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Events as Triggers for Behavioral Diversity in Multi-Agent Reinforcement Learning
Hannes Büchi, Manon Flageat, Eduardo Sebastián, Amanda Prorok
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Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 93%
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Summary
The paper introduces an event-driven framework for Multi-Agent Reinforcement Learning (MARL) that decouples agent identity from behavior. It proposes Neural Manifold Diversity (NMD) as a metric to quantify diversity on a continuous behavior manifold, independent of agent identities. An event-based hypernetwork generates Low-Rank Adaptation (LoRA) modules over a shared team policy to dynamically reconfigure agent behaviors in response to system events. The framework ensures diversity constraints align with reward maximization, outperforming baselines and enabling zero-shot generalization for tasks requiring sequential behavior reassignment.
Entities (11)
Relation Signals (9)
Neural Manifold Diversity → measures → Behavioral Diversity
confidence 96% · Neural Manifold Diversity (NMD), a formal distance metric that remains well-defined when behaviors are transient and agent-agnostic.
Multi-Agent Reinforcement Learning → requires → Behavioral Diversity
confidence 95% · Effective multi-agent cooperation requires agents to adopt diverse behaviors as task conditions evolve
Framework → decouples → Agent identity
confidence 94% · decouples agent identity from behavior
Events → trigger → Behavioral Transitions
confidence 94% · Events are changes in the state of the system that induce qualitative changes in the task.
Neural Manifold Diversity → definedon → Behavior Manifold
confidence 93% · NMD, which measures diversity in the behavior manifold directly, independent of the agents currently executing those behaviors.
Event-based Hypernetwork → generates → Low-Rank Adaptation
confidence 92% · event-based hypernetwork that generates Low-Rank Adaptation (LoRA) modules over a shared team policy
Low-Rank Adaptation → adapts → Shared Team Policy
confidence 91% · enabling on-the-fly agent-policy reconfiguration in response to events.
Partial Observable Markov Games → →
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Abstract
Abstract:Effective multi-agent cooperation requires agents to adopt diverse behaviors as task conditions evolve-and to do so at the right moment. Yet, current Multi-Agent Reinforcement Learning (MARL) frameworks that facilitate this diversity are still limited by the fact that they bind fixed behaviors to fixed agent identities. Consequently, they are ill-equipped for tasks where agents need to take on different roles at very specific moments in time. We argue that, to define these behavioral transitions, the missing ingredient is $\textbf{events}$. Events are changes in the state of the system that induce qualitative changes in the task. Based on this view, we introduce a framework that decouples agent identity from behavior, capturing a continuous manifold from which agents instantiate their behaviors in response to events. This framework is based on two elements. First, to build an expressive behavior manifold, we introduce Neural Manifold Diversity (NMD), a formal distance metric that remains well-defined when behaviors are transient and agent-agnostic. Second, we use an event-based hypernetwork that generates Low-Rank Adaptation (LoRA) modules over a shared team policy, enabling on-the-fly agent-policy reconfiguration in response to events. We prove that this construction ensures that diversity does not interfere with reward maximization by design. Empirical results demonstrate that our framework outperforms established baselines across benchmarks while exhibiting zero-shot generalization, and being the only method that solves tasks requiring sequential behavior reassignment.
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- Source: https://arxiv.org/abs/2605.12388v2
- Canonical: https://arxiv.org/abs/2605.12388v2
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Events as Triggers for Behavioral Diversity in Multi-Agent Reinforcement Learning Hannes Büchi Manon Flageat Eduardo Sebastián11footnotemark: 1 Amanda Prorok Department of Computer Science and Technology, University of Cambridge, UK hmb71, mf873, es2121, asp45@cam.ac.uk Equal contribution. Abstract Effective multi-agent cooperation requires agents to adopt diverse behaviors as task conditions evolve—and to do so at the right moment. Yet, current Multi-Agent Reinforcement Learning (MARL) frameworks that facilitate this diversity are still limited by the fact that they bind fixed behaviors to fixed agent identities. Consequently, they are ill-equipped for tasks where agents need to take on different roles at very specific moments in time. We argue that, to define these behavioral transitions, the missing ingredient is events. Events are changes in the state of the system that induce qualitative changes in the task. Based on this view, we introduce a framework that decouples agent identity from behavior, capturing a continuous manifold from which agents instantiate their behaviors in response to events. This framework is based on two elements. First, to build an expressive behavior manifold, we introduce Neural Manifold Diversity (NMD), a formal distance metric that remains well-defined when behaviors are transient and agent-agnostic. Second, we use an event-based hypernetwork that generates Low-Rank Adaptation (LoRA) modules over a shared team policy, enabling on-the-fly agent-policy reconfiguration in response to events. We prove that this construction ensures that diversity does not interfere with reward maximization by design. Empirical results demonstrate that our framework outperforms established baselines across benchmarks while exhibiting zero-shot generalization, and being the only method that solves tasks requiring sequential behavior reassignment. 1 Introduction Figure 1: Proposed Framework. (a) Rather than shifting behaviors on a fixed timestep or episode schedule, our framework triggers behavioral transitions in response to task events. We realize this through two components: (b) NMD, a diversity metric defined directly on the behavior manifold and independent of which agent executes which behavior; and (c) an event-driven hypernetwork that generates LoRA (Hu et al., 2022) modules over a shared policy to allocate behaviors from the manifold. (d) We visualize the behavior manifold via a two-dimensional t-SNE projection across 128128 replications of the Pressure Plate environment, alongside agent trajectories for one rollout. Effective cooperation in multi-agent systems hinges less on agent identity than on agent behavior at any given moment. As tasks unfold, optimal team performance demands that agents adapt their behaviors dynamically, not uniformly across the team nor on a synchronized schedule, but at the right moment and under the right circumstances. When a quadrotor in a search team loses power, the remaining agents expand their coverage to compensate; or a defender in football pushes forward only once possession changes to the other team. These transitions are not arbitrary but triggered by events. Events are changes in the system’s state—e.g., team composition and environment configuration—that induce qualitative changes in the task and demand a behavioral response. Behavioral diversity, or behavioral heterogeneity, is a key driver of performance in cooperative Multi-Agent Reinforcement Learning (MARL) (Li et al., 2021), and recent approaches have proposed various ways to introduce it within agent teams (Bettini et al., 2023). Yet few existing approaches capture dynamic behavior adaptation. Most either assign behaviors statically, fixing each agent’s role for the entire episode, or periodically, switching every few time steps regardless of context. We argue that this gap reflects a deeper conceptual issue. Whether diversity is enforced via unique per-agent heterogeneous components in the policy (Bettini et al., 2024a), agent-specific weights generated by a hypernetwork (Tessera et al., 2025), or partitioned action spaces (Wang et al., 2020b), existing approaches share the same implicit assumption: behaviors are tied to specific agents. We challenge this assumption. We view behaviors as a property of the task—a continuous manifold the team draws from—whose traversal is dictated by events rather than by agent identity. Decoupling behavior and identity reframes the central question of diversity in MARL: rather than asking how to differentiate agents, we ask how to populate, structure and navigate the behavior manifold. This brings three requirements. First, to ensure the manifold is genuinely diverse, we need a metric defined directly on behaviors rather than on agents. We introduce Neural Manifold Diversity (NMD), which measures diversity in the behavior manifold directly, independent of the agents currently executing those behaviors. Second, we need an architecture that can both learn such a manifold and adapt agents as events unfold. We achieve this through an event-based hypernetwork that generates Low-Rank Adaptation (LoRA) (Hu et al., 2022) modules over a shared team policy, a computationally efficient alternative to full policy regeneration. Third, we need an optimization procedure that jointly maximizes team reward while preserving manifold diversity. We adopt a diversity-constrained objective (Bettini et al., 2024a), and prove that this construction does not interfere with reward maximization by design. Fig. 1 illustrates our framework. In summary, our contributions are: • We introduce an event-centric paradigm in which agents draw their behaviors dynamically from a continuous manifold, adapting in response to events. • We propose a behavior-centric metric, NMD, that models diversity directly within this manifold, independent of which agent executes which behavior. • We realize the framework through an event-driven hypernetwork generating LoRA modules. We prove that this architecture aligns task-objective gradients with NMD constraints through diversity control, guaranteeing joint reward maximization and diversity preservation. 2 Related Work A variety of MARL approaches investigate the allocation of diverse behaviors (also frequently referred to as “roles”) within a team. We review the most closely related work along three axes, and refer the reader to Appendix A for an extended discussion. Temporal Behavioral Assignment. A core question for behavior allocation is the assignment timeline. Existing approaches fall into two categories: methods that fix one behavior per agent for the entire episode (Tessera et al., 2025; Bettini et al., 2024a; Jiang and Lu, 2021; Li et al., 2021), and methods that adapt behaviors on a fixed temporal schedule regardless of task conditions (Fu et al., 2025; Qi et al., 2026; Mahajan et al., 2019; Wang et al., 2020a; Goel et al., 2025). We advocate for a third category: event-based allocation, in which behavioral shifts are triggered by state changes rather than fixed schedules or episode boundaries. Representations of Behaviors. Behavioral specialization is fundamentally constrained by the underlying behavior representation. Existing approaches include learning explicit per-behavior policies or policy deviations (Jiang and Lu, 2021; Li et al., 2021; Bettini et al., 2024a; Christianos et al., 2021; Kim and Sung, 2023; Li et al., 2024), conditioning a shared policy on agent identity or capabilities (Tessera et al., 2025; Fu et al., 2025; Deka and Sycara, 2021), and learning a latent behavior space to condition a shared policy or hypernetwork (Mahajan et al., 2019; Wang et al., 2020a). Our framework builds on similar representations but conditions behavior generation on task events, enabling dynamic allocation that prior approaches cannot support. Quantifying and Enforcing Behavioral Diversity. Existing methods to encourage diversity fall into two categories: auxiliary rewards or losses (Wang et al., 2020a; Li et al., 2021; Mahajan et al., 2019; Jiang and Lu, 2021), which provide no formal guarantee on final team diversity; and hard constraints (Bettini et al., 2024a; Tan et al., 2023), which provide stronger guarantees but require a target diversity value a priori. Diversity is commonly quantified via information-theoretic objectives (Wang et al., 2020a; Li et al., 2021; Mahajan et al., 2019; Jiang and Lu, 2021) or explicit metrics such as System Neural Diversity (SND) (Bettini et al., 2025). We propose NMD, a new metric inspired by SND that remains well-defined under dynamic behavior allocation, and use it within a constrained optimization framework to enforce target diversity levels. 3 Background Cooperative Multi-Agent Reinforcement Learning. We model cooperative multi-agent tasks as Partially Observable Markov Games (POMGs) (Shapley, 1953), defined by the tuple ℳ=⟨,,,,,Ω,ℛ,γ,ℬ⟩M= ,S,A,O,P, ,R,γ,B . Here, N is the set of agents, S the state space, A and O the joint action and observation space, P the transition dynamics, Ω the observation function, ℛR the shared reward function, and γ∈[0,1)γ∈[0,1) is the discount factor. We additionally let ℬB denote the behavior manifold of ℳM, i.e., the space of admissible policies π:→()π:O (A) that the task supports. In this sense, ℬB is a property of the task—its structure and coverage are determined by what strategies ℳM admits—and is not given a priori; how to represent and learn it is the central concern of Section 4. Each agent i∈i executes a policy πi(ai,t|oi,t)∈ℬ _i(a_i,t|o_i,t) , and the team objective is to find a joint policy ∗ π^* maximizing the expected discounted return R=[∑t=0∞γt∑i∈ri,t]R=E[ _t=0^∞γ^t _i r_i,t]. Note that a consequence of this formulation is that a behavior is equivalent to a policy, so we use these terms as interchangeable. We also remark that the observation vector oi∈o_i includes a physical capability descriptor when relevant for the task, in contrast to existing approaches where physical capabilities are defined as independent quantities. We do this to emphasize that our focus is on behavioral diversity, in which agents display different action distributions given identical inputs, rather than physical diversity. Events as Triggers of Behavioral Change. Standard POMGs treat all transitions as instances of the same dynamics P, leaving no formal distinction between qualitative changes in task progress. To capture such moments, we augment the POMG with a discrete event space Ξ and an event-detection function ℰ:→Ξ∪∅E:S→ ∪\ \ that maps the current state of the system to either a triggered event ξt∈Ξ _t∈ or the null symbol ∅ , i.e., no event happened. Roughly speaking, an event is a change in the system state—including team composition and environment configuration—that induces a qualitative shift in the optimal joint behavior. Events may originate from distinct sources. In our empirical study, we consider the following cases: changes in an agent’s own observations (e.g., a LiDAR detecting a target); changes in the team configuration (e.g., an agent being added); or environmental signals (e.g., a door opening). When an event triggers, it induces a non-stationary transition (st,t,t)→ξt(st+1,t+1,t+1)(s_t,N_t,b_t) _t(s_t+1,N_t+1,b_t+1) that displaces the system onto a new optimal trajectory manifold, where t=(π1,t,…,π|t|,t)∈ℬ|t|b_t=( _1,t,…, _|N_t|,t) ^|N_t|. The challenge for the team is to respond to ξt _t by adapting the behaviors of its members. We refer to the resulting setting as an Event-Augmented POMG, defined, with an abuse of notation, as ℳ=⟨,,,,,Ω,ℛ,γ,ℬ,Ξ,ℰ⟩M= ,S,A,O,P, ,R,γ,B, ,E . Problem Formulation. Given ℳM, our goal is to learn a mechanism that decouples agent identity from executed behavior. Concretely, we seek to learn (1) the continuous behavior manifold ℬB that captures the space of behaviors admitted in a task, and (2) an event-conditioned mechanism f:×Ξ→ℬf:O× that maps each agent’s observation and most recent event to a behavior πi,t∈ℬ _i,t . Beyond maximizing cumulative reward, if behaviors are truly properties of the task only, the mechanism should generalize zero-shot as any of the following vary: (1) number of agents, (2) physical capabilities, and (3) event sequences, including disruptions unseen during training (e.g., abrupt teammate removal). 4 A Framework for Event-Driven Behavioral Adaptation To describe our framework, we proceed in three steps. First, we introduce NMD (Neural Manifold Diversity), the metric that characterizes the behavior manifold ℬB (Section 4.1). Second, we present its architectural realization: an event-driven hypernetwork that generates LoRA modules over a shared policy backbone that allows us to learn ℬB and allocate policies to agents according to it (Section 4.2). Third, we describe how diversity is maintained and controlled on ℬB and we state the theoretical guarantees that follow from this construction, deferring proofs to Appendix B (Section 4.3). 4.1 Modeling the Behavior Manifold: Neural Manifold Diversity The behavior manifold ℬB is the central object our framework manipulates: agents draw from it, events trigger movement within it, and its geometry determines what coordination strategies the team has available. To make this manifold a tractable object—one we can measure and control—we begin by quantifying the diversity of policies drawn from it. Definition 1. Let ℬB denote the continuous behavior manifold induced by the task ℳM and let p(π)p(π) denote a probability density over ℬB. The Neural Manifold Diversity is the expected pairwise divergence between behaviors sampled from ℬB, evaluated over the observation distribution p(o)p(o): NMD(ℬ,)=∫ℬ∫ℬ∫W2(πm(o),πn(o))p(o)op(πm)πmp(πn)πn,NMD(B,O)= _B _B _OW_2\! ( _m(o), _n(o) )p(o)\,do\,p( _m)\,d _m\,p( _n)\,d _n, (1) where W2W_2 is the 2-Wasserstein distance between action distributions. A property that justifies treating NMD as a principled diversity measure rather than an arbitrary statistic is that the pairwise term in Eq. (1) defines a proper distance on ℬB. Proposition 2. Let d:ℬ×ℬ→ℝ≥0d:B×B _≥ 0 be d(πm,πn):=o∼p(o)[W2(πm(o),πn(o))]d( _m, _n):=E_o p(o)\! [W_2( _m(o), _n(o)) ]. Then d is a pseudometric on ℬB, and a metric whenever policies are distinguishable on the support of p(o)p(o). The proposition guarantees that d satisfies non-negativity, symmetry, identity of indiscernibles (up to p(o)p(o) not belonging to a null set), and the triangle inequality. As a consequence, NMD(ℬ,)NMD(B,O) admits a clear interpretation: it is the expected d-distance between two behaviors drawn independently from ℬB. The problem with Eq. (1) is that it is intractable because ℬB and O are continuous and high-dimensional. In practice, we instantiate a finite collection of policies πmm=1B\ _m\_m=1^B of behaviors and we observe a finite set of episodic observations ep⊂O_ep , substituting the integrals with empirical averages to yield to an estimator: NMD^(πmm=1B)=2B(B−1)|ep|∑m=1B∑n=m+1B∑o∈epW2(πm(o),πn(o)). NMD\! (\ _m\_m=1^B )= 2B(B-1)\,|O_ep| _m=1^B _n=m+1^B _o _epW_2\! ( _m(o), _n(o) ). (2) This is an unbiased statistic for NMD(ℬ,)NMD(B,O) whenever the sampled behaviors are i.i.d. In our implementation, πmm=1B\ _m\_m=1^B is the set of behaviors in a parallel batch of environments and epO_ep is the set of observations encountered along the rollout. Structurally, Eqs. (1) and (2) resemble that of SND (Bettini et al., 2025) because it also poses diversity as a distance in action distribution. However, the most important aspect of NMD is what is absent from it: agent indices. Existing metrics, including SND, compute pairwise distances between policies indexed by agent identities, while NMD computes distances between behaviors π∈ℬπ . This allows NMD to remain well-defined under the dynamics of our framework: behaviors that are transient, shared across agents, and adapted in response to events. 4.2 Architectural Realization To capture, populate and navigate ℬB, we require an architecture satisfying three requirements: (i) it must produce behaviors as a function of an agent’s local observation and the most recent event; (i) it must do so cheaply enough to avoid prohibitive per-timestep computation; and (i) it must allow to introduce mechanisms to preserve the diversity of the behavior manifold ℬB. We achieve this with an event-driven hypernetwork that emits LoRA modules over a shared policy backbone. Event-driven hypernetwork. Let gθg_θ be a transformer-based hypernetwork. Given the most recent event ξt _t, gθg_θ ingests a token per agent from their local observations oi,to_i,t, the event encoding ete_t, and the global target diversity scalar NMDdesNMD_des, and emits a LoRA pair (Cm,Dm)(C_m,D_m) per agent. Note that we use index m instead of i to highlight that the hypernetwork might predict the same (Cm,Dm)(C_m,D_m) pair for various agents. The hypernetwork is queried only when an event fires (and once at initialization); between events, the pair is held fixed. This avoids the static brittleness of episode-initial generation (Tessera et al., 2025) and the cost and instability of per-step regeneration (Fu et al., 2025). LoRA over a shared backbone. The team shares a policy backbone WsharedW_shared and applies the assigned LoRA adapters DmCmD_mC_m to the final linear layer of the policy. Given the penultimate feature ϕ(ot)φ(o_t), the pre-activation output of behavior m is: zm(ot)=Wsharedϕ(ot)+αDmCmϕ(ot),z_m(o_t)=W_shared\,φ(o_t)\;+\;α\,D_mC_m\,φ(o_t), (3) where α is a scalar that is used to preserve diversity, which will be explained in Section 4.3. The final action distribution πm(ot) _m(o_t) is obtained by applying an activation function (squashed Gaussian for continuous control) to zm(ot)z_m(o_t). Three design choices deserve note. First, restricting the hypernetwork to emit Cm∈ℝr×dC_m ^r× d, Dm∈ℝda×rD_m ^d_a× r with r≪dr d stabilizes optimization relative to generating dense matrices; it also reduces the hypernetwork’s output dimensionality (see Appendix D.2). Second, applying the LoRA update to the final linear layer ensures that the heterogeneous component enters the policy linearly in ϕ(ot)φ(o_t), the condition that makes diversity maintenance possible and gradient projections analytically tractable (Section 4.3). Third, WsharedW_shared accumulates task-general representations, while (Cm,Dm)(C_m,D_m) encodes only the deviation needed to instantiate a specific behavior, keeping the hypernetwork output small and the solution memory efficient. Overall, the event-based hypernetwork with LoRA modules allows us to learn a representation of ℬB through the weights θ of gθg_θ and, at the same time, assign behaviors to agents by generating (Cm,Dm)(C_m,D_m) pairs, effectively allowing to capture, populate and navigate the behavior manifold ℬB. 4.3 Diversity Control on the Behavior Manifold To ensure that the hypernetwork learns a well-structured behavior manifold, we opt to enforce a diversity constraint inspired by Diversity Control (DiCo) (Bettini et al., 2024a). We express each behavior as the sum of a shared component Wsharedϕ(ot)W_sharedφ(o_t) and a per-behavior deviation um(ot)=DmCmϕ(ot)u_m(o_t)=D_mC_mφ(o_t), as described in Eq. (3). To ensure diversity constraint, the scalar α is set as: α=NMDdesNMD^(um(ot)m=1B),α= NMD_des NMD\! (\u_m(o_t)\_m=1^B ), (4) where the denominator is computed over the B behaviors present in the parallel batch of environments, and NMDdesNMD_des is the target diversity level. Because the LoRA update enters Eq. (3) linearly, scaling the deviations by α scales the resulting behavioral distance by α exactly, so the realized diversity equals NMDdesNMD_des by construction. This linearity is the technical reason for the final-layer linear placement in Section 4.2: any nonlinearity between the LoRA update and the policy mean would forfeit this reduction. Our formulation is an extension of DiCo (Bettini et al., 2024a), which expresses each agent’s policy as πi(o)=πh(o)+λπh,i(o) _i(o)= _h(o)+λ\, _h,i(o), where πh _h is a shared component, πh,i _h,i is a per-agent deviation, and λ scales the deviation to match a target SND value. However, unlike ours, DiCo is agent-centric: each πh,i _h,i is permanently bound to an agent and the matched diversity is measured between fixed agent indices. Together, Eqs. (3) and (4) generalize DiCo from a fixed set of indexed agents to the time-varying set of behaviors instantiated on the manifold. This brings to the main result that justifies the architectural choice in Section 4.2 and the diversity control mechanism: we do not merely enforce a target diversity level, but also make reward optimization and diversity preservation jointly stable without auxiliary losses or trust regions. Theorem 3. The gradient of the expected return R with respect to the deviation um(ot)u_m(o_t) satisfies ∇umR=αPum(∇zmR)⊤ _u_mR=α\,P_u_m\,( _z_mR) , where Pum=I−um⊗∇umNMD^NMD^P_u_m=I- u_m _u_m NMD NMD is an idempotent projection matrix. The theorem shows that diversity control does not enter as an external regularizer, but as a projector on the reward-maximizing gradient: any component of ∇zmR _z_mR that would alter NMD NMD is filtered out before the LoRA parameters are updated. The reward objective and the diversity constraint are therefore aligned by construction—the optimized architecture can move freely along directions that preserve realized diversity, and is gated along directions that would violate it. Corollary 4. The projection PumP_u_m has two limiting regimes. Vanishing deviation: when ‖um‖→0\|u_m\|→ 0, Pum→IP_u_m→ I and the gradient is unprojected, allowing the behavior to escape a degenerate niche. Dominant deviation: when ‖um‖\|u_m\| is large, PumP_u_m contracts strongly along umu_m, suppressing updates that would further amplify ‖um‖\|u_m\| and collapse the heterogeneity of the remaining behaviors. 5 Experiments We empirically address four questions. First, we assess whether our framework constitutes a competitive MARL approach: (Q1) How does our framework perform relative to established baselines across benchmarks? Second, if behaviors are truly properties of the task, our approach should generalize without retraining across agent counts and capabilities. Our constrained formulation further enables generalization to a third axis, target diversity levels: (Q2) Does the framework generalize zero-shot across varying agent counts, capabilities, and diversity levels? Third, we verify that our event-based framework triggers the intended behavioral shifts: (Q3) Can our framework solve tasks that require sequential behavior allocation in response to events? Fourth, if behaviors are task properties, the framework should also generalize to unseen event sequences, such as the removal of an agent mid-episode: (Q4) Does our framework generalize to unseen event sequences? 5.1 Experimental Setup Baselines. We evaluate our framework against four baselines. HyperMARL (Tessera et al., 2025) allocates behaviors episode-wise via a hypernetwork conditioned on agent identities. Capability-Aware Shared Hypernetworks (CASH) (Fu et al., 2025) allocates behaviors at every timestep via a hypernetwork conditioned on capabilities when available, or agent identities. Diversity Control (DiCo) (Bettini et al., 2024a) inspired our constrained diversity optimization, but operates on fixed agent indices using cross-agent SND as the constraint; for each task, we perform a grid search over target SND values and report the best-performing variant. Finally, Parameter Sharing (PS) (Gupta et al., 2017) is a baseline with no behavior allocation, where all agents share the same policy. Environments. We consider two sets of experiments. First, to address (Q1) and (Q2), we evaluate our approach on four benchmarks established in recent studies on behavioral diversity (Tessera et al., 2025; Bettini et al., 2024a, b): (i) Navigation (local-sensing exploration), (i) Dispersion (reaching M targets from a central point), (i) Reverse Transport (collective package pushing), and (iv) Football (competitive multi-team benchmark inspired by Google Research Football (Kurach et al., 2020)). Second, to address (Q3) and (Q4), we introduce two custom tasks designed to highlight the benefits of our event-driven formulation. Pressure Plate requires a team to coordinate sequentially by holding down plates to open a locked door separating them from their goal; this task can only be solved with sequential event-based behavior allocation. Wind Flocking requires the team to maintain a formation where a an agent must provide shielding for another agent against wind resistance; this task allows for direct visualization of the effects of our proposed NMD. All environments are implemented in VMAS (Bettini et al., 2022), a popular and scalable MARL simulator. Details are in Appendix C and source code at https://anonymous.4open.science/r/hyperscale-48D6/. Events. A key dimension of our framework is the event. In our experiments, an event is defined by the removal of an agent, a modification in agent capabilities, a change in the target diversity, or a change in an element of the environment (e.g., door open). In tasks featuring partial observability, such as Navigation, a change in the local LiDAR reading is likewise categorized as an event. 5.2 Q1: Performance compared to establish baselines To address (Q1), we evaluate the performance of all approaches across the four primary tasks. Fig. 2 reports the task-specific completion rate, the average reward, and the episode length. Our proposed framework systematically ranks top-1, achieving a completion rate of at least 95%95\% across all tasks and significantly outperforming all baselines on the more complex Football task. Examining each panel reveals how the performance gap scales with task complexity. On Reverse Transport, all methods reach competitive completion rates except PS. The clear separation in this task is in episode length, where our method consistently terminates earliest, closely followed by DiCo but with a wide gap with respect ot the other methods. This indicates that event-driven behavior allocation produces more direct trajectories rather than merely succeeding more often. On Navigation, only our framework and CASH reach high completion with low episode length. On Dispersion, the gap widens. Our framework retains a near-perfect completion rate while CASH drops to roughly 50%, DiCo to roughly 80%, and HyperMARL and PS collapse. This is the first task where static behavior assignment becomes a clear liability: with M targets and no per-target specialization unlocked at the right moment, agents converge on overlapping subsets of targets. The most pronounced gap appears on Football, where every baseline collapses to near-zero completion, while our framework remains close to 100%. We attribute this to the task’s reliance on event-driven role transitions—possession changes, ball proximity, opponent positioning—which static or scheduled allocations cannot track. (a) Reverse Transport (b) Navigation (c) Dispersion (d) Football Figure 2: Performance Comparison. Final completion rate, average reward, and episode length across 55 seeds. Colors represent methods: green (Ours), orange (HM: HyperMARL), yellow (CH: CASH), blue (DC: DiCo), purple (PS: Parameter Sharing). 5.3 Q2: Zero-shot Generalization across Number of Agents, Capabilities and Target Diversity To address (Q2), we evaluate generalization across three axes. For agent count generalization, we exclude HyperMARL as its architecture cannot accommodate changes in team size, and replicate learned policies for DiCo and PS to enable comparison. For capability generalization, the considered tasks are homogeneous by default, so we run independent training runs for multiple capability values, concatenating capabilities to observations as inputs to the hypernetworks for our framework and CASH. The varied capabilities are: LiDAR range for Navigation, agent speed for Dispersion and Football, and agent force for Reverse Transport. Diversity target generalization is enabled directly by our constrained formulation and requires no additional training runs. The results in Fig. 3 show that our proposed approach maintains performance across these axes of zero-shot generalization, while baselines either stagnate or lose performance. Along the agent-count axis, our framework remains stable across the entire in-distribution and out-distribution ranges for all tasks, evidence that the hypernetwork has learned a behavior manifold in which changing the number of agents simply implies sampling a different amount of points in ℬB. The same conclusion can be drawn from the capability axis. Variations in LiDAR range, agent speed, and pushing force leave the high completion rate of our method largely unchanged, consistently outperforming baselines. Along the target NMD axis, our completion rate is stable across roughly two order of magnitude of NMD values. Interestingly, the absolute value of these ranges vary across tasks, which implies that there is actually a “good” region of NMD values. The stability of this last axis is the empirical counterpart to Theorem 3: because diversity is enforced through a projection, the achievable reward does not change as NMDdesNMD_des is varied within reasonable ranges. Overall, this also helps to explain the success of our approach, as diversity control maintains a sufficient level of diversity to span all necessary behaviors to solve the task. (a) Reverse Transport (b) Navigation (c) Dispersion (d) Football Figure 3: Generalization. Final completion rate across 55 seeds for the three generalization axes. White areas denote training distribution; gray areas denote zero-shot evaluation. Colors represent methods: green (Ours), yellow (CH: CASH), blue (DC: DiCo), purple (PS: Parameter Sharing). 5.4 Q3: Adaptive Behaviors in Response to Events To address (Q3), we introduce the Pressure Plate task, which requires accurate event-based behavior allocation to succeed. In this scenario, agents must adopt multiple behaviors sequentially; for example, one agent must hold a pressure plate until teammates pass through a door, then transition to a new behavior once a second plate is secured on the opposite side. The results in Fig. 4(a) and 4(b) show that our approach is the only evaluated method that successfully completes the task. To verify that this performance is driven by the event mechanism, we compare our approach to an ablation without event re-querying. This ablation fails because agents are statically assigned a single behavior, leaving them unable to adapt to the sequential requirements of the task, which results in the final agent remaining stuck behind the door. Interestingly, our event-based formulation enables sequential strategies without explicit time awareness, an emergent property from the decoupling of agent identity and behavior. This is further studied in Fig. 1, that visualizes agent trajectories and a t-SNE projection of the behavior space ℬB. Across multiple stochastic runs, distinct clusters emerge, demonstrating that event-based re-querying induces specialized and consistent behavioral modes. (a) Pressure Plate - Completion [%][\%] (b) Pressure Plate - Reward Environment Agent Removal Completion Rate Median−(M-Q1)+(Q3-M)^+(Q3-M)_-(M-Q1) (%) Pressure Plate No 98.4−0.8+1.498.4^+1.4_-0.8 Pressure Plate Yes 81.2−0.2+0.981.2^+0.9_-0.2 Football No 98.2−0.8+1.098.2^+1.0_-0.8 Football Yes 88.6−0.7+6.688.6^+6.6_-0.7 (c) Agent Removal Event (d) Qualitative visualization in Wind Flocking at steps 20, 70, 120, and 170. Figure 4: Study of Events. (a) Completion rate and (b) average episode reward on Pressure Plate. Colors: green (Ours and SQ: Single Query), orange (HM: HyperMARL), yellow (CH: CASH), blue (DC: DiCo), purple (PS: Parameter Sharing). (c) Agent removal on Pressure Plate and Football. (d) Visualization for four NMD values in Wind Flocking. The arrow indicates wind direction. 5.5 Q4: Generalization to Unseen Event Sequences To address (Q4), we evaluate the Football, Pressure Plate, and Wind Flocking environments by introducing unseen event sequences. Specifically, in Football, we remove an agent at a random point; in Pressure Plate, we remove the agent that first reaches the second pressure plate; and in Wind Flocking, we randomly change the target diversity mid-episode. Results in Table 4(c) demonstrate that our method is robust to these additional events and successfully recovers, highlighting its ability to assign relevant behaviors following such events. Specifically, neither perturbation pushes performance into the regime occupied by the baselines in Section 5.2, suggesting that the framework’s response to unseen events is qualitatively close to redistributing roles. Fig. 4(d) visualizes the impact of diversity on the Wind Flocking environment: as the requested diversity decreases, the agents form tighter flocks, highlighting the direct effect of NMD on agent behavior. The four snapshots correspond to NMD targets that change mid-episode. The agent geometry tracks each change visibly: the inter-agent distance grows monotonically with NMD between steps 2020 and 120120, and contracts again at step 170170 once NMD is reduced below the initial value. Because the target diversity is supplied to the hypernetwork as an input scalar, this transition does not require retraining and confirms that NMDdesNMD_des acts as a direct, interpretable control on the realized formation rather than as a soft regularization target. Quantitative results are reported in Appendix D.1. 6 Conclusion, Limitations, and Future Work This work argues that behavioral adaptation in MARL is best understood as event-driven: triggered by task-relevant transitions rather than by agent identity or fixed temporal cadence. From this premise follows a structural decoupling in which behaviors are no longer tied to agents but instantiated from a continuous manifold the team shares. As a consequence, the meaning of diversity is shifted to a property of the task, not of the agents that populate it. We made this concrete through Neural Manifold Diversity, which we proved is a valid distance on behaviors, and through an event-driven hypernetwork that generates LoRA modules over a shared policy. We also proved that this implementation aligns reward maximization with diversity preservation by design. Our empirical results demonstrated that our framework facilitates more effective coordination and generalization compared to existing approaches, outperforming baselines. The benefits of an event-based formulation are further validated in a pressure plate task, in which sequential assignment emerges to solve the task despite the absence of a memory mechanism (e.g., RNN) that learns a state-machine representation of the task status. Although this work establishes the foundation for an event-based view of behavioral adaptation in MARL, it presents limitations. The societal implications of these are discussed Appendix E, while here we focus on the technical limitations. First, the expressivity of behaviors is inherently constrained by the choice of representation. We use an implicit behavior typing representation where behaviors are inferred from local observations and attention to teammate observations via a hypernetwork, which results in diversity expressed within the space of policy parameters. Consequently, the richness of the strategy space is tied to this specific representation. As an initial effort to investigate this constraint, we provide experiments regarding the dimensionality of our LoRA adapter in Appendix D.2. Second, we note that in our current formulation, events are defined a priori using domain knowledge. A compelling extension for future work involves inferring these events either end-to-end or through latent discovery. This logic similarly applies to the diversity target, as our method generalizes across a range of NMD values, yet the identification of optimal values for this metric could likely be automated through an additional optimization objective. Lastly, the realization of our framework through a hypernetwork is currently centralized, requiring access to the observations from all agents to assign behaviors. It would be interesting to derive an extension of our solution that is decentralized, in which agents decide on their behavior based only on neighboring information. Acknowledgements This work is supported by the European Research Council (ERC) Project 949940 (gAIa), by a Leverhulme Trust Research Project Grant, and by the EPSRC funded INFORMED-AI project EP/Y028732/1. We gratefully acknowledge their support. References M. Bettini, R. Kortvelesy, J. Blumenkamp, and A. Prorok (2022) VMAS: a vectorized multi-agent simulator for collective robot learning. In International Symposium on Distributed Autonomous Robotic Systems, p. 42–56. Cited by: §5.1. M. Bettini, R. Kortvelesy, and A. Prorok (2024a) Controlling behavioral diversity in multi-agent reinforcement learning. 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The first one comprises methods that fix one behavior per agent for the entire episode [Tessera et al., 2025, Bettini et al., 2024a, Jiang and Lu, 2021, Li et al., 2021, Liu, 2025, Christianos et al., 2021], either because they learn per-agent policies [Bettini et al., 2024a, Jiang and Lu, 2021, Christianos et al., 2021], or because they do not allow dynamically changing behavior [Tessera et al., 2025, Li et al., 2021, Liu, 2025]. The second category comprises methods that adapt behaviors on a fixed temporal schedule—either every timestep or every c steps—regardless of task conditions [Fu et al., 2025, Qi et al., 2026, Mahajan et al., 2019, Wang et al., 2020a, b, Goel et al., 2025]. Representations of Behaviors. Behavioral specialization is fundamentally constrained by the underlying behavior representation. A common paradigm is to learn explicit policies for each behavior, with methodologies ranging from learning deviations from a common policy [Li et al., 2021, Bettini et al., 2024a] to selective parameter sharing [Christianos et al., 2021, Kim and Sung, 2023, Li et al., 2024, Jiang and Lu, 2021]. However, these approaches do not easily accommodate dynamic behavior allocation as the learned behaviors remain fixed and tied to specific agents. Another frequent approach is to use a shared policy conditioned on agent identity, typically defined as one-hot encoding [Tessera et al., 2025], or capabilities when available [Fu et al., 2025]. This method, also known as behavior typing [Bettini et al., 2023], has been applied through both direct policy conditioning [Deka and Sycara, 2021] and conditioned hypernetworks that indirectly predict policy parameters [Tessera et al., 2025, Fu et al., 2025]. However, it inherently limits scalability. A more flexible extension involves learning a latent behavior space (or role space) from which behaviors are sampled, either to condition a shared policy [Mahajan et al., 2019, Qi et al., 2026, Goel et al., 2025] or a shared hypernetwork [Wang et al., 2020a, Liu, 2025]. While this has shown promising results, it relies on training additional encoder networks and requires prior knowledge on the dimensionality and structure of the latent space to identify distinct behaviors. Alternatively, Wang et al. [2020b] achieves specialization by partitioning the action space, though this requires fixing the decomposition early to maintain tractability. In contrast to these explicitly conditioned or constrained methods, inferred behavior typing [Gupta et al., 2017, Bettini et al., 2023] derives behavior solely from environmental observations. Quantifying and Enforcing Behavioral Diversity. While the previous section focuses on enabling diversity through representation, this is often insufficient to ensure functional divergence in practice. Consequently, while some works do not explicitly reward diversity [Wang et al., 2020b, Christianos et al., 2021], many works have proposed various methods to encourage diversity within teams. These methods generally fall into two categories: those that encourage diversity through auxiliary rewards or losses [Wang et al., 2020a, Li et al., 2021, Mahajan et al., 2019, Jiang and Lu, 2021, Liu, 2025, Qi et al., 2026, Goel et al., 2025], and those that enforce diversity as a constraint in the learning process [Bettini et al., 2024a, Tan et al., 2023]. While the first type of approach is easier to construct and optimize, it offers only weak guarantees on final team diversity. The second type provides a strong guarantee that diversity constraints will be satisfied, although it requires a reference target value to be known a priori. Additionally, researchers have proposed different ways to quantify this diversity. Many papers focus on information-theoretic objectives where the trajectories of a policy should carry maximum information about its behavior [Wang et al., 2020a, Li et al., 2021, Mahajan et al., 2019, Jiang and Lu, 2021, Qi et al., 2026, Goel et al., 2025], requiring training models to differentiate trajectories. Alternatively, [Liu, 2025] proposes to use a contrastive loss to distinguish behavior, which also requires a separate model to differentiate trajectories. Finally, Bettini et al. [2025] proposed System Neural Diversity (SND), an explicit diversity metric based on the action distribution of the policies. Appendix B Proofs of Theoretical Results B.1 Notation and Assumptions (A1) Behavior manifold. The behavior manifold ℬB is a measurable space of conditional action distributions π:→()π:O (A), where ()P(A) denotes the set of Borel probability measures on the action space A. We assume ⊆ℝdaA ^d_a is equipped with the Euclidean norm ∥⋅∥\|·\|, and that for every π∈ℬπ and every o∈o , the action distribution π(o)π(o) has finite second moment. (A2) Observation distribution. Observations o∈o are drawn from a probability density p(o)p(o), and behaviors π∈ℬπ are drawn from a density p(π)p(π). All integrals over O and ℬB are taken with respect to these densities. (A3) Gaussian policy parameterization. When invoked (Lemma 6 onward), each behavior πm _m is a Gaussian policy πm(⋅∣o)=(μm(o),Σ) _m(· o)=N( _m(o), ), with mean μm(o)∈ℝda _m(o) ^d_a and a covariance Σ≻0 0 that is shared across behaviors and independent of o. (A4) Linear LoRA parameterization. The mean of behavior m is decomposed as μm(o)=Wsharedϕ(o)+αum(o) _m(o)=W_shared\,φ(o)+α\,u_m(o), um(o):=DmCmϕ(o)u_m(o):=D_mC_m\,φ(o), where ϕ(o)∈ℝdφ(o) ^d is a shared feature, Wshared∈ℝda×dW_shared ^d_a× d is the shared backbone, Cm∈ℝr×dC_m ^r× d and Dm∈ℝda×rD_m ^d_a× r are the LoRA factors, and α∈ℝ>0α _>0 is the diversity scalar. The deviation um(o)u_m(o) is deterministic. (A5) Differentiability. The expected return R is differentiable in the LoRA parameters Cm,Dm\C_m,D_m\ and in the shared parameters via ϕφ and WsharedW_shared, and the empirical NMD estimator NMD NMD defined in Eq. (2) of the main text is differentiable in the deviations um\u_m\ on the open set NMD^>0 NMD>0. For a fixed observation o, we abbreviate um:=um(o)u_m:=u_m(o), μm:=μm(o) _m:= _m(o), ϕ:=ϕ(o)φ:=φ(o), and zm:=μmz_m:= _m (the policy mean coincides with the pre-activation in (A3)–(A4)). B.2 Proof of Proposition 2 Proof. We verify the four metric axioms. Non-negativity. For every o, the 2-Wasserstein distance satisfies W2(πm(o),πn(o))≥0W_2( _m(o), _n(o))≥ 0 (it is a metric on 2()P_2(A) by [Villani and others, 2009] Thm. 6.18). The expectation of a non-negative integrand is non-negative, so d(πm,πn)≥0d( _m, _n)≥ 0. Symmetry. W2W_2 is symmetric in its arguments, hence W2(πm(o),πn(o))=W2(πn(o),πm(o))W_2( _m(o), _n(o))=W_2( _n(o), _m(o)) point-wise in o, and the expectation preserves equality. Triangle inequality. Fix any πm,πn,πk∈ℬ _m, _n, _k . Point-wise in o, the triangle inequality for W2W_2 gives W2(πm(o),πk(o))≤W2(πm(o),πn(o))+W2(πn(o),πk(o)).W_2( _m(o), _k(o))≤ W_2( _m(o), _n(o))+W_2( _n(o), _k(o)). Taking expectation under p(o)p(o) and using linearity and monotonicity of expectation: d(πm,πk)≤d(πm,πn)+d(πn,πk).d( _m, _k)≤ d( _m, _n)+d( _n, _k). Identity. If πm=πn _m= _n in ℬB, then πm(o)=πn(o) _m(o)= _n(o) for all o, hence W2(πm(o),πn(o))=0W_2( _m(o), _n(o))=0 point-wise and d(πm,πn)=0d( _m, _n)=0. Conversely, d(πm,πn)=0d( _m, _n)=0 implies W2(πm(o),πn(o))=0W_2( _m(o), _n(o))=0 for p-almost every o, hence πm(o)=πn(o) _m(o)= _n(o) for p-almost every o. This gives identity up to p-null sets, which establishes the pseudometric property. Under the additional separation assumption the converse is strengthened to πm=πn _m= _n, yielding a genuine metric. ∎ Corollary 5. Under the separation assumption, NMD(ℬ)=0NMD(B)=0 if and only if p(π)p(π) concentrates on a single behavior. The forward direction follows from d(πm,πn)=0⇔πm=πnd( _m, _n)=0 _m= _n; the converse from non-negativity of d and the integral definition of NMDNMD. B.3 Proof of Theorem 3 Before proving the main result of the theorem, we state and prove a Lemma that helps to derive the connection between diversity in the behavior manifold and reward maximization. Lemma 6. Let πm _m be a Gaussian policy with mean μm _m and covariance Σ shared across behaviors. If the heterogeneous component um(ot)u_m(o_t) enters μm _m linearly and is deterministic, then W2(πm,πn)=‖μm−μn‖2W_2( _m, _n)=\| _m- _n\|_2. Proof. The closed-form expression for the 2-Wasserstein distance between two non-degenerate Gaussians on ℝdaR^d_a is W22((μm,Σm),(μn,Σn))=‖μm−μn‖22+tr(Σm+Σn−2(Σm1/2ΣnΣm1/2)1/2),W_2^2\! (N( _m, _m),N( _n, _n) )=\| _m- _n\|_2^2+tr\! ( _m+ _n-2\,( _m^1/2 _n _m^1/2)^1/2 ), a result due to Olkin and Pukelsheim [1982]. Under (A3), Σm=Σn=Σ _m= _n= , so Σm+Σn−2(Σm1/2ΣnΣm1/2)1/2=2Σ−2(Σ1/2ΣΣ1/2)1/2=2Σ−2Σ=0, _m+ _n-2( _m^1/2 _n _m^1/2)^1/2=2 -2( ^1/2 \, ^1/2)^1/2=2 -2 =0, where we used that for Σ≻0 0, Σ1/2ΣΣ1/2=Σ2 ^1/2 \, ^1/2= ^2 and (Σ2)1/2=Σ( ^2)^1/2= . The trace term vanishes, leaving W22(πm(o),πn(o))=‖μm(o)−μn(o)‖22.W_2^2( _m(o), _n(o))=\| _m(o)- _n(o)\|_2^2. Taking square roots (both sides are non-negative) gives the claim. ∎ Substituting Lemma 2 into Eq. (2) of the main text and using μm(o)−μn(o)=α(um−un) _m(o)- _n(o)=α\,(u_m-u_n) from (A4), we obtain NMD^(πmi=1b)=2αb(b−1)|ep|∑i<j∑o∈ep‖um−un‖2. NMD\! (\ _m\_i=1^b )= 2\,αb(b-1)\,|O_ep| _i<j _o _ep\|u_m-u_n\|_2. (5) This makes explicit that NMD NMD is a homogeneous function of degree 1 in the deviations um\u_m\, result that leads to the proof of Theorem 3. Proof. We compute ∇umR _u_mR via the chain rule, treating α as a function of unj=1b\u_n\_j=1^b through Eq. (5). Step 1. From (A4), zm=Wsharedϕ+αumz_m=W_sharedφ+α u_m. The dependence of zmz_m on umu_m has two contributions: the explicit αumα u_m term, and the implicit dependence of α on umu_m through NMD NMD. Treating umu_m as a vector in ℝdaR^d_a: ∂zm∂um=αI+um⊗∇umα, ∂ z_m∂ u_m=α\,I+u_m _u_mα, where ⊗ denotes the outer product (we use the convention (a⊗b)v=a(b⊤v)(a b)\,v=a\,(b v). Substituting α=NMDdes/NMD^α=NMD_des/ NMD and applying the quotient rule: ∇umα=−NMDdesNMD 2∇umNMD^=−αNMD^∇umNMD^. _u_mα=- NMD_des NMD^\,2\, _u_m NMD=- α NMD\, _u_m NMD. Therefore, ∂zm∂um=αI−αNMD^um⊗∇umNMD^=αPum, ∂ z_m∂ u_m=α\,I- α NMD\,u_m _u_m NMD=α\,P_u_m, with PumP_u_m as in Theorem 3. Step 2. By the chain rule, ∇umR=(∂zm∂um)⊤∇zmR=αPum⊤∇zmR. _u_mR= ( ∂ z_m∂ u_m ) _z_mR=α\,P_u_m \, _z_mR. Because PumP_u_m is symmetric in the dyadic structure that matters for the projection identity below (we verify idempotency on the same form), we drop the transpose for notational convenience and write ∇umR=αPum∇zmR. _u_mR=α\,P_u_m\, _z_mR. Step 3. Using Eq. (5), NMD^0 NMD_0 is positively homogeneous of degree 1 in umu_m. By Euler’s homogeneous function theorem, um⊤∇umNMD^=NMD^.u_m \, _u_m NMD= NMD. (6) Now compute Pum2P_u_m^2. Writing v:=∇umNMD^v:= _u_m NMD and N:=NMD^N:= NMD for brevity, Pum2=(I−umv⊤N)(I−umv⊤N)=I−2umv⊤N+um(v⊤um)v⊤N2.P_u_m^2= (I- u_mv N )\! (I- u_mv N )=I- 2\,u_mv N+ u_m(v u_m)\,v N^2. By Eq. (6), v⊤um=Nv u_m=N, so the last term simplifies: um(v⊤um)v⊤N2=umNv⊤N2=umv⊤N. u_m(v u_m)\,v N^2= u_m\,N\,v N^2= u_mv N. Substituting back, Pum2=I−2umv⊤N+umv⊤N=I−umv⊤N=Pum.P_u_m^2=I- 2\,u_mv N+ u_mv N=I- u_mv N=P_u_m. Thus PumP_u_m is idempotent. ∎ B.4 Proof of Corollary 4 Proof. (i) The empirical NMD is the symmetric pairwise sum (Eq. (5)) NMD^∝∑j≠k‖un−uk‖2/[normalization]. NMD _j≠ k\|u_n-u_k\|_2 /\, [normalization ]. Differentiating with respect to umu_m: ∇umNMD^=c∑j≠ium−un‖um−un‖2, _u_m NMD=c _j≠ i u_m-u_n\|u_m-u_n\|_2, for a positive constant c depending only on B and |ep||O_ep|. As ‖um‖→0\|u_m\|→ 0 (with all ‖un‖\|u_n\| for j≠ij≠ i bounded away from zero), the summand vectors (um−un)/‖um−un‖2(u_m-u_n)/\|u_m-u_n\|_2 remain bounded in norm by 11, but the second-moment contribution to PumP_u_m is the rank-one term um⊗∇umNMD^/NMD^u_m _u_m NMD/ NMD. The numerator factor um→0u_m→ 0 drives this rank-one term to zero in operator norm, while the denominator NMD NMD remains bounded below by a positive constant (since the remaining behaviors maintain non-trivial pairwise distances). Hence Pum→IP_u_m→ I. (i) Write um=tu^mu_m=t u_m with t→∞t→∞ and u^m u_m fixed. For large t, the dominant pairs (um,un)(u_m,u_n) in NMD NMD are those involving umu_m, since ‖um−un‖∼t\|u_m-u_n\| t while ‖un−uk‖\|u_n-u_k\| for j,k≠ij,k≠ i remains O(1)O(1). Asymptotically, NMD^=c(B−1)B(B−1)/2t+O(1)=2cBt+O(1), NMD= c\,(B-1)\,B(B-1)/2\,\,t+O(1)= 2cB\,t+O(1), and ∇umNMD^=c∑j≠ium−un‖um−un‖2=c(B−1)u^m+O(1/t). _u_m NMD=c _j≠ i u_m-u_n\|u_m-u_n\|_2=c\,(B-1)\, u_m+O(1/t). Substituting into the expression for PumP_u_m in Theorem 3 and using um=tu^mu_m=t u_m: Pum=I−(tu^m)(c(B−1)u^m+O(1/t))⊤(2c/B)t+O(1)=I−c(B−1)tu^mu^m⊤(2c/B)t+O(1/t).P_u_m=I- (t u_m)\,(c(B-1) u_m+O(1/t)) (2c/B)\,t+O(1)=I- c(B-1)\,t\, u_m u_m (2c/B)\,t+O(1/t). Canceling t in the leading term: Pum⟶I−B(B−1)2u^mu^m⊤as t→∞.P_u_m\; \;I- B(B-1)2\, u_m u_m t→∞. This is the orthogonal projector onto u^m⟂ u_m . ∎ B.5 Bounded Action Spaces and Saturated Activations Lemma 6 assumes a Gaussian policy with linear mean. In practice, continuous-control policies typically apply a squashing nonlinearity such as tanh to enforce bounded actions, yielding a=tanh(z)a= (z) with z∼(μ,Σ)z (μ, ). We discuss the impact on the preceding results. Effect on Lemma 6. The Wasserstein distance between two squashed Gaussians no longer admits a closed form: the push-forward of a Gaussian under tanh is non-Gaussian, and W2W_2 between such distributions must be computed numerically. However, in the unsaturated regime a standard Lipschitz inequality gives W2(πm∘tanh−1,πn∘tanh−1)≤LtanhW2(πm,πn)=Ltanh‖μm−μn‖2,W_2( _m ^-1, _n ^-1)≤ L_ \,W_2( _m, _n)=L_ \,\| _m- _n\|_2, with Ltanh=1L_ =1. NMD computed in pre-activation space therefore upper-bounds NMD computed in action space, and the relative ordering of behaviors is preserved. Effect on Theorem 3. The chain rule of Step 1 acquires an extra Jacobian factor diag(sech2(zm))diag(sech^2(z_m)). In the unsaturated regime this Jacobian is close to the identity and the projection structure of PumP_u_m is preserved with multiplicative error O(‖zm‖2)O(\|z_m\|^2). In the saturated regime the gradient flow ∇zmR _z_mR along those coordinates is suppressed by the activation rather than the projector, so the projection PumP_u_m contributes no further regulation in those directions. The corollary’s qualitative regimes (vanishing/dominant deviation) carry over coordinate-wise, with the saturation acting as an additional, complementary gating mechanism rather than as a violation of the analysis. Effect on Corollary 4. Vanishing-deviation behavior (regime (i)) is unaffected, since μm→Wsharedϕ _m→ W_sharedφ remains in the unsaturated regime for typical feature scales. Dominant-deviation behavior (regime (i)) is strengthened: as ‖um‖→∞\|u_m\|→∞, the squashing nonlinearity contributes additional suppression on top of the projector, ensuring that pathologically large LoRA outputs cannot escape regulation through the action nonlinearity. Appendix C Experimental Details We evaluate our approach across six distinct multi-agent tasks that challenge various aspects of coordination, observability, and physical interaction. Dispersion: all agents spawn at a the center of the arena while goals are distributed randomly. The complexity lies in the fact that with total observability and a shared starting point, agents must coordinate to ensure they do not all converge to the same closest goal. The optimal policy requires high behavioral diversity to disperse and tackle different goals simultaneously (Fig. 5(a)). Navigation: introduces limited observability, requiring agents to utilize LiDAR sensors to explore the area and locate specific goals. Agents must navigate efficiently under time constraints, relying on LiDAR to locate their specific targets in an environment where information is sparse during the initial exploration phase (Fig. 5(b)). Reverse Transport: agents spawn inside a physical package and must exert collective force to push it toward a goal. Because one agent is barely able to move the mass alone, the team must synchronize their velocities to move the package effectively, receiving rewards proportional to the reduction in distance to the goal (Fig. 5(c)). Football: features a team of three agents playing against a programmed heuristic. Beyond coordinating an attack to score a goal (sparse reward), the random spawn locations force agents to switch between defensive positioning and counter-attacking. The challenge is amplified by the need to manage ball control and shooting lanes while competing against a consistent heuristic opponent (Fig. 5(d)). Pressure Plate: a logic-based task where three agents must navigate a locked door by utilizing two pressure plates. This necessitates a three-phase strategy: (i) one agent presses the first plate to allow the others to cross, (i) those agents must then find and hold a second plate on the opposite side so the first agent can also cross, and (i) all agents move to the final goal (Fig. 5(e)). Wind Flocking: explores energy-efficient collective movement among heterogeneous agents exposed to a north-to-south wind. The agents must discover a formation where the larger agent shields the smaller one from the wind to optimize their performance and minimize energy expenditure (Fig. 5(f)). (a) Dispersion (b) Navigation (c) Reverse Transport (d) Football (e) Pressure Plate (f) Wind Flocking Figure 5: Overview of the six tasks: (a) Dispersion, (b) Navigation, (c) Reverse Transport, (d) Football, (e) Pressure Plate, and (f) Wind Flocking. Table 1: Hyperparameters used in the experiments. Hyperparameter Value / Configuration Shared Environment & Execution Setup Total Training Steps 10710^7 Total Training Steps for Football 150×106150× 10^6 Parallel Environments 128 Time steps for Navigation & Dispersion 200 Time steps for Reverse Transport & Football 300 Total Number of Mini-batches 8 Shared MAPPO & Optimization Parameters Optimizer Adam Adam β1 _1 0.90.9 Adam β2 _2 0.9990.999 Adam ϵε 1×10−51× 10^-5 Discount Factor (γ) 0.990.99 GAE Parameter (λ) 0.950.95 Clipping Parameter (ϵε) 0.20.2 Entropy Coefficient 0.010.01 Value Function Coefficient 0.50.5 Max Gradient Norm 0.50.5 Proposed Framework Homogeneous Network Size [128, 128] Critic Hidden Dimension [128, 128] Learning Rate 6×10−46× 10^-4 Transformer Layers (Blocks) 22 Attention Heads 22 Embedding Dimension 6464 MLP Hidden Dimension 256256 LoRA Rank 88 DiCo Baseline Bettini et al. [2024a] Homogeneous Network Size [128, 128] Heterogeneous Network Size (Per-Agent) [128, 128] Learning Rate 1×10−41× 10^-4 SND Values Tested (Grid Search) 0.1,0.2,0.5,0.9,1.0,1.1,1.5,2.00.1,0.2,0.5,0.9,1.0,1.1,1.5,2.0 Best SND for Navigation and Dispersion 1.0 Best SND for Reverse Transport 0.1 Best SND for Football 0.2 HyperMARL Baseline Tessera et al. [2025] Network Size (Hypernetwork) 64 Network Size (Agents) [64, 64] Learning Rate 5×10−45× 10^-4 Embedding Dimension (etie_t^i) 8 CASH Baseline Fu et al. [2025] Policy Hidden Dimensions [128, 128] Critic Hidden Dimension [128, 128] GRU Hidden Dimension 128 (1 layer) Learning Rate 2×10−32× 10^-3 Hypernetwork Hidden Dimension 6464 Hypernetwork Layers 44 Decoder Hidden Dimension 6464 Use Two-Layer Decoder True For initial experimentation, we required around 500 compute hours using an NVIDIA GeForce RTX 2080 Ti GPU and an Intel(R) Xeon(R) Gold 6248R CPU @ 3.00GHz. For the final multi-seed evaluation, we invested additional 500 compute hours on the same hardware. All the hyperparameters used in the paper can be found in Table 1. Appendix D More Results D.1 Average Episode Rewards for Main Experiments For all reported results in the main text, we evaluated the models using the best-performing checkpoints saved during the training process. Fig. 6 illustrates the learning progress across our six evaluation environments: Navigation, Dispersion, Reverse Transport, Football, Pressure Plate, and Wind Flocking. The plots compare our method against the four baselines: Full Parameter-Sharing. A homogeneous baseline with no behavior allocation, where all agents share the same policy. HyperMARL [Tessera et al., 2025]. Allocates behaviors episode-wise at timestep 0 via a hypernetwork conditioned on agent identities. Diversity Control (DiCo) [Bettini et al., 2024a] Operates on fixed agent indices using cross-agent SND as the constraint; for each task, we perform a grid search over target SND values and report the best-performing variant. Capability-Aware Shared Hypernetworks (CASH) [Fu et al., 2025]. Allocates behaviors at every timestep via a hypernetwork conditioned on capabilities when available, or agent identities In nearly all environments, our method and the majority of baselines exhibit stable convergence toward high mean rewards, demonstrating that training is well-configured. The exception is CASH (Fig. 6(a) and 6(b)), which exhibits “shaky” performance and, in some independent runs, fails to converge to a competitive reward (Fig. 6(c)). We attribute this behavior to the policy brittleness inherent to its architecture. Because CASH requires re-querying the hypernetwork at every timestep to generate agent-specific weights, the resulting policy can be highly sensitive to minor fluctuations in the hypernetwork’s output. For DiCo, we evaluated several SND values and picked the one with best performance to ensure a fair comparison; the specific values used for each environment are detailed in Table 1. (a) Navigation (b) Dispersion (c) Reverse Transport (d) Football (e) Pressure Plate (f) Wind Flocking Figure 6: Comparison of mean rewards across all evaluation tasks. Learning curves for all the tasks. Colors denote the different methods evaluated: Ours (green), Full Parameter-Sharing (purple), CASH [Fu et al., 2025] (yellow), HyperMARL [Tessera et al., 2025] (orange), and DiCo [Bettini et al., 2024a] (blue). Performance is averaged over 5 seeds. D.2 LoRA Rank Ablation We conducted an ablation study on the navigation task across five independent seeds to determine the optimal rank for the Low-Rank Adaptation (LoRA) modules. Based on these results, illustrated in Fig. 7, we utilized a rank of r=8r=8 for all subsequent experiments. This configuration proved to be the “sweet spot” for our architecture, providing sufficient expressivity while maintaining the stability of the hypernetwork during training. In this sense, we recall that the dimensionality of the LoRA rank plays a critical role in the underlying optimization landscape. Restricting the hypernetwork to emit matrices Cm∈ℝr×dC_m ^r× d and Dm∈ℝda×rD_m ^d_a× r, where r≪dr d, significantly stabilizes optimization relative to generating full dense matrices. Figure 7: LoRA rank ablation in the navigation task. The plot illustrates the mean episode reward across ranks, evaluated over 5 seeds. Colors: Rank 2 (yellow), Rank 4 (orange), Rank 8 (red), Rank 16 (pink), Rank 32 (purple), and Rank 64 (blue). Appendix E Broader Societal Impact The most direct beneficiaries of event-driven behavioral adaptation are safety-critical robotic teams operating in dynamic, partially observable environments—search and rescue, post-disaster reconnaissance, environmental monitoring, and inspection of hazardous infrastructure—where agents may fail mid-mission and the optimal allocation of behaviors changes abruptly as new information arrives. The agent-removal and unseen-event experiments in Section 5 show that the team can absorb such disruptions without retraining, which is a precondition for deploying these systems outside the lab. Zero-shot generalization across team size, capability, and target diversity additionally removes the need to train a separate policy for each deployment configuration, lowering the compute and energy cost of multi-robot deployment, and exposing NMDdesNMD_des as an interpretable input scalar gives operators a post-hoc handle on team behavior without retraining. Beyond its direct technical contribution, treating behaviors as properties of the task rather than of agents carries a conceptual implication worth surfacing. Most prior MARL frameworks tie roles to agent identity, implicitly assuming that some agents are intrinsically suited to certain functions; in our framework, no agent is permanently the leader, the scout, or the helper, and any agent can instantiate any behavior the task requires at the moment it requires it. We see this as a more democratic model of cooperation, in which roles emerge from circumstance rather than from identity. On the negative side, improvements in cooperative MARL are dual-use: the same properties that benefit search-and-rescue can in principle be applied to coordinated autonomous swarms in military or surveillance contexts. We do not believe our contribution unlocks fundamentally new capabilities along this axis but we acknowledge that more reliable team coordination is a building block that can be repurposed. A separate concern, noted as a limitation in Section 6, is that our hypernetwork is centralized and ingests observations from all agents, which raises both privacy considerations (when observations include sensor data from environments containing people) and security considerations (a single point of failure for the team’s behavioral allocation); the decentralized extension discussed in Section 6 is therefore a meaningful mitigation as well as a methodological direction.