Monday, May 19, 2008

The Mathematics of Tournament Poker

I am probably taking on way more than I should here, but what can I really lose. You guys never agree with me anyway so let the hating begin.

I have a few topics to cover as a prelude to my attempt at a deterministic model for NL Holdem MTTs. I am basically trying to show that if you know what hole cards you will get (and in what order), how you personally play those hole cards in various situations, what hole cards your opponents will get and how they play them in general, then your MTT results are entirely determined in advance. Since you have no control of your opponents at the table and how they play their hands in general, and you also have no control of the hole cards you get and the order that you receive them in an MTT, you can't possibly control your own destiny in a large MTT. You are completely at the whims of the cards that you receive and the opponents and situations that you face. You can't win a MTT with air, and even the best MTT players need luck, cards, and situations on their side to actually get the win. This is something that I have felt for a longtime, but I am just starting to see the mathematical foundations for it. This does not mean at all that MTTs can't be beat for a profit, it just means that you can't beat them on skill alone.

While you can't control your starting cards or your opponents, you can control how you play them. This does not mean that you are inventing how to play your cards on the fly in an MTT. If you are a skilled MTT player, you know what you are doing and have a plan for nearly all situations that arise. Sure you will run into some strange and borderline situations from time to time, but even then you are relying on your experience to make the ultimate decision. You bring your poker game (hopefully your A-game) to an MTT. You do not create your game as a tournament progresses.

What this means is that when situations arise in an MTT you will play your cards as best as you can play them, based on what has been successful for you in the past (a rational model). Each unique starting hand has a weighted range of possible outcomes, based on how you play that hand and the various situations you will face with it. When you play a hand in an MTT, you are at the mercy of this range of possible outcomes. I will use expectation value graphs as a way of visualizing the EV ranges for specific hands. Rather than average all results into a total "Expectation Value" for a hand, these graphs show what really happens when you play a hand. If you double up 50% the time and lose your stack 50% of the time with a certain starting hand it will have an EV = 0, while the same hand will make or break you in an MTT. Raw EV values do not show this. What is much more important is the range of things that can happen when dealt a hand. And this range of possible outcomes is something that is stamped into your poker game before the MTT begins. A few more foundational ideas are below.

MTT stack sizes in large MTTs tend to form a Normal Gaussian Distribution

Above is a classic example of a normal distribution. This is a distribution of values where most values lie near the median value for the series, and values away from the median in both directions are progressively rarer to find. This chart puts things in terms of standard deviations, but that is not really important. What is important is that MTT stack size ranges will tend to follow this distribution throughout an MTT. Most stacks will be near the median stack size, and only a few will have a very large or near zero stack size. This was always obvious to me, but the reason why is not so obvious. If you started out 1000 players playing a limit holdem cash game with mega deepstacks, you would absolutely expect a distribution like this. Everybody starts with the same median amount, and your wins and losses overtime make your stack wonder around this median. Only a rare stack up of wins or losses can put you very far off of the median, and a nice perfect distribution would form naturally. But MTTs do not operate this way. The blinds are going up. People are getting eliminated. The average stack size is marching up as time goes by, yet the stack sizes still tend to follow a Gaussian distribution. Why is this? You would actually expect a lopsided saddle like distribution with the guys who have won some large pots on one side, and the guys who have bled down on the low side, and just a few people managing to stay near the median. This is not what happens. It becomes easier to "double-up" as your stack size shrinks relative to the median, while it gets tougher to double up as your stack size grows relative to the median. You get more action near the bottom, and you either get knocked out or pushed towards the median. From the perspective of the guy in the lead, the median is constantly growing with eliminations, but there are not enough chips in play at his table for him to easily keep pace. You simply can't fully double through when you are in the chip lead. So there is this force acting on everyone in the MTT pulling their stack sizes towards the median, and a Gaussian distribution overall. The main point here is that as you progress through an MTT and avoid elimination, you are most likely to stay near the median chip stack size. This will help to simplify a deterministic MTT model.

NL Holdem MTTs elimination rates are very predictable

If you know the blind schedule of an MTT and the average chip counts, you can easily predict how many people will be eliminated between breaks with a high level of accuracy. If you play an hour of NL Holdem a certain percentage of the players will be eliminated. Leaving you with a new blind schedule and average chip count to determine what percentage will go out the next round. Because the blinds march up in a steady way, and people are eliminated in a steady way, if you leave out the differences due to starting stack sizes (double or triple stacks), the same percentage of players will be eliminated in just about every hour of play. This is normally somewhere between 1/2 to 2/3s of the players who start the hour, or about 60% on average.

You can lose your entire stack or double through on any hand in an MTT

Since we will assume that in general you will be near an average stack while playing an MTT you will normally have a chance of doubling though or getting eliminated on any hand you are dealt. (based on your Expectation Value Graph for that hand)

Expectation value curves flatten out as the blinds go up in relation to your stack.

The EV graphs that I will use are based on database of actual hands at 1/2 NL holdem cash, with a $200 starting stack. The stack size is 100x the BB. In an MTT you will rarely be this deep. You will be playing for larger percentages of your stack when playing these same hands in an MTT, and this will tend to flatten out the graphs making larger percentage swings possible. The flattening of the graph will be entirely determined by the Stack size to BB ratio prior to being dealt a hand. You will see how this works when I show my first MTT example.

When you get to push or fold mode in an MTT the EV graphs will not be the same.

Good MTT players will be making adjustments as an MTT progresses and playing hands radically differently when necessary including going into "push or fold" mode. I do not dispute this, and my cash game EV graphs can't show this. However, you can make assumptions on what the changes will be and what the effect will be on overall expectation values. I will say that cash game play is near optimal, and that radical departures from a cash game style for a hand will lead to a reduction in overall expectation value for the hand. When you go into push or fold mode you are sacrificing some expectation value, by forcing the EV graph to change in a way that increases the chances for you to both chip up and be eliminated.

Good MTT players will adjust to the table and gain some additional EV.

While you can't choose who you will play against, good MTT players will exploit certain types of players at their table and raise the expectation value of certain hands as a result. While this is true, this is also fully reflected in a players EV graphs for their hands. Overtime they will play others of various styles and to the extent that those types of players are present and can be exploited an EV graph will adjust to this. So the idea of exploiting other specific playing styles to further your MTT chances is already fully baked into your A-game, and not adjusted for on the fly in an MTT. You either already now how to exploit the situation, or you don't. You do not learn how to exploit the situation during the MTT.

Expectation Value Graph Examples

Below are several Expectation Value Graphs for various NL Holdem starting hands. All based on a database of 100x deep 1/2 NL holdem hands. The graphs show the likelihood of increasing or decreasing your stack by various percentages based on your starting hand. The X axis represents winning or losing your stack in 5% increments. They are centered on EV=0 which most likely means that you folded preflop while not in a blind (or chopped heads-up). To the left is win 0-5% of your stack, to the right is lose 0-5% of your stack. The X-axis legend is offset a bit to the left. The Y-Axis shows the percentage likelihood of that result. A red line indicates the overall EV for the hand. During an MTT when you get dealt a hand, your starting stack size, plus the EV chart, plus an element of chance (which determines where you land on the chart) fully determines what your ending stack would be. This ending stack becomes the input for the next starting hand, and away you go until you lose all of your chips, or win all chips in play.

AA Expectation Value

77-99 Expectation Value (Middle Pocket Pairs)

22-66 Expectation Value (Small Pocket Pairs)
65s to T9s Expectation Value (Middle Suited Connectors)

AKo & AKs Expectation Value

Non-Suited, Non-Connected, No Paint (Throwaway Hand)

I think I need to stop now. I will put up my first example of an MTT progression fully determined by starting cards later in the week. All of this will start to come together at that point I hope.

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At 5:04 PM, Blogger $mokkee said...

am i reading this right? throw away hands have higher EV than suited connectors?

this just tells me you're not playing suited connectors often enough and you got lucky on occasion with throw away hands in the blinds.

that would be my interpretation if i'm reading your graph correctly.

At 5:31 PM, Blogger Blinders said...

The charts are correct. What it means is that I would be more profitable if I treated 65s to T9s as a throwaway hand. Trying to play these hands costs me money. There are probaly a handful of players who play these for a profit, but most lose money with medium to small suited connectors. They can be a nice leak. Check PokerTracker and you will see what I mean.

At 2:26 AM, Blogger TanOrpheus said...

Very interesting, but aren't you making an assumption that you are just playing your hole cards and all hands are going to showdown ?

More than half the pots I win in any MTT are without a showdown and often without the cards to back them up.

I am probably misunderstanding something like usual.

If cards and EV solely determined MTT progression how can we explain players like Ivey or the online pros getting so deep so often? Are they just on a steak.

At 9:06 AM, Blogger Blinders said...

The EV curves include all hands played, showdown or not, folded preflop or not. If you make tons of moves with air in MTTs your personal EV curve will show this for the throwaway type hands.

To answer your question, players like Ivey have better looking EV curves (higher overall EV, and wider distributions) for the majority of the hands they choose to play in MTTs. This is what gives him an edge over other MTT players and allows him to do consistantly better. Lets say Ivey enters a 1000 player MTT. His EV edge because of the way he plays specific hands gives him a 400-1 shot of winning which is a HUGE edge on the field. He is still a 400-1 shot. How his hole cards fall will determine which one of the 400 trials will be a win, and which ones will not. As good as Ivey is, he does not win all MTTs that he enters.


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