Solution Concepts in Hierarchical Games under Bounded Rationality with Applications to Autonomous Driving

TL;DR

This work formalizes hierarchical game theory to model bounded rationality in autonomous driving, proposing four behavioural metamodels (three Quantal Level-k variants and a Nash-equilibrium with quantal errors) and evaluating 30 concrete models on a large naturalistic intersection dataset. The two-level framework captures high-level manoeuvres and low-level trajectories, with trajectory sampling schemes and multiobjective utilities reflecting safety, progress, and pedestrian considerations. Empirical results show that a Quantal Level-k model with level-0 rule-following best explains manoeuvre choices, while trajectory-level decisions benefit from bounded sampling and non-strategic solutions, with situational factors significantly impacting performance. The findings offer practical guidance for AV planners on selecting solution concepts and sampling schemes to closely match human driving behavior, enabling safer and more efficient autonomous navigation in complex traffic.

Abstract

With autonomous vehicles (AV) set to integrate further into regular human traffic, there is an increasing consensus on treating AV motion planning as a multi-agent problem. However, the traditional game-theoretic assumption of complete rationality is too strong for human driving, and there is a need for understanding human driving as a \emph{bounded rational} activity through a behavioural game-theoretic lens. To that end, we adapt four metamodels of bounded rational behaviour: three based on Quantal level-k and one based on Nash equilibrium with quantal errors. We formalize the different solution concepts that can be applied in the context of hierarchical games, a framework used in multi-agent motion planning, for the purpose of creating game theoretic models of driving behaviour. Furthermore, based on a contributed dataset of human driving at a busy urban intersection with a total of approximately 4k agents and 44k decision points, we evaluate the behaviour models on the basis of model fit to naturalistic data, as well as their predictive capacity. Our results suggest that among the behaviour models evaluated, at the level of maneuvers, modeling driving behaviour as an adaptation of the Quantal level-k model with level-0 behaviour modelled as pure rule-following provides the best fit to naturalistic driving behaviour. At the level of trajectories, bounds sampling of actions and a maxmax non-strategic models is the most accurate within the set of models in comparison. We also find a significant impact of situational factors on the performance of behaviour models.

Figures (10)

• Figure 1: An example of a two level hierarchical game with action level game 1 being the game of manoeuvres and action level game 2 is the game of trajectories. Different solution concepts can be used at different levels to find a game solution.
• Figure 2: Illustration of two instances of hierarchical games. (a) As a Stackelberg game modelling a lane change maneuver and (b) simultaneous move game modelling intersection navigation. A hierarchical game is instantiated every $\Delta t_{p}$ seconds with action plan of $\Delta t_{h}$ seconds.
• Figure 3: A snapshot of the intersection traffic scene. Representative trajectories based on the three sampling schemes over a $R^3$. The figure shows the path ($R^2$) projection of the trajectories and the dimension of time not represented in the figure.
• Figure 4: Representative trajectories based on the 3 sampling schemes over a $R^3$ lattice showing the spatial representation of the (a) path and (b) velocity profiles. Lattice points are connected with cubic splines.
• Figure 5: Utility function that maps a) minimum distance gap between trajectories to an utility interval [-1,1] (inhibitory utility for vehicle-vehicle interactions), and b) trajectory length to the utility interval [0,1] (excitatory utility).