Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2607.08692

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

Arthur F. Ramos·Tulio Soria
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

cs.GTcs.FL

Keep this discovery

BibTeXRIS

Arthur F. Ramos, Tulio Soria. 2026-07-09. From Rules to Nash Equilibria: A Lean 4 Case Study in Game-Theoretic Analysis of a Competitive Trading Card Game. https://arxiv.org/abs/2607.08692

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

MMS Allocation for Chores with Online Agent Arrivals

We study the fair allocation of $m$ indivisible chores to $n$ agents with subadditive cost functions arriving online in an arbitrary order. Upon an agent's arrival, we are informed of her cost function and must irrevocably assign her a set of chores. We focus on the Maximin Share (MMS) fairness notion and aim to compute an allocation in which all items are assigned, and no agent incurs a cost more than $\alpha$ times her MMS. Without any prior information about the instance (other than $n$ and $m$), we design an algorithm with a competitive ratio of $O(\min\{n, k\log^{1+\epsilon}k, \log m\})$ for any constant $\epsilon > 0$, where $k$ denotes the number of cost function types. Our bound matches the best known offline approximation guarantees for MMS under subadditive costs and is nearly optimal with respect to all three parameters: we show that even for binary additive cost functions, no online algorithm can achieve a competitive ratio of $o(\min\{n, k\log k, \log m\})$. We then consider the setting in which the $k$ cost function types are known in advance (though the realized types of arriving agents are not). For additive cost functions, we provide an algorithm with a competitive ratio of $O(\min\{\log k, \log(kn)/\log\log(kn)\})$, and show that constant-competitive algorithms do not exist for general $k$, even for the binary additive setting. For binary additive functions when $k \le n$, we propose a $3$-competitive algorithm and establish a lower bound of $2$.

cs.GT↗
arXiv2026

Truncated Noisy Best-Response Algorithms: Toward Game Theoretic Learning with Safety Guarantees

We consider a game theoretic approach to solve multi-agent coordination problems with submodular maximization objectives. It is known for such problems that the Nash equilibria for the corresponding game are always within 50% of the optimal, but that the equilibria which achieve this worst-case bound are not stable. To exploit this instability, we propose a family of algorithms which we call Truncated Noisy Best-Response (TNBR) Algorithms. These algorithms are flexibly characterized by agents asynchronously and stochastically selecting actions from a neighbourhood of their best response payoffs. We compute bounds on the recurrent classes of TNBR algorithms' associated Markov chains. Our bounds fall into two categories: first, "Performance" bounds ensure that TNBR algorithms always have a high-value recurrent state; second, "Safety" bounds ensure that TNBR algorithms never have arbitrarily-bad recurrent states. Furthermore, these two types of bounds are linked by a waterbed-like effect: every game with a poor Safety guarantee necessarily has a favorable Performance guarantee.

cs.GT↗
arXiv2026

Existence of the Core in Approval-Based Committee Elections

We settle the main open question in the theory of approval-based multi-winner elections: we show that there always exists a committee in the core. The core is a stability and group fairness concept. The proof introduces a new voting rule that optimizes an entropy-like objective function over committees and payment systems. All local optima of this objective function lie in the core, which implies that a core committee can be found in polynomial time.

cs.GT↗