Equilibrium solutions of three player Kuhn poker with $N>3$ cards: A new numerical method using regularization and arc-length continuation

TL;DR

This paper introduces a novel numerical method combining regularization and arc-length continuation to analyze complex equilibrium solutions in three-player Kuhn poker with more than three cards, revealing new strategic behaviors.

Contribution

The paper develops a new computational approach for solving polynomial systems in game theory and uncovers complex equilibrium structures in extended Kuhn poker variants.

Findings

Equilibrium solution curves are highly complex even for small card numbers.

N>4 card games differ qualitatively from the standard N=4 case.

Identifies a new equilibrium bet type with midrange cards acting as bluff and value bets.

Abstract

We study the equilibrium solutions of three player Kuhn poker with $N>3$ cards. We compute these solutions as a function of the initial pot size, $P$, using a novel method based on regularizing the system of polynomial equations and inequalities that defines the solutions, and solving the resulting system of nonlinear, algebraic equations using a combination of Newton's method and arc-length continuation. We find that the structure of the equilibrium solution curve is very complex, even for games with a small number of cards. Standard three player Kuhn poker, which is played with $N=4$ cards, is qualitatively different from the game with $N>4$ cards because of the simplicity of the structure of the value betting and bluffing ranges of each player. When $N>5$, we find that there is a new type of equilibrium bet with midrange cards that acts as a bluff against one player and a value bet against the other.