Learning Probabilistic Responsibility Allocations for Multi-Agent Interactions

Isaac Remy; Caleb Chang; Karen Leung

cs.MAApr 13, 2026

Learning Probabilistic Responsibility Allocations for Multi-Agent Interactions

Authors: Isaac Remy, Caleb Chang, Karen Leung

Organizations: University of Washington, Department of Aeronautics and Astronautics · NVIDIA

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.

Figures & tables

Fig. 1 : Bottom left: If all agents behave selfishly, executing their desired control, a collision occurs. So who is “responsible” for deviating from their desired control? Our responsibility generative model encodes the scene context into a latent space (top left), decodes it to a distribution over responsibility allocations (top right), which pass through a responsibility-aware safety filter, producing a multimodal distribution of agent behaviors (bottom right).

Fig. 2 : High-level diagram of the responsibility CVAE with continuous latent space. A distribution over responsibility allocation $p(\boldsymbol{\gamma}\mid\mathbf{s})$ conditioned on scene $\mathbf{s}$ is computed as the pushforward distribution of the latent distribution $p(z\mid\mathbf{s})$ under the initial part of the decoder $g_{\phi}$ . Responsibility allocation values $\boldsymbol{\gamma}$ are then passed through the responsibility safety filter to produce predicted agent controls $\mathbf{u}$ .

Panel of Figure 3 — Fig. 3 : Learning synthetic $p(\boldsymbol{\gamma}\mid\mathbf{s})$ . (a) The ground truth distribution of $\gamma_{1}$ with respect to agents’ relative $x-$ position; purple represents low densities and yellow represents high density. (b) Estimated distribution of $\gamma_{1}$ from a responsibility CVAE with Gaussian latent space. (c) Agent 1’s ground truth $x$ -acceleration vs. $x$ -accel. from conditionally sampled $\gamma_{1}$ .

Fig. 4 : Responsibility allocations $\boldsymbol{\gamma}$ 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 $\gamma$ for red vs. green and red vs. blue over time; more yellow means higher density.

Method	ADE (m)	Miss Rate (%)
Direct $\mathbf{u}$ prediction	$0.140_{\pm 9.10\text{e-}3}$	$1.27\%_{\pm 2.85\text{e-}4}$
Following $\mathbf{u}_{\mathrm{desired}}$	$0.406_{\pm 2.26\text{e-}3}$	$49.3\%_{\pm 7.27\text{e-}3}$
(1) $\mathrm{none}$	$0.221_{\pm 9.44\text{e-}3}$	$9.31\%_{\pm 2.83\text{e-}3}$
(2) $\mathrm{softmax}$	$\mathbf{0.178_{\pm 9.91\text{e-}3}}$	$10.1\%_{\pm 3.47\text{e-}3}$
(3) $\mathrm{clip}$ at $0$	$\mathbf{0.185_{\pm 9.76\text{e-}3}}$	$9.80\%_{\pm 2.83\text{e-}3}$
(4) $\mathrm{clip}$ at $-\beta$	$0.216_{\pm 9.30\text{e-}3}$	$\mathbf{8.74\%_{\pm 2.87\text{e-}3}}$

TABLE I : Predicted trajectory performance on INTERACTION data over

1

second prediction horizon. Lower is better.

Multi-Agent Systems

Multi-Agent Systems studies how autonomous agents coordinate, compete, communicate, and learn in shared or interdependent tasks.

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When multiple agents share a cost budget, a common Lagrange multiplier can enforce the aggregate constraint but does not determine how its penalty should be allocated across agents. Uniform penalties ignore heterogeneity in the rewards agents sacrifice, while agent-specific multipliers may still rely on the same aggregate cost signal. We introduce Lagrangian Responsibility Allocation (LiRA), which learns each agent's share of a common multiplier by optimizing social welfare over a finite training horizon. The multiplier enforces the aggregate budget, while responsibility shares redistribute its influence without modifying the original rewards or constraints. For convex games under standard regularity conditions, varying these shares induces a smooth family of normalized generalized Nash equilibria in which active constraints remain at their budgets while welfare varies. To optimize responsibility before convergence, we derive a welfare gradient that accounts for both learning updates and the induced change in data distribution. Across CityLearn, MABIM, Harvest, and MetaDrive, spanning 3 to 400 agents, LiRA improves average social welfare by up to 29% over uniform and agent-specific multiplier baselines. Grid and driving costs remain within budget, inventory violations decrease, and Harvest makes more effective use of available budget.

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