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

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.