The Nash-equilibrium push-fold chart is a good resource for new HU SNG players. Advanced heads-up players can improve on 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 would tell us to fold. Hence, there is a maximally exploitative all-in range that is different from the Nash ranges. To solve for a more pareto-optimal equilibrium, the only two variables that need to be considered are your opponents calling range and your effective stack size.
1 - 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?"
Fortunately for us, poker is 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 a rational equilibrium strategy. For more information about Nash, game theory, and how they relate to poker, click on over to my article about Game Theory.
2 - Why do we use the Nash Equilibrium?
3 - Why is the Nash Equilibrium insufficient?
Nash supposes several things that are simply not true about poker. Nash's assumptions:
- The players will do their utmost to maximize their expected payoff. -- This should be the case in poker, but emotions, fears, and irrationality still exist and can get in the way. Additionally, we have to assume in the Nash equilibrium that every game is independent of each other, but this is not the case in poker. An otherwise rational player might be correctly hesitant to take a big risk with a large portion of his bankroll, for instance.
- The players are flawless in execution. -- No human player can be flawless in their execution. Even if we assume they were, read on...
- The players have sufficient intelligence to deduce the solution. -- 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.
- 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, here is a strategy for beating a player who is using SAGE.
- 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.
- 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.
4 - Better-Than-Nash Solution
First, 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.
Chart #0: Opponent Calls Your All-in 30% Of The Time
Teal: Push <20BB
Red: Push <15BB
Purple: Push <10BB
Blue: Push <7.5BB
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:
- Because you cannot know with any accuracy if your opponent's calling range is 25% or 35%. It seems like a good, middling range.
- 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"
- Because many players have a real issue calling a shove with hands like Q5, even though it may be optimal.
For the rest of these charts, the following legend applies.
Blue: Always Push These No Matter What (Unexploitable)
White: Push If Opponent's Folding Range Includes a SINGLE Hand Colored in Red
Chart #1: 12BB Effective Stacks
Chart #2: 11BB Effective Stacks
Chart #3: 10BB Effective Stacks
Chart #4: 9BB Effective Stacks
Chart #5: 8BB Effective Stacks
Chart #6: 7BB Effective Stacks
Chart #7: 6BB Effective Stacks
White: Push If Opponent's Folding Range Includes a SINGLE Hand Colored in Red
Chart #1: 12BB Effective Stacks
Chart #2: 11BB Effective StacksChart #3: 10BB Effective StacksChart #4: 9BB Effective StacksChart #5: 8BB Effective StacksChart #6: 7BB Effective StacksChart #7: 6BB Effective StacksBlue: Always Push These No Matter What (Unexploitable)
White: Push If Opponent's Folding Range Includes a SINGLE Hand Colored in Red
White: Push If Opponent's Folding Range Includes a SINGLE Hand Colored in Red
To clarify, if we're 6BB deep with our opponent, we shouldn't shove 32o unless our opponent folds a hand like Q8. And 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.
Update: I'd like to remind everyone that this information is licensed under the Creative Commons license. If you want to reproduce this information, please click the link in the footer for information.
[[ Roll The Dice ]]Update: I'd like to remind everyone that this information is licensed under the Creative Commons license. If you want to reproduce this information, please click the link in the footer for information.
bigsaje October 24, 2009 12:21 AM
I think people are calling wider than 30% especially at 10/7.5 bb.