von Neumann and Newman Pokers with Finite Decks
Tipaluck Krityakierne, Thotsaporn Aek Thanatipanonda, Doron Zeilberger
TL;DR
The paper investigates finite-deck analogues of von Neumann's two-player poker and Newman’s poker, along with a three-player extension, bridging classic infinite-deck results and realistic finite settings. It develops and implements LP/NLP-based methods (slow and fast variants) to compute Nash equilibria, enabling large-scale exploration of per-card strategies and their bluffing structures. Key contributions include explicit pure and mixed NE results across a range of deck sizes and bet sizes, convergence to continuous-deck limits (e.g., $1/7$ for Newman and $1/9$ for certain two-player settings), and a demonstration that three-player dynamics can yield higher values for the first mover. The work provides practical Maple tools and detailed numerical evidence, offering insights into equilibrium design, bluffing strategies, and the relationship between finite and continuous poker in game-theoretic analysis.
Abstract
John von Neumann studied a simplified version of poker where the "deck" consists of infinitely many cards, in fact, all real numbers between $0$ and $1$. We harness the power of computation, both numeric and symbolic, to investigate analogs with finitely many cards. We also study finite analogs of a simplified poker introduced by D.J. Newman, and conclude with a thorough investigation, fully implemented in Maple, of the three-player game, doing both the finite and the infinite versions. This paper is accompanied by two Maple packages and numerous output files.