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.
Billingham, J. (2017). Full street simplified three player Kuhn poker. ArXiv