Bivariate Trajectory and Undercurrent Theory (BTUT) - A Solution for Scalable Game Theory

BTUT: Bivariate Trajectory–Undercurrent Theory A PDE-Free, Scalable Framework for Large-Scale Game Dynamics — Addressing DARPA Mathematical Challenge 13 Overview BTUT is a novel, kernel-based game-theoretic framework that directly tackles DARPA Mathematical Challenge 13 (2008): "Creating a Game Theory that Scales — What new scalable mathematics is needed to replace the traditional Partial Differential Equations (PDE) approach to differential games?" Instead of PDEs or static Nash equilibria, BTUT introduces dynamic flow equilibria balancing observable trajectories (agent actions) and latent undercurrents (network pressures). Operating on scale-free networks with hub-weighted influence wi=(ki/kmax⁡)τw_i = (k_i / k_{\max})^\tauwi=(ki/kmax)τ, it models mixed interactions (Prisoner's Dilemma, Hawk-Dove, Stag Hunt) via local rules and momentum updates. Key Innovations & Results PDE-free scalability: Strict O(N) complexity, verified up to 1e6+ agents. Ultra-robust convergence: Equilibrium p∗≈1.000p^* \approx 1.000p∗≈1.000 (std ≈ 4.9×10⁻¹⁰) in <20 iterations. Rich phase behavior: Sharp transitions (see Phase Map A staircase at γ=1.45, c_A^SH=0.40). Scale-invariant continuous-action extension: Full cooperation in 2–3 iterations (no dense matrices). Runtime: ~0.12s for 30 iterations at N=1,000 (~8.3M agents/sec equivalent). Repository Contents Core: btut_grok_test.py, btut_continuous.py Tools: btut_animate.py, btut_diagnostics.py, btut_scaling_test.py, btut_random_sweep.py Theory: btut_math.pdf (full derivation) Interactive: BTUT.html Requirements: requirements.txt Applications Scalable modeling of emergent cooperation in AI agent fleets, cyber-physical systems, resource allocation, and alignment scenarios. Developed by Diren Kumaratilleke (UNC Chapel Hill freshman, Class of 2029) AI-assisted development via Grok-4 (xAI) and GPT models. License: MIT