  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.
