In this blog post, we are going to discuss and prove a 100% long term statistically unexploitable strategy in a heads up (2-max) No Limit Texas Hold’em Sit & Go game. Note how I said the strategy is “unexploitable” – meaning in the long run the strategy is statistically unbeatable, but it might not be “optimal” in the sense that when more information becomes available to a good player, the strategy can be modified to become more optimal. Poker is a very mathematically complex game, for that very reason, we are going to keep the scenario simple for analysis’ sake.
This post is the third in the series, if you have not read the previous posts, it will be a good time to go back and read them. I am going to assume you already understand the concept about Nash Equilibrium which was thoroughly covered and explained in the second post of the series, without that knowledge it might be hard for you to fully understand the proof.
Why Poker is a mathematically complex game and why you need math to win?
Let’s start by revisiting the specific scenario we are trying to solve. The game is a heads up No Limit Texas Hold’em Sit & Go game involving only two players. Each player starts with 100 big blinds, one winner will take the entire price pool for each game, and they can play for as long as they wish, they can play on however many tables as they wish. Let’s keep the scenario simple for now by assuming both players have no prior knowledge/information about each other.
In a heads up scenario, total possible preflop hand combination from two players are:
1.6 million is a large number for human but not for a computer program, we will be able to run through all possible scenarios and calculate all hand equities in a few seconds.
On the other hand, in a 10-handed scenario, that number becomes:
This number, however, is too large even for the most powerful computer to iterate through. Roughly it’s going to take 10,000,000,000,000 years. There is a Monte Carlo method to work around this limitation by obtaining not the exact calculation but a fairly accurate estimation, however that is outside the scope of this post. Well, the point is, poker is a mathematically complex game, it becomes more complicated when involving more players. With the right amount of calculated aggression and a strategical mind, heads up game could easily be the most beatable game. I am going to show you exactly that.
Hand Equity Table and Unexploitable Nash Equilibrium Strategy
No Limit Texas Hold’em has 1326 unique starting hands,
to reduce the number of total starting hands for analysis purposes, we can categorize them into 169 starting hand groups by counting all the hands with same rank into suited, off-suited and paired groups. For example,
- There are C(4,2) = 6 hands for all pairs of a particular rank. i.e. (Ah As, Ah Ac, Ah Ad, As Ac, As Ad, Ac Ad)
- There are 4 hands for all suited non pair hands. i.e. (Ah Kh, Ac Kc, Ad Kd, As Ks)
- There are 12 hands for all off-suited non pair hands.
See here for a complete list of all possible starting hand groups.
I wrote a brutal force program to calculate preflop hand equity by iterating through all possible 1.6 million hand combinations and assign the hand equity values to the 169 hand groups. Note the hand equity percentage will change in a different game setup, the calculation is done only for heads up scenarios.
Below is a subset of all calculated values sorted by equity in descending order. The equity percentage means when picking up a particular hand, the hand has x% chance winning against a random hand on showdown.
This information alone is not enough for us to derive an unexploitable strategy, but it gave us some idea about relative hand strength.
What we really need to know is when picking up a particular hand, exactly how many hands out of 1326 all possible hands is beating(has a higher preflop equity than) the one we have.
Obviously, considering only preflop:
AA has 0 hands beating it.
KK is beat by 6 hands (AA hand group).
QQ is beat by 12 hands (AA & KK)
A9s is beat by 91 hands, if you look at the above table, that is all hand groups from AA, KK to 66, excluding hands may contain the A and 9 of the suit we have. By doing this through the entire 169 hand groups, we are able to calculate a jamming probability table showing below:
Now, if we incorporate Nash Equilibrium concept(IMPORTANT: if you are unfamiliar with Nash Equilibrium make sure you read the second post of this series) with all these information here, we can derive a completely unexploitable strategy – unexploitable in this case really means even if the opponent knows exactly what our strategy is, he will not be able to change his strategy accordingly to take advantage of the information.
The strategy is a simple preflop jam or fold decision. According to above table, when picking up AA, we’ll jam 100% of the time, when picking up KK, we’ll jam 99.55% of the time, …..when picking up 32o, which is the worst starting hand in HU situation, we’ll only jam 8.144% of the time.
Interestingly enough, below is a graph of hand groups with their corresponding preflop equity and jamming probability. There are a lot more information revealed in the graph, but that’ll be another post to write.
You may ask me why there is nobody playing like this out there, for couple of reasons: this strategy is very simple based on Game Theory Nash Equilibrium, it is pretty easy to be figured out by your opponent, once they do, they can choose not to play with you – unless your opponent is a complete fish Another reason is a lot of good players are already using this strategy but with a modified version that takes a lot more information into consideration, things can get quite complicated when you have information such as your opponent’s preflop hand range, check-raising range, 3-barrel bluff percentage etc. This strategy can be modified to cope with all that, but it is indeed complicated. Enjoy!
Excel spreadsheet: Preflop Equity and Jamming Probability
I would strongly recommend all readers to carefully study the above spreadsheet, you will find a lot of interesting information you would otherwise not able to obtain – even in real games. Feel free to leave me comments/questions regarding any scenario that we might be able to discuss & solve together.