Nash Equilibrium and GTO in Poker

From Push or Fold Charts to GTO

Everyone knows the Push or Fold charts, including the ones by Slansky Chubukov.

These charts are based on a concept known as Nash equilibrium. However, this concept is not limited to these charts alone, as some players might believe. In fact, it’s not limited to poker either. Let’s break it down a bit:

Nash Equilibrium is a concept from game theory invented by John Forbes Nash (1928/2015), an American mathematician and economist (not a poker player, as some might think). This equilibrium concept applies to "non-cooperative" games, like poker, and also to other fields such as economics. This is why Nash received a Nobel Prize in economics for his work.

Mathematically, a Nash equilibrium is a fixed point in a process where each player seeks to maximize their gains based on the choices of the other players.

Behind this definition, which may seem a bit complex, lies a fairly simple concept that few players truly grasp. Understanding it will give you an advantage over others, and by the end of this article, you should have a complete grasp of the subject.

In poker, the concept of Nash equilibrium or GTO play is widely used. Push or fold charts, jennifear charts, solvers, and numerous software tools available on this site all rely on Nash equilibrium. Mastering this concept is crucial for improvement, as it’s at the core of all strategies in this game.

To better understand Nash equilibrium and GTO in poker, let’s start with a simple example often used in game theory:

The Ice Cream Vendor Example

There are many examples of Nash equilibrium, one of the most famous being the ice cream vendor example: Imagine two ice cream vendors on a beach, each looking for the best spot to set up and maximize their number of customers. Initially, they each set up randomly on the beach, as shown in the following diagram:

the red vendor is located on the right side of the beach — Since customers are lazy, they each go to the nearest vendor. Here, the red vendor would benefit by moving to the right, so we're far from equilibrium.

Now consider the case where both vendors are symmetrically positioned with equal gains, as shown in the diagram below:

the two vendors are equidistant from the center — At first glance, this might seem like an equilibrium since both vendors have equal gains. However, the definition of Nash equilibrium reminds us that equilibrium is reached only if neither player can change their strategy to gain more. Here, if the red vendor moves right or the blue vendor moves left, they cover a larger portion of the beach and optimize their gain. This is not equilibrium, and neither vendor’s strategy is GTO here.

The Nash equilibrium in this game is shown in the diagram below, where each vendor maximizes their gain and has no incentive to move. This is called a Nash equilibrium:

here both vendors are located at the center of the beach — Try to imagine a better spot for each vendor; it’s impossible.

Nash Equilibrium in Poker

You might say, “This is all well and good with your ice cream stories, but I want to improve my poker game, not become an ice cream vendor on a beach.” Patience, we’re getting there! This example lets us introduce Nash equilibrium simply, so let’s get back to poker.

In poker, solvers are programmed to find this equilibrium point, and a player who plays at equilibrium is said to be playing GTO (Game Theory Optimal).

However, playing GTO or at equilibrium, i.e., game-theory-optimal, is not always optimal in the “real world,” i.e., at the tables. Playing GTO means that if the other opponent also plays optimally, then our strategy is optimal. But if the opponent does not play optimally, there are other strategies that are even more optimal and, therefore, more profitable. This is known as exploitative play.

In poker, if an opponent folds more than GTO suggests against a push, it will be optimal to push more than GTO recommends, thus exploiting the opponent's tendency to fold too much.

This is where “reading” opponents and the pool comes in; if we know an opponent deviates from GTO and how they deviate, we can also deviate to exploit this and maximize our gains.

The ideal poker strategy alternates between GTO and exploitative play. We want to play GTO and like solvers against a player who plays GTO and play exploitatively against a player who does not. Just as we would position ourselves at the center of the beach against a vendor who is centered but position right next to them if they are off-center, to always maximize our gains.