arxiv: 2604.13128 · v1 · submitted 2026-04-13 · 💻 cs.MA · cs.LG· cs.RO· cs.SY· eess.SY

Recognition: unknown

Learning Probabilistic Responsibility Allocations for Multi-Agent Interactions

Isaac Remy , Caleb Chang , Karen Leung

Authors on Pith no claims yet

Pith reviewed 2026-05-10 14:59 UTC · model grok-4.3

classification 💻 cs.MA cs.LGcs.ROcs.SYeess.SY

keywords responsibility allocationmulti-agent interactionsconditional variational autoencodertrajectory forecastingautonomous drivingprobabilistic modelingdifferentiable optimizationINTERACTION dataset

0 comments

The pith

A conditional variational autoencoder learns distributions over responsibility allocations in multi-agent interactions by mapping them to observable controls.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper seeks to capture how agents in shared environments divide responsibility for safety and accommodation, which shapes their actual paths away from individual goals. It builds a probabilistic model that outputs a distribution over possible responsibility values conditioned on the scene and each agent's context. This matters for autonomous systems because explicit responsibility signals can guide socially compliant planning and increase trustworthiness in human-machine interactions. The approach avoids the need for direct labels by training through a layer that converts responsibility choices into induced controls whose match to real data provides the learning signal.

Core claim

Responsibility allocations in multi-agent settings can be represented as samples from the latent space of a conditional variational autoencoder trained on trajectory data; the CVAE is conditioned on scene and agent information, and a differentiable optimization layer converts each sampled allocation into the control signals that would result, allowing the model to be optimized directly against observed trajectories even though explicit responsibility labels do not exist.

What carries the argument

Conditional variational autoencoder whose latent variables represent responsibility allocations, paired with a differentiable optimization layer that converts allocations into induced controls for supervision.

If this is right

The model produces multiple plausible responsibility allocations for any given scene rather than a single point estimate.
Downstream planners can sample from the learned distribution to generate behaviors that explicitly trade off individual goals against accommodation.
Analysis of the learned distributions on the INTERACTION dataset reveals recurring patterns in how drivers yield or assert priority.
The same architecture can be applied to any multi-agent dataset where trajectories but not responsibility labels are recorded.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The responsibility lens could be used to diagnose failures in existing multi-agent predictors by checking whether low-probability allocations correspond to observed collisions or near-misses.
One could test whether conditioning the model on additional context such as weather or time of day further reduces uncertainty in the responsibility distribution.
If the induced-control matching remains accurate across domains, the same CVAE-plus-differentiable-layer structure might transfer to non-driving settings such as pedestrian crowds or robot teams.

Load-bearing premise

That responsibility allocations are recoverable from the controls they induce, so that matching induced controls to observed trajectories supplies enough training signal without ground-truth responsibility labels.

What would settle it

A controlled experiment in which human observers annotate responsibility levels for the same scenes; if the model's sampled distributions show no statistical alignment with those annotations, or if replacing the responsibility layer with a direct trajectory predictor yields equal or better control matching, the central claim would be falsified.

Figures

Figures reproduced from arXiv: 2604.13128 by Caleb Chang, Isaac Remy, Karen Leung.

**Figure 2.** Figure 2: High-level diagram of the responsibility CVAE with [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: Learning synthetic p(γ | s). (a) The ground truth distribution of γ1 with respect to agents’ relative x−position; purple represents low densities and yellow represents high density. (b) Estimated distribution of γ1 from a responsibility CVAE with Gaussian latent space. (c) Agent 1’s ground truth x-acceleration vs. x-accel. from conditionally sampled γ1. VII. EXPERIMENTS AND DISCUSSION The first objective o… view at source ↗

**Figure 4.** Figure 4: Responsibility allocations γ in a traffic intersection from the INTERACTION dataset. Left: The blue car arrives in the interaction first, and is leaving (dashed lines denote past trajectories), while the red car has arrived before green. Right: Estimated probability densities of γ for red vs. green and red vs. blue over time; more yellow means higher density. TABLE I: Predicted trajectory performance on IN… view at source ↗

read the original abstract

Human behavior in interactive settings is shaped not only by individual objectives but also by shared constraints with others, such as safety. Understanding how people allocate responsibility, i.e., how much one deviates from their desired policy to accommodate others, can inform the design of socially compliant and trustworthy autonomous systems. In this work, we introduce a method for learning a probabilistic responsibility allocation model that captures the multimodal uncertainty inherent in multi-agent interactions. Specifically, our approach leverages the latent space of a conditional variational autoencoder, combined with techniques from multi-agent trajectory forecasting, to learn a distribution over responsibility allocations conditioned on scene and agent context. Although ground-truth responsibility labels are unavailable, the model remains tractable by incorporating a differentiable optimization layer that maps responsibility allocations to induced controls, which are available. We evaluate our method on the INTERACTION driving dataset and demonstrate that it not only achieves strong predictive performance but also provides interpretable insights, through the lens of responsibility, into patterns of multi-agent interaction.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a CVAE-plus-differentiable-optimizer way to learn distributions over responsibility allocations in multi-agent scenes, but the mapping from allocations to controls looks under-constrained for identifiability.

read the letter

The core contribution is a conditional VAE that outputs a distribution over responsibility vectors for each agent, then routes those vectors through a differentiable optimization layer so the model can be trained on observed controls without any ground-truth responsibility labels. They run it on the INTERACTION driving dataset and report that the resulting forecasts are competitive while also yielding interpretable responsibility breakdowns. That combination of latent modeling and the optimization trick is the genuinely new piece; prior trajectory work has used similar layers for other purposes, but not framed around responsibility allocation this way. The approach is technically clean on the surface and the lack of labels is handled practically rather than ignored. The soft spot is exactly the one the stress-test flags: nothing in the description shows that distinct responsibility vectors cannot produce identical equilibrium controls under the chosen cost functions. If that many-to-one case is common, the posterior over responsibility can be shaped more by the optimizer than by any independent signal of accommodation, which weakens the claim that the model is learning responsibility rather than just another latent that fits the trajectories. The abstract also stays light on quantitative details, so the “strong predictive performance” claim needs the full tables and baselines to judge. This is aimed at people building socially aware planners for driving or robotics who want probabilistic, context-conditioned responsibility models. It is coherent enough and technically grounded enough to deserve a serious referee, even though the identifiability question will probably require extra experiments or analysis in revision.

Referee Report

2 major / 2 minor

Summary. The paper introduces a CVAE-based approach to learn a conditional distribution over responsibility allocations in multi-agent interactions. Responsibility vectors are passed through a differentiable optimization layer to produce induced controls that are matched to observed trajectories, enabling training without ground-truth labels. The method combines this with multi-agent trajectory forecasting techniques and is evaluated on the INTERACTION driving dataset, with claims of strong predictive performance and interpretable insights into interaction patterns.

Significance. If the responsibility allocations can be shown to be identifiable and not merely artifacts of the optimization layer, the work would provide a useful probabilistic framework for interpreting shared constraints in multi-agent settings, with potential value for designing socially compliant autonomous systems. The use of a latent-space CVAE to capture multimodality is a positive aspect, but the absence of identifiability guarantees limits the immediate impact.

major comments (2)

[Method] Method (differentiable optimization layer): The central construction defines responsibility allocations r and maps them to induced controls u(r) via the differentiable layer, then matches u(r) to data. No analysis, proof, or regularization is described to ensure the map r → u(r) is injective or that the posterior selects a unique mode; multiple distinct r can induce identical equilibrium controls under standard multi-agent costs, so the learned p(r | context) risks being shaped by the layer rather than independent evidence of responsibility.
[Experiments] Evaluation: The abstract and results claim 'strong predictive performance' and 'interpretable insights' but supply no quantitative metrics (e.g., ADE/FDE, log-likelihood), baselines, ablation studies, or error analysis on the INTERACTION dataset. Without these, it is impossible to assess whether the model outperforms standard trajectory predictors or whether the responsibility lens adds explanatory power beyond the forecasting component.

minor comments (2)

[Abstract] Abstract: The phrase 'strong predictive performance' is used without any accompanying numbers or comparison; this should be replaced with concrete metrics or removed.
[Method] Notation: The responsibility vector r and the conditioning context are introduced without an explicit equation defining their dimensionality or the precise form of the CVAE encoder/decoder; adding these would improve clarity.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the constructive and detailed review. We appreciate the recognition of the CVAE-based probabilistic framework and its potential for interpreting multi-agent interactions. We address each major comment below with clarifications and planned revisions to improve the manuscript.

read point-by-point responses

Referee: [Method] Method (differentiable optimization layer): The central construction defines responsibility allocations r and maps them to induced controls u(r) via the differentiable layer, then matches u(r) to data. No analysis, proof, or regularization is described to ensure the map r → u(r) is injective or that the posterior selects a unique mode; multiple distinct r can induce identical equilibrium controls under standard multi-agent costs, so the learned p(r | context) risks being shaped by the layer rather than independent evidence of responsibility.

Authors: We agree that the mapping from responsibility allocations r to induced controls u(r) is not guaranteed to be injective under general multi-agent cost functions, and that this could influence the learned posterior. The differentiable optimization layer is intended to provide a tractable training signal by matching induced controls to observed trajectories, thereby grounding the latent responsibility distribution in data without requiring labels. In the revised version, we will expand the method section with an analysis of the optimization layer's properties (including cases of non-uniqueness), add a regularization term to promote distinct responsibility modes, and include empirical checks on the diversity of sampled r values. We cannot, however, provide a general proof of identifiability without further assumptions on the underlying costs. revision: partial
Referee: [Experiments] Evaluation: The abstract and results claim 'strong predictive performance' and 'interpretable insights' but supply no quantitative metrics (e.g., ADE/FDE, log-likelihood), baselines, ablation studies, or error analysis on the INTERACTION dataset. Without these, it is impossible to assess whether the model outperforms standard trajectory predictors or whether the responsibility lens adds explanatory power beyond the forecasting component.

Authors: The evaluation on the INTERACTION dataset in the manuscript includes both predictive performance demonstrations and qualitative analysis of responsibility patterns. To directly address the concern, we will revise the abstract, results, and evaluation sections to explicitly report quantitative metrics (ADE, FDE, and log-likelihood where relevant), detail the baseline trajectory predictors used for comparison, incorporate ablation studies isolating the responsibility allocation component, and expand error analysis to quantify the added explanatory value of the responsibility lens. revision: yes

standing simulated objections not resolved

Formal identifiability guarantees or proofs for the responsibility allocations given the differentiable optimization layer.

Circularity Check

0 steps flagged

No significant circularity; latent inference uses observable controls as external supervision

full rationale

The paper trains a CVAE to output a distribution over responsibility allocations r conditioned on context, then routes r through a differentiable optimization layer whose output (induced controls) is matched to observed trajectories from the INTERACTION dataset. This is standard variational latent-variable modeling with a reconstruction loss on observables; the learned p(r|context) is not equivalent to the input data by construction, nor is responsibility redefined as the output of the layer. Evaluation on held-out predictive performance supplies an independent benchmark. No self-citation, ansatz smuggling, or uniqueness theorem is invoked to close the loop. The acknowledged lack of ground-truth labels is handled by the data-driven loss rather than by tautological redefinition.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only access prevents identification of concrete free parameters, axioms, or invented entities; the responsibility allocation appears as a learned latent concept grounded via optimization but no explicit counts or details are given.

pith-pipeline@v0.9.0 · 5472 in / 1105 out tokens · 67909 ms · 2026-05-10T14:59:00.706035+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

32 extracted references · 8 canonical work pages · 1 internal anchor

[1]

Learning responsibility allocations for multi-agent interactions: A differentiable optimization approach with control barrier functions,

I. Remy, D. Fridovich-Keil, and K. Leung, “Learning responsibility allocations for multi-agent interactions: A differentiable optimization approach with control barrier functions,” inAmerican Control Confer- ence, 2025

2025
[2]

Learning Responsibil- ity Allocations for Safe Human-Robot Interaction with Applications to Autonomous Driving,

R. Cosner, Y . Chen, K. Leung, and M. Pavone, “Learning Responsibil- ity Allocations for Safe Human-Robot Interaction with Applications to Autonomous Driving,” inProc. IEEE Conf. on Robotics and Automation, 2023

2023
[3]

Social Coordination and Altruism in Autonomous Driving,

B. Toghi, R. Valiente, D. Sadigh, R. Pedarsani, and Y . Fallah, “Social Coordination and Altruism in Autonomous Driving,”IEEE Transac- tions on Intelligent Vehicles, vol. 23, no. 12, pp. 24 791–24 804, 2022

2022
[4]

Legible and Proactive Robot Planning for Prosocial Human-Robot Interactions,

J. Geldenbott and K. Leung, “Legible and Proactive Robot Planning for Prosocial Human-Robot Interactions,” inProc. IEEE Conf. on Robotics and Automation, 2024

2024
[5]

Courteous Au- tonomous Cars,

L. Sun, W. Zhan, M. Tomizuka, and A. Dragan, “Courteous Au- tonomous Cars,” inIEEE/RSJ Int. Conf. on Intelligent Robots & Systems, 2018

2018
[6]

Social behavior for autonomous vehicles,

W. Schwarting, A. Pierson, J. Alonso-Mora, S. Karaman, and D. Rus, “Social behavior for autonomous vehicles,”Proceedings of the Na- tional Academy of Sciences, vol. 116, no. 50, pp. 24 972–24 978, 2019

2019
[7]

Cooperative Autonomous Vehicles that Sympathize with Human Drivers,

B. Toghi, R. Valiente, D. Sadigh, R. Pedarsani, and Y . P. Fallah, “Cooperative Autonomous Vehicles that Sympathize with Human Drivers,” inIEEE/RSJ Int. Conf. on Intelligent Robots & Systems, 2021

2021
[8]

Human motion trajectory prediction: A survey,

A. Rudenko, L. Palmieri, M. Herman, K. M. Kitani, D. M. Gavrila, and K. O. Arras, “Human motion trajectory prediction: A survey,”Int. Journal of Robotics Research, vol. 39, no. 8, pp. 895–935, 2020

2020
[9]

Lang, Sourabh V ora, Venice Erin Liong, Qiang Xu, Anush Krishnan, Yu Pan, Giancarlo Baldan, and Oscar Beijbom

H. Caesar, V . Bankiti, A. H. Lang, S. V ora, V . E. Liong, Q. Xu, A. Krishnan, Y . Pan, G. Baldan, and O. Beijbom, “nuScenes: A multimodal dataset for autonomous driving,”Available at https: //arxiv.org/abs/1903.11027, 2019

work page arXiv 1903
[10]

Scalability in Perception for Autonomous Driving: Waymo Open Dataset,

P. Sun, H. Kretzschmar, X. Dotiwalla, A. Chouard, V . Patnaik, P. Tsui, J. Guo, Y . Zhou, Y . Chai, B. Caine, V . Vasudevan, W. Han, J. Ngiam, H. Zhao, A. Timofeev, S. Ettinger, M. Krivokon, A. Gao, A. Joshi, Y . Zhang, J. Shlens, Z. Chen, and D. Anguelov, “Scalability in Perception for Autonomous Driving: Waymo Open Dataset,” inIEEE Conf. on Computer Vis...

2020
[11]

PRECOG: PREdiction Conditioned On Goals in Visual Multi-Agent Settings,

N. Rhinehart, R. McAllister, K. Kitani, and S. Levine, “PRECOG: PREdiction Conditioned On Goals in Visual Multi-Agent Settings,” in IEEE Int. Conf. on Computer Vision, 2019

2019
[12]

INTENT: Trajectory Prediction Framework with Intention-Guided Contrastive Clustering,

Y . Tang and W. Ma, “INTENT: Trajectory Prediction Framework with Intention-Guided Contrastive Clustering,”Available at https://arxiv. org/abs/2503.04952, 2025

work page arXiv 2025
[13]

Multimodal Probabilistic Model-Based Planning for Human-Robot Interaction,

E. Schmerling, K. Leung, W. V ollprecht, and M. Pavone, “Multimodal Probabilistic Model-Based Planning for Human-Robot Interaction,” in Proc. IEEE Conf. on Robotics and Automation, 2018

2018
[14]

Interpretable Trajectory Prediction for Autonomous Vehicles Via Counterfactual Re- sponsibility,

K.-C. Hsu, K. Leung, Y . Chen, J. Fisac, and M. Pavone, “Interpretable Trajectory Prediction for Autonomous Vehicles Via Counterfactual Re- sponsibility,” inIEEE/RSJ Int. Conf. on Intelligent Robots & Systems, 2023, paper https://ieeexplore.ieee.org/document/10341712

work page arXiv 2023
[15]

Gpt-driver: Learning to drive with gpt.arXiv preprint arXiv:2310.01415,

J. Mao, Y . Qian, J. Ye, H. Zhao, and Y . Wang, “GPT-Driver: Learning to Drive with GPT,”Available at https://arxiv.org/abs/2310.01415, 2023

work page arXiv 2023
[16]

Driving with LLMs: Fusing Object-Level Vector Modality for Explainable Autonomous Driving,

L. Chen, O. Sinavski, J. H ¨uermann, A. Karnsund, A. J. Willmott, and D. Birch, “Driving with LLMs: Fusing Object-Level Vector Modality for Explainable Autonomous Driving,” inProc. IEEE Conf. on Robotics and Automation, 2024

2024
[17]

Embodied Understanding of Driving Scenarios,

Y . Zhou, L. Huang, Q. Bu, J. Zeng, T. Li, H. Qiu, H. Zhu, M. Guo, Y . Qiao, and H. Li, “Embodied Understanding of Driving Scenarios,” inEuropean Conf. on Computer Vision, 2024

2024
[18]

Trajectory Prediction Meets Large Language Models: A Survey,

Y . Xu, R. Yang, Y . Zhang, and Y . Wang, “Trajectory Prediction Meets Large Language Models: A Survey,”Available at https://arxiv.org/ abs/2506.03408, 2025

work page arXiv 2025
[19]

Control barrier function based quadratic programs with application to adaptive cruise control,

A. D. Ames, J. W. Grizzle, and P. Tabuada, “Control barrier function based quadratic programs with application to adaptive cruise control,” inProc. IEEE Conf. on Decision and Control, 2014

2014
[20]

Control Barrier Function Based Quadratic Programs for Safety Critical Systems,

A. D. Ames, X. Xu, J. W. Grizzle, and P. Tabuada, “Control Barrier Function Based Quadratic Programs for Safety Critical Systems,” IEEE Transactions on Automatic Control, vol. 62, no. 8, pp. 3861– 3876, 2017

2017
[21]

Responsibility-associated Multi-agent Collision Avoidance with Social Preferences,

Y . Lyu, W. Luo, and J. Dolan, “Responsibility-associated Multi-agent Collision Avoidance with Social Preferences,” inProc. IEEE Int. Conf. on Intelligent Transportation Systems, 2022

2022
[22]

Multimodal Deep Generative Models for Trajectory Prediction: A Conditional Variational Autoencoder Approach,

B. Ivanovic, K. Leung, E. Schmerling, and M. Pavone, “Multimodal Deep Generative Models for Trajectory Prediction: A Conditional Variational Autoencoder Approach,”IEEE Robotics and Automation Letters, vol. 6, no. 2, pp. 295–302, 2021

2021
[23]

Tutorial on

C. Doersch, “Tutorial on variational autoencoders,”Available at https: //arxiv.org/abs/1606.05908, 2016

work page arXiv 2016
[24]

Auto-Encoding Variational Bayes

D. P. Kingma and M. Welling, “Auto-Encoding Variational Bayes,” Available at https://arxiv.org/abs/1312.6114, 2022

work page internal anchor Pith review Pith/arXiv arXiv 2022
[25]

Categorial reparameterization with gumbel-softmax,

E. Jang, S. Gu, and B. Poole, “Categorial reparameterization with gumbel-softmax,” inInt. Conf. on Learning Representations, 2017

2017
[26]

AgentFormer: Agent-Aware Transformers for Socio-Temporal Multi-Agent Forecasting,

Y . Yuan, X. Weng, Y . Ou, and K. Kitani, “AgentFormer: Agent-Aware Transformers for Socio-Temporal Multi-Agent Forecasting,” inIEEE Int. Conf. on Computer Vision, 2021

2021
[27]

Adam: A Method for Stochastic Optimization,

D. Kingma and J. Ba, “Adam: A Method for Stochastic Optimization,” inInt. Conf. on Learning Representations, 2015

2015
[28]

beta-V AE: Learning Basic Visual Concepts with a Constrained Variational Framework,

I. Higgins, L. Matthey, A. Pal, C. Burgess, X. Glorot, M. Botvinick, S. Mohamed, and A. Lerchner, “beta-V AE: Learning Basic Visual Concepts with a Constrained Variational Framework,” inInt. Conf. on Learning Representations, 2017

2017
[29]

JAX: composable transformations of Python+NumPy programs,

J. Bradbury, R. Frostig, P. Hawkins, M. J. Johnson, C. Leary, D. Maclaurin, G. Necula, A. Paszke, J. VanderPlas, S. Wanderman- Milne, and Q. Zhang, “JAX: composable transformations of Python+NumPy programs,”Available at http://github.com/google/jax, 2018

2018
[30]

Equinox: neural networks in JAX callable PyTrees and filtered transformations,

P. Kidger and C. Garcia, “Equinox: neural networks in JAX callable PyTrees and filtered transformations,” inConf. on Neural Information Processing Systems, 2021

2021
[31]

On the Differentiability of the Primal- Dual Interior-Point Method,

K. Tracy and Z. Manchester, “On the Differentiability of the Primal- Dual Interior-Point Method,”Available at https://arxiv.org/abs/2406. 11749, 2024

2024
[32]

Interaction dataset: An international, adversarial and cooperative motion dataset in interactive driving scenarios with semantic maps

W. Zhan, L. Sun, D. Wang, H. Shi, A. Clausse, M. Nau- mann, J. K ¨ummerle, H. K ¨onigshof, C. Stiller, A. de La Fortelle, and M. Tomizuka, “INTERACTION Dataset: An INTERna- tional, Adversarial and Cooperative moTION Dataset in Inter- active Driving Scenarios with Semantic Maps,”Available at https://arxiv.org/abs/1910.03088, 2019

work page arXiv 1910