Effective Game-Theoretic Motion Planning via Nested Search

TL;DR

This work tackles game-theoretic motion planning for multiple kinodynamic robots by introducing Game-Theoretic Nested Search (GTNS), a nested outer-inner search that computes a global Nash equilibrium (NE) while pruning non-NE trajectories. GTNS operates on per-robot kinodynamic graphs and a tensor-product joint graph, solving a best-response inner problem to enforce NE and using an admissible heuristic to guide an optimal outer search. It additionally enables explicit NE selection via a user-defined global objective, enabling controllable equilibria in driving and racing scenarios, demonstrated with real-time-like performance on commodity hardware and strong theoretical guarantees (NE monotonicity and GTNS optimality). Relative to prior methods, GTNS achieves substantial speedups in complex, high-branching settings, while maintaining convergence guarantees and general applicability to non-convex dynamics and hard collision constraints, making it practical for safety-critical autonomous systems and AI-data generation for training models.

Abstract

To facilitate effective, safe deployment in the real world, individual robots must reason about interactions with other agents, which often occur without explicit communication. Recent work has identified game theory, particularly the concept of Nash Equilibrium (NE), as a key enabler for behavior-aware decision-making. Yet, existing work falls short of fully unleashing the power of game-theoretic reasoning. Specifically, popular optimization-based methods require simplified robot dynamics and tend to get trapped in local minima due to convexification. Other works that rely on payoff matrices suffer from poor scalability due to the explicit enumeration of all possible trajectories. To bridge this gap, we introduce Game-Theoretic Nested Search (GTNS), a novel, scalable, and provably correct approach for computing NEs in general dynamical systems. GTNS efficiently searches the action space of all agents involved, while discarding trajectories that violate the NE constraint (no unilateral deviation) through an inner search over a lower-dimensional space. Our algorithm enables explicit selection among equilibria by utilizing a user-specified global objective, thereby capturing a rich set of realistic interactions. We demonstrate the approach on a variety of autonomous driving and racing scenarios where we achieve solutions in mere seconds on commodity hardware.

Figures (10)

• Figure 1: Multi-lane merge: Robot 1 (blue) merges across Robot 2's (orange) lane into Robot 3's (green) lane. Existing algorithms typically restricted to a single homotopy class, whereas our method---via tuning of the global objective (here, decreasing Robot 1's priority (larger $\alpha$ in $J{=}(1{-}\alpha)J^1{+}\alpha(J^2{+}J^3)$) and increasing proximity penalty weight $\lambda_{\mathrm{prox}}$)---yields distinct NE behaviors: (a) aggressive zip-merge; (b) zip-merge; (c) yield-then-merge; (d) over-cautious.
• Figure 2: Kinodynamic grid graph. At each $(x,y)$ lattice coordinate (blue dots) there are various $\theta,\ v$ and $\delta$ values. Graph edges are gray; a randomly selected subset is emphasized and colored by the source state.
• Figure 3: Racetrack overtake. Prioritized robot takes inside lane at final turn and wins the race. Robot 1 (blue) always blocked at first turn by Robot 2 (orange) despite priority.
• Figure 4: Opposing-lane overtake. Robot 1 (blue) tries to overtake Robot 2 (orange) via Robot 3's (green) opposing lane. Different NE overtake strategies chosen as priority and proximity penalty are modulated.
• Figure 5: Following distance. Semi-collaborative setting, such as two cars on the same team, with social welfare cost $J=\sum_i J^i$ and $\lambda_{\mathrm{prox}}\in[0,1]$. The cars are attempting to cross the finish line, with the lowest possible summed cost. The addition of a proximity penalty to the cost, affects the following distance that Robot 1 (blue) maintains from Robot 2 (orange): A small $\lambda_{\mathrm{prox}}$ value yields close drafting, and larger values increase the time-gap as Robot 1 deliberately backs off from Robot 2, visible at the first corner.

Theorems & Definitions (6)

• definition thmcounterdefinition: Nash Equilibrium
• definition thmcounterdefinition: Graph Nash Equilibrium
• lemma thmcounterlemma: Monotonicity of Nash Equilibrium
• proof
• theorem thmcountertheorem
• proof