The Mathematics of Poker: Extending the Nash-Shapley Model

In 1938, John von Neumann proposed his now-famous mathematical model of poker. Over the years, other poker models have been proposed and studied by mathematicians (Borel, Bellman, Blackwell), economists (Kuhn, McAdams), and even professional poker players (Chris Ferguson). However, all of these models focus on the two-player case. It was not until 1950 that the first three-player poker model was proposed by John Nash and Lloyd Shapley, who derived optimal probabilistic strategies for each of the players. Their model remains one of the only mathematical poker models involving more than two players. The Nash-Shapley model assumes there are only two kinds of cards, high and low. At the beginning of the game, each player is dealt a card, chosen at random from the two kinds. The game then proceeds for up to five rounds of betting or passing actions. Nash and Shapley derived optimal betting probabilities for each player and each round of this game. We implemented the Nash-Shapley model in Mathematica in order to have the ability to extend the model in new directions. In particular, we explored different player profiles (Random, Optimal, Semi-Optimal, Naive, Tight/Loose, Contrarian) in an effort to uncover the effects of different strategies on player profits. We did so by running simulations, as well as working out the relevant probabilities and expected values theoretically.Ope