Trajectory Entropy: Modeling Game-State Stability from Multimodal Trajectory Prediction
IEEE Transactions on Intelligent Transportation Systems 2026
Abstract
Multi-agent interaction and joint decision-making are central challenges in autonomous driving. Existing level-k game-based methods refine multimodal trajectory predictions through several reasoning stages, but they process every agent at every stage, even after some agents have already reached a stable state. This creates redundant computation and may introduce unnecessary prediction noise.
In this paper, we introduce Trajectory Entropy (TE), a confidence-weighted dispersion measure that quantifies the stability of each agent's game state directly from multimodal trajectory predictions. Based on TE, we propose a lightweight Trajectory Entropy Gate that enables agent-wise early exit: stable agents are frozen and propagated to later stages, while uncertain agents continue deeper interaction reasoning. Experiments on Waymo Open Motion Dataset and nuPlan show improvements in trajectory prediction and planning performance, together with reduced inference cost.
Background
Autonomous driving systems must anticipate the future motion of surrounding vehicles, pedestrians, and cyclists, then use those forecasts to plan an ego trajectory that is safe, comfortable, and compliant with traffic rules. This requirement gives rise to the joint task of multi-agent trajectory prediction and ego-vehicle planning.
Driving unfolds in a dynamic and strongly interactive environment. One vehicle's acceleration, braking, turn, or lane change alters the feasible space of other agents, whose responses then change the ego plan. Road users may have several plausible futures, and their importance can change over time. A practical system therefore needs to allocate enough reasoning to unresolved interactions while avoiding repeated computation for agents whose behavior has already stabilized.
From Level-k Reasoning to Agent-wise Early Exit
GameFormer addresses coupled multi-agent motion by organizing joint prediction and ego planning as a hierarchy of Transformer blocks inspired by level-k game theory. Level 0 produces multimodal trajectories for all agents; each subsequent level refines those predictions through cross-attention to the previous level's outputs.
However, GameFormer treats every agent as equally important and processes all of them at every game level. Agents whose states have already stabilized are decoded repeatedly, wasting computation and potentially injecting noise into deeper layers. This motivates an agent-wise early-exit mechanism in which different agents can stop at different reasoning depths according to their own stability.
Method
1. Trajectory Entropy
The multimodal trajectories produced at every game level contain a direct signal of agent-state stability. Concentrated, confident modes indicate a stable state, while dispersed alternatives indicate unresolved behavior. For each agent, TE computes pairwise Euclidean distances between trajectory modes, uses squared distance to represent spatial dispersion, and weights each pair by the confidence of its two modes.
Trajectory modes and agent-state stability. Concentrated modes correspond to stable predictions, while spatially separated high-confidence modes indicate unresolved behavior.
The confidence-weighted dispersion is accumulated over the prediction horizon and normalized by the expected distance traveled per unit time, reducing scale differences caused by speed and horizon length. A prediction dominated by one high-confidence mode produces low TE, whereas several high-confidence modes that remain far apart produce high TE.
Trajectory Entropy formulation. TE combines modal confidence, pairwise trajectory distances, and the prediction horizon into a stability indicator.
2. Trajectory Entropy Gate
Before each game decoder, the Trajectory Entropy Gate evaluates every active agent and compares its TE with the threshold for that level. If TE falls below the threshold, the agent is considered stable: its current trajectory prediction is propagated directly to later levels and the agent becomes inactive. Agents above the threshold remain active and continue interaction reasoning.
Agent-wise early exit with the TE Gate. Stable agents are frozen, while uncertain agents enter the next level-k decoder.
Results
We evaluate TE on the Waymo Open Motion Dataset (WOMD) and nuPlan across trajectory prediction, open-loop planning, and closed-loop planning. The TE Gate improves prediction quality, planning performance, and inference efficiency while adding little overhead for entropy computation and gating.
On nuPlan trajectory prediction, TE reduces minFDE by 32.88% and miss rate by 44.62%. On Waymo closed-loop planning, the success rate improves from 82.25% to 87.25%. Inference time is reduced by 14.24% for nuPlan prediction and 18.54% for Waymo open-loop planning. The aggregate nuPlan reactive closed-loop score improves by as much as 16.48%.
Planning performance. The TE Gate improves closed-loop planning performance while reducing computation spent on stabilized agents.
TE decreases over successive level-k game stages, providing empirical evidence that it tracks the stabilization of agent states. Incorrect freezing occurs mainly at highly interactive intersections and accounts for approximately 5.12% of 2,800 Waymo open-loop scenes. Practical deployment should combine TE gating with conservative thresholds, cross-level consistency checks, agent reactivation, and an independent safety monitor.
Citation
@article{zhang2026trajectoryentropy,
title={Trajectory Entropy: Modeling Game State Stability From Multimodal Trajectory Prediction},
author={Zhang, Yesheng and Sun, Wenjian and Chen, Yuheng and Liu, Qingwei and Lin, Qi and Zhang, Rui and Zhao, Xu},
journal={IEEE Transactions on Intelligent Transportation Systems},
year={2026}
}