A Data Driven Structural Decomposition of Dynamic Games via Best Response Maps

Dynamic games are powerful tools to model multi-agent decision-making, yet computing Nash (generalized Nash) equilibria remains a central challenge in such settings. Complexity arises from tightly coupled optimality conditions, nested optimization structures, and poor numerical conditioning. Existing game-theoretic solvers address these challenges by directly solving the joint game, typically requiring explicit modeling of all agents’ objective functions and constraints, while learning-based approaches often decouple interaction through prediction or policy approximation, sacrificing equilibrium consistency. This paper introduces a conceptually novel formulation for dynamic games by restructuring the equilibrium computation. Rather than solving a fully coupled game or decoupling agents through prediction or policy approximation, a data-driven structural reduction of the game is proposed that removes nested optimization layers and derivative coupling by embedding an offline-compiled best-response map as a feasibility constraint. Under standard regularity conditions, when the best-response operator is exact, any converged solution of the reduced problem corresponds to a local open-loop Nash (GNE) equilibrium of the original game; with a learned surrogate, the solution is approximately equilibrium-consistent up to the best-response approximation error. The proposed formulation is supported by mathematical proofs, accompanying a large-scale Monte Carlo study in a two-player open-loop dynamic game motivated by the autonomous racing problem. Comparisons are made against state-of-the-art joint game solvers, and results are reported on solution quality, computational cost, and constraint satisfaction.

💡 Research Summary

The paper tackles the long‑standing difficulty of computing Nash or generalized Nash equilibria (GNE) in finite‑horizon dynamic games, especially when the agents’ objectives and constraints are tightly coupled, the optimality conditions are nested, and the resulting nonlinear programs are poorly conditioned. Traditional approaches fall into two camps. Joint‑solvers (e.g., augmented‑Lagrangian, SQP, or MCP formulations) treat the whole game as a single large KKT or complementarity system. While conceptually straightforward, they require explicit knowledge of every player’s cost and constraints, lead to a dramatic increase in dimensionality, and often suffer from numerical fragility and slow convergence in realistic robotic settings. Iterative best‑response (IBR) methods avoid forming the full system by alternating between solving each player’s optimal control problem while holding the other fixed. However, IBR still requires solving a full constrained optimal control problem at every iteration, can diverge in non‑convex or constrained games, and, when differentiated through, may unintentionally compute a Stackelberg equilibrium rather than a Nash equilibrium.

The authors propose a fundamentally different structural reduction. They observe that the Nash equilibrium can be characterized as a fixed point of the best‑response operators. Instead of evaluating the opponent’s best response online (as in IBR) or embedding it in a joint solver, they compile an offline, data‑driven approximation of the opponent’s best‑response map (B_2(\cdot)) from historical interaction data. This map is then imposed as an explicit feasibility constraint (Z_2 = B_2(Z_1)) inside the ego player’s (player 1) optimal planning problem. Consequently, the online problem consists of (i) the ego player’s KKT conditions with the opponent’s trajectory treated as a parameter, and (ii) the best‑response feasibility block. The resulting reduced KKT system, \