GTO in poker and beyond: the Nash Equilibrium

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Game Theory Optimal, or GTO, has become one of the most commonly used terms in poker in recent years. Unfortunately, the understanding of this concept among poker players is rather poor, usually boiling down to "good game" or "bad game vs fish" level explanations. In this article, we will discuss the theoretical basis of GTO.
Poker GTO

Game theory is a branch of mathematics. Games in this context have a very broad meaning. It is essentially a theory that analyses any situation in which rational actors (usually human beings) act according to a fixed (or predictable) strategy. From this definition alone it is easy to identify GTO concepts in poker adaptation.

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One of the best known and most important principles in poker game theory is the Nash equilibrium. Most poker players only know this concept from the tables that are Push/Fold All in preflop. These tables usually indicate at how many BB Available at shoot or call All in against one player. Of course, it is possible to make such tables for a larger number of players, but it is usually not so easy (and useful).

GTO poker - Nash equilibrium

Nash equilibrium made its earliest appearance in poker in the well-known "preflop All in" tables, as this is the easiest state to calculate in the game. The fewer chips and players remain, the fewer possible solutions (theoretically) each player can implement.

To better understand how the calculation of this situation came about, it is necessary to understand what state is considered a Nash equilibrium. Nash equilibrium is the state of the game that is reached when the following conditions exist:

  1. There are at least two players.
  2. A non-cooperative game (one in which players can win something at the expense of others).
  3. Known strategies for all players.
  4. Strategies are formed in such a way that no player can win more by changing only his strategy.

The fourth point is the most important in this case. Nash equilibrium describes a strategy in which all players play optimally. In poker terms, it is a plan of action such that it is impossible to win more EV by changing it. In poker, Nash Equilibrium would mean 0 EV for all players.

An example of the GTO not in poker
Prisoner's dilemma - the most common example of game theory

John Forbes Nashthe discoverer of this equilibrium, has proved one very important phenomenon: the Nash equilibrium theorem. This theorem states that the Nash Equilibrium can be found in all games where the number of decisions is not infinite and where the payoffs correlate with the actions of the players (or provide an incentive to develop a strategy based on some principles). This means that the Nash Equilibrium exists not only in the tables, but also in the entire poker game from preflop to the last river decision. For every poker strategy with at least one rule, there exists a GTO solution that would manifest itself as a one-sided Nash equilibrium. We will discuss the search for such Game Theory Optimal poker solutions in the next part of this article. It is already possible to discover such strategies with the proposed solvers. Thus, GTO is not just for "top top regs".

GTO and Nash Equilibrium myths

As mentioned earlier, in poker we usually hear about Nash Equilibrium only in the context of preflop tables, and GTO only in the context of solvers or some very distant and abstract strategy. In fact, these concepts apply to all levels of poker.

Why were the Nash Tables developed first, before more complex GTO strategies were developed and discussed? First of all, as is well known, these tables are best applied at very low effect stack sizes. The fewer the chips, the fewer the opportunities for both pre and postflop. This is compounded by the importance of the postflop at deep. Many players do not like to open shove AA at every opportunity.

The second, and much more important reason, is to simplify the game. Nash equilibrium requires an understanding of at least one player's strategy Fullwhich is not realistic in most poker situations. Many recreational players probably only have a semblance of strategy instead of rational rules of play. For this reason Nash's balance in poker could only be achieved by simplifying the game.

GTO solvers and other modern applications allow us to calculate a strategy that approaches Nash's balance. The best solvers are still not perfect, but for many players this is not important. The average exploitability per pot is less than 0.1%, so it means nothing in poker practice, as players will not be able to memorise all possible GTO strategies. The most important practical and theoretical aspect of GTOs and solvers is the formulation of the perfect counter-strategy. For any poker strategy, it is possible to find an optimal, maximum EV strategy. This strategy will be maximally profitable until another player changes it.

Example of a Poker GTO solver
Example of a Poker GTO solver

Of course, even the simplest optimal strategies require knowledge of the opponent's playing style to get started. It would be best to know the specific rules, as these can make a big difference in the final result. For example, if we have two players, one who in position 100% flopped top pair cbettina ยฝ pot size, and the other who in position bettina only 50% flopped cbet (with a better kicker), even though all the rest of the rules of the game are identical, the optimal strategies will differ greatly.

GTO Poker - Nash's rebalance

One of the most important sentences that accompanies all these Nash tables is that it is a simplified poker game. It allows a Small Blind to play only All in or Fold, and a Big Blind to play only Call or Fold. Players cannot make any other actions such as a Raise and can never go postflop. This is the main reason why these tables always show Small Blind All in 20+ BB or even higher decisions, even though they are usually not applicable in practice.

Finally, many of these tables and solutions are based on the calculations in Mathematics of Poker. Contrary to what many players believe, we should not blindly apply these tables to our own game, as we will never achieve the desired Nash Equilibrium with these tables, especially against recreational players.

A very simple, but well illustrated example is a player who shoves any two cards 25 BB deep. In this case, the tables would neither be close to the Nash Equilibrium nor close to the maximum EV, since the optimal strategy would be different. The optimal strategy would allow for a wider range of calls, since Small Blind would be shooting many more bad hands than the solutions.

Changes in Nash equilibrium

Of course, in reality there will be many more minor deviations from the strategy described in the tables. Spin n Go and HUSNG players know that Heads Up will play very differently in the 14-9 BB range. Some opponents will limp a lot, some will have more minraise in their ranks and so on. Because of these differences, there will always be a big difference in the optimal shove/fold range, so you should always review these situations on ICMizer, especially from a BB perspective. Of course, in situations where a player is only met once, it may be necessary to use intuition or other simplifications of the situation.

This is a very old video about the Nash changes, but it illustrates very well how an optimal game should change.

The push/fold tables can be relied upon as the game gets closer and closer to 1 BB effective stacks, as all possible strategies approach the Nash Equilibrium. With the exception of the all-fold strategy, most strategies in the 2, 3, 4 or 5 big blinds will be almost identical or very slightly different.

So, in this article, we have discussed the basics of GTO in poker and the most famous concept of this theory, the Nash Equilibrium. The next articles will focus more on finding the optimal strategy and the principles of solvers.