricting the domain ofvvto subsets of each coalition\.

Restricted game on\{1,2\}\\\{1,2\\\}\.

Restricted game on\{3\}\\\{3\\\}\.

Thus, relative to the coalition structureTT, the original gamevvdecomposes into two independent subgames: a two\-player game on\{1,2\}\\\{1,2\\\}and a singleton game on\{3\}\\\{3\\\}\.

###### Definition 3\.2\(Aumann–Drèze value\)\.

The*Aumann–Drèze value*ofvvrelative to the coalition structureTTis the payoff vectorΩ​\(v;T\)=\(Ωi​\(v;T\)\)i∈N\\Omega\(v;T\)=\\bigl\(\\Omega\_\{i\}\(v;T\)\\bigr\)\_\{i\\in N\}defined componentwise by

Ωi​\(v;T\)=ϕi​\(v\|CT\(i\)\),i∈N,\\Omega\_\{i\}\(v;T\)=\\phi\_\{i\}\\\!\\left\(v\_\{\|C\_\{T\}\(i\)\}\\right\),\\qquad i\\in N,\(3\.2\)whereϕ​\(⋅\)\\phi\(\\cdot\)denotes the Shapley value\.

Equivalently, for any coalitionTj∈TT\_\{j\}\\in Tand any playeri∈Tji\\in T\_\{j\},

Ωi​\(v;T\)=∑S⊆Tj∖\{i\}\|S\|\!​\(\|Tj\|−\|S\|−1\)\!\|Tj\|\!​\[v​\(S∪\{i\}\)−v​\(S\)\]\.\\Omega\_\{i\}\(v;T\)=\\sum\_\{S\\subseteq T\_\{j\}\\setminus\\\{i\\\}\}\\frac\{\|S\|\!\\,\(\|T\_\{j\}\|\-\|S\|\-1\)\!\}\{\|T\_\{j\}\|\!\}\\bigl\[v\(S\\cup\\\{i\\\}\)\-v\(S\)\\bigr\]\.\(3\.3\)
###### Example 3\.3\(Computation of the Aumann–Drèze value\)\.

Using \([3\.3](https://arxiv.org/html/2606.19683#S3.E3)\) for the partitionT=\{\{1,2\},\{3\}\}T=\\\{\\\{1,2\\\},\\\{3\\\}\\\}, we compute the Aumann–Drèze value\.

For agent11,

Ω1​\(v;T\)=∑S⊆\{2\}\|S\|\!​\(2−\|S\|−1\)\!2\!​\[v​\(S∪\{1\}\)−v​\(S\)\],\\Omega\_\{1\}\(v;T\)=\\sum\_\{S\\subseteq\\\{2\\\}\}\\frac\{\|S\|\!\\,\(2\-\|S\|\-1\)\!\}\{2\!\}\\bigl\[v\(S\\cup\\\{1\\\}\)\-v\(S\)\\bigr\],which yields

SSWeightv​\(S∪\{1\}\)−v​\(S\)v\(S\\cup\\\{1\\\}\)\-v\(S\)Contribution∅\\varnothing12\\frac\{1\}\{2\}00\{2\}\\\{2\\\}12\\frac\{1\}\{2\}4422HenceΩ1​\(v;T\)=2\\Omega\_\{1\}\(v;T\)=2\. By symmetry,Ω2​\(v;T\)=2\\Omega\_\{2\}\(v;T\)=2\. SinceCT​\(3\)=\{3\}C\_\{T\}\(3\)=\\\{3\\\}is a singleton coalition,Ω3​\(v;T\)=0\\Omega\_\{3\}\(v;T\)=0\. ThusΩ​\(v;T\)=\(2,2,0\)\\Omega\(v;T\)=\(2,2,0\)\.

###### Proposition 3\.4\(Coalitional efficiency\)\.

For every coalitionTj∈TT\_\{j\}\\in T,

∑i∈TjΩi​\(v;T\)=v​\(Tj\)\.\\sum\_\{i\\in T\_\{j\}\}\\Omega\_\{i\}\(v;T\)=v\(T\_\{j\}\)\.\(3\.4\)

###### Proof\.

For eachTj∈TT\_\{j\}\\in T, the vector\(Ωi​\(v;T\)\)i∈Tj\(\\Omega\_\{i\}\(v;T\)\)\_\{i\\in T\_\{j\}\}coincides with the Shapley value of the restricted gamev\|Tjv\_\{\|T\_\{j\}\}\. Since the Shapley value is efficient, the sum equalsv​\(Tj\)v\(T\_\{j\}\)\. ∎

###### Example 3\.5\(Coalitional efficiency in the running example\)\.

For the coalition\{1,2\}\\\{1,2\\\},Ω1​\(v;T\)\+Ω2​\(v;T\)=2\+2=4=v​\(\{1,2\}\)\\Omega\_\{1\}\(v;T\)\+\\Omega\_\{2\}\(v;T\)=2\+2=4=v\(\\\{1,2\\\}\), and for the singleton coalition\{3\}\\\{3\\\},Ω3​\(v;T\)=0=v​\(\{3\}\)\\Omega\_\{3\}\(v;T\)=0=v\(\\\{3\\\}\)\. Thus \([3\.4](https://arxiv.org/html/2606.19683#S3.E4)\) holds for every coalition inTT\.

## 4Exit–and–Join Rules under Aumann–Drèze

### 4\.1State space and notation

LetN=\{1,…,n\}N=\\\{1,\\dots,n\\\}be a finite set of agents and letv:2N→ℝv:2^\{N\}\\to\\mathbb\{R\}be a transferable–utility \(TU\) cooperative game withv​\(∅\)=0v\(\\varnothing\)=0\. The system state at timettis a coalition structureTt∈Π​\(N\)T\_\{t\}\\in\\Pi\(N\), whereΠ​\(N\)\\Pi\(N\)denotes the set of all partitions ofNN\. For each agenti∈Ni\\in N, letCTt​\(i\)∈TtC\_\{T\_\{t\}\}\(i\)\\in T\_\{t\}denote the unique coalition containingiiat timett\. Payoffs are evaluated using the Aumann–Drèze value:Ωi​\(v;Tt\)=ϕi​\(v\|CTt​\(i\)\)\\Omega\_\{i\}\(v;T\_\{t\}\)=\\phi\_\{i\}\\\!\\left\(v\\big\|\_\{C\_\{T\_\{t\}\}\(i\)\}\\right\), that is, agentiireceives its Shapley value in the game restricted to its current coalition\. The evolution of the system is driven by unilateral exit–and–join decisions of individual agents\.

### 4\.2Exit–and–join actions

###### Definition 4\.1\(Exit–and–join action\)\.

LetTt∈Π​\(N\)T\_\{t\}\\in\\Pi\(N\)be a coalition structure at timettand leti∈Ni\\in Nbe an agent\. An*exit–and–join action*of agentiiat timettconsists of the selection of a destination

D∈𝒟i​\(Tt\):=\(Tt∖\{CTt​\(i\)\}\)∪\{∅\},D\\in\\mathcal\{D\}\_\{i\}\(T\_\{t\}\):=\\bigl\(T\_\{t\}\\setminus\\\{C\_\{T\_\{t\}\}\(i\)\\\}\\bigr\)\\cup\\\{\\varnothing\\\},whereCTt​\(i\)C\_\{T\_\{t\}\}\(i\)denotes the unique coalition ofTtT\_\{t\}containing agentii\.

###### Definition 4\.2\(Exit–and–join transition\)\.

Given a coalition structureTt∈Π​\(N\)T\_\{t\}\\in\\Pi\(N\), an agenti∈Ni\\in N, and a destinationD∈𝒟i​\(Tt\)D\\in\\mathcal\{D\}\_\{i\}\(T\_\{t\}\), the*exit–and–join transition*induced by agentiiand destinationDDis the coalition structure

Tt\+1=Fi​\(Tt,D\):=\(Tt∖\{CTt​\(i\),D\}\)∪\{CTt​\(i\)∖\{i\}\}∗∪\{D∪\{i\}\},T\_\{t\+1\}=F\_\{i\}\(T\_\{t\},D\):=\\bigl\(T\_\{t\}\\setminus\\\{C\_\{T\_\{t\}\}\(i\),D\\\}\\bigr\)\\cup\\bigl\\\{C\_\{T\_\{t\}\}\(i\)\\setminus\\\{i\\\}\\bigr\\\}^\{\\\!\*\}\\cup\\\{D\\cup\\\{i\\\}\\\},\(4\.1\)where\{⋅\}∗\\\{\\cdot\\\}^\{\\\!\*\}denotes inclusion only if the set is nonempty, and whereD∪\{i\}=\{i\}D\\cup\\\{i\\\}=\\\{i\\\}whenD=∅D=\\varnothing\.

The mappingFi:Π​\(N\)×𝒟i​\(Tt\)→Π​\(N\)F\_\{i\}:\\Pi\(N\)\\times\\mathcal\{D\}\_\{i\}\(T\_\{t\}\)\\to\\Pi\(N\)defines a well–posed discrete–time state transition on the space of coalition structures\. IfD=∅D=\\varnothing, the transition corresponds to agentiiexiting its current coalition and forming a singleton\. IfD∈Tt∖\{CTt​\(i\)\}D\\in T\_\{t\}\\setminus\\\{C\_\{T\_\{t\}\}\(i\)\\\}, the transition corresponds to agentiiexiting its current coalition and joining the destination coalition\. If agentiiis initially a singleton and joins a nonempty destination, the singleton coalition disappears\. In all cases, the transition modifies at most two coalitions and preserves the partition structure ofΠ​\(N\)\\Pi\(N\)\.

### 4\.3Acceptance rule

Exit–and–join moves may be subject to acceptance by the destination coalition\.

###### Definition 4\.3\(Acceptance rule\)\.

Given a proposed moveTt→Fi​\(Tt,D\)T\_\{t\}\\to F\_\{i\}\(T\_\{t\},D\), the destination coalitionD∈TtD\\in T\_\{t\}accepts agentiiif

Ωℓ​\(v;Fi​\(Tt,D\)\)≥Ωℓ​\(v;Tt\),∀ℓ∈D\.\\Omega\_\{\\ell\}\(v;F\_\{i\}\(T\_\{t\},D\)\)\\geq\\Omega\_\{\\ell\}\(v;T\_\{t\}\),\\qquad\\forall\\ell\\in D\.

A costly\-admission variant is useful when incumbents incur onboarding, coordination, or congestion costs from admitting a new member\. In that case a scalar acceptance costκ≥0\\kappa\\geq 0modifies the rule toΩℓ​\(v;Fi​\(Tt,D\)\)−κ≥Ωℓ​\(v;Tt\)\\Omega\_\{\\ell\}\(v;F\_\{i\}\(T\_\{t\},D\)\)\-\\kappa\\geq\\Omega\_\{\\ell\}\(v;T\_\{t\}\)for everyℓ∈D\\ell\\in D\. The baseline acceptance rule corresponds toκ=0\\kappa=0, while larger values ofκ\\kappamake coalitions more selective even when entry would not lower their gross Aumann–Drèze payoffs\. A particularly simple benchmark is*automatic consent*, in which all destination coalitions accept all entrants\. This corresponds to the case in which coalition membership is unrestricted and agents are free to join any coalition unilaterally\. Formally, the acceptance condition is vacuous and every proposed exit–and–join move is admissible subject only to the deviating agent’s own incentive\.

### 4\.4Switching costs

While exit–and–join actions describe the set of feasible coalition reconfigurations, actual coalition changes may involve frictions that affect an agent’s willingness to deviate\. Such frictions may arise from coordination effort, renegotiation of internal transfers, loss of institutional capital, or adjustment delays associated with entering a new coalition\. To capture these effects, we associate a switching cost with each exit–and–join transition\.

###### Definition 4\.4\(Switching cost\)\.

For any coalition structureTt∈Π​\(N\)T\_\{t\}\\in\\Pi\(N\), agenti∈Ni\\in N, and destinationD∈𝒟i​\(Tt\)D\\in\\mathcal\{D\}\_\{i\}\(T\_\{t\}\), the switching cost incurred by agentiiwhen executing the exit–and–join transitionTt⟶Fi​\(Tt,D\)T\_\{t\}\\longrightarrow F\_\{i\}\(T\_\{t\},D\)is given by a functionci​\(Tt,Fi​\(Tt,D\)\)≥0c\_\{i\}\\bigl\(T\_\{t\},F\_\{i\}\(T\_\{t\},D\)\\bigr\)\\geq 0\.

The switching cost is allowed to depend on the current coalition structure, the identity of the moving agent, and the resulting coalition structure, thereby accommodating heterogeneous frictions and history–dependent adjustment costs\. A fundamental benchmark is the frictionless case in which switching is costless\. Throughout much of the analysis, we therefore consider the zero–cost regime defined by

ci​\(Tt,Fi​\(Tt,D\)\)≡0for all​i∈N,Tt∈Π​\(N\),D∈𝒟i​\(Tt\)\.c\_\{i\}\\bigl\(T\_\{t\},F\_\{i\}\(T\_\{t\},D\)\\bigr\)\\equiv 0\\qquad\\text\{for all \}i\\in N,\\;T\_\{t\}\\in\\Pi\(N\),\\;D\\in\\mathcal\{D\}\_\{i\}\(T\_\{t\}\)\.This regime isolates the incentive effects induced purely by the Aumann–Drèze payoff and yields the simplest form of exit–and–join dynamics\. Positive switching costs will be introduced subsequently to refine equilibrium selection and to strengthen stability and convergence properties of the induced dynamical system\.

### 4\.5Decision rule

Given a coalition structureTt∈Π​\(N\)T\_\{t\}\\in\\Pi\(N\)at timett, each agent evaluates potential exit–and–join actions by comparing its payoff under the current coalition structure with the payoff induced by the corresponding transition, net of any switching costs\. The evaluation is unilateral in the sense that each agent treats the induced coalition structure as fixed when assessing the consequences of a deviation\.

###### Definition 4\.5\(Aumann–Drèze exit–and–join rule\)\.

At timett, agenti∈Ni\\in Naccepts a destinationD∈𝒟i​\(Tt\)D\\in\\mathcal\{D\}\_\{i\}\(T\_\{t\}\)if the induced exit–and–join transitionTt→Fi​\(Tt,D\)T\_\{t\}\\to F\_\{i\}\(T\_\{t\},D\)satisfies

Ωi​\(v;Fi​\(Tt,D\)\)−ci​\(Tt,Fi​\(Tt,D\)\)\>Ωi​\(v;Tt\),\\Omega\_\{i\}\\bigl\(v;F\_\{i\}\(T\_\{t\},D\)\\bigr\)\-c\_\{i\}\\bigl\(T\_\{t\},F\_\{i\}\(T\_\{t\},D\)\\bigr\)\>\\Omega\_\{i\}\(v;T\_\{t\}\),and the destination coalitionDDsatisfies the acceptance condition\. If multiple destinations satisfy this inequality, agentiiselects a destination that maximizes its net payoffΩi​\(v;Fi​\(Tt,D\)\)−ci​\(Tt,Fi​\(Tt,D\)\)\\Omega\_\{i\}\(v;F\_\{i\}\(T\_\{t\},D\)\)\-c\_\{i\}\(T\_\{t\},F\_\{i\}\(T\_\{t\},D\)\)\. If no destination is admissible, the agent remains in its current coalition and the state remains unchanged, so thatTt\+1=TtT\_\{t\+1\}=T\_\{t\}\.

The resulting behavior is myopic in the sense that agents optimize their own Aumann–Drèze payoff at each step without anticipating future coalition reconfigurations or strategic responses by other agents\. Consequently, the exit–and–join rule induces a discrete–time, asynchronous best–response dynamics on the finite state spaceΠ​\(N\)\\Pi\(N\)\.

## 5Exit–and–Join Equilibrium

We now formalize the equilibrium notion induced by unilateral exit–and–join behavior under the Aumann–Drèze value and examine its structural properties\.

###### Definition 5\.1\(Exit–and–join equilibrium\)\.

A coalition structureT⋆∈Π​\(N\)T^\{\\star\}\\in\\Pi\(N\)is an*exit–and–join equilibrium*under the Aumann–Drèze value if there exists no agenti∈Ni\\in Nand no destinationD∈𝒟i​\(T⋆\)D\\in\\mathcal\{D\}\_\{i\}\(T^\{\\star\}\)such that the exit–and–join transitionT⋆→Fi​\(T⋆,D\)T^\{\\star\}\\to F\_\{i\}\(T^\{\\star\},D\)is admissible and

Ωi​\(v;Fi​\(T⋆,D\)\)−ci​\(T⋆,Fi​\(T⋆,D\)\)\>Ωi​\(v;T⋆\)\.\\Omega\_\{i\}\\\!\\left\(v;F\_\{i\}\(T^\{\\star\},D\)\\right\)\-c\_\{i\}\\\!\\left\(T^\{\\star\},F\_\{i\}\(T^\{\\star\},D\)\\right\)\>\\Omega\_\{i\}\(v;T^\{\\star\}\)\.

Equivalently, no agent can strictly improve its Aumann–Drèze payoff by a unilateral exit–and–join deviation that is accepted by the destination coalition\.

###### Example 5\.2\(Exit–and–join deviation in the running example\)\.

We continue with the transferable–utility game, coalition structure, and Aumann–Drèze payoff vector introduced in Examples[3\.1](https://arxiv.org/html/2606.19683#S3.Thmtheorem1),[3\.3](https://arxiv.org/html/2606.19683#S3.Thmtheorem3), and[3\.5](https://arxiv.org/html/2606.19683#S3.Thmtheorem5)\. At timett, the coalition structure isTt=\{\{1,2\},\{3\}\}T\_\{t\}=\\bigl\\\{\\\{1,2\\\},\\\{3\\\}\\bigr\\\}, with associated Aumann–Drèze payoffsΩ​\(v;Tt\)=\(2,2,0\)\\Omega\(v;T\_\{t\}\)=\(2,2,0\)\. We examine unilateral exit–and–join actions under zero switching cost and automatic acceptance\.

Consider first agent22\. SinceCTt​\(2\)=\{1,2\}C\_\{T\_\{t\}\}\(2\)=\\\{1,2\\\}, the feasible destinations are𝒟2​\(Tt\)=\{\{3\},∅\}\\mathcal\{D\}\_\{2\}\(T\_\{t\}\)=\\\{\\\{3\\\},\\varnothing\\\}\. ChoosingD=\{3\}D=\\\{3\\\}yieldsTt\+1=F2​\(Tt,\{3\}\)=\{\{1\},\{2,3\}\}T\_\{t\+1\}=F\_\{2\}\(T\_\{t\},\\\{3\\\}\)=\\bigl\\\{\\\{1\\\},\\\{2,3\\\}\\bigr\\\}\. The restricted game on\{2,3\}\\\{2,3\\\}has zero worth, hence

Ω2​\(v;F2​\(Tt,\{3\}\)\)=0<Ω2​\(v;Tt\)=2\.\\Omega\_\{2\}\\bigl\(v;F\_\{2\}\(T\_\{t\},\\\{3\\\}\)\\bigr\)=0<\\Omega\_\{2\}\(v;T\_\{t\}\)=2\.Thus this deviation is not profitable\. Next consider agent33\. SinceCTt​\(3\)=\{3\}C\_\{T\_\{t\}\}\(3\)=\\\{3\\\}, agent33may chooseD=\{1,2\}D=\\\{1,2\\\}, yieldingTt\+1=F3​\(Tt,\{1,2\}\)=\{\{1,2,3\}\}T\_\{t\+1\}=F\_\{3\}\(T\_\{t\},\\\{1,2\\\}\)=\\bigl\\\{\\\{1,2,3\\\}\\bigr\\\}\. For this coalition structure, the Aumann–Drèze value coincides with the Shapley value of the grand coalition, and therefore

Ω3​\(v;F3​\(Tt,\{1,2\}\)\)=ϕ3​\(v\)\>Ω3​\(v;Tt\)=0\.\\Omega\_\{3\}\\bigl\(v;F\_\{3\}\(T\_\{t\},\\\{1,2\\\}\)\\bigr\)=\\phi\_\{3\}\(v\)\>\\Omega\_\{3\}\(v;T\_\{t\}\)=0\.Hence agent33admits a profitable exit–and–join deviation, andTt=\{\{1,2\},\{3\}\}T\_\{t\}=\\\{\\\{1,2\\\},\\\{3\\\}\\\}is not an exit–and–join equilibrium\.

### 5\.1Best–Response and Nash Equilibrium Interpretation

Exit–and–join equilibrium admits a precise noncooperative interpretation as a best–response fixed point and, equivalently, as a pure–strategy Nash equilibrium of an induced, state–dependent game defined at each coalition structure\.

###### Definition 5\.3\(Induced exit–and–join game\)\.

Fix a coalition structureT∈Π​\(N\)T\\in\\Pi\(N\)\. The*induced exit–and–join game*atTTis the noncooperative game

𝒢​\(T\)=\(N,\{Ai​\(T\)\}i∈N,\{ui​\(⋅;T\)\}i∈N\),\\mathcal\{G\}\(T\)=\\bigl\(N,\\\{A\_\{i\}\(T\)\\\}\_\{i\\in N\},\\\{u\_\{i\}\(\\cdot;T\)\\\}\_\{i\\in N\}\\bigr\),where the player set isNN\. The action set of playeriiis the accepted\-action set

Ai​\(T\):=\{CT​\(i\)\}∪\{D∈𝒟i​\(T\):D=∅​or​D​accepts​i\};A\_\{i\}\(T\):=\\\{C\_\{T\}\(i\)\\\}\\cup\\Bigl\\\{D\\in\\mathcal\{D\}\_\{i\}\(T\):D=\\varnothing\\text\{ or \}D\\text\{ accepts \}i\\Bigr\\\};choosingD=CT​\(i\)D=C\_\{T\}\(i\)is the stay action, withFi​\(T,CT​\(i\)\):=TF\_\{i\}\(T,C\_\{T\}\(i\)\):=Tandci​\(T,T\):=0c\_\{i\}\(T,T\):=0\. The payoff to playeriifrom choosingD∈Ai​\(T\)D\\in A\_\{i\}\(T\)is

ui​\(D;T\):=Ωi​\(v;Fi​\(T,D\)\)−ci​\(T,Fi​\(T,D\)\)\.u\_\{i\}\(D;T\):=\\Omega\_\{i\}\\bigl\(v;F\_\{i\}\(T,D\)\\bigr\)\-c\_\{i\}\\bigl\(T,F\_\{i\}\(T,D\)\\bigr\)\.Actions chosen by other agents are not jointly implemented and therefore do not enter the payoff function\.

The best–response correspondence of agentiiat coalition structureTTis

BRi​\(T\):=arg⁡maxD∈Ai​\(T\)⁡ui​\(D;T\)\.\\mathrm\{BR\}\_\{i\}\(T\):=\\arg\\max\_\{D\\in A\_\{i\}\(T\)\}u\_\{i\}\(D;T\)\.
###### Lemma 5\.4\(Best–response characterization\)\.

A coalition structureT⋆T^\{\\star\}is an exit–and–join equilibrium if and only if, for every agenti∈Ni\\in N,

CT⋆​\(i\)∈BRi​\(T⋆\)\.C\_\{T^\{\\star\}\}\(i\)\\in\\mathrm\{BR\}\_\{i\}\(T^\{\\star\}\)\.

###### Proof\.

\(⇒\\Rightarrow\) Suppose thatT⋆T^\{\\star\}is an exit–and–join equilibrium\. By Definition[5\.1](https://arxiv.org/html/2606.19683#S5.Thmtheorem1), there exists no destinationD∈𝒟i​\(T⋆\)D\\in\\mathcal\{D\}\_\{i\}\(T^\{\\star\}\)such that the exit–and–join transitionT⋆→Fi​\(T⋆,D\)T^\{\\star\}\\to F\_\{i\}\(T^\{\\star\},D\)is admissible and yields a strictly higher payoff for agentii\. Equivalently,

ui​\(D;T⋆\)≤ui​\(CT⋆​\(i\);T⋆\),∀D∈Ai​\(T⋆\)\.u\_\{i\}\(D;T^\{\\star\}\)\\leq u\_\{i\}\\bigl\(C\_\{T^\{\\star\}\}\(i\);T^\{\\star\}\\bigr\),\\qquad\\forall\\,D\\in A\_\{i\}\(T^\{\\star\}\)\.Since choosingD=CT⋆​\(i\)D=C\_\{T^\{\\star\}\}\(i\)leaves the coalition structure unchanged and incurs zero switching cost, it attains the maximum ofui​\(⋅;T⋆\)u\_\{i\}\(\\cdot;T^\{\\star\}\)\. HenceCT⋆​\(i\)∈BRi​\(T⋆\)C\_\{T^\{\\star\}\}\(i\)\\in\\mathrm\{BR\}\_\{i\}\(T^\{\\star\}\)\.

\(⇐\\Leftarrow\) Conversely, suppose that for every agentii,CT⋆​\(i\)∈BRi​\(T⋆\)C\_\{T^\{\\star\}\}\(i\)\\in\\mathrm\{BR\}\_\{i\}\(T^\{\\star\}\)\. Then no destinationD∈𝒟i​\(T⋆\)D\\in\\mathcal\{D\}\_\{i\}\(T^\{\\star\}\)yields a strictly higher payoff than remaining in the current coalition\. In particular, there exists no admissible exit–and–join deviation that strictly improves agentii’s payoff\. Since this holds for every agent,T⋆T^\{\\star\}admits no profitable admissible exit–and–join deviation and is therefore an exit–and–join equilibrium\. ∎

###### Proposition 5\.5\(Exit–and–join equilibrium as Nash equilibrium\)\.

A coalition structureT⋆∈Π​\(N\)T^\{\\star\}\\in\\Pi\(N\)is an exit–and–join equilibrium if and only if the action profile

Di⋆:=CT⋆​\(i\),i∈N,D\_\{i\}^\{\\star\}:=C\_\{T^\{\\star\}\}\(i\),\\qquad i\\in N,is a pure–strategy Nash equilibrium of the induced game𝒢​\(T⋆\)\\mathcal\{G\}\(T^\{\\star\}\)\.

###### Proof\.

Fix the coalition structureT⋆T^\{\\star\}and consider the induced game𝒢​\(T⋆\)\\mathcal\{G\}\(T^\{\\star\}\)\. By construction, the payoff of agentiidepends only on its own actionDi∈Ai​\(T⋆\)D\_\{i\}\\in A\_\{i\}\(T^\{\\star\}\)and not on the actions of other agents\.

An action profile\(Di⋆\)i∈N\(D\_\{i\}^\{\\star\}\)\_\{i\\in N\}is therefore a pure–strategy Nash equilibrium of𝒢​\(T⋆\)\\mathcal\{G\}\(T^\{\\star\}\)if and only if, for every agentii,

Di⋆∈arg⁡maxD∈Ai​\(T⋆\)⁡ui​\(D;T⋆\),D\_\{i\}^\{\\star\}\\in\\arg\\max\_\{D\\in A\_\{i\}\(T^\{\\star\}\)\}u\_\{i\}\(D;T^\{\\star\}\),that is,Di⋆∈BRi​\(T⋆\)D\_\{i\}^\{\\star\}\\in\\mathrm\{BR\}\_\{i\}\(T^\{\\star\}\)\.

SubstitutingDi⋆=CT⋆​\(i\)D\_\{i\}^\{\\star\}=C\_\{T^\{\\star\}\}\(i\), this condition is equivalent, by Lemma[5\.4](https://arxiv.org/html/2606.19683#S5.Thmtheorem4), toT⋆T^\{\\star\}being an exit–and–join equilibrium\. ∎

### 5\.2Grand Coalition and Efficiency

###### Proposition 5\.6\(Grand coalition stability under convexity\)\.

If the cooperative gamevvis convex and switching costs are nonnegative, then the grand coalition\{N\}\\\{N\\\}is an exit–and–join equilibrium under the Aumann–Drèze value\.

###### Proof\.

Consider the coalition structureT=\{N\}T=\\\{N\\\}\. Under the Aumann–Drèze value, each agent’s payoff coincides with its Shapley value in the full game:

Ωi​\(v;\{N\}\)=ϕi​\(v\),i∈N\.\\Omega\_\{i\}\(v;\\\{N\\\}\)=\\phi\_\{i\}\(v\),\\qquad i\\in N\.
WhenT=\{N\}T=\\\{N\\\}, the only exit–and–join destination available to agentiiisD=∅D=\\varnothing, which makesiia singleton\. The resulting payoff isv​\(\{i\}\)v\(\\\{i\\\}\)\. For a convex cooperative game, marginal contributions are increasing with coalition size\. Therefore, for every predecessor setS⊆N∖\{i\}S\\subseteq N\\setminus\\\{i\\\},

v​\(S∪\{i\}\)−v​\(S\)≥v​\(\{i\}\)−v​\(∅\)=v​\(\{i\}\)\.v\(S\\cup\\\{i\\\}\)\-v\(S\)\\geq v\(\\\{i\\\}\)\-v\(\\varnothing\)=v\(\\\{i\\\}\)\.The Shapley valueϕi​\(v\)\\phi\_\{i\}\(v\)is an average of these marginal contributions, soϕi​\(v\)≥v​\(\{i\}\)\\phi\_\{i\}\(v\)\\geq v\(\\\{i\\\}\)\. Nonnegative switching costs can only reduce the net payoff from exit\. Hence no agent has a strictly profitable unilateral deviation from the grand coalition\. Hence the grand coalition\{N\}\\\{N\\\}is an exit–and–join equilibrium\. ∎

###### Lemma 5\.7\(Efficiency of the grand coalition\)\.

Ifvvis convex, then the grand coalition\{N\}\\\{N\\\}maximizes total coalition surplus

V​\(T\):=∑C∈Tv​\(C\)\.V\(T\):=\\sum\_\{C\\in T\}v\(C\)\.If, in addition,v​\(A∪B\)\>v​\(A\)\+v​\(B\)v\(A\\cup B\)\>v\(A\)\+v\(B\)for every pair of nonempty disjoint coalitionsA,B⊆NA,B\\subseteq N, then the maximizer is unique\.

###### Proof\.

Convexity implies superadditivity of the cooperative game: for any disjoint coalitionsA,B⊆NA,B\\subseteq N,

v​\(A∪B\)≥v​\(A\)\+v​\(B\)\.v\(A\\cup B\)\\;\\geq\\;v\(A\)\+v\(B\)\.
LetT=\{C1,…,Cm\}T=\\\{C\_\{1\},\\dots,C\_\{m\}\\\}be an arbitrary coalition structure\. By repeated application of superadditivity,

v​\(N\)=v​\(⋃k=1mCk\)≥∑k=1mv​\(Ck\)=V​\(T\)\.v\(N\)=v\\\!\\left\(\\bigcup\_\{k=1\}^\{m\}C\_\{k\}\\right\)\\;\\geq\\;\\sum\_\{k=1\}^\{m\}v\(C\_\{k\}\)=V\(T\)\.ThusV​\(T\)≤V​\(\{N\}\)=v​\(N\)V\(T\)\\leq V\(\\\{N\\\}\)=v\(N\)for everyT∈Π​\(N\)T\\in\\Pi\(N\)\.

If the displayed strict superadditivity condition holds, then every nontrivial partition admits a merge that strictly increases total surplus\. Repeating the merge argument givesV​\(T\)<V​\(\{N\}\)V\(T\)<V\(\\\{N\\\}\)for everyT≠\{N\}T\\neq\\\{N\\\}\. ∎

Convexity therefore guarantees efficiency of the grand coalition, but not uniqueness of exit–and–join equilibrium\.

###### Example 5\.8\(Nonuniqueness in a convex game\)\.

Letv​\(S\)=∑i∈Saiv\(S\)=\\sum\_\{i\\in S\}a\_\{i\}withai∈ℝa\_\{i\}\\in\\mathbb\{R\}\. This game is modular and hence convex\. For every coalition structureTTand every playerii, the Aumann–Drèze payoff isΩi​\(v;T\)=ai\\Omega\_\{i\}\(v;T\)=a\_\{i\}, independent of the coalition containingii\. Under zero switching costs, no exit–and–join move gives a strict improvement\. Thus every partition is an exit–and–join equilibrium\. This example shows that convexity alone does not imply uniqueness or dynamic selection of the grand coalition\.

## 6Lyapunov Analysis of Aumann–Drèze Exit–and–Join Dynamics

This section studies the dynamic properties of the Aumann–Drèze exit–and–join process\. While the previous section characterized exit–and–join equilibria and their game–theoretic interpretation, it did not address whether such equilibria are reached by the induced dynamics\. Here we analyze the evolution of coalition structures under unilateral exit–and–join moves and establish convergence properties using Lyapunov arguments\.

### 6\.1Exit–and–Join Dynamics with Acceptance

We formulate the Aumann–Drèze exit–and–join process as a discrete–time, asynchronous dynamical system on the space of coalition structures\. Algorithmic representations are introduced afterward to make the dynamics explicit\.

#### 6\.1\.1Dynamical system formulation

LetΠ​\(N\)\\Pi\(N\)denote the finite set of coalition structures onNN\. At each time stept∈ℕt\\in\\mathbb\{N\}, the system state is a coalition structureTt∈Π​\(N\)T\_\{t\}\\in\\Pi\(N\)\. GivenTtT\_\{t\}, an agenti∈Ni\\in Nmay select a destination coalitionD∈𝒟i​\(Tt\)D\\in\\mathcal\{D\}\_\{i\}\(T\_\{t\}\)and induce the exit–and–join transitionTt\+1=Fi​\(Tt,D\)T\_\{t\+1\}=F\_\{i\}\(T\_\{t\},D\), as defined in Definition[4\.2](https://arxiv.org/html/2606.19683#S4.Thmtheorem2)\. Payoffs are evaluated using the Aumann–Drèze valueΩ​\(v;⋅\)\\Omega\(v;\\cdot\)\.

An exit–and–join action\(i,D\)\(i,D\)is said to be*profitable*at stateTtT\_\{t\}if

Ωi​\(v;Fi​\(Tt,D\)\)−ci​\(Tt,Fi​\(Tt,D\)\)\>Ωi​\(v;Tt\)\.\\Omega\_\{i\}\\bigl\(v;F\_\{i\}\(T\_\{t\},D\)\\bigr\)\-c\_\{i\}\\bigl\(T\_\{t\},F\_\{i\}\(T\_\{t\},D\)\\bigr\)\>\\Omega\_\{i\}\(v;T\_\{t\}\)\.\(6\.1\)
The destination coalitionD∈TtD\\in T\_\{t\}is said to*accept*agentiiif

Ωℓ​\(v;Fi​\(Tt,D\)\)≥Ωℓ​\(v;Tt\),∀ℓ∈D\.\\Omega\_\{\\ell\}\\bigl\(v;F\_\{i\}\(T\_\{t\},D\)\\bigr\)\\geq\\Omega\_\{\\ell\}\(v;T\_\{t\}\),\\qquad\\forall\\ell\\in D\.\(6\.2\)
An exit–and–join transitionTt→Fi​\(Tt,D\)T\_\{t\}\\to F\_\{i\}\(T\_\{t\},D\)is called*admissible*if both \([6\.1](https://arxiv.org/html/2606.19683#S6.E1)\) and \([6\.2](https://arxiv.org/html/2606.19683#S6.E2)\) hold\. If multiple admissible actions exist, one is selected according to a specified activation rule \(deterministic or stochastic\)\. If no admissible action exists, the state remains unchanged\.

This defines a discrete–time, asynchronous dynamical systemTt\+1∈ℱ​\(Tt\)T\_\{t\+1\}\\in\\mathcal\{F\}\(T\_\{t\}\), whereℱ:Π​\(N\)⇉Π​\(N\)\\mathcal\{F\}:\\Pi\(N\)\\rightrightarrows\\Pi\(N\)is the set–valued transition map collecting all admissible Aumann–Drèze exit–and–join moves fromTtT\_\{t\}\.

#### 6\.1\.2Algorithmic description

The above dynamics can be summarized procedurally by the following algorithmic template\.

Algorithm 1Aumann–Drèze Exit–and–Join Dynamics with Acceptance1:Initialize coalition structure

T∈Π​\(N\)T\\in\\Pi\(N\)
2:Specify an agent activation rule

3:whilethere exists an admissible exit–and–join action at

TTdo

4:Select an active agent

i∈Ni\\in Naccording to the activation rule

5:Compute the feasible destination set

𝒟i​\(T\)\\mathcal\{D\}\_\{i\}\(T\)
6:Initialize admissible destination set

𝒜i←∅\\mathcal\{A\}\_\{i\}\\leftarrow\\varnothing
7:foreach

D∈𝒟i​\(T\)D\\in\\mathcal\{D\}\_\{i\}\(T\)do

8:Compute candidate coalition structure

T′←Fi​\(T,D\)T^\{\\prime\}\\leftarrow F\_\{i\}\(T,D\)
9:if

Ωi​\(v;T′\)−ci​\(T,T′\)\>Ωi​\(v;T\)\\Omega\_\{i\}\(v;T^\{\\prime\}\)\-c\_\{i\}\(T,T^\{\\prime\}\)\>\\Omega\_\{i\}\(v;T\)then

10:if

Ωℓ​\(v;T′\)≥Ωℓ​\(v;T\)​∀ℓ∈D\\Omega\_\{\\ell\}\(v;T^\{\\prime\}\)\\geq\\Omega\_\{\\ell\}\(v;T\)\\ \\forall\\,\\ell\\in Dthen

11:

𝒜i←𝒜i∪\{D\}\\mathcal\{A\}\_\{i\}\\leftarrow\\mathcal\{A\}\_\{i\}\\cup\\\{D\\\}
12:endif

13:endif

14:endfor

15:if

𝒜i≠∅\\mathcal\{A\}\_\{i\}\\neq\\varnothingthen

16:Select a destination

D∈𝒜iD\\in\\mathcal\{A\}\_\{i\}
17:Update coalition structure

T←Fi​\(T,D\)T\\leftarrow F\_\{i\}\(T,D\)
18:endif

19:endwhile

20:returnterminal coalition structure

TT

The terminal coalition structure reached by the Aumann–Drèze exit–and–join dynamics is generally not unique and may depend on the initial condition\. When a strict Lyapunov or potential certificate exists, convergence occurs in finite time, but the terminal partition is determined by the particular sequence of admissible local improvements realized along the trajectory\. Different initial coalition structures may therefore lead to distinct exit–and–join equilibria, each locally stable but potentially yielding different aggregate surplus levels\.

### 6\.2Marginal Alignment and Scalar Lyapunov Monotonicity

We now identify conditions under which the local incentive structure induced by the Aumann–Drèze value admits a scalar representation in terms of aggregate coalition surplus\. Convexity of the cooperative game is important for efficiency of the grand coalition, but it does not by itself imply that every individually profitable exit–and–join move increases aggregate coalition surplus\. A scalar Lyapunov argument requires an additional alignment condition between individual incentives and surplus changes\.

###### Definition 6\.2\(Ordinal marginal alignment\)\.

The Aumann–Drèze exit–and–join incentives satisfy*ordinal marginal alignment*with coalition surplus if, for every coalition structureT∈Π​\(N\)T\\in\\Pi\(N\), every agenti∈Ni\\in N, and every destinationD∈𝒟i​\(T\)D\\in\\mathcal\{D\}\_\{i\}\(T\),

sign⁡\(Ωi​\(v;Fi​\(T,D\)\)−Ωi​\(v;T\)\)\\displaystyle\\operatorname\{sign\}\\\!\\left\(\\Omega\_\{i\}\\bigl\(v;F\_\{i\}\(T,D\)\\bigr\)\-\\Omega\_\{i\}\(v;T\)\\right\)\(6\.3\)=sign⁡\(v​\(D∪\{i\}\)−v​\(D\)−\[v​\(CT​\(i\)\)−v​\(CT​\(i\)∖\{i\}\)\]\)\.\\displaystyle\\quad=\\operatorname\{sign\}\\\!\\left\(v\(D\\cup\\\{i\\\}\)\-v\(D\)\-\\bigl\[v\(C\_\{T\}\(i\)\)\-v\(C\_\{T\}\(i\)\\setminus\\\{i\\\}\)\\bigr\]\\right\)\.

#### 6\.2\.1Coalition surplus as a scalar Lyapunov function

Recall the coalition–surplus functionV:Π​\(N\)→ℝV:\\Pi\(N\)\\to\\mathbb\{R\},V​\(T\):=∑C∈Tv​\(C\)V\(T\):=\\sum\_\{C\\in T\}v\(C\)\.

###### Theorem 6\.3\(Scalar Lyapunov monotonicity under marginal alignment\)\.

Suppose Aumann–Drèze exit–and–join incentives satisfy ordinal marginal alignment with coalition surplus and switching costs are nonnegative\. Then, along any admissible exit–and–join transitionT→Fi​\(T,D\)T\\to F\_\{i\}\(T,D\), the coalition–surplus function strictly increases:V​\(Fi​\(T,D\)\)\>V​\(T\)V\\bigl\(F\_\{i\}\(T,D\)\\bigr\)\>V\(T\)\.

###### Proof\.

LetC=CT​\(i\)C=C\_\{T\}\(i\)be the coalition containing agentiiunderTT\. Only the coalitionsCCandDDare affected by the transition, so

V​\(Fi​\(T,D\)\)−V​\(T\)=\[v​\(D∪\{i\}\)−v​\(D\)\]−\[v​\(C\)−v​\(C∖\{i\}\)\]\.V\\bigl\(F\_\{i\}\(T,D\)\\bigr\)\-V\(T\)=\\bigl\[v\(D\\cup\\\{i\\\}\)\-v\(D\)\\bigr\]\-\\bigl\[v\(C\)\-v\(C\\setminus\\\{i\\\}\)\\bigr\]\.\(6\.4\)
If the transition is admissible, thenΩi​\(v;Fi​\(T,D\)\)−ci​\(T,Fi​\(T,D\)\)\>Ωi​\(v;T\)\\Omega\_\{i\}\\bigl\(v;F\_\{i\}\(T,D\)\\bigr\)\-c\_\{i\}\\bigl\(T,F\_\{i\}\(T,D\)\\bigr\)\>\\Omega\_\{i\}\(v;T\)\. Becauseci≥0c\_\{i\}\\geq 0, the deviating agent’s Aumann–Drèze payoff strictly increases\. Ordinal marginal alignment therefore impliesv​\(D∪\{i\}\)−v​\(D\)\>v​\(C\)−v​\(C∖\{i\}\)v\(D\\cup\\\{i\\\}\)\-v\(D\)\>v\(C\)\-v\(C\\setminus\\\{i\\\}\)\. Substituting into \([6\.4](https://arxiv.org/html/2606.19683#S6.E4)\) yieldsV​\(Fi​\(T,D\)\)\>V​\(T\)\.V\(F\_\{i\}\(T,D\)\)\>V\(T\)\.∎

#### 6\.2\.2Grand coalition and efficiency

We now formalize the relationship between convexity, efficiency, and equilibrium under exit–and–join dynamics\.

###### Proposition 6\.4\(Efficiency of the grand coalition under convexity\)\.

Suppose the cooperative gamevvis convex\. Then the coalition–surplus functionV​\(T\)=∑C∈Tv​\(C\)V\(T\)=\\sum\_\{C\\in T\}v\(C\)is maximized overΠ​\(N\)\\Pi\(N\)by the grand coalitionTgc=\{N\}T^\{\\mathrm\{gc\}\}=\\\{N\\\}\. The maximizer is unique ifv​\(A∪B\)\>v​\(A\)\+v​\(B\)v\(A\\cup B\)\>v\(A\)\+v\(B\)for every pair of nonempty disjoint coalitionsA,B⊆NA,B\\subseteq N\.

###### Proof\.

Convexity implies superadditivity: for any disjoint coalitionsS,T⊆NS,T\\subseteq N,v​\(S∪T\)≥v​\(S\)\+v​\(T\)v\(S\\cup T\)\\geq v\(S\)\+v\(T\)\. By repeated application, any partitionT=\{C1,…,Cm\}T=\\\{C\_\{1\},\\dots,C\_\{m\}\\\}ofNNsatisfiesV​\(T\)=∑k=1mv​\(Ck\)≤v​\(N\)=V​\(\{N\}\)V\(T\)=\\sum\_\{k=1\}^\{m\}v\(C\_\{k\}\)\\leq v\(N\)=V\(\\\{N\\\}\), so the grand coalition is globally efficient\. Under the stated strict superadditivity condition, every nontrivial partition can be improved by at least one merge, which gives uniqueness\. ∎

###### Proposition 6\.5\(Grand coalition as an exit–and–join equilibrium\)\.

Suppose the cooperative gamevvis convex and switching costs are nonnegative\. Then the grand coalition\{N\}\\\{N\\\}is an exit–and–join equilibrium under the Aumann–Drèze value\.

###### Proof\.

This is Proposition[5\.6](https://arxiv.org/html/2606.19683#S5.Thmtheorem6)\. UnderT=\{N\}T=\\\{N\\\}, an agent’s only exit–and–join destination is the singleton optionD=∅D=\\varnothing\. Convexity implies that the Shapley value in the grand coalition weakly dominates the singleton payoff, and nonnegative switching costs preclude a strictly profitable exit\. ∎

### 6\.3Analysis for General Cooperative Games

We now develop the Lyapunov analysis for exit–and–join dynamics in the setting of a general transferable–utility cooperative game\. In the absence of convexity or other surplus–alignment assumptions, scalar Lyapunov functions need not exist\. Acceptance still gives a local monotonicity property for the deviating agent and the destination coalition, but global convergence requires an additional no\-cycle certificate, such as a scalar Lyapunov or potential function\.

#### 6\.3\.1Restricted monotonicity under admissible transitions

Under the acceptance rule, admissible exit–and–join transitions satisfy a*restricted monotonicity*property with respect to the Aumann–Drèze payoff vector\. This property is local to the deviating agent and the destination coalition and does not imply Pareto improvement of the full payoff vector\.

###### Lemma 6\.7\(Restricted payoff improvement\)\.

IfT→Fi​\(T,D\)T\\to F\_\{i\}\(T,D\)is an admissible exit–and–join transition, then the deviating agentiistrictly improves its payoff,Ωi​\(v;Fi​\(T,D\)\)\>Ωi​\(v;T\)\\Omega\_\{i\}\\bigl\(v;F\_\{i\}\(T,D\)\\bigr\)\>\\Omega\_\{i\}\(v;T\), and every agentℓ∈D\\ell\\in Dweakly improves its payoff,Ωℓ​\(v;Fi​\(T,D\)\)≥Ωℓ​\(v;T\)\\Omega\_\{\\ell\}\\bigl\(v;F\_\{i\}\(T,D\)\\bigr\)\\geq\\Omega\_\{\\ell\}\(v;T\)\. No restriction is imposed on the payoffs of agents outside\{i\}∪D\\\{i\\\}\\cup D, including members of the origin coalitionCT​\(i\)∖\{i\}C\_\{T\}\(i\)\\setminus\\\{i\\\}\.

###### Proof\.

The strict improvement of agentiifollows from admissibility of the move\. Weak improvement of agents inDDfollows from the acceptance rule\. No conditions are imposed on other agents\. ∎

Lemma[6\.7](https://arxiv.org/html/2606.19683#S6.Thmtheorem7)explicitly shows that admissible exit–and–join transitions do*not*preserve the Pareto order on the full payoff vector\. Because members of the origin coalition need not be protected, the restricted improvement property alone does not provide a global acyclicity argument\. The next result states the certificate needed for finite termination\.

#### 6\.3\.2Absence of cycles and finite termination

###### Proposition 6\.8\(Acyclicity under a strict Lyapunov certificate\)\.

Suppose there exists a functionL:Π​\(N\)→ℝL:\\Pi\(N\)\\to\\mathbb\{R\}such thatL​\(Fi​\(T,D\)\)\>L​\(T\)L\\bigl\(F\_\{i\}\(T,D\)\\bigr\)\>L\(T\)for every admissible exit–and–join transitionT→Fi​\(T,D\)T\\to F\_\{i\}\(T,D\)\. Then the induced state transition graph onΠ​\(N\)\\Pi\(N\)is acyclic\. In particular, every trajectory terminates in finite time at an exit–and–join equilibrium\.

###### Proof\.

Suppose, by contradiction, that the dynamics admit a cycleT0⟶T1⟶⋯⟶TK=T0T\_\{0\}\\longrightarrow T\_\{1\}\\longrightarrow\\cdots\\longrightarrow T\_\{K\}=T\_\{0\}\. The strict Lyapunov condition givesL​\(T0\)<L​\(T1\)<⋯<L​\(TK\)=L​\(T0\)L\(T\_\{0\}\)<L\(T\_\{1\}\)<\\cdots<L\(T\_\{K\}\)=L\(T\_\{0\}\), a contradiction\. Thus no directed cycle exists\. SinceΠ​\(N\)\\Pi\(N\)is finite, every trajectory reaches a state with no outgoing admissible transition\. By Definition[5\.1](https://arxiv.org/html/2606.19683#S5.Thmtheorem1), such a state is an exit–and–join equilibrium\. ∎

###### Theorem 6\.9\(Finite termination under marginal alignment\)\.

If Aumann–Drèze exit–and–join incentives satisfy ordinal marginal alignment with coalition surplus and switching costs are nonnegative, then the exit–and–join dynamics with acceptance terminate in finite time at a coalition structure admitting no admissible exit–and–join deviation\.

###### Proof\.

By Theorem[6\.3](https://arxiv.org/html/2606.19683#S6.Thmtheorem3), the coalition\-surplus functionV​\(T\)=∑C∈Tv​\(C\)V\(T\)=\\sum\_\{C\\in T\}v\(C\)strictly increases along every admissible transition\. Applying Proposition[6\.8](https://arxiv.org/html/2606.19683#S6.Thmtheorem8)withL=VL=Vgives acyclicity and finite termination\. ∎

Thus, acceptance identifies admissible local improvements, while marginal alignment or an exact potential structure supplies the global monotonicity needed to guarantee convergence\.

#### 6\.3\.3Equilibrium Characterization

###### Proposition 6\.10\(Equilibrium and Nash optimality\)\.

LetT⋆∈Π​\(N\)T^\{\\star\}\\in\\Pi\(N\)be a coalition structure\. The following conditions are equivalent:T⋆T^\{\\star\}is an exit–and–join equilibrium; the action profileDi⋆:=CT⋆​\(i\)D\_\{i\}^\{\\star\}:=C\_\{T^\{\\star\}\}\(i\),i∈Ni\\in N, is a pure–strategy Nash equilibrium of the induced game𝒢​\(T⋆\)\\mathcal\{G\}\(T^\{\\star\}\); and there exists no admissible exit–and–join transitionT⋆→Fi​\(T⋆,D\)T^\{\\star\}\\to F\_\{i\}\(T^\{\\star\},D\)such thatΩi​\(v;Fi​\(T⋆,D\)\)−ci​\(T⋆,Fi​\(T⋆,D\)\)\>Ωi​\(v;T⋆\)\\Omega\_\{i\}\\bigl\(v;F\_\{i\}\(T^\{\\star\},D\)\\bigr\)\-c\_\{i\}\\bigl\(T^\{\\star\},F\_\{i\}\(T^\{\\star\},D\)\\bigr\)\>\\Omega\_\{i\}\(v;T^\{\\star\}\)\.

###### Proof\.

*\(1\)⇔\\Leftrightarrow\(2\)\.*By Proposition[5\.5](https://arxiv.org/html/2606.19683#S5.Thmtheorem5),T⋆T^\{\\star\}is an exit–and–join equilibrium if and only if each agent’s current coalitionCT⋆​\(i\)C\_\{T^\{\\star\}\}\(i\)is a best response in the induced game𝒢​\(T⋆\)\\mathcal\{G\}\(T^\{\\star\}\)\. This is precisely the definition of a pure–strategy Nash equilibrium\.

*\(2\)⇔\\Leftrightarrow\(3\)\.*A deviationD∈𝒟i​\(T⋆\)D\\in\\mathcal\{D\}\_\{i\}\(T^\{\\star\}\)is profitable in the induced game if and only ifΩi​\(v;Fi​\(T⋆,D\)\)−ci​\(T⋆,Fi​\(T⋆,D\)\)\>Ωi​\(v;T⋆\)\\Omega\_\{i\}\\bigl\(v;F\_\{i\}\(T^\{\\star\},D\)\\bigr\)\-c\_\{i\}\\bigl\(T^\{\\star\},F\_\{i\}\(T^\{\\star\},D\)\\bigr\)\>\\Omega\_\{i\}\(v;T^\{\\star\}\)\. Hence the Nash equilibrium condition is equivalent to the absence of admissible exit–and–join deviations that strictly improve the deviating agent’s payoff, which is exactly the definition of exit–and–join equilibrium\. ∎

###### Definition 6\.11\(Exact marginal alignment\)\.

A cooperative gamevvsatisfies*exact marginal alignment*under the Aumann–Drèze value if, for every coalition structureTT, every agenti∈Ni\\in N, and every destination coalitionD∈𝒟i​\(T\)D\\in\\mathcal\{D\}\_\{i\}\(T\),

Ωi​\(v;Fi​\(T,D\)\)−Ωi​\(v;T\)=v​\(D∪\{i\}\)−v​\(D\)−\[v​\(CT​\(i\)\)−v​\(CT​\(i\)∖\{i\}\)\]\.\\Omega\_\{i\}\\bigl\(v;F\_\{i\}\(T,D\)\\bigr\)\-\\Omega\_\{i\}\(v;T\)=v\(D\\cup\\\{i\\\}\)\-v\(D\)\-\\bigl\[v\(C\_\{T\}\(i\)\)\-v\(C\_\{T\}\(i\)\\setminus\\\{i\\\}\)\\bigr\]\.

###### Example 6\.12\(Exact marginal alignment in a linear public–good game\)\.

LetN=\{1,…,n\}N=\\\{1,\\dots,n\\\}and consider the transferable–utility gamev​\(S\)=∑i∈Sai\+β​\|S\|v\(S\)=\\sum\_\{i\\in S\}a\_\{i\}\+\\beta\\,\|S\|, whereai∈ℝa\_\{i\}\\in\\mathbb\{R\}andβ≥0\\beta\\geq 0\. For anyS⊆T⊆N∖\{i\}S\\subseteq T\\subseteq N\\setminus\\\{i\\\},v​\(S∪\{i\}\)−v​\(S\)=ai\+β=v​\(T∪\{i\}\)−v​\(T\)v\(S\\cup\\\{i\\\}\)\-v\(S\)=a\_\{i\}\+\\beta=v\(T\\cup\\\{i\\\}\)\-v\(T\); marginal contributions are therefore constant, and the game is convex \(indeed, modular\)\. For any coalitionC⊆NC\\subseteq N, the restricted gamev\|Cv\|\_\{C\}is additive, so the Aumann–Drèze value coincides with the Shapley value and is given byΩi​\(v;C\)=ai\+β\\Omega\_\{i\}\(v;C\)=a\_\{i\}\+\\betafori∈Ci\\in C\. To verify exact marginal alignment, fix a coalition structureT∈Π​\(N\)T\\in\\Pi\(N\), letC=CT​\(i\)C=C\_\{T\}\(i\), and letD∈𝒟i​\(T\)D\\in\\mathcal\{D\}\_\{i\}\(T\)\. The change in total coalition surplus induced by the exit–and–join moveT→Fi​\(T,D\)T\\to F\_\{i\}\(T,D\)is

v​\(D∪\{i\}\)−v​\(D\)−\[v​\(C\)−v​\(C∖\{i\}\)\]\\displaystyle v\(D\\cup\\\{i\\\}\)\-v\(D\)\-\\bigl\[v\(C\)\-v\(C\\setminus\\\{i\\\}\)\\bigr\]=\(ai\+β\)−\(ai\+β\)=0\.\\displaystyle\\qquad=\(a\_\{i\}\+\\beta\)\-\(a\_\{i\}\+\\beta\)=0\.The corresponding change in agentii’s Aumann–Drèze payoff is also zero, so exact marginal alignment holds\. Each agent contributes a private valueaia\_\{i\}and a modular participation termβ\\beta\. Because the game is additive, each agent fully internalizes its marginal effect\. The induced exit–and–join game is therefore an exact potential game with potentialΦ​\(T\)=V​\(T\)=∑C∈Tv​\(C\)\\Phi\(T\)=V\(T\)=\\sum\_\{C\\in T\}v\(C\)\.

###### Example 6\.13\(Exact potential game without convexity\)\.

LetN=\{1,…,n\}N=\\\{1,\\dots,n\\\}and define the cooperative gamev​\(S\)=∑i∈Sai−γ​\(\|S\|2\)v\(S\)=\\sum\_\{i\\in S\}a\_\{i\}\-\\gamma\\binom\{\|S\|\}\{2\}, whereai∈ℝa\_\{i\}\\in\\mathbb\{R\}andγ\>0\\gamma\>0\. The marginal contribution of agentiito a coalitionSSisv​\(S∪\{i\}\)−v​\(S\)=ai−γ​\|S\|v\(S\\cup\\\{i\\\}\)\-v\(S\)=a\_\{i\}\-\\gamma\|S\|, which decreases with coalition size; hence the game is submodular and is not convex in the cooperative\-game sense\. For any coalitionCCwithi∈Ci\\in C, the pairwise congestion term is symmetric, and the Aumann–Drèze value isΩi​\(v;C\)=ai−γ2​\(\|C\|−1\)\\Omega\_\{i\}\(v;C\)=a\_\{i\}\-\\frac\{\\gamma\}\{2\}\\bigl\(\|C\|\-1\\bigr\)\. Fix a coalition structureTT, letC=CT​\(i\)C=C\_\{T\}\(i\)andD∈𝒟i​\(T\)D\\in\\mathcal\{D\}\_\{i\}\(T\)\. ThenΩi​\(v;Fi​\(T,D\)\)−Ωi​\(v;T\)=γ2​\(\|C\|−1−\|D\|\)\\Omega\_\{i\}\\bigl\(v;F\_\{i\}\(T,D\)\\bigr\)\-\\Omega\_\{i\}\(v;T\)=\\frac\{\\gamma\}\{2\}\\bigl\(\|C\|\-1\-\|D\|\\bigr\)\.

DefineΦ​\(T\):=−γ2​∑C∈T\(\|C\|2\)\\Phi\(T\):=\-\\frac\{\\gamma\}\{2\}\\sum\_\{C\\in T\}\\binom\{\|C\|\}\{2\}\. ThenΦ​\(Fi​\(T,D\)\)−Φ​\(T\)=γ2​\(\|C\|−1−\|D\|\)\\Phi\(F\_\{i\}\(T,D\)\)\-\\Phi\(T\)=\\frac\{\\gamma\}\{2\}\\bigl\(\|C\|\-1\-\|D\|\\bigr\)\.

HenceΩi​\(v;Fi​\(T,D\)\)−Ωi​\(v;T\)=Φ​\(Fi​\(T,D\)\)−Φ​\(T\)\\Omega\_\{i\}\\bigl\(v;F\_\{i\}\(T,D\)\\bigr\)\-\\Omega\_\{i\}\(v;T\)=\\Phi\(F\_\{i\}\(T,D\)\)\-\\Phi\(T\), and the induced exit–and–join game𝒢​\(T\)\\mathcal\{G\}\(T\)is an*exact potential game*with potentialΦ\\Phi, despite the underlying cooperative gamevvbeing submodular and non\-convex\.

###### Theorem 6\.14\(Potential structure and Lyapunov alignment under exact marginal alignment\)\.

Suppose the cooperative gamevvsatisfies exact marginal alignment under the Aumann–Drèze value\. Assume switching costs are zero and acceptance is automatic\.

Then the induced exit–and–join game𝒢​\(T\)\\mathcal\{G\}\(T\)is an*exact potential game*with potential functionΦ​\(T\)=V​\(T\):=∑C∈Tv​\(C\)\\Phi\(T\)=V\(T\):=\\sum\_\{C\\in T\}v\(C\)\.

Moreover, the coalition–surplus functionVVis simultaneously an exact potential function for the induced exit–and–join game and a strict scalar Lyapunov function for the exit–and–join dynamics\.

In particular, for every admissible exit–and–join transitionT→T′=Fi​\(T,D\)T\\to T^\{\\prime\}=F\_\{i\}\(T,D\),ui​\(D;T\)−ui​\(CT​\(i\);T\)=Φ​\(T′\)−Φ​\(T\)=V​\(T′\)−V​\(T\)\>0u\_\{i\}\(D;T\)\-u\_\{i\}\(C\_\{T\}\(i\);T\)=\\Phi\(T^\{\\prime\}\)\-\\Phi\(T\)=V\(T^\{\\prime\}\)\-V\(T\)\>0\.

###### Proof\.

Consider an admissible exit–and–join transitionT→T′=Fi​\(T,D\)T\\to T^\{\\prime\}=F\_\{i\}\(T,D\)\. Only two coalitions are affected by the move: the origin coalitionCT​\(i\)C\_\{T\}\(i\)and the destination coalitionDD\. Therefore, the change in coalition surplus is

V​\(T′\)−V​\(T\)=\[v​\(D∪\{i\}\)−v​\(D\)\]−\[v​\(CT​\(i\)\)−v​\(CT​\(i\)∖\{i\}\)\]\.V\(T^\{\\prime\}\)\-V\(T\)=\\bigl\[v\(D\\cup\\\{i\\\}\)\-v\(D\)\\bigr\]\-\\bigl\[v\(C\_\{T\}\(i\)\)\-v\(C\_\{T\}\(i\)\\setminus\\\{i\\\}\)\\bigr\]\.\(6\.5\)
By exact marginal alignment, the deviating agent’s Aumann–Drèze payoff change satisfiesΩi​\(v;T′\)−Ωi​\(v;T\)=V​\(T′\)−V​\(T\)\\Omega\_\{i\}\(v;T^\{\\prime\}\)\-\\Omega\_\{i\}\(v;T\)=V\(T^\{\\prime\}\)\-V\(T\)\. Since switching costs are zero, the payoff difference in the induced game𝒢​\(T\)\\mathcal\{G\}\(T\)coincides with the Aumann–Drèze payoff difference:ui​\(D;T\)−ui​\(CT​\(i\);T\)=Ωi​\(v;T′\)−Ωi​\(v;T\)u\_\{i\}\(D;T\)\-u\_\{i\}\(C\_\{T\}\(i\);T\)=\\Omega\_\{i\}\(v;T^\{\\prime\}\)\-\\Omega\_\{i\}\(v;T\)\. Combining with \([6\.5](https://arxiv.org/html/2606.19683#S6.E5)\) yieldsui​\(D;T\)−ui​\(CT​\(i\);T\)=V​\(T′\)−V​\(T\)u\_\{i\}\(D;T\)\-u\_\{i\}\(C\_\{T\}\(i\);T\)=V\(T^\{\\prime\}\)\-V\(T\), establishing thatV=ΦV=\\Phiis an exact potential function for the induced exit–and–join game\.

Finally, admissibility implies that the deviating agent strictly improves its payoff\. By exact marginal alignment, this is equivalent toV​\(T′\)\>V​\(T\)V\(T^\{\\prime\}\)\>V\(T\)\. SinceΠ​\(N\)\\Pi\(N\)is finite,VVis a strict Lyapunov function and guarantees finite–time convergence of the dynamics to a local maximizer ofVV\. ∎

###### Theorem 6\.15\(Exact potential structure for exit–and–join dynamics\)\.

Consider a transferable–utility cooperative gamevvand the associated Aumann–Drèze exit–and–join dynamics with zero switching costs and acceptance\. Suppose there exists a functionΦ:Π​\(N\)→ℝ\\Phi:\\Pi\(N\)\\to\\mathbb\{R\}such that for every coalition structureT∈Π​\(N\)T\\in\\Pi\(N\), every agenti∈Ni\\in N, and every destinationD∈𝒟i​\(T\)D\\in\\mathcal\{D\}\_\{i\}\(T\),

Ωi​\(v;Fi​\(T,D\)\)−Ωi​\(v;T\)=Φ​\(Fi​\(T,D\)\)−Φ​\(T\)\.\\Omega\_\{i\}\\bigl\(v;F\_\{i\}\(T,D\)\\bigr\)\-\\Omega\_\{i\}\(v;T\)=\\Phi\\bigl\(F\_\{i\}\(T,D\)\\bigr\)\-\\Phi\(T\)\.\(6\.6\)
Then the induced exit–and–join game𝒢​\(T\)\\mathcal\{G\}\(T\)is an*exact potential game*with potential functionΦ\\Phi\. The functionΦ\\Phialso provides an*exact scalar representation of unilateral incentive changes*: for every admissible exit–and–join transitionT→Fi​\(T,D\)T\\to F\_\{i\}\(T,D\),Ωi​\(v;Fi​\(T,D\)\)\>Ωi​\(v;T\)\\Omega\_\{i\}\\bigl\(v;F\_\{i\}\(T,D\)\\bigr\)\>\\Omega\_\{i\}\(v;T\)if and only ifΦ​\(Fi​\(T,D\)\)\>Φ​\(T\)\\Phi\\bigl\(F\_\{i\}\(T,D\)\\bigr\)\>\\Phi\(T\)\. A coalition structureT⋆T^\{\\star\}is an exit–and–join equilibrium if and only if it is a pure–strategy Nash equilibrium of the induced game𝒢​\(T⋆\)\\mathcal\{G\}\(T^\{\\star\}\)\. Along any admissible exit–and–join transition, the potential strictly increases,Φ​\(Fi​\(T,D\)\)\>Φ​\(T\)\\Phi\\bigl\(F\_\{i\}\(T,D\)\\bigr\)\>\\Phi\(T\), and the exit–and–join dynamics converge in finite time to a coalition structure that locally maximizesΦ\\Phiwith respect to accepted exit–and–join deviations\.

These conclusions hold independently of convexity, superadditivity, or any efficiency property of the cooperative gamevv\.

###### Proof\.

We prove each claim in turn\.

For the exact potential structure, fix a coalition structureT∈Π​\(N\)T\\in\\Pi\(N\)\. In the induced exit–and–join game𝒢​\(T\)\\mathcal\{G\}\(T\), the payoff of agentiifrom choosing destinationD∈𝒟i​\(T\)D\\in\\mathcal\{D\}\_\{i\}\(T\)isui​\(D;T\)=Ωi​\(v;Fi​\(T,D\)\)u\_\{i\}\(D;T\)=\\Omega\_\{i\}\\bigl\(v;F\_\{i\}\(T,D\)\\bigr\), since switching costs are zero\. Condition \([6\.6](https://arxiv.org/html/2606.19683#S6.E6)\) implies that for any two destinationsD,D′∈𝒟i​\(T\)D,D^\{\\prime\}\\in\\mathcal\{D\}\_\{i\}\(T\),ui​\(D′;T\)−ui​\(D;T\)=Φ​\(Fi​\(T,D′\)\)−Φ​\(Fi​\(T,D\)\)u\_\{i\}\(D^\{\\prime\};T\)\-u\_\{i\}\(D;T\)=\\Phi\\bigl\(F\_\{i\}\(T,D^\{\\prime\}\)\\bigr\)\-\\Phi\\bigl\(F\_\{i\}\(T,D\)\\bigr\)\. Thus unilateral payoff differences coincide exactly with differences in the functionΦ\\Phi, and𝒢​\(T\)\\mathcal\{G\}\(T\)is an exact potential game with potential functionΦ\\Phi\.

The same identity gives a scalar representation of incentives\. For any admissible exit–and–join transitionT→Fi​\(T,D\)T\\to F\_\{i\}\(T,D\), admissibility requires that the deviating agent strictly improves its Aumann–Drèze payoff:Ωi​\(v;Fi​\(T,D\)\)\>Ωi​\(v;T\)\\Omega\_\{i\}\\bigl\(v;F\_\{i\}\(T,D\)\\bigr\)\>\\Omega\_\{i\}\(v;T\)\. By \([6\.6](https://arxiv.org/html/2606.19683#S6.E6)\), this inequality holds if and only ifΦ​\(Fi​\(T,D\)\)\>Φ​\(T\)\\Phi\\bigl\(F\_\{i\}\(T,D\)\\bigr\)\>\\Phi\(T\)\. HenceΦ\\Phiprovides an exact scalar representation of the deviating agent’s incentive change\.

For the equilibrium equivalence, recall that in an exact potential game, a pure–strategy Nash equilibrium is equivalent to a local maximizer of the potential function with respect to the feasible unilateral deviations encoded in the action sets\. By part \(i\), this condition is equivalent to the absence of admissible exit–and–join deviations that strictly improve the deviating agent’s payoff, which is precisely the definition of an exit–and–join equilibrium\.

Finally, along every admissible exit–and–join transition, the potentialΦ\\Phistrictly increases\. Since the state spaceΠ​\(N\)\\Pi\(N\)is finite, no infinite strictly increasing sequence of potential values can exist\. Therefore, the exit–and–join dynamics converge in finite time to a coalition structure that locally maximizesΦ\\Phiwith respect to accepted exit–and–join deviations\. ∎

## 7Matrix Representation of the Aumann–Drèze Exit–Join Dynamics

The matrix formulation of the Aumann–Drèze value admits a natural iterative interpretation, which makes it particularly suitable for dynamic exit–join coalition formation models\.

### 7\.1Matrix Form of the Aumann–Drèze Value

LetN=\{1,…,n\}N=\\\{1,\\dots,n\\\}be the set of players andv:2N→ℝv:2^\{N\}\\to\\mathbb\{R\}a transferable\-utility game withv​\(∅\)=0v\(\\varnothing\)=0\. Fix an ordering of the nonempty coalitions ofNN, and define the vectorized gamevec​\(v\)∈ℝ2n−1\\mathrm\{vec\}\(v\)\\in\\mathbb\{R\}^\{2^\{n\}\-1\}\. LetT=\{T1,…,Tm\}T=\\\{T\_\{1\},\\dots,T\_\{m\}\\\}be a coalition structure onNN, and denotetj:=\|Tj\|t\_\{j\}:=\|T\_\{j\}\|\. The following block representation is written in coalition\-block player order; an implicit permutation maps the resulting vector back to the canonical ordering of agents inNN\. For each coalitionTjT\_\{j\}, define the restricted gamevj​\(S\):=v​\(S\)v\_\{j\}\(S\):=v\(S\)for all nonemptyS⊆TjS\\subseteq T\_\{j\}, and letvec​\(vj\)∈ℝ2tj−1\\mathrm\{vec\}\(v\_\{j\}\)\\in\\mathbb\{R\}^\{2^\{t\_\{j\}\}\-1\}denote its vectorization\. The restriction fromvvtovjv\_\{j\}is implemented by a binary matrix

RTj∈\{0,1\}\(2tj−1\)×\(2n−1\),vec​\(vj\)=RTj​vec​\(v\),R\_\{T\_\{j\}\}\\in\\\{0,1\\\}^\{\(2^\{t\_\{j\}\}\-1\)\\times\(2^\{n\}\-1\)\},\\qquad\\mathrm\{vec\}\(v\_\{j\}\)=R\_\{T\_\{j\}\}\\,\\mathrm\{vec\}\(v\),which selects precisely those coalition values contained inTjT\_\{j\}\. LetA\(tj\)∈ℝtj×\(2tj−1\)A^\{\(t\_\{j\}\)\}\\in\\mathbb\{R\}^\{t\_\{j\}\\times\(2^\{t\_\{j\}\}\-1\)\}denote the Shapley value matrix for atjt\_\{j\}\-player game\. The Aumann–Drèze value of\(v,T\)\(v,T\)admits the linear representation

AD​\(v,T\)=\(A\(t1\)​RT1⋮A\(tm\)​RTm\)​vec​\(v\)=BTAD​vec​\(v\)\.\\mathrm\{AD\}\(v,T\)=\\begin\{pmatrix\}A^\{\(t\_\{1\}\)\}R\_\{T\_\{1\}\}\\\\ \\vdots\\\\ A^\{\(t\_\{m\}\)\}R\_\{T\_\{m\}\}\\end\{pmatrix\}\\mathrm\{vec\}\(v\)=B\_\{T\}^\{\\mathrm\{AD\}\}\\,\\mathrm\{vec\}\(v\)\.\(7\.1\)The operatorBTADB\_\{T\}^\{\\mathrm\{AD\}\}is block\-diagonal across coalitions, reflecting the locality of the Aumann–Drèze value\.

### 7\.2Exit–Join Dynamics as a Matrix System

LetTkT^\{k\}denote the coalition structure at iterationkk, and define the payoff vector

xk=AD​\(v,Tk\)=BTkAD​vec​\(v\)∈ℝn\.x^\{k\}=\\mathrm\{AD\}\(v,T^\{k\}\)=B\_\{T^\{k\}\}^\{\\mathrm\{AD\}\}\\,\\mathrm\{vec\}\(v\)\\in\\mathbb\{R\}^\{n\}\.\(7\.2\)
Suppose that at iterationkk, an agentiiexits coalitionTakT\_\{a\}^\{k\}and joins coalitionTbkT\_\{b\}^\{k\}, yieldingTak\+1=Tak∖\{i\}T\_\{a\}^\{k\+1\}=T\_\{a\}^\{k\}\\setminus\\\{i\\\}andTbk\+1=Tbk∪\{i\}T\_\{b\}^\{k\+1\}=T\_\{b\}^\{k\}\\cup\\\{i\\\}, with all other coalitions unchanged\.

BecauseBTADB\_\{T\}^\{\\mathrm\{AD\}\}is block\-diagonal, only the blocks associated withTakT\_\{a\}^\{k\}andTbkT\_\{b\}^\{k\}are affected\. The operator update can therefore be written as

BTk\+1AD=BTkAD\+Δ​Bak\+Δ​Bbk,B\_\{T^\{k\+1\}\}^\{\\mathrm\{AD\}\}=B\_\{T^\{k\}\}^\{\\mathrm\{AD\}\}\+\\Delta B\_\{a\}^\{k\}\+\\Delta B\_\{b\}^\{k\},\(7\.3\)whereΔ​Bak\\Delta B\_\{a\}^\{k\}andΔ​Bbk\\Delta B\_\{b\}^\{k\}replace the corresponding coalition blocks, and all other block rows are zero\.

Multiplying \([7\.3](https://arxiv.org/html/2606.19683#S7.E3)\) byvec​\(v\)\\mathrm\{vec\}\(v\)yields the payoff dynamics

xk\+1=xk\+\(Δ​Bak\+Δ​Bbk\)​vec​\(v\)\.x^\{k\+1\}=x^\{k\}\+\\left\(\\Delta B\_\{a\}^\{k\}\+\\Delta B\_\{b\}^\{k\}\\right\)\\mathrm\{vec\}\(v\)\.\(7\.4\)
Equivalently, for each playerℓ\\ell,

xℓk\+1=\{Φℓ​\(v\|Tak\+1\),ℓ∈Tak\+1,Φℓ​\(v\|Tbk\+1\),ℓ∈Tbk\+1,xℓk,otherwise,x\_\{\\ell\}^\{k\+1\}=\\begin\{cases\}\\Phi\_\{\\ell\}\\\!\\left\(v\|\_\{T\_\{a\}^\{k\+1\}\}\\right\),&\\ell\\in T\_\{a\}^\{k\+1\},\\\\\[4\.0pt\] \\Phi\_\{\\ell\}\\\!\\left\(v\|\_\{T\_\{b\}^\{k\+1\}\}\\right\),&\\ell\\in T\_\{b\}^\{k\+1\},\\\\\[4\.0pt\] x\_\{\\ell\}^\{k\},&\\text\{otherwise\},\\end\{cases\}whereΦℓ​\(w\)\\Phi\_\{\\ell\}\(w\)denotes the Shapley value component of playerℓ\\ellin the gameww\.

### 7\.3Event–Triggered Switching Interpretation

The update \([7\.4](https://arxiv.org/html/2606.19683#S7.E4)\) defines a discrete\-time, state\-dependent switched linear system\. An exit–join move\(a,b,i\)\(a,b,i\)is*admissible*if it yields a strict payoff improvement for agentii,

Φi​\(v\|Tbk∪\{i\}\)\>Φi​\(v\|Tak\)\.\\Phi\_\{i\}\\\!\\left\(v\|\_\{T\_\{b\}^\{k\}\\cup\\\{i\\\}\}\\right\)\>\\Phi\_\{i\}\\\!\\left\(v\|\_\{T\_\{a\}^\{k\}\}\\right\)\.\(7\.5\)
Let𝒮​\(xk\)\\mathcal\{S\}\(x^\{k\}\)denote the set of admissible exit–join moves at statexkx^\{k\}\. The system evolves according to

xk\+1=\{xk\+\(Δ​Bak\+Δ​Bbk\)​vec​\(v\),if​𝒮​\(xk\)≠∅,xk,otherwise\.x^\{k\+1\}=\\begin\{cases\}x^\{k\}\+\\left\(\\Delta B\_\{a\}^\{k\}\+\\Delta B\_\{b\}^\{k\}\\right\)\\mathrm\{vec\}\(v\),&\\text\{if \}\\mathcal\{S\}\(x^\{k\}\)\\neq\\varnothing,\\\\\[6\.0pt\] x^\{k\},&\\text\{otherwise\}\.\\end\{cases\}
If𝒮​\(xk\)=∅\\mathcal\{S\}\(x^\{k\}\)=\\varnothing, the dynamics terminate at a partitionT⋆T^\{\\star\}such that no agent can profitably deviate via an exit–join move under the Aumann–Drèze value\.

## 8Numerical Experiments

This section reports a broader numerical study of the Aumann–Drèze exit–and–join dynamics\. The experiments test finite termination, monotonicity of the coalition\-surplus Lyapunov function, sensitivity to switching and acceptance costs, and behavior in a special convex game\. All figures and tables are generated by a reproducible Python script in the repository\.

We use clustered pairwise transferable\-utility games with

v​\(S\)=∑i∈Sai\+∑\{i,j\}⊆Swi​j\.v\(S\)=\\sum\_\{i\\in S\}a\_\{i\}\+\\sum\_\{\\\{i,j\\\}\\subseteq S\}w\_\{ij\}\.For such games, the Aumann–Drèze payoff of agentiiin coalitionCCis

Ωi​\(v;C\)=ai\+12​∑j∈C∖\{i\}wi​j\.\\Omega\_\{i\}\(v;C\)=a\_\{i\}\+\\frac\{1\}\{2\}\\sum\_\{j\\in C\\setminus\\\{i\\\}\}w\_\{ij\}\.The sign of the moving agent’s payoff change is therefore aligned with the sign of the coalition\-surplus change\. This class provides a controlled testbed for the Lyapunov results while still allowing heterogeneous attraction and repulsion across agents\. Weights are drawn from a clustered random model: within\-cluster weights are positive on average, while cross\-cluster weights are mixed and may be negative\. Initial partitions are deliberately fragmented, active agents are sampled in random order, and a move is executed only if it strictly improves the moving agent’s net Aumann–Drèze payoff and is accepted by every member of the destination coalition\. The baseline acceptance rule usesκ=0\\kappa=0\. In the costly\-admission experiments, a nonempty destination accepts a mover only when each incumbent’s payoff gain from admitting that mover is at leastκ\\kappa, which iswi​ℓ/2≥κw\_\{i\\ell\}/2\\geq\\kappain the pairwise games\.

As a representative trajectory, we usen=30n=30agents, five latent clusters, and switching costc=0\.05c=0\.05\. Starting from a fragmented partition, the process executes 31 accepted exit–and–join moves, increases coalition surplus by 62\.67, and terminates at a partition with seven coalitions\. Figure[1](https://arxiv.org/html/2606.19683#S8.F1)shows both the membership trajectory and the Lyapunov trace, with ten repeated terminal checks appended after the final accepted move\. The coalition labels are constant over this terminal tail, while the scalar surplus trajectory is strictly increasing before termination and flat after no admissible deviation remains\.

![Refer to caption](https://arxiv.org/html/2606.19683v1/x1.png)

![Refer to caption](https://arxiv.org/html/2606.19683v1/x2.png)

Figure 1:Representative Aumann–Drèze exit–and–join trajectory\. Left: coalition membership over accepted moves followed by terminal checks\. Right: coalition\-surplus Lyapunov function along the same trajectory\. The dashed marker indicates the first terminal state; the flat tail makes convergence visible\.To test robustness, we simulate 180 independently generated games withn=24n=24agents and four latent clusters at switching costc=0\.05c=0\.05\. Every run terminates at an exit–and–join equilibrium, and no run exhibits a violation of Lyapunov monotonicity\. The mean number of accepted moves is 21\.9, the median is 22, and the 90th percentile is 25\.1\. The mean surplus gain is 52\.64, and terminal partitions contain 4\.9 coalitions on average\. Figure[2](https://arxiv.org/html/2606.19683#S8.F2)summarizes the empirical distribution of termination times, surplus gains, and terminal partition sizes\.

![Refer to caption](https://arxiv.org/html/2606.19683v1/x3.png)Figure 2:Monte Carlo summary over 180 random clustered pairwise games\. All runs terminate with zero observed Lyapunov monotonicity violations\.We next vary the switching cost while holding the random game instances and initial partitions fixed across cost levels\. This paired design isolates the effect of mobility frictions\. As shown in Figure[3](https://arxiv.org/html/2606.19683#S8.F3)and Table[2](https://arxiv.org/html/2606.19683#S8.T2), low to moderate costs preserve most of the surplus improvement, while high costs sharply reduce mobility and leave the terminal partition more fragmented\. Atc=1\.50c=1\.50, the mean number of accepted moves falls to 2\.3 and the mean terminal number of coalitions rises to 11\.7, compared with 22\.9 moves and 4\.9 coalitions at zero cost\.

![Refer to caption](https://arxiv.org/html/2606.19683v1/x4.png)Figure 3:Sensitivity to switching costs under a paired Monte Carlo design\. Higher costs reduce accepted mobility, reduce realized surplus gains, and increase terminal fragmentation\.We also vary the acceptance costκ\\kappawhile holding the switching cost fixed atc=0\.05c=0\.05and reusing the same paired\-game design\. Acceptance costs act on the destination coalition rather than on the moving agent: a higherκ\\kappamakes incumbents more selective about admitting a newcomer\. As shown in Figure[4](https://arxiv.org/html/2606.19683#S8.F4), this admission friction has a visible effect even though all runs still terminate and preserve Lyapunov monotonicity\. Whenκ\\kappaincreases from0to0\.250\.25, the mean surplus gain falls from53\.1653\.16to35\.9035\.90, the mean number of accepted moves falls from21\.821\.8to18\.418\.4, and the mean terminal number of coalitions rises from4\.74\.7to8\.28\.2\. Figure[5](https://arxiv.org/html/2606.19683#S8.F5)reports the joint effect of switching and acceptance costs; the high\-cost region leaves the process more fragmented because fewer profitable moves pass both the mover’s net\-gain test and the destination’s admission test\.

![Refer to caption](https://arxiv.org/html/2606.19683v1/x5.png)Figure 4:Sensitivity to acceptance costs at switching costc=0\.05c=0\.05\. Higher acceptance costs make destination coalitions more selective, reducing realized surplus gains and increasing terminal fragmentation\.![Refer to caption](https://arxiv.org/html/2606.19683v1/x6.png)Figure 5:Joint sensitivity to switching and acceptance costs under a paired Monte Carlo design\. The left panel reports mean surplus gains, and the right panel reports mean final coalition counts\.Table 2:Numerical diagnostics for Aumann–Drèze exit–and–join dynamics\. Monte Carlo results use 180 random clustered pairwise games; the switching\- and acceptance\-cost sweeps use 80 paired random games per cost level\.All Monte Carlo runs in the reported randomized sweeps terminated with zero Lyapunov monotonicity violations\. The convex benchmark uses the symmetric convex gamev​\(S\)=0\.45​\(\|S\|2\)v\(S\)=0\.45\\binom\{\|S\|\}\{2\}and reaches the grand coalition\.

Finally, we isolate a special convex game for which the grand coalition is both efficient and dynamically selected from a singleton initial partition\. Letv​\(S\)=0\.45​\(\|S\|2\)v\(S\)=0\.45\\binom\{\|S\|\}\{2\}\. This symmetric game is convex because marginal contributions increase with coalition size\. With switching costc=0\.05c=0\.05and acceptance costκ=0\.02\\kappa=0\.02, every accepted move joins a larger coalition, incumbents strictly prefer admission, and the process reaches the grand coalition after 17 accepted moves\. Figure[6](https://arxiv.org/html/2606.19683#S8.F6)shows the number of coalitions decreasing to one and the coalition\-surplus trajectory reaching the grand\-coalition valuev​\(N\)=68\.85v\(N\)=68\.85\. This benchmark does not overturn the earlier nonuniqueness example; rather, it shows that strict complementarity in a convex game can make the efficient coalition structure dynamically attractive under the exit–and–join protocol\.

![Refer to caption](https://arxiv.org/html/2606.19683v1/x7.png)Figure 6:Special convex benchmarkv​\(S\)=0\.45​\(\|S\|2\)v\(S\)=0\.45\\binom\{\|S\|\}\{2\}\. Starting from singletons, the exit–and–join dynamics reach the grand coalition and attain the grand\-coalition surplus\.Taken together, the experiments support the theoretical picture\. In aligned pairwise games, accepted Aumann–Drèze exit–and–join moves produce monotone surplus improvement and finite termination across a large set of randomized instances\. Switching costs reduce the moving agent’s net incentive to deviate, while acceptance costs make destination coalitions more selective; both frictions can stabilize more fragmented terminal partitions\. The convex benchmark complements the randomized experiments by showing a case in which strong positive complementarities select the grand coalition through local accepted moves\.

## 9Conclusion and Extensions

This paper developed a decentralized model of coalition formation driven by local exit–and–join decisions\. By using the Aumann–Drèze value as the payoff rule, the model preserves the cooperative interpretation of within\-coalition surplus allocation while allowing coalition structures to evolve through noncooperative unilateral deviations\. The resulting equilibrium concept is a local stability condition: a terminal partition admits no accepted exit–and–join move that strictly improves the deviating agent’s net payoff\.

The analysis separates three issues that are often conflated in coalition formation models\. Cooperative convexity supports efficiency of the grand coalition, but does not by itself guarantee that decentralized exit–and–join dynamics will select it\. Acceptance rules provide local protection for members of destination coalitions, but global convergence requires a no\-cycle certificate\. Exact or ordinal marginal alignment supplies such a certificate by turning unilateral incentives into a scalar potential or Lyapunov function\. The numerical experiments show that mover\-side switching costs and destination\-side acceptance costs stabilize different forms of local organization, and that a convex game with strict complementarities can select the grand coalition through local accepted moves\.

Several extensions are natural\. One direction is to study strategic admission policies in which incumbent coalitions use transfers, asymmetric sharing rules, or switching costs to attract entrants and deter departures\. Such mechanisms can generate lock\-in or monopoly\-like outcomes even when the underlying surplus function is not globally efficient\. Another direction is to combine the present model with learning, so that agents estimate coalition values from partial observations rather than observing the relevant payoff comparisons directly\. Both directions preserve the central theme of the paper: coalition structure is not imposed by a planner, but emerges from local incentives interacting with coalition\-level rules\.

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