The Topology of Poker
Laurent Bartholdi, Roman Mikhailov
TL;DR
This work introduces a topological framework for analyzing Texas Hold'em by constructing a simplicial complex $\mathscr K_X$ on the set of two-card hands, where a simplex corresponds to a subset on which the beating relation $r$ is transitive and $r((i,j),(k,\ell))$ holds when $(i,j)$ wins with probability $>50\%$ after averaging over board cards. The authors prove that $\mathscr K_X$ can contain nontrivial topology, including an induced $S^4$, implying that one-player vs. another-player win probabilities do not fully determine the game state and that the notion of bluffing can be ill-defined in certain configurations. They implement the approach in Julia, compute the relation $r$ exhaustively, and use $Oscar$OSCAR and Polymake to analyze homology, providing concrete examples (e.g., a subset $Y$ where $\mathscr K_Y$ contains $S^1$ and its join forms $S^4$). The paper highlights future directions such as sequential reveal of cards, persistent homology to study data sensitivity, and applications to other intransitive games, indicating a rich, computable topological invariant for complex multi-player strategy contexts with potential implications for strategic thinking in poker and beyond.
Abstract
We examine the complexity of the ``Texas Hold'em'' variant of poker from a topological perspective. We show that there exists a natural simplicial complex governing the multi-way winning probabilities between various hands, and that this simplicial complex contains $4$-dimensional spheres as induced subcomplexes. We deduce that evaluating the strength of a pair of cards in Texas Hold'em is an intricate problem, and that even the notion of who is bluffing against whom is ill-defined in some situations.