1 Introduction↩︎

Coalition processes are inherently multiscale. Agents form, leave, and reconfigure coalitions at a strategic level, while the members of each coalition execute tasks, exchange information, and coordinate at a tactical level. These two layers are coupled: coalition structure determines who interacts with whom, and the outcomes of interaction reshape the incentives for future coalition formation. The result is a strategic–tactical dynamical architecture in which structure and performance co-evolve [1]–[4]. At the strategic level, agents decide whether to remain in their current coalition, exit, or join another coalition, and these decisions reshape the interaction network [1], [5]. At the tactical level, coalition members repeatedly communicate, aggregate information, and often reach internal agreement through consensus-like dynamics [6], [7]. Strategic reconfiguration therefore changes the tactical interaction pattern, while tactical outcomes change the strategic payoff landscape [2], [4].

This coupling appears in many multi-agent systems. In robotic teams, autonomous robots form coalitions for exploration, surveillance, transportation, or other tasks, and distributed control protocols coordinate behavior within each team [2]. In multi-agent AI systems, agents may collaborate temporarily for distributed inference or learning and then reorganize partnerships according to the utility of previous collaborations [3], [4]. In social networks, individuals form professional circles, online communities, or friendship groups; repeated interaction within these groups can produce shared views, while restricted cross-group interaction can sustain echo chambers and polarization [6]–[9]. Segregated interaction patterns can be measured using network-based segregation indices [10], and large-scale polarization dynamics have been documented empirically [11].

Movement between coalitions is rarely frictionless. Switching can incur explicit costs, such as time, resources, renegotiation, or institutional barriers. It can also incur implicit costs, such as cognitive discomfort, social resistance, or reluctance to leave familiar environments. Behavioral mechanisms such as confirmation bias reinforce these frictions [8]. We refer to these impediments collectively as coalition barriers. Such barriers regulate the amount of mixing between groups and therefore shape the global connectivity of the interaction network over time [9], [10].

The exit–and–join dynamics studied in this paper provide the elementary strategic move through which coalition structure changes. At a strategic date, an agent identifies its current coalition as the origin coalition and considers joining either another existing coalition or an empty destination, the latter corresponding to the formation of a singleton. This candidate move changes only the origin and destination coalitions: the agent is removed from the origin and inserted into the destination, while all other coalitions remain fixed. The move is implemented only when it passes two gates. First, the agent’s Aumann–Drèze payoff after the move, net of switching costs, must strictly improve. Second, the destination coalition must accept the entrant. Thus, the process is neither unrestricted mobility nor centralized coalition optimization; it is a local, incentive-driven reconfiguration rule constrained by frictions and acceptance.

1.1 Poiesis and Praxis in Coalition Systems↩︎

The strategic–tactical framework admits a natural interpretation through Aristotle’s distinction between poiesis and praxis. In the Nicomachean Ethics, poiesis denotes productive activity oriented toward bringing something into being, whereas praxis denotes action whose end lies in the activity itself [12]. Coalition formation corresponds to poiesis. At the strategic level, agents constitute, dissolve, and reconfigure coalitions. This activity produces the interaction structure within which collective behavior occurs; it is structural and generative because it determines the topology of relations [1]. Tactical execution corresponds to praxis. Within a formed coalition, agents interact, deliberate, exchange information, and coordinate. Consensus formation and collaborative task performance unfold inside the structure that has been produced [6], [7]. In this layer, the activity of interaction and coordination is itself the locus of realization. The two dimensions are inseparable. Poiesis determines who interacts with whom; praxis determines how states, beliefs, or outputs evolve under that structure. The outcomes of praxis feed back into subsequent poiesis, since performance and agreement influence future coalition decisions [4], [5]. Coalition systems therefore exhibit a recurring cycle: structure is produced, action unfolds within it, and action reshapes future structure.

1.2 Related Work↩︎

The strategic layer of the present model is rooted in cooperative game theory and coalition formation. Classical value theory studies how a transferable surplus should be allocated among agents, beginning with the Shapley value and continuing through coalition-structure values such as the Aumann–Drèze value and the Owen value [1], [13], [14]. Related stability notions appear in hedonic coalition formation, stable partitions, and endogenous coalition-structure models [15]–[18]. Dynamic coalition formation further emphasizes that coalition structures may be the outcome of a sequential adjustment process rather than a static allocation [19]–[23]. In multi-agent systems, coalition methods have been developed for task allocation, distributed problem solving, and coalitional control [2], [24], [25]. These literatures provide the strategic vocabulary used here, but they typically treat the characteristic function or payoff primitives as exogenous. The present paper instead derives coalition value from the tactical consensus outcome generated inside each coalition.

The tactical layer builds on consensus and opinion dynamics. DeGroot averaging, social-influence models, and distributed agreement protocols characterize how agents aggregate information over a fixed or time-varying interaction graph [26]–[30]. Work on bounded confidence, antagonistic interactions, echo chambers, and polarization shows how network constraints and selective interaction can produce fragmentation rather than unanimity [7]–[9], [31], [32]. In contrast with models that take the communication graph as primitive, the framework here makes the graph endogenous: strategic exit–and–join decisions determine which agents interact tactically, while tactical consensus changes the payoffs that drive future reconfiguration. The closest methodological connection is therefore to work on dynamic and learning-based coalitional games, where incentives, state evolution, and allocation rules interact over time [4], [33]. The distinction is that our state variable is not only a game payoff or resource level, but also a slow opinion profile shaped by repeated within-coalition averaging. This yields a multiscale mechanism in which strategic instability can either obstruct coalition convergence or, through temporal mixing, support tactical consensus.

1.3 Contributions↩︎

This work asks what global patterns emerge when strategic coalition formation is coupled with tactical within-coalition consensus dynamics. Depending on connectivity patterns and barrier levels, the system may converge to global unanimity, stabilize in segregated structures, or sustain polarization across persistent clusters [7], [9]. The paper makes four main contributions. First, it introduces a fast–slow model in which transferable coalition value is generated endogenously by within-coalition consensus. Second, it formulates state-dependent exit–and–join dynamics with Aumann–Drèze payoffs, switching frictions, and acceptance constraints. Third, it characterizes tactical unanimity, strategic stability, segregation, and polarization in terms of temporal connectivity, coalition barriers, and the geometry of the performance landscape. Fourth, it uses numerical experiments to illustrate a central phenomenon: strategic instability can induce tactical unanimity.

The last point is the main conceptual message. When coalition barriers are low, agents frequently reconfigure partnerships. Although the coalition structure may remain strategically unstable, the union of interactions over time can become connected, and repeated cross-group mixing can drive global consensus [6]. Conversely, high barriers stabilize coalition structure but can preserve disconnected interaction patterns, allowing segregation or polarization to persist [8], [10]. Coalition barriers therefore become a design and control variable. Reducing barriers promotes mixing and unanimity, while increasing barriers stabilizes structure at the risk of fragmentation. By tuning incentives, mobility costs, or communication constraints, one can influence the emergent strategic–tactical dynamics [2], [4]. The framework developed here provides a mathematical basis for analyzing these multiscale coalition phenomena and for understanding how strategic decisions and tactical consensus jointly determine long-run collective outcomes.

2 Consensus–Induced Coalition Value with Separated Time Scales↩︎

A central modeling step of our framework is to specify how the transferable utility \(v(S)\) of a coalition \(S\subseteq N\) is generated endogenously from internal (tactical-level) consensus dynamics, while coalition formation itself evolves on a slower strategic time scale through exit–and–join deviations. This fast–slow separation allows coalition value to emerge from information aggregation and coordination processes internal to each coalition, while preserving a decentralized noncooperative mechanism for coalition reconfiguration.

2.1 Tactical states and consensus representation↩︎

Each agent \(i\in N\) possesses an intrinsic opinion (or internal state) \(x_i^0(\tau)\in\mathbb{R}^d,\) which represents the agent’s private information, belief, preference, or local estimate of a task-relevant quantity at strategic time \(\tau\). At the beginning of the tactical phase associated with \(\tau\), these intrinsic opinions initialize the within-coalition dynamics. When an agent exits one coalition and joins another at strategic time \(\tau+1\), its tactical state in the new coalition is initialized using the updated intrinsic opinion \(x_i^0(\tau+1)\). Thus, intrinsic opinions evolve slowly across strategic iterations, while tactical states are transient.

Fix a strategic time \(\tau\) and a coalition \(S\in T(\tau)\). Each agent \(i\in S\) holds a tactical opinion state \(x_i(\tau,k)\in\mathbb{R}^d\), \(k=0,1,\dots,K_\tau\), and we define the stacked coalition state \(x_S(\tau,k):=\mathrm{col}\bigl(x_i(\tau,k)\bigr)_{i\in S} \in\mathbb{R}^{d|S|}\).

The interaction topology within coalition \(S\) is encoded by a row-stochastic matrix \(P_S=[p_{ij}^S]\), with \(p_{ij}^S\ge0\) and \(\sum_{j\in S}p_{ij}^S=1\), which is assumed fixed over the tactical window \(\{0,\dots,K_\tau\}\).

The within-coalition consensus dynamics follow a DeGroot-type update: \[\label{eq:degroot95tau95k} x_i(\tau,k+1) = \sum_{j\in S} p_{ij}^S\, x_j(\tau,k), \qquad i\in S,\tag{1}\] with initial condition \[\label{eq:tactical95init} x_i(\tau,0)=x_i^0(\tau), \qquad i\in S.\tag{2}\] These dynamics evolve independently across coalitions in \(T(\tau)\).

2.2 Slow adaptation and induced coalition value↩︎

At the end of the tactical phase, intrinsic opinions are updated according to a slow adaptation rule that blends prior beliefs with coalition-level consensus. For each agent \(i\in S\subseteq T(\tau)\), define \[\label{eq:intrinsic95update} x_i^0(\tau+1) = (1-\gamma_i)\,x_i^0(\tau) + \gamma_i\,\bar x_S(\tau), \qquad \gamma_i\in[0,1].\tag{3}\]

This update captures learning, belief revision, or assimilation of collective information. When \(\gamma_i=0\), agents retain fixed intrinsic opinions; when \(\gamma_i=1\), intrinsic opinions fully reset to the coalition consensus. The updated values \(x_i^0(\tau+1)\) serve as initial conditions for the tactical dynamics at strategic time \(\tau+1\).

3 State-Dependent Exit–and–Join Dynamics↩︎

We now formalize the exit–and–join dynamics and characterize the induced state dependence of coalition values and individual incentives. Coalition-level consensus evolves endogenously as a fast process nested within a slower strategic reconfiguration of coalition membership. The resulting dynamics define a state-dependent cooperative game whose payoff allocations are determined by the Aumann–Drèze value. The mechanism is local and sequential. At each strategic time, a single active agent compares its current payoff with the payoff it would receive after moving from its current coalition to a candidate destination. The destination may be an existing coalition or the empty destination, which represents forming a singleton. A candidate move first generates a trial coalition structure; it is then evaluated using the consensus-induced characteristic function and the Aumann–Drèze payoff allocation in that trial structure. The move is executed only if the entrant obtains a strict net payoff improvement after switching costs and the destination coalition accepts the entrant. Once an implementable move is executed, the next tactical phase recomputes coalition consensus under the new membership structure.

3.1 Strategic state, moves, and payoffs↩︎

At each strategic time \(\tau = 0,1,2,\dots,\) the system state is the pair \[\label{eq:system95state} \mathcal{S}(\tau) = \bigl(T(\tau), \{\bar x_S(\tau)\}_{S\in T(\tau)}\bigr),\tag{5}\] where \(T(\tau)\in\Pi(N)\) is a coalition structure and \(\bar x_S(\tau)\in\mathbb{R}^d\) denotes the steady-state consensus outcome of coalition \(S\). For any agent \(i\in N\), let \[\label{eq:current95coalition} C_{T(\tau)}(i)\in T(\tau)\tag{6}\] denote the unique coalition containing agent \(i\) at time \(\tau\). This coalition is the agent’s origin coalition. An exit–and–join decision is specified by choosing a destination coalition different from the origin, or by choosing the empty destination when the agent forms a singleton.

The operator \(F_i\) should be interpreted as a trial transition rather than as an automatic update. It describes what the partition would become if the move were carried out. The subsequent payoff and acceptance tests determine whether this trial transition is actually implemented.

Coalition surplus at time \(\tau\) is generated by consensus outcomes according to \[\label{eq:consensus95value} v(S;\tau) = V\!\left(\bar x_S(\tau)\right), \qquad v(\varnothing;\tau)=0.\tag{10}\]

Given a coalition structure \(T\), individual payoffs are allocated using the Aumann–Drèze value. For each agent \(i\in N\), \[\label{eq:ad95payoff} \Omega_i\bigl(v(\cdot;\tau);T\bigr) = \phi_i\!\left(v_{|C_T(i)}(\cdot;\tau)\right),\tag{11}\] where \(\phi_i\) denotes the Shapley value and \(v_{|C_T(i)}\) is the restriction of \(v\) to agent \(i\)’s current coalition.

Equivalently, if \(S=C_T(i)\), \[\label{eq:ad95expansion} \Omega_i = \sum_{R\subseteq S\setminus\{i\}} \frac{|R|!\,(|S|-|R|-1)!}{|S|!} \Bigl[ v(R\cup\{i\};\tau)-v(R;\tau) \Bigr].\tag{12}\]

Agent \(i\) evaluates a move to destination \(D\) according to the net payoff \[\label{eq:exit95join95payoff} u_i\bigl(D;T(\tau)\bigr) = \Omega_i\!\left(v(\cdot;\tau);F_i(T(\tau),D)\right) - c_i\!\left(T(\tau),D\right),\tag{13}\] where \(c_i(T,D)\ge 0\) is a switching cost.

The move is admissible if \[\label{eq:admissibility} u_i\bigl(D;T(\tau)\bigr) > \Omega_i\!\left(v(\cdot;\tau);T(\tau)\right),\tag{14}\] and may additionally be subject to an acceptance constraint by the destination coalition (e.g., Pareto non-decrease of incumbents’ AD payoffs). Admissibility is therefore the entrant’s private incentive test. It asks whether the move is profitable for the moving agent after the switching cost is paid, before considering whether the destination coalition is willing to admit the entrant.

3.2 Acceptance, recomputation, and state dependence↩︎

In addition to individual payoff improvement, exit–and–join moves may be subject to acceptance by the destination coalition. Acceptance rules capture institutional, technological, or social frictions that prevent coalitions from being freely entered, even when unilateral incentives signal improvement.

A canonical acceptance rule requires Pareto non-decrease of Aumann–Drèze payoffs for incumbents: \[\label{eq:pareto95acceptance} \Omega_j\!\left(v(\cdot;\tau);F_i(T,D)\right) \ge \Omega_j\!\left(v(\cdot;\tau);T\right), \qquad \forall j \in D.\tag{15}\] This rule ensures that no incumbent agent is made strictly worse off by admitting the entrant.

More permissive or restrictive rules can be accommodated, including majority approval, weighted consent, or threshold-based acceptance. The singleton destination \(D=\varnothing\) is assumed to accept all entrants by convention.

Switching costs \(c_i(T,D)\) capture frictions associated with leaving a coalition and integrating into a new one. These costs may represent coordination delays, renegotiation overhead, loss of trust, or reinitialization of tactical processes. We allow \(c_i(T,D)\) to depend on both the origin and destination coalitions, and to be heterogeneous across agents.

Switching costs affect admissibility but not acceptance. In particular, even if a move is accepted by the destination coalition, it is implemented only if the agent’s net payoff strictly improves. Conversely, a payoff-improving move may still be blocked if it fails the acceptance rule.

This separation between individual incentives, switching frictions, and coalition acceptance clarifies the strategic structure of exit–and–join dynamics and allows the analysis to distinguish unilateral profitability from collective admissibility. In the terminology used below, an admissible move is individually profitable, an accepted move passes the destination’s entry rule, and an implementable move satisfies both requirements. Only implementable moves change the coalition structure.

Following an implementable exit–and–join move at strategic time \(\tau\), the coalition structure updates to \(T(\tau+1)\). The subsequent tactical phase (stage \(\tau+1\)) starts from the intrinsic profile \(x^0(\tau+1)\) produced by the reset map 3 . Thus, consensus at stage \(\tau+1\) is computed from \(x^0(\tau+1)\), not from the previous intrinsic profile \(x^0(\tau)\).

Specifically, let agent \(i\) move from its origin coalition \(C=C_{T(\tau)}(i)\) to a destination coalition \(D\) at time \(\tau\). Then \(T(\tau+1)=F_i(T(\tau),D)\) and the coalitions whose membership changes are \(C' := C\setminus\{i\}\) and \(D' := D\cup\{i\}\). The coalition-wise consensus values at strategic time \(\tau+1\) are given by \[\label{eq:consensus95recompute95revised} \begin{align} \bar x_{C'}(\tau+1) &= \sum_{j\in C'} \pi_j^{\,C'}\, x_j^0(\tau+1),\\[0.5em] \bar x_{D'}(\tau+1) &= \sum_{j\in D'} \pi_j^{\,D'}\, x_j^0(\tau+1), \end{align}\tag{17}\] where \(\pi^{\,C'}\) and \(\pi^{\,D'}\) denote the stationary distributions associated with the within-coalition interaction matrices for \(C'\) and \(D'\).

For every unaffected coalition \(S \in T(\tau)\setminus\{C,D\}\), the membership of \(S\) is unchanged across the strategic transition, that is, \(S\in T(\tau+1)\) and \(S\) contains the same set of agents as at time \(\tau\). Accordingly, the within-coalition interaction pattern governing the DeGroot dynamics 1 remains the same.

However, the consensus value at stage \(\tau+1\) is computed from the updated intrinsic profile \(x^0(\tau+1)\) rather than \(x^0(\tau)\). Thus, even for unaffected coalitions, \[\label{eq:consensus95unaffected95revised} \bar x_S(\tau+1) = \lim_{k\to K_{\tau+1}} x_i(\tau+1,k), \qquad \forall i\in S,\tag{18}\] where the tactical dynamics at stage \(\tau+1\) are initialized according to \(x_i(\tau+1,0)=x_i^0(\tau+1)\).

Consequently, an implementable exit–and–join move induces a discrete transition in the strategic state \(\mathcal{S}(\tau)=\bigl(T(\tau),x^0(\tau)\bigr)\mapsto \mathcal{S}(\tau+1)=\bigl(T(\tau+1),x^0(\tau+1)\bigr)\), and the tactical dynamics in stage \(\tau+1\) evolve according to the DeGroot updates 1 with initial condition 2 applied to the coalition structure \(T(\tau+1)\).

In the present framework, coalition values are generated endogenously by within-coalition consensus dynamics. At strategic time \(\tau\), the induced transferable-utility characteristic function is defined by \[\label{eq:state95dependent95game} v(S;\tau) = V\!\left(\bar x_S(\tau)\right), \qquad S \subseteq N,\tag{19}\] where each coalition consensus state \(\bar x_S(\tau)\) is determined by the current coalition structure \(T(\tau)\).

As a result, the cooperative game is intrinsically state dependent: different coalition structures induce different collections of consensus states, and hence different characteristic functions, even though the underlying agent set \(N\) remains fixed. Coalition formation therefore gives rise to a sequence of cooperative games whose payoff structure evolves endogenously with coalition membership through the consensus mechanism.

Although all agents within a coalition \(S\) converge to a common tactical consensus state \(\bar x_S(\tau)\), their strategic incentives are generally heterogeneous. Under the Aumann–Drèze value, the payoff allocated to an agent \(i \in S\) is determined by its expected marginal contribution across all subcoalitions of \(S\).

For any subset \(Q \subseteq S \setminus \{i\}\), the marginal contribution of agent \(i\) is given by \[\label{eq:marginal95contribution} v(Q \cup \{i\};\tau) - v(Q;\tau) = V\!\left(\bar x_{Q \cup \{i\}}(\tau)\right) - V\!\left(\bar x_Q(\tau)\right).\tag{20}\] This quantity captures how the intrinsic opinion of agent \(i\) alters the consensus outcome of the subcoalition \(Q\), and thus quantifies the influence of agent \(i\) on coalition-level performance.

Consequently, exit–and–join incentives are driven by comparative marginal influence on consensus outcomes across coalitions, rather than by disagreement within a coalition. Even when tactical consensus is fully achieved, strategic incentives for coalition reconfiguration may persist due to asymmetries in agents’ contributions to collective performance.

3.3 Algorithmic summary↩︎

Algorithm 1 summarizes the coupled evolution of coalition structure, tactical consensus, and individual incentives under the proposed fast–slow framework. The algorithm explicitly separates tactical-level information aggregation from strategic-level coalition reconfiguration, while allowing slow intertemporal learning through updates of intrinsic opinions. Strategic transitions are governed jointly by individual payoff improvement, switching frictions, and coalition acceptance constraints.

At the beginning of each strategic iteration \(\tau\), the coalition structure \(T(\tau)\) and intrinsic opinions \(\{x_i^0(\tau)\}_{i\in N}\) are treated as fixed. The algorithm first executes a tactical consensus phase, during which within-coalition DeGroot dynamics aggregate private information and yield coalition-level consensus states \(\{\bar x_S(\tau)\}_{S \in T(\tau)}\). Once tactical consensus is formed, the algorithm proceeds to the strategic evaluation phase. Coalition values are computed by applying the performance functional \(V\) to each consensus outcome, and individual payoffs are allocated using the Aumann–Drèze value. A single agent is then selected to consider a unilateral exit–and–join deviation. A deviation is implemented only if it yields a strictly higher net payoff after accounting for switching costs and is accepted by the destination coalition. If no such admissible and accepted deviation exists, the process terminates at an exit–and–join equilibrium. Following any strategic update, the algorithm performs a slow intrinsic-opinion update. Each agent partially assimilates the coalition consensus into its intrinsic opinion, introducing intertemporal dependence across coalition formations. Through the repeated interaction of fast tactical consensus dynamics, strategic incentive-driven coalition reconfiguration subject to acceptance constraints, and slow belief adaptation, the algorithm provides an operational realization of consensus-induced coalition formation under Aumann–Drèze incentives.

4 Time-Dependent Induced Exit–and–Join Game↩︎

We now define the noncooperative game induced by coalition formation when coalition surplus is generated endogenously through tactical consensus dynamics and evolves jointly with intrinsic agent beliefs over strategic time. While the overall coalition-formation process is dynamic, the induced game at each strategic time is a static normal-form game whose payoff structure depends on the prevailing strategic state. The resulting formulation is therefore both state dependent and time indexed.

4.1 Induced game at a strategic state↩︎

Let \(\tau \in \mathbb{N}\) denote the strategic time index. At time \(\tau\), the strategic state of the system is given by \(\bigl(T(\tau), \{x_i^0(\tau)\}_{i\in N}\bigr),\) where \(T(\tau)\in\Pi(N)\) is a coalition structure and \(x_i^0(\tau)\in\mathbb{R}^d\) is the intrinsic opinion of agent \(i\).

Conditional on \(T(\tau)\) and \(\{x_i^0(\tau)\}\), we assume that tactical consensus dynamics within each coalition have fully converged. Accordingly, each coalition \(S\in T(\tau)\) is represented by a consensus state \(\bar x_S(\tau)\in\mathbb{R}^d,\) defined as the unique fixed point of the within-coalition consensus dynamics. This fixed point admits the weighted-average representation \[\label{eq:GT95consensus95map} \bar x_S(\tau) = \sum_{i\in S} \pi_i^S\, x_i^0(\tau),\tag{21}\] where \(\pi^S\in\Delta^{|S|}\) is the stationary distribution associated with the interaction matrix of coalition \(S\).

The tactical consensus outcomes induce a transferable-utility characteristic function at strategic time \(\tau\), \[\label{eq:GT95characteristic} v(S;\tau) = V\!\left(\bar x_S(\tau)\right), \qquad v(\varnothing;\tau)=0,\tag{22}\] where \(V:\mathbb{R}^d\to\mathbb{R}\) is a coalition performance functional. Because the consensus states \(\bar x_S(\tau)\) depend on both the coalition structure \(T(\tau)\) and intrinsic opinions \(\{x_i^0(\tau)\}\), the characteristic function \(v(\cdot;\tau)\) evolves endogenously over strategic time.

The player set is the agent set \(\mathcal{N} := N.\) For each agent \(i\in N\), let \(C_{T(\tau)}(i)\in T(\tau)\) denote the unique coalition containing \(i\). At coalition structure \(T(\tau)\), the admissible action set of agent \(i\) is \[\label{eq:GT95action95set} A_i\bigl(T(\tau)\bigr) := \bigl(T(\tau)\setminus\{C_{T(\tau)}(i)\}\bigr)\;\cup\;\{\varnothing\}.\tag{23}\] An action \(D\in A_i(T(\tau))\) represents a unilateral exit–and–join decision, with \(D=\varnothing\) corresponding to the formation of a singleton coalition.

Fix a strategic time \(\tau\) and a coalition structure \(T(\tau) \in \Pi(N)\). For each agent \(i \in N\), let \(C_{T(\tau)}(i) \in T(\tau)\) denote the unique coalition containing \(i\), and let the admissible destination set be \(A_i\bigl(T(\tau)\bigr):= \bigl(T(\tau)\setminus\{C_{T(\tau)}(i)\}\bigr)\cup\{\varnothing\}\).

For \(D \in A_i\bigl(T(\tau)\bigr)\), the unilateral exit–and–join operator is defined by \[\label{eq:GT95exit95join95operator95clean} F_i\bigl(T(\tau),D\bigr) = \bigl(T(\tau)\setminus\{C_{T(\tau)}(i),D\}\bigr) \;\cup\; \{C_{T(\tau)}(i)\setminus\{i\}\} \;\cup\; \{D\cup\{i\}\},\tag{24}\] with empty coalitions removed by convention.

Thus \(F_i(T,D)\) denotes the coalition structure obtained when agent \(i\) exits its current coalition \(C_{T}(i)\) and joins the destination coalition \(D\). Only the origin and destination coalitions are modified; all other coalitions remain unchanged.

The consensus state under a proposed deviation is updated locally. Let \(C := C_{T(\tau)}(i)\). Under the deviated structure \(T' = F_i(T(\tau),D)\), only the coalitions \(C\setminus\{i\}\) and \(D\cup\{i\}\) require recomputation of their consensus states. These are given by \[\label{eq:GT95consensus95update95clean} \begin{align} \bar x_{C\setminus\{i\}}(\tau) &= \sum_{j\in C\setminus\{i\}} \pi_j^{\,C\setminus\{i\}}\, x_j^0(\tau),\\ \bar x_{D\cup\{i\}}(\tau) &= \sum_{j\in D\cup\{i\}} \pi_j^{\,D\cup\{i\}}\, x_j^0(\tau). \end{align}\tag{25}\] For every other coalition \(S \in T(\tau)\) with \(S \notin \{C,D\}\), the consensus state remains unchanged.

Let \(\Omega_i(v(\cdot;\tau);T)\) denote the Aumann–Drèze value of agent \(i\) under coalition structure \(T\) for the characteristic function \(v(\cdot;\tau)\).

Although each game \(G_\tau(T(\tau))\) is static, the family \(\{G_\tau\}_{\tau\in\mathbb{N}}\) evolves endogenously as intrinsic opinions, consensus states, and coalition structures change over time. Coalition formation therefore unfolds as a sequence of state-dependent, time-indexed noncooperative games coupled through fast tactical consensus dynamics and slow belief adaptation.

4.2 Equilibrium and fixed-point characterization↩︎

The equilibrium concept must jointly account for tactical consensus formation, strategic coalition stability, and slow belief adaptation across time scales.

We characterize joint tactical–strategic equilibria as fixed points of a discrete-time operator defined on an augmented state space.

At such a state, \(x_i^{0,\star}=\bar x_{C_{T^\star}(i)}\) for all \(i\), \(T^\star\) admits no profitable unilateral exit–and–join deviation, and intrinsic opinions are invariant under the belief-update rule. Hence no further evolution occurs at either time scale.

4.3 Existence↩︎

We now give a sufficient condition for the composite operator \(\Phi\) to admit a fixed point.

5 Emergent Behavior↩︎

5.1 Tactical and Strategic Unanimity↩︎

Let \((T^\star,\{x_i^{0,\star}\}_{i\in N})\) be a joint tactical–strategic equilibrium. For each coalition \(S \in T^\star\), define its consensus value \(\bar x_S^\star=\sum_{i\in S}\pi_i^S x_i^{0,\star}\).

5.1.1 Strategic Unanimity↩︎

Consider the frictionless case with zero switching costs and automatic acceptance. If the grand coalition \(N\) is strategically stable, then no agent \(i\in N\) prefers to exit and form a singleton. Hence \(\Omega_i(v;N)\ge v(\{i\})\) for all \(i\).

Under the Shapley allocation, \(\Omega_i(v;N)\) is a convex combination of marginal contributions \(v(S\cup\{i\})-v(S)\) over \(S\subseteq N\setminus\{i\}\), with strictly positive weights. In particular, \(\Omega_i(v;N)\le v(N)-v(N\setminus\{i\})\). Therefore stability of \(N\) requires \(v(N)\ge v(N\setminus\{i\})+v(\{i\})\) for all \(i\in N\). Equivalently, each agent’s marginal contribution to the grand coalition must be at least as large as its standalone value.

5.1.2 Tactical Unanimity↩︎

The tactical dynamics defined in 1 –3 can be embedded into a single global switching-system representation. Recall that at each strategic time \(\tau\) the tactical dynamics 1 evolve for \(k=0,\dots,K_\tau-1\). Define cumulative switching times \(T_0:=0\) and \(T_{\tau+1}:=T_\tau+K_\tau\). Introduce a global discrete time index \(t = 0,1,2,\dots\) such that the tactical phase corresponding to strategic time \(\tau\) occupies the interval \(t \in \{T_\tau, \dots, T_{\tau+1}-1\}.\) Let \(x(t) := \operatorname{col}\bigl(x_i(t)\bigr)_{i\in N} \in \mathbb{R}^{dn}.\) At the switching instant \(t=T_\tau\), we identify \(x(T_\tau)=x^0(\tau),\) where \(x^0(\tau)\) is the intrinsic profile defined in 2 .

Fix a strategic time \(\tau\) and let \(T(\tau)\in\Pi(N)\) be the current coalition structure. Recall that for each coalition \(S\in T(\tau)\), the within-coalition tactical dynamics are governed by the DeGroot update 1 with interaction matrix \(P_S=[p_{ij}^S]_{i,j\in S}\), where \(p_{ij}^S\ge0\) and \(\sum_{j\in S}p_{ij}^S=1\). Thus \(P_S\) is a row-stochastic matrix defined on the index set \(S\). The coalition structure \(T(\tau)\) induces a block-diagonal matrix \[\widetilde{P}(\tau) = \operatorname{blkdiag}\bigl(P_S\bigr)_{S\in T(\tau)} \in\mathbb{R}^{n\times n},\] where each block \(P_S\) occupies the rows and columns corresponding to the agents in \(S\), and all off-block entries are zero.

Equivalently, for \(i,j\in N\), \[\widetilde{p}_{ij}(\tau) = \begin{cases} p_{ij}^S, & \text{if } i,j\in S \text{ for some } S\in T(\tau),\\ 0, & \text{otherwise}. \end{cases}\]

Because each block \(P_S\) is row-stochastic, \(\widetilde{P}(\tau)\) is also row-stochastic: \(\widetilde{p}_{ij}(\tau)\ge0\) and \(\sum_{j\in N}\widetilde{p}_{ij}(\tau)=1\) for all \(i\in N\). Hence \(\widetilde{P}(\tau)\mathbf{1}=\mathbf{1}\), where \(\mathbf{1}\) denotes the all-ones vector in \(\mathbb{R}^n\).

The associated directed graph \(\mathcal{G}(\tau)=(N,E(\tau))\) has edge \((j,i)\) whenever \(\widetilde{p}_{ij}(\tau)>0\). Then \(\mathcal{G}(\tau)\) is the disjoint union of the subgraphs induced by the coalitions in \(T(\tau)\). There are no inter-coalition edges, reflecting the fact that tactical interaction occurs only within coalitions.

If each \(P_S\) is primitive, then for each coalition \(S\) there exists a unique stationary distribution \(\pi^S\in\Delta^{|S|}\) satisfying \((\pi^S)^\top P_S = (\pi^S)^\top.\) The spectrum of \(\widetilde{P}(\tau)\) is the union of the spectra of its blocks. In particular, the eigenvalue \(1\) has multiplicity equal to \(|T(\tau)|\), and the corresponding right eigenspace is spanned by vectors that are constant on each coalition. Consequently, \[\widetilde{P}(\tau)^k \;\longrightarrow\; \operatorname{blkdiag} \bigl( \mathbf{1} \, (\pi^S)^\top \bigr)_{S\in T(\tau)} \quad \text{as } k\to\infty,\] which recovers the coalition-wise consensus limit ?? .

The switching signal is \(\sigma(t)=\tau \quad \text{whenever } t\in[T_\tau,T_{\tau+1}-1].\) Then 1 can be written globally as the switching linear system \[\label{eq:global95switching} x(t+1) = \widetilde{P}(\sigma(t))\,x(t).\tag{32}\]

For \(t\in[T_\tau,T_{\tau+1}-1]\), \(x(t) = \widetilde{P}(\tau)^{\,t-T_\tau} x^0(\tau).\) Under the primitivity assumption on each \(P_S\), the limit ?? becomes \[\label{eq:global95consensus} \lim_{t\uparrow T_{\tau+1}} x(t) = \bar x(\tau),\tag{33}\] where \(\bar x(\tau)\) stacks the coalition consensus values \(\bar x_S(\tau)\) defined in ?? .

At \(t=T_{\tau+1}\), intrinsic opinions are updated according to 3 , which in stacked form reads \[\label{eq:global95intrinsic95update} x^0(\tau+1) = (I-\Gamma)x^0(\tau) + \Gamma \bar x(\tau),\tag{34}\] where \(\Gamma=\operatorname{diag}(\gamma_i)\).

5.2 Polarization and Segregation↩︎

5.2.1 Segregation↩︎

5.3 Polarization and Performance Landscape↩︎

5.3.1 Segregation and Polarization↩︎

Segregation and polarization describe distinct mechanisms behind multi-cluster equilibria. An equilibrium is segregated if the set of consensus values \(\mathcal{X}^\star := \{\bar x_S^\star : S\in T^\star\}\) contains at least two distinct elements. Segregation is therefore a structural property of the equilibrium partition: different coalitions stabilize at different consensus states. It does not impose any condition on the geometry of \(V\). Polarization imposes an additional landscape requirement. For distinct coalitions \(S_1,S_2\in T^\star\) with consensus values \(x_1\) and \(x_2\), polarization requires \(v(S_1\cup S_2) < v(S_1)+v(S_2)\). By Theorem 12, this is equivalent to the existence of \(\lambda\in(0,1)\) such that \(V(\lambda x_1+(1-\lambda)x_2) < V(x_1)+V(x_2)\). Thus polarization reflects a geometric property of \(V\) along the segment connecting cluster consensuses. Segregation may arise purely from strategic frictions such as switching costs or acceptance rules, even when merging would improve aggregate performance. Polarization, by contrast, requires that the performance landscape itself discourage merging. Hence polarization implies segregation, but segregation does not imply polarization.

5.4 Cognitive Barriers↩︎

The previous subsection characterized segregation geometrically, through directional derivatives of the performance functional \(V\) along the line segment connecting equilibrium consensus states. We now introduce a complementary payoff-based measure of stability. While equilibrium ensures that no agent has a profitable unilateral deviation, it does not quantify how robust the segregation is. An equilibrium may be strictly stable, marginal, or nearly unstable. To measure the stability margin of a segregated configuration, we introduce the notion of a cognitive barrier: the minimal compensating incentive required to induce an agent to abandon its current coalition. This concept captures rational resistance embedded in the equilibrium. Even when multiple coalitions coexist, agents may require strictly positive incentives to reconsider their alignment. The magnitude of this resistance quantifies the robustness of segregation. Let \((T^\star,\{x_i^{0,\star}\}_{i\in N})\) be a joint tactical–strategic equilibrium. For each coalition \(S\in T^\star\), denote its equilibrium consensus state by \(\bar x_S^\star=\sum_{j\in S}\pi_j^S x_j^{0,\star}\). Fix distinct coalitions \(S,R \in T^\star\). For agent \(i\in S\), define the switching gain \(\Delta_i^{SR}:=\Omega_i\bigl(v;F_i(T^\star,R)\bigr)-\Omega_i(v;T^\star)\), where \(F_i(T^\star,R)\) denotes the partition obtained by moving \(i\) from \(S\) to \(R\). The switching gain measures the net payoff advantage of migrating from \(S\) to \(R\). Strategic stability implies \(\Delta_i^{SR}\le0\) for all \(i\in S\) and \(R\neq S\), since no profitable unilateral deviation exists.

The quantity \(B_i^{SR}\) represents the minimal compensating reward required to induce agent \(i\) to abandon its current coalition \(S\) and join \(R\). It quantifies the directional stability margin of agent \(i\) with respect to migration from \(S\) to \(R\). In particular, \(B_i^{SR}=0\) if and only if agent \(i\) is indifferent between the two coalitions.

The quantity \(B^{SR}_{\min}\) is the smallest incentive required to trigger at least one migration from \(S\) to \(R\). The quantity \(B^{SR}_{\max}\) is the uniform incentive required to relocate all members of \(S\) to \(R\). Thus \(B^{SR}_{\min}\) measures the weakest link in coalition stability, while \(B^{SR}_{\max}\) measures its strongest resistance.

The scalar \(B_{\min}(T^\star)\) provides a global measure of the robustness of segregation. If \(B_{\min}(T^\star)>0\), the equilibrium partition is strictly stable: every cross-coalition deviation requires a strictly positive incentive. If \(B_{\min}(T^\star)=0\), the equilibrium is marginal: at least one agent is indifferent to switching. If a negative switching gain were to arise (i.e., \(B_i^{SR}<0\)), strategic stability would fail and segregation would collapse. Geometrically, let \(\Delta = \bar x_S^\star - \bar x_R^\star\) denote the displacement between two equilibrium consensus states. When \(V\) is differentiable, the switching gain admits a line-integral representation along the segment connecting \(\bar x_R^\star\) and \(\bar x_S^\star\): \[\Delta_i^{SR} = \int_0^{\alpha_{i}} \langle \nabla V(\bar x_R^\star + t\Delta), \Delta \rangle dt - \int_0^{\tilde{\alpha}_{i}} \langle \nabla V(\bar x_S^\star - t\Delta), \Delta \rangle dt,\] where \(\alpha_i\) and \(\tilde{\alpha}_i\) are influence coefficients determined by the consensus weights.

Thus cognitive barriers are governed by directional derivatives of \(V\) along the segregation vector. When the directional derivative vanishes along the connecting segment, barriers shrink. When \(V\) bends sharply against \(\Delta\), barriers grow.

If \(V\) is strictly concave, the directional derivative \(\langle\nabla V(\gamma(t)),\Delta\rangle\) is strictly monotone along the segment. In this case segregation is incompatible with strict stability, and the global barrier \(B_{\min}(T^\star)\) vanishes.

Conversely, large positive barriers require sufficient curvature of \(V\) to penalize motion toward alternative consensus states. Hence cognitive barriers encode second-order information about the geometry of the performance landscape.

6 Numerical Experiments↩︎

This section provides a computational illustration of the mechanisms developed above. The experiments are not intended as a separate proof of convergence. Rather, they show how the strategic exit–and–join rule, the tactical consensus map, switching frictions, and slow belief adaptation interact along representative trajectories.

6.1 Baseline protocol↩︎

The baseline comparison varies only the switching cost \(c\in\{0,0.01,0.02\}\). The plotted diagnostics are the coalition-label trajectory, the aggregate coalition score \(W(\tau):=\sum_{S\in T(\tau)}v(S;\tau)\), and the intrinsic-opinion trajectory. Together these quantities separate the strategic state, the coalitional objective, and the slow belief state.

6.2 Nonnegative switching costs↩︎

Figure 2 shows the evolution of coalition membership under the three nonnegative switching-cost regimes. Rows correspond to agents, columns to strategic time, and colors indicate coalition labels.

The costless regime exhibits frequent early reconfiguration: agents enter and leave coalitions until the payoff-improving moves are exhausted. Positive switching costs shrink the admissible transition set because a move must now generate enough Aumann–Drèze improvement to offset the friction. As \(c\) increases, the observed partitions become more persistent and the transient reorganization phase shortens. This is the computational analogue of the cognitive-barrier interpretation in Subsection 5.4: a larger switching cost raises the payoff threshold required to cross from one coalitional arrangement to another.

Figure 3 reports the total coalition surplus \(W(\tau)=\sum_{S\in T(\tau)}v(S;\tau)\) as a function of strategic time. The curves are best read as an empirical potential diagnostic for the displayed trajectories. Implemented moves are payoff-improving after switching costs and must be accepted by the destination coalition; in these simulations, the aggregate score increases during the active reconfiguration phase and then plateaus once the partition becomes strategically stable.

Higher switching costs reduce the number of strategic moves and therefore produce flatter surplus trajectories. This does not mean that the tactical layer is inactive; rather, after the strategic partition stabilizes, additional tactical averaging occurs within a fixed coalitional architecture. Figure 4 depicts the evolution of intrinsic opinions for all agents. Despite heterogeneous initial conditions and changing coalition memberships, opinions contract toward a common range because each slow update pulls an agent toward the consensus state of its current coalition.

The opinion panels clarify the fast–slow structure of the model. Tactical consensus acts inside the currently formed coalitions, while the slow update transfers coalition-level information back into intrinsic opinions. Larger switching costs indirectly slow global opinion mixing because agents encounter fewer coalition partners, but the baseline regimes still display contraction of opinion dispersion.

6.3 Mobility subsidy and instability-induced mixing↩︎

The preceding experiments restrict \(c\) to be nonnegative, so switching is never subsidized. We next reverse the sign of this friction and set \(c_{\mathrm{switch}}=-0.01\), with \(\gamma=0.6\) and the nonconcave performance landscape \(V(x)=-(x^2-0.5)^2\). A negative switching cost is a mobility subsidy. The individual improvement condition becomes \(\Omega_i(T_{\mathrm{new}})>\Omega_i(T)+c_{\mathrm{switch}} =\Omega_i(T)-0.01\), so a move may be implemented even when it slightly lowers the agent’s immediate Aumann–Drèze payoff. The admissible transition set is therefore enlarged, and the strategic layer acquires an exploratory component.

Figure 5 displays the coalition evolution. Coalition labels fluctuate persistently over time and agents repeatedly enter and leave different coalitions; no stationary partition emerges.

The contrast between Figures 5 and 6 is the main numerical point of this regime. The coalition partition does not converge, yet the intrinsic opinions approach a common value. This is not a contradiction. At each strategic time, the partition \(T(\tau)\) induces an averaging operator \(A(\tau)=(I-\Gamma)+\Gamma\mathcal{C}(T(\tau))\), so \(x^0(\tau+1)=A(\tau)x^0(\tau)\). A fixed partition may average only within disconnected coalitions, but persistent reshuffling changes the interaction graph over time. When the union of these graphs is repeatedly connected, the standard consensus mechanism for products of stochastic matrices applies, and the intrinsic profile contracts toward unanimity.

The numerical experiments therefore support the conceptual separation developed in the theory. Nonnegative switching costs operate as cognitive barriers that stabilize coalition structures and slow cross-coalition information exchange. Negative switching costs remove this barrier, producing persistent strategic motion; nevertheless, the tactical layer can still converge because repeated coalition reshuffling supplies the temporal connectivity required for global averaging.

7 Conclusion↩︎

This paper has developed a multiscale model of coalition formation in which strategic exit–and–join decisions are coupled to tactical consensus dynamics inside coalitions. The framework treats coalition value as an endogenous object: agents first aggregate information through within-coalition DeGroot dynamics, and the resulting consensus state determines the transferable utility that drives Aumann–Drèze payoffs, acceptance decisions, and switching incentives. This construction connects two levels of collective behavior that are often studied separately. The tactical layer describes how agents coordinate and produce a coalition-level state, while the strategic layer describes how agents reconfigure the coalitional architecture in response to the payoff implications of that state.

The analysis identifies conditions under which the coupled process admits stable configurations and clarifies when tactical agreement and strategic coalition formation reinforce or obstruct one another. Switching costs play a particularly central role. When they are nonnegative, they act as cognitive barriers that restrict admissible moves, enlarge viability regions, and promote strategic persistence. Under additional convexity or supermodularity conditions, the induced coalition game supports stronger forms of strategic unanimity, while nonconcave performance landscapes and acceptance constraints can sustain segregation, local stability, or persistent disagreement. These results show that consensus at the tactical level is neither automatic nor equivalent to grand-coalition formation at the strategic level; the two notions of unanimity are linked through the induced value function, payoff allocation rule, and frictions governing coalition mobility.

The numerical experiments illustrate the same separation in finite trajectories. Increasing nonnegative switching costs stabilizes coalition membership and slows cross-coalition information exchange, whereas a mobility subsidy can destroy strategic convergence while still promoting global tactical consensus through repeated temporal mixing. This instability–consensus separation suggests that strategic volatility is not always detrimental: when it creates sufficiently rich interaction patterns, it can improve information aggregation even in the absence of a stable partition. Future work may extend the model to heterogeneous learning rates, stochastic acceptance rules, network-constrained mobility, and mechanism-design problems in which switching frictions or subsidies are chosen to balance institutional stability against global information integration.

References↩︎