Nash Equilibrium in Poker
https://math.mit.edu/~apost/courses/18.204_2018/Jingyu_Li_paper.pdf
In Exploitability and Game Theory Optimal Play in Poker, Jen Li introduces and explores the theory and briefly touches on the mathematics involved in understanding Nash equilibriums present in a 2-player game of no limit Texas hold’em. To understand the rules of this variation of poker, Li provides a brief overview of the rules. This blog post will assume you are familiar with the rules and some common terms that are easily looked up either in Li’s work or online.
The main focus of Li’s paper is to show how Nash equilibriums influence heads-up no limit Texas hold’em (HUNLHE). If you are familiar with any variation of poker, you know that your objective is to make bets in order to maximize profit. However, it is known that all multi-player finite payout matrices have at least one Nash equilibrium strategy. We know that HUNLHE is a zero-sum game, so we can infer that the Nash equilibrium strategies could be used by both players which would result in both players gaining no profit in the long run. This strategy is known as the game theory optimal strategy or GTO strategy. However, because of the complexity of this variation of poker, this equilibrium is exceptionally difficult to implement and beginning players’ intuition often leads them away from this optimal strategy. This is what leads Li and others to explore the concept of exploitability. Exploitability is defined as the rate at which strategy A will lose money to strategy B when strategy B is designed to maximize profit against strategy A, given that strategy A is entirely known to B. If a strategy is GTO, exploitability will be zero. Thus, one may assume that the mark of a good poker player is one which plays as close to a GTO strategy as possible in order to minimize their exploitability. While this strategy guarantees that the player implementing the GTO strategy will never lose money in the long run, it is not always the most profitable strategy. Due to the complexity of a perfectly GTO strategy, players rarely play anywhere near a GTO strategy and therefore have an exploitability that is greater than zero. This leads better poker players to stray away from their most GTO strategy in order to maximize their profit against an imperfect opponent.
To better understand how a player might implement an exploitative strategy, it is easiest to examine the simplest part of the HUNLHE game tree. The simplest part of the game tree is when there are the least number of future actions, which in HUNLHE is the final round of betting known as the river. If a good poker player knows that his opponent only bets on the river when he has a strong hand, his exploitative strategy would be to fold all of his hands except for his very strong hands. While this strategy of folding all but very strong hands would not be GTO, it would be far more profitable than if he decided to call with his moderately good hands. This leads us to the concept of balance. As seen in the example above, it would be far less exploitable if a strategy bets not only when it holds a strong hand but also with some of its weaker holdings as well. A strategy that implements this is considered balanced and is a major distinction between advanced and beginning players.
There is much more to discuss when it comes to the game theory of poker, but understanding the basics of how Nash equilibriums can be applied to poker is a great way to better understand how these equilibrium strategies manifest in the world around us.