From Rules to Nash Equilibria: A Lean 4 Case Study in Game-Theoretic Analysis of a Competitive Trading Card Game

Abstract

We present a metagame analysis of the competitive Pokemon Trading Card Game, machine-checked in Lean 4 over real tournament data. The headline game-theoretic results, including Nash equilibrium, replicator dynamics, and the matrix-level type-bridge computation, rely on native_decide, which trusts Lean's compiler rather than its kernel; the trust boundary is made explicit. The artifact spans approximately 31,900 lines, 87 files, and 2,627 theorems, of which roughly 200 directly verify empirical claims, with no sorry, admit, or custom axioms. Analyzing Trainer Hill data from January to February 2026 for events with at least 50 players, over 14 archetypes and their full pairwise matchup matrix, we prove a popularity paradox: the most played deck, Dragapult, with 15.5% metagame share, has only 46.7% expected win rate, while Grimmsnarl, with 5.1% share, achieves 52.7%. A machine-checked Nash equilibrium of the raw game assigns Dragapult 0% weight; exhaustive enumeration over all nonempty support subsets confirms a unique symmetric Nash equilibrium of the constant-sum symmetrization with seven-deck support. Against this equilibrium mix, Dragapult falls 40.4 permil below the game value. Single-step replicator dynamics indicate downward fitness pressure on Dragapult, upward pressure on Grimmsnarl, and strongest extinction pressure on Alakazam. A 10,000-iteration sensitivity analysis confirms qualitative stability, with core support decks appearing in more than 96% of resampled equilibria. The primary contribution is methodological: a reproducible case study showing how formal verification can turn qualitative metagame narratives into machine-checkable, re-runnable strategic science.