Fish N Chips

basic combinatorics

Combinatorics, or counting combos, is a tool that help you dissect your opponent's range. It will give you a better understanding of the nature of other peoples ranges, how often they include draws, made hands or pure bluffs. A combo is simply one specific hand, for instance AcAs (AcAd is a different combo). Before the flop it's 6 possible combos of every pair (3+2+1), four combos of any suited hand (4x1) and 12 combos of all unsuited (and unpaired) hands (4x3).

Post-flop we can find the possible combos of a given unpaired hand by multiplying the reminding cards. If you for instance want to know how many combos there are of AK on a A82 flop, you multiply 3(aces)x4(kings)=12 combos. There are 3 suited (3x1) and 9 (3x3) unsuited combinations left. If we want to compare how often villain will have AK versus AA on this flop, we have to count the combos of AA first, which is 3 (2+1). If villains range were only AK and AA here, we would now know that he is 80% (12/15=0.8) likely to have AK and 20% (3/15=0.2) likely to have AA. By breaking our opponent's range down in this way, it's far easier to figure out the best possible line to take.

To give a few basic numbers of combos:

One pair (AQ on AK2) = 12 combos

Two pair (AK on AK2) = 9 combos

Straight (AK on QJT) = 16 combos

Full house (AK on AAK) = 6 combos

Quads (AK on KKK) = 4 combos

Set (KK on AK2) = 3 combos

Quads (KK on KK2) =1 combo

Remember to not include dead combos if you hold a relevant card in your own hand. If you have AQ on a AQ6 flop for instance, it's just one combo of AA, one combo of QQ and 8 combos of AK left. I think this covers the basics of what combinatorics in poker is about. If we have a good estimate of our opponents range and frequencies, we should be able to relatively fast (after some practice) be able to estimate how big part of it is made up of draws, different made hands and air. For more complex problems, and to get a better feel for different situations, I suggest you download Stox Combo (free). Play a little bit around with it and use it in your session analysis. It works in about the same way as PokerStove, but counts and compare combos rather than calculating equity.

Relevant posts/articles:

decision making

value and polarized ranges

decision making

I want to talk a little bit about a thought process you should go through in every situation at the table. Poker is a dynamic game, and one of the most costly and common pitfalls is to start playing on auto pilot. If it's something you doesn't do already, you have to start asking yourself the following questions before making a decision:

1. What is my opponent's range?
2. What is my perceived range?
3. How does my actual hand do?

Being able to answer these questions well is one of the most important skills in poker, and you will be surprised how often you can find more profitable and creative lines, rather than to throw out a "standard" bet. I'm not advocating any FPS, but to actually understand why you make the moves you do. I doesn't agree with a lot of other regulars these days, who says NLH is a dying game and close to "solved". Yes, the games are tougher than a few years ago, but still the games changes and there will always be room for beating them. A lot of these regs have been used to "standard lines" that used to work out well, but because of approaching the game in a lazy way, they have problems following the trends and changes in the dynamics. To survive as a poker player you have to stay ahead of the marked and being able to think for yourself, no matter how many instructional sites that pops up. Well, let's get started:

What is my opponent's range?
To answer this question is often not as easy as it sounds. Based on your observations of your opponent(s) tendencies, you have to make an educated guess. Is he likely to have a polarized range in this situation, and in that case; how well balanced is it between bluffs and value hands? Is he betting thin for value? It's equally important to know your opponent(s) range when he call or check to you. What does this mean - is he most likely to check-raise, being trappy or weak? Most of these questions can't be answered by stats, you actually have to pay attention and take notes.

What is my perceived range?

This is all about how your opponent(s) answers the first question - what you are able to represent. This is for obvious reasons extremely important against strong players. If a good hand reader has a reason to think a big part of your range is draws and the river bricks off, you are far more likely to get called by a weak hand than if a scary card did fall. Another case is if a "scary" card fall, but it doesn't help your range too much; be prepared to be called down light by a good player! To be conscious about your perceived range is important against weak players too. The secret is that also bad players thinks, just not in same way as you. Their answer could be something as simple as "he got to have AK", "he got to have a flush", "he never has anything" or something along those lines. To be able to get into your opponent's head has always been a mantra in poker, but is not such a romantic idea as it often sounds like - it's all about being able to use sound reasoning. One of the easiest and biggest mistakes many players do, is to transfer their own way of thinking to other players. They doesn't think in the same way as you! So don't blame them when they call your third barrel bluff (on a board that crush your range) with second pair.

How does my actual hand do?

This is an important question, and something many players struggles with. Maybe your K-high actually is quite likely to win a showdown or maybe your two pair-never is good. To be able to truly think in terms of relative and not absolute hand strength, is the key to finding the best moves. The tool you use to figure out when to go for pot-control and when to consider making a made hand into a bluff. How does your hand fit into the bigger picture? This is what you should base your decision on.

Putting it together

I think that the biggest problem for most is to put all of this together. You might say "he is weak most of the time, so I will get him of his hand with a few barrels" or "I look super-strong if I check-raise here". In the first case you might not think closely enough about what you are representing, in the second one you might ignore how strong your opponent's range is. You have to take everything into consideration before making your decision.

Relevant posts/articles:

combinatorics

value and polarized ranges

practical introduction to game theory

G-Bucks

value and polarized ranges

I assume most of you are familiar with the terms polarized and value. To give a short explanation; when you mostly bet the top of your range, you are betting for value and/or protection. The wider the range of hands you bet for value, the thinner your value range is (ref. G Bucks). When you polarize your range, you still bet the top of your range, but often not so thin. Instead you include the hands that are just a little bit too weak to check/call with, in your bet/raising range (gets a bit different on the river). The main idea of value betting thin is to protect and push your thin edges, while polarizing your range mostly is about pot-control; to not being blown of the best hand. In many ways you can say polarizing your range is a defencive idea. You will see many of the strongest players disagree in these marginal spots. A player like Brian Townsend is usually a proponent of pushing your value edges, even if you risk being blown of your hand, while Phil Galfond is more of the "pot-control" and "range balancing" school. We are of course talking about marginal situations here, but it illustrates that poker is a game far from being solved.

This post is mostly going to be about when to apply the two in general. I will just discuss the use at single streets, as it gets a lot more complicated with future streets of betting in mind. I will write about more specific situations, multiple streets and ways to misrepresent your range (range-merging) in later posts and hand analyzes. I will refer to value range and polarized range as VR and PR from here on.

Of course neither a VR nor a PR is superior to the other in principle, and you should usually mix it up according to game theory. I think there are some guidelines that direct the best course of action with your medium-strength hands in a given situation though. After all, our goal should usually be to play as exploitative as possible, not as optimal as possible (see a practical introduction to game theory). These criteria are based on how your opponent(s) is likely to think and respond. This has a lot to do with not just how the player usually responds, but also the dynamics of the table and the history between the players (the meta game).

The arguments for using a VR:
- Your opponent(s) is likely to respond loose  
- Your opponent(s) is likely to respond passive
- It’s dangerous to give away free-cards
- Your opponent(s) is likely to read you for a PR
- You want to isolate a player with a made hand
- You want multi-way action with a draw
- Against a low intensity player

The arguments for using a PR: 
- Your opponent(s) is likely to respond tight 
- Your opponent(s) is likely to respond aggressive
- It’s not dangerous to give away free-cards
- You can easily represent a wide VR
- Your semi-bluffs got high fold-equity
- Your opponent(s) is likely to have used a thin VR with his previous bet
- Against a high intensity player

There are probably more points to add to the list, but these are the most important arguments. Maybe not all of them make sense yet, and that you see some paradoxes. It's not unusual that you have a "conflict of interest" between two arguments, and the situation dictates which is the most important. There is also something to be said about position; often I prefer a VR oop, while a PR ip, but this is a more situational concept and doesn't fit in as a general rule. I will talk in greater depth about this later. This short article is by no means comprehensive, but I hope it at least will stimulate some thoughts around the topic.

a practical introduction to game theory

I will try to explain some game theory concepts that apply to poker in this post. It's a whole branch of mathematics, but I'm not going into much depth here. I will cover some of the mathematics later on. The important thing for now, is that you understand the concepts/terms before reading my upcoming articles. The truth is that I'm not a game theory expert, not even much of a mathematician. I only know the concepts which apply to poker. What I want you to know after reading this post is the difference between a pure and a mixed strategy, between an optimal and an exploitative strategy, and how one strategy can dominate another.

Pure strategy

A pure strategy simply mean that you use the same action every time. That you doesn't mix it up at all. Say you got AA in the big blind and one player has gone directly all-in before it's you to act. The obvious correct strategy is to call 100% of the time in this scenario. This is a pure strategy. It's no need to fold some percentage of the time to "mix it up". This is also an example of a dominant strategy. To call dominates all other options (more profitable every time), and is both the most exploitative and optimal strategy at once.

Mixed strategy

Say you got 85s on the button. All players are 100bb deep. One player raise 3xbb from the cut-off. Against a strong player it's probably not profitable to raise or call 100% of the time here, but to do so a small percentage of the time probably is. Say you fold 60% of the time, raise 30% of the time and call 10% of the time. This would be a mixed strategy, as you doesn't take the same action every time. In this case folding isn't a dominant strategy, as (you think) it's more profitable to raise or call some percentage of the time.

Optimal strategy

An optimal strategy is maybe not quite what you expect it to be. It's usually not the most profitable strategy (unless your opponent also is using an optimal strategy), but an un-exploitable strategy. In this case it's better to use an easier game as an example. In "scissor, rock, paper" the optimal strategy will be to use each option 1/3 of the time. Your opponent simply can't beat this strategy, the best he can do is to do the same. In poker an example is to bluff at an optimal frequency versus the times you value-bet on the river, to make your opponent indifferent to what action he takes (the options are equally profitable for him).

Exploitative strategy

Say in the "scissor, rock, paper" example, that your opponent only plays paper, no matter what you do. The optimal strategy is still to use each option 1/3 of the time, but it's clearly not the most profitable. It will still beat his strategy, but you will win far more often if you always play scissor. In this case you exploit the weakness in his strategy. The problem is that if he realize what you are doing, he can now exploit your strategy. In poker an example would be to bluff far more frequently in a river spot than what's optimal if you know your opponent folds too often. Sometimes you might hear a player say a play is +EV in a vacuum. What he mean is that the play will show a profit given the conditions (your opponent will fold too often for instance), but not something you should do every single time (because you can easily be "re-exploited").

Practical use of game theory in poker

In hold'em no one plays optimally. It's a far too complex a game. What we often do though, is to make "single street games" and solve them. Meaning very simplified versions of poker, where it's possible to find the optimal strategy. They still learn us a lot and can often be applied very well to real situations.

If we feel an opponent has an edge on us in one spot, it's a good idea to play with as optimal frequencies as possible. The point being to cancel out his edge and rather exploit him in other spots. A common example would be against a player who is far too loose pre-flop, but very solid post-flop. You would generally try to get as much money in pre-flop against him as possible (with a stronger range). After the flop though, it could be an idea to mix it up as close to (what you think is) optimal as possible, instead of trying to out-play and out-guess him.