Inferring Occluded Agent Behavior in Dynamic Games from Noise Corrupted Observations

1. Problem Statement

In autonomous driving and mobile robotics, agents (e.g., vehicles) interact within a dynamic environment often modeled as a noncooperative dynamic game where trajectories are Nash equilibria. Existing game-theoretic inference and planning methods rely on observing all agents to accurately estimate their intentions (cost function parameters) and predict trajectories.

However, a critical limitation in real-world scenarios is occlusion. Sensors (LiDAR, cameras) have range constraints and blind spots, meaning agents may be hidden behind other objects or vehicles.

• The Challenge: If an agent is occluded, standard methods cannot observe its state. Consequently, they fail to infer the reason why visible agents are behaving in certain ways (e.g., a visible car decelerating might be reacting to an invisible car).
• The Consequence: Ignoring occluded agents leads to inaccurate estimation of visible agents' objectives and suboptimal, potentially unsafe navigation decisions (e.g., proceeding into an intersection when a hidden vehicle is about to cross).

2. Methodology

The authors propose a unified framework combining game-theoretic inference with contingency planning to handle occlusions.

A. Occlusion-Aware Game-Theoretic Estimator

The core inference problem is formulated as a Maximum Likelihood Estimation (MLE) task.

• Goal: Estimate the weighting parameters ($w$) of all agents' cost functions and the trajectories ($x$) of both visible and occluded agents, using only noisy position observations of the visible agents ($y_V$).
• Formulation:
  • The system assumes agents play an Open-Loop Nash Equilibrium (OLNE).
  • The Nash condition is encoded using the Karush-Kuhn-Tucker (KKT) conditions (first-order necessary conditions), creating a set of equality constraints $G(x, u, \lambda; w) = 0$.
  • The objective is to minimize the least-squares error between observed noisy positions and the predicted positions of visible agents, subject to the Nash equilibrium constraints.
  • Key Insight: By solving for the parameters that best explain the behavior of visible agents, the system simultaneously infers the existence and trajectory of occluded agents that are influencing the visible ones.

B. Occlusion-Aware Contingency Game Planner

To navigate safely given the uncertainty of occluded agents, the authors propose a Receding Horizon Contingency Game.

• Hypothesis Space: The planner considers two hypotheses ($\theta \in \Theta$):
  1. $\theta_1$: Occluded agents exist.
  2. $\theta_2$: Occluded agents do not exist.
• Branching Time ($t_b$): A specific time horizon is defined after which occluded agents are guaranteed to become visible.
• Strategy Structure:
  • Before $t_b$ ($k < t_b$): The agent must choose a single control input that balances the risk of both hypotheses. It minimizes the expected cost weighted by the belief probability $b(\theta)$ of each scenario.
  • After $t_b$ ($k \geq t_b$): The agent knows the true state (occluded agents are visible). It switches to the specific control input corresponding to the realized hypothesis.
• Implementation: The agent constructs a larger game with "copies" of other agents for each hypothesis. It solves for a Nash equilibrium in this expanded space, enforcing the constraint that control inputs must be identical before $t_b$.

C. Receding Horizon Pipeline

The estimator and planner are integrated into a loop:

1. Estimation: At each time step $k$, the robot uses the last $K$ observations of visible agents to estimate the parameters and states of all agents (visible + occluded) via the MLE solver.
2. Planning: Using these estimates, the robot solves the contingency game to generate a control sequence.
3. Execution: The robot executes the first control input and updates its state, repeating the process.

3. Key Contributions

1. Occlusion-Aware Estimator: A novel method to simultaneously infer the cost function parameters and trajectories of occluded agents based solely on noisy observations of visible agents, leveraging the coupling of agent behaviors in a Nash game.
2. Contingency Game Planner: A receding-horizon planning strategy that explicitly models the uncertainty of occluded agents, allowing the robot to balance risk between "occlusion exists" and "occlusion does not exist" until the occlusion clears.
3. Integrated Pipeline: A complete system combining inference and planning that operates effectively in dynamic, noisy environments.

4. Experimental Results

The authors validated their approach using Monte Carlo simulations in three scenarios:

• 3-Agent Interaction (Estimation):
  • Setup: One robot observes two visible agents and one occluded agent.
  • Result: The proposed method significantly outperformed an "occlusion-ignorant" baseline (which assumes a 2-agent game). It achieved higher accuracy in estimating the parameters and trajectories of visible agents and successfully reconstructed the trajectory of the occluded agent.
• Collision Avoidance (Planning):
  • Setup: Multi-agent scenarios (4, 6, 8 agents) with random occlusions.
  • Result: The contingency planner maintained larger minimum distances between agents compared to the baseline. Agents equipped with the proposed method behaved more conservatively before the branching time ($t_b$), actively avoiding potential collisions with hidden agents.
• Crossing-Road Scenario (Realistic Driving):
  • Setup: A complex intersection where a green vehicle is occluded from horizontal traffic by a red vehicle.
  • Result:
    • Baseline: Ignored occluded vehicles, assumed a 2-agent game, and entered the intersection blindly, leading to near-accidents.
    • Proposed Method: Inferred the existence of occluded vehicles from the red vehicle's deceleration. It formulated a contingency plan that slowed down before the intersection, avoiding a collision.
    • Metrics: The proposed method reduced the Average Displacement Error (ADE) for observed agents and increased the minimum distance to occluded agents by approximately 1 meter compared to the baseline.

5. Significance

• Safety: The primary contribution is a significant enhancement in navigation safety. By reasoning about "what might be hidden" rather than just "what is seen," robots can avoid catastrophic collisions in occluded environments.
• Behavioral Understanding: The method demonstrates that occluded agents leave "traces" in the behavior of visible agents. Inferring these traces allows for a more accurate understanding of the entire traffic ecosystem, not just the visible subset.
• Robustness: The approach is robust to sensor noise, making it suitable for real-world deployment where sensor data is rarely perfect.
• Future Impact: This work bridges the gap between theoretical game theory and practical autonomous navigation, providing a pathway for robots to operate safely in complex, partially observable urban environments. Future work aims to integrate real sensor data (LiDAR/Camera) for deployment in physical systems.