The design of voting rules is traditionally guided by desirable axioms.

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As such, several crucial questions arise: What are the game-theoretic implications of the no-regret guarantees of FTRL? Do the dynamics of FTRL converge to an equilibrium of the underlying game? A folk answer to this question is that " no-regret learning converges to equilibrium in all games "

We first introduce some additional notations for convenience. Our proof mainly consists of the following steps: 1. Helper lemmas and a crude bound. See A.2, and more precisely, Lemmas A.9 and A.10. 3. Final bound for null -approximate NE value. See A.3. 4. Final bounds for null -NE policy. See A.5. 14 A.1 Important Lemmas We start with the component-wise error bounds.

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Abstract--Trajectory prediction is a fundamental task in Autonomous V ehicles (A Vs) and Intelligent Transportation Systems (ITS), supporting efficient motion planning and real-time traffic safety management. Diffusion models have recently demonstrated strong performance in probabilistic trajectory prediction, but their large model size and slow sampling process hinder real-world deployment. This paper proposes Collaborative-Distilled Diffusion Models (CDDM), a novel method for real-time and lightweight trajectory prediction. Built upon Collaborative Progressive Distillation (CPD), CDDM progressively transfers knowledge from a high-capacity teacher diffusion model to a lightweight student model, jointly reducing both the number of sampling steps and the model size across distillation iterations. A dual-signal regularized distillation loss is further introduced to incorporate guidance from both the teacher and ground-truth data, mitigating potential overfitting and ensuring robust performance. Extensive experiments on the ETH-UCY pedestrian benchmark and the nuScenes vehicle benchmark demonstrate that CDDM achieves state-of-the-art prediction accuracy. The well-distilled CDDM retains 96.2% and 95.5% of the baseline model's ADE and FDE performance on pedestrian trajectories, while requiring only 231K parameters and 4 or 2 sampling steps, corresponding to 161 compression, 31 acceleration, and 9 ms latency. Qualitative results further show that CDDM generates diverse and accurate trajectories under dynamic agent behaviors and complex social interactions. By bridging high-performing generative models with practical deployment constraints, CDDM enables resource-efficient probabilistic prediction for A Vs and ITS. As the rapid development of Autonomous V ehicles (A Vs) and Intelligent Transportation Systems (ITS), an increasing trend of research advancement in trajectory prediction has emerged. Trajectory prediction refers to the predictive estimation of traffic agents' future motion or states (e.g., vehicles, pedestrians) in complex surrounding environments.

B. Algorithmically Heterogeneous MAMP T echniques Unlike algorithmically homogeneous MAMP methods, al-gorithmically heterogeneous MAMP methods do not require each agent run the same solver. To our surprise, we could not find any published work that addresses this problem setting. In particular, existing MAMP methods for heterogeneous teams focus on robots with different capabilities but use algorithmi-cally homogeneous solutions (e.g., [7], [11], [16]). On the other hand, existing multi-agent task planning/coordination methods focus on heterogeneous behaviors or task assignment and not on collision-free movement [27], [28]. Thus, part of this paper's goal is to introduce / bring attention to the Algorithmically Heterogeneous MAMP (AH-MAMP) problem setting. AH-MAMP tries to achieve collision-free motion planning for heterogeneous single-agent solvers without being able to modify the solvers. Solutions for AH-MAMP instead require designing multi-agent protocols with well-defined single-agent APIs, with the protocol/API abstraction enabling using heterogeneous single-agent solvers.

The emergence of large language models has enabled sophisticated multi-agent systems, yet coordinating their reasoning capabilities through prompt engineering remains challenging. We present a theoretically-grounded framework for dynamic prompt orchestration that enhances reasoning across multiple specialized agents. This framework addresses three core challenges: logical consistency preservation during agent transitions, reasoning-aware prompt adaptation, and scalable coordination of distributed inference. Our approach formalizes agent states using prompt templates, reasoning context vectors, and capability matrices. We prove system convergence to stable coordination patterns when step sizes satisfy $α< \frac{1}{2L}$ where $L$ is the Lipschitz constant of the state transition function. We implement this through a distributed architecture that dynamically routes reasoning tasks while maintaining semantic coherence. Experimental results on 1,000 synthetic multi-agent conversations demonstrate a 42% reduction in reasoning latency, a 23% improvement in logical consistency measured by ROUGE-L score, and an 89% success rate for task completion without context loss across agent transitions. Ablation studies identify the consensus mechanism as the primary performance driver, while revealing limitations: performance degrades beyond 10 agent transitions, and the system requires 76.5GB memory for 1,000 concurrent agents. These findings establish a new paradigm for scalable reasoning in multi-agent systems, providing theoretical foundations for understanding reasoning emergence across coordinated language models.