Full street simplified three player Kuhn poker

TL;DR

This paper analyzes a simplified three-player Kuhn poker variant, deriving all equilibrium solutions analytically, and studies the dynamics of repeated play, revealing complex behaviors like periodicity and chaos.

Contribution

It introduces a simplified model of three-player Kuhn poker that allows analytical solutions for all equilibria and explores the dynamic stability of these equilibria in repeated play.

Findings

Three distinct equilibrium solutions exist depending on pot size.

Equilibria are not asymptotically stable in the differential equation model.

Player profits over time align with equilibrium predictions.

Abstract

We study a simplified version of full street, three player Kuhn poker, in which the weakest card, J, must be checked and/or folded by a player who holds it. The number of nontrivial betting frequencies that must be calculated is thereby reduced from 23 to 11, and all equilibrium solutions can be found analytically. In particular, there are three ranges of values of the pot size, $P$, for which there are three distinct, coexisting equilibrium solutions. We also study an ordinary differential equation model of repeated play of the game, which we expect to be at least qualitatively accurate when all players both adjust their betting frequencies sufficiently slowly and have sufficiently short memories. We find that none of the equilibrium solutions of the game is asymptotically stable as a solution of the ordinary differential equations. Depending on the pot size, the solution may be periodic, close to periodic or have long chaotic transients. In each case, the rates at which the players accumulate profit closely match those associated with one of the equilibrium solutions of the game.