Better Than Nash Equilibriums for Poker (Game Theory)

What if you were to find out the Nash Equilibrium for Poker that you've been using all this time was... wrong? Who actually did the math originally? Do you know? I surely don't. I've done the math and found that the Nash equilibriums for poker chart that so many new players use is actually wrong. It's true, the chart touted by thousands of poker players around the globe is bunk. It tells you to push hands you shouldn't. What is a Nash Equilibrium? It is a chart that brings you to 0EV. Playing according to Nash equilibriums for poker guarantees that you won't lose money, but the goal of poker is to win money, not avoid losing it.

Clearly, against a player who folds 100% of their hands, even at very large stack sizes we could profit by shoving hands that the Nash equilibrium charts would tell us to fold. This proves that there exists maximally exploitative all-in range (called a "best response") that is different from the Nash ranges. To solve for a better solution, I considered the three variables that actually matter: your hand range, your opponents calling range, and your effective stack size.

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1 - Beating the Nash Equilibriums for Poker

In this first chart, I will posit the assumption that our opponent will call with no more than 30% of their hands. If this is the case, then the following chart illustrates the hands you should push with based on your effective stack size.

A 30% Calling range is very loose and means the opponent knows that you are pushing light and is trying to call light in order to exploit you. Why 30%? A few reasons:

  1. Because you cannot know with any accuracy if your opponent's calling range is 25% or 35%. It seems like a good, middling range.
  2. Because it is the optimal calling range for someone shoving 40-50% of their hands. I believe people intuitively settle on something like 30% for somebody who is opening "very loose"
  3. Because many players have a real issue calling a shove with hands like Q5, even though it may be optimal.


Chart #0: Opponent Calls Your All-in 30% Of The Time
nash equilibriums for poker

Teal: Push <20BB
Red: Push <15BB
Purple: Push <10BB
Blue: Push <7.5BB

Unless our opponent is willing to call extremely light, we are pushing 100% of our range under 7.5BB.

For the rest of these charts, the following legend applies.


Blue/Red: Always Push These Hands No Matter What (Unexploitable)
White: Push 100% Of Hands If Opponent's Folding Range Includes a SINGLE Hand Colored in Red
Chart #1: 12BB Effective Stacks
better than nash equilibriums for poker 12bb

11BB chart removed. Log in or register for free to access it.

Chart #3: 10BB Effective Stacks
better than nash equilibriums for poker 10bb

Chart #4: 9BB Effective Stacks
better than nash equilibriums for poker 9bb
8BB chart removed. Log in or register for free to access it.

Chart #6: 7BB Effective Stacks
better than nash equilibriums for poker 7bb

6BB chart removed. Log in or register for free to access it.

To clarify, if we're 7BB deep with our opponent, we shouldn't shove 32o unless our opponent folds a hand like Q8, but if he does, we should be happy to push all in with 32o. If we were to use the Nash equilibrium charts, we would fold 32o at 6BB no matter what our opponents strategy. In fact, we would even fold hands significantly stronger than 32o, like J4o. This accounts for another 30% of our range that we could be profitably shoving, instead of folding.

Against an opponent who will not fold a hand in red, you play hands in white according to the Nash Equilibrium strategy found here.

2 - What is a Nash Equilibrium?

The brilliant economist, John Nash, in the 1950s, developed a system by which zero-sum games can be solved. He put forward the question, "If everybody is trying to maximize the amount of money they win, what is the strategy that each player should rationally adopt?"

3 - Why do we use the Nash Equilibrium?

A sit and go is very similar to a zero-sum game where each player is trying to rationally win more money. Therefore, some enterprising poker minds, originally, the Austrian Helmuth Melcher, have used Nash's equations and assumptions to develop an equilibrium strategy for play. You can find the Nash Equilibrium developed by Mr. Melcher here. Unfortunately, an equilibrium strategy results in an expected value of ZERO, which means you lose money to the rake by playing this strategy.

4 - Why is the Nash Equilibrium insufficient?

Nash supposes several things that are simply not true about poker. Nash's assumptions:

  1. The players will do their utmost to maximize their expected payoffThis should be the case in poker, but emotions, fears, and irrationality still exist and can get in the way. Additionally, when using Nash equilibriums for poker, we have to assume that every game is independent of each other, but this is not the case in poker. An otherwise rational player might be hesitant to take a big risk with a large portion of his bankroll, for instance. We have no way of knowing what other factors are involved in our opponents' decisions.
  2. The players are flawless in execution. -- No human player can be flawless in their execution of any strategy. Even if we assume they were, read on...
  3. The players have sufficient intelligence to deduce the solution. Ultimately, poker is too complex to be solved. We can reach some solutions for specific questions, like, "should we go all-in preflop," but in these cases our opponent has yet to act. Once the opponent acts, the permutations of possibilities are endless. To deduce the solution, we would need to know exactly how he plays every hand.
  4. The players know the planned equilibrium strategy of all the other players. We cannot know this, ever. Best case scenario, we're playing an opponent that we know is using a specific system, exactly. A good example of this is the SAGE system. If we know this, we can define an optimal strategy, but it is not a Nash equilibrium.
  5. The players believe that a deviation in their own strategy will not cause deviations by any other players. If I deviate from my strategy, other players should and will deviate from theirs to exploit me. Why does Nash require this assumption? It's very simple. If I deviate from my strategy, and it causes my opponent to deviate in order to exploit me, he has unbalanced his strategy. If I know that he has unbalanced his strategy, I should find an exploit for his strategy. Then, he should find an exploit for mine. According to Nash, all of these levels must happen between hands, in an instant, essentially bringing us back to a Nash equilibrium. This idea that every player engages in Nth-level metagame on every hand is only possible in theory.
  6. There is common knowledge that all players meet these conditions. So, not only must each player know that the other players meet the conditions, but they must know that they all know that they meet them, and know that they know that they know that they meet them, and so on. This is not the way poker works. Nash is giving us an academic solution to a problem. It simply does not apply as well to poker as many players think it does.

 

8 Comments

  • bigsaje says:

    I think people are calling wider than 30% especially at 10/7.5 bb.

  • Rozzos says:

    Hello Mister Smith, first i’m french so sorry for my poor english!
    I would like to answer you something about heads up. I’m not a good player i play in 11.50 on stars (Rozzos), I want to be a better player and in the future a winner player in 50/100 $ levels. But now i need to work my game and my principal problem is when blinds are 25/50 with effective stack size 15/30BB! It’s difficult for me because i don’t really put vilain on a range that’s why lot of my decision are EV-… Can you have advices to progress in this part of the game? Thank you for your job, I like your posts and good luck a the tables!

  • JuniorMontana01 says:

    What shall I do if I would like to translate it to an other language?
    How can I contact you?

  • bigsaje says:

    Check out the “Derivative Works” link in the footer.

  • vald3s says:

    hey saje, in the last tables you didn’t included suited connectors, 45s, 86s type hands that in standard nash equilibrium they are included for shoving >20bb, what do you think about that?

  • bigsaje says:

    If you have 25BB, and you shove 45s, you also have to shove AA at 25BB. This is because the Nash Equilibrium says, “How deep can I shove this range { 22+, 45s+, etc. }.” Now, it’s not profitable to shove 45s 20BB deep, but it’s very marginally profitable if you also have 22+ and all the other hands in your range.
    I think, equity-wise, you lose far too much shoving AA at 25BB, and gain far too little shoving 45s. As such, I have opted to avoid establishing a range-based equilibrium. The hands in those charts are each unequivocally individually profitable to shove.
    Finally, the red-white-blue charts are not all sides to the story. They illustrate the “bounds” of the equation.
    In one case, the blue hands, you can always shove no matter your opponents calling range. This chart is similar to the Sklansky-Chubokov rankings that I discuss here (http://www.riskoriented.com/multitabling-part-3.html) — but it is not identical. S-C requires that you turn your hand face up to your opponent, but my “blue hand” calculations do not.
    In the other case, the red hands illustrate what the opponents calling range would look like, if you were to shove any two cards. If he folds any hands that are in the red, you can open at those blinds with ATC. But, if he folds some hands that are not in the red, but not much weaker, you can open suited connectors as well. But, that gets complicated and is not solvable on the fly.

  • birchy says:

    If we are the BB and our opponent shoves, can we still use these ranges or do we need to tighten up?

  • bigsaje says:

    No charts needed! If your opponent pushes into you, you should call with the top 66% his range. Meaning, if he’s pushing 100% of his hands, you should call with 66% of your hands. If he’s pushing 30%, you should call with 20%. Just estimate his range, and call with the best two-thirds of it.

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