Game Theory in Formula 1: From Physical to Strategic Interactions

TL;DR

This work develops a holistic, two-agent optimization framework for Formula 1 racing that integrates wake-induced aerodynamics, trajectory planning, and hybrid energy management within dynamic games. By formulating both Stackelberg and Nash equilibria and employing KKT-based reformulations, the approach yields tractable single-level nonlinear programs that reveal how wake effects shape optimal trajectories and overtaking decisions. Case studies illustrate how drag and downforce reductions depend on relative positioning and how energy budgets influence overtaking locations, demonstrating the framework’s ability to reproduce realistic racing behavior. The method offers practical potential for motorsport engineering and autonomous racing, providing a principled way to analyze and optimize multi-agent strategies under complex physical coupling.

Abstract

This paper presents an optimization framework to model Formula 1 racing dynamics, where multiple cars interact physically and strategically. Aerodynamic wake effects, trajectory optimization, and energy management are integrated by means of physical models. We describe the minimum lap time problem with two agents as either a Nash or a Stackelberg game, and by employing the Karush-Kuhn-Tucker conditions during the problem formulation, we recover the structure of a nonlinear program. In addition, we introduce an algorithm to refine local Stackelberg solutions, using the Nash costs as upper bounds. The resulting strategies are analyzed through case studies. We examine the impact of slipstreaming on trajectory selection in corners, straights, and high-speed sections, while also identifying optimal overtaking locations based on energy allocation strategies. Exploiting the structural similarities of the game formulations, we are able to compare symmetric and hierarchical strategies to analyze competitive racing dynamics. By incorporating a physically accurate interaction model and accounting for the optimal responses of competing agents, our approach reveals typical Formula 1 strategic behaviors. The proposed methodology closes the gap between theoretical game theory and real-world racing, with potential applications in motorsport engineering and autonomous racing.

Figures (9)

• Figure 1:
• Figure 2: Schematic of the . The on-board energy storages are the fuel tank and the battery. The prime movers are the and the turbocharged engine.
• Figure 3: Schematic of the trajectory's model. The centerline curvilinear coordinate is represented by $s$, $r_\mathrm{c}$ is the curvature radius, $y$ the lateral displacement of the agent w.r.t. the centerline and $\varphi$ the heading angle.
• Figure 4: Schematic example of the agents in the coordinate system. We show two locations along the track, with lateral deviations and time. As an example, for the time instant $t = \qty{8}{\second}$, $A$ is at $s_{2}$, ahead of $B$, which is still at $s_{1}$. The lateral deviation is computed when the cars are at the same location, e.g., $s_{1}$.
• Figure 5:

Theorems & Definitions (3)

• Definition 1
• Definition 2
• Definition 3