Simplified three player Kuhn poker

TL;DR

This paper analyzes a simplified three-player Kuhn poker game, finding all equilibrium solutions analytically and exploring the dynamics of strategy adjustments through differential equations and repeated play models.

Contribution

It provides the first analytical solutions for all equilibria in a three-player Kuhn poker variant and examines the dynamic behavior of strategies over repeated interactions.

Findings

Multiple equilibrium solutions depending on pot size.

Oscillatory dynamics in strategy adjustment models.

Complex behavior in repeated play with noisy oscillations.

Abstract

We study a very small three player poker game (one-third street Kuhn poker), and a simplified version of the game that is interesting because it has three distinct equilibrium solutions. For one-third street Kuhn poker, we are able to find all of the equilibrium solutions analytically. For large enough pot size, $P$, there is a degree of freedom in the solution that allows one player to transfer profit between the other two players without changing their own profit. This has potentially interesting consequences in repeated play of the game. We also show that in a simplified version of the game with $P>5$, there is one equilibrium solution if $5 < P < P^* \equiv (5+\sqrt{73})/2$, and three distinct equilibrium solutions if $P > P^*$. This may be the simplest non-trivial multiplayer poker game with more than one distinct equilibrium solution and provides us with a test case for theories of dynamic strategy adjustment over multiple realisations of the game. We then study a third order system of ordinary differential equations that models the dynamics of three players who try to maximise their expectation by continuously varying their betting frequencies. We find that the dynamics of this system are oscillatory, with two distinct types of solution. We then study a difference equation model, based on repeated play of the game, in which each player continually updates their estimates of the other players' betting frequencies. We find that the dynamics are noisy, but basically oscillatory for short enough estimation periods and slow enough frequency adjustments, but that the dynamics can be very different for other parameter values.