Game theory explained, finally. The legendary mathematician John von Neumann is credited with pioneered game theory about 70 years ago. Also sometimes called the theory of games, it doesn’t just apply to poker. In fact, it’s most powerful application today is not to what you and I think of games at all. AS YOU MIGHT HAVE GUESSED, “GAME THEORY” IS THE MATHEMATICAL STUDY OF GAMES. The same goes for poker hands. The variance for 100 poker hands in NLH 6max is, say, 10000 (100 squared). To calculate the variance of the sum for 10 thousand hands you have to think how many groups of 100 hands does 10000 have? Well, 10000/100 = 100.

1. Game Theory Applied To Poker
2. Game Theory Applied To Poker Tournaments
3. Game Theory Applied To Poker Chips
4. Game Theory Applied To Poker

Poker Game Theory And Math Are Actually Very Easy To Use, Here We Look At
Using Game Theory to Improve Your Online Tournament Results.

Game theory is a branch of applied math which looks at competitive situations where 2 or more people have conflicting interests. This has been applied to global politics, economics and of course to poker. David Sklansky covered poker game theory in his seminal work ‘The Theory of Poker’ and Chris ‘Jesus’ Ferguson made this famous by using poker game theory rather than experience to achieve his phenomenal success.

In game theory, a cooperative game (or coalitional game) is a game with competition between groups of players ('coalitions') due to the possibility of external enforcement of cooperative behavior (e.g. Through contract law).

The key to application of poker game theory is that there are ways to vary your poker strategy depending on your opponents and the individual situation. After all, any poker strategy which does exactly the same thing each time is very easy to exploit.

So we have 2 dynamics at work – on one side a mixed vs static strategy, on the other an Unexploitable vs exploitable strategy. Unexploitable strategies are defensive in nature, they will prevent you losing chips and will win small amounts. In order to win you need to play in such a way as to exploit your opponents weakness – this of course involves deviating from Unexploitable strategy yourself and is where Poker Game Theory really starts to get interesting.

Below are 4 common ways in which poker game theory can be put to use at the table.

Poker Game Theory Application #1 – Optimal Bluffing Frequency

As made famous by David Sklansky the idea is that you should bluff with a busted hand on the river at the same frequency as the odds you are offering your opponent from the pot. Thus if the pot offers 4/1 at the river bluffing 25% of the time will mean your opponent loses the same amount whether he calls or folds, all else being equal. A great way to do this is to calculate the odds you will offer on the end of the hand when you bet and then randomize your bluffing by choosing a number of extra cards which do not make your hand. For example if you are offering 4/1 and have 8 real 'outs' to make your hand then choose 2 bust cards to bluff with.

Of course – this is the Unexploitable strategy. If you feel that your opponent either calls or folds more frequently than is correct you can adjust your bluffing frequency accordingly... adaping to expliot your opponents tendencies.

Poker Game Theory Application #2 – The Independent Chip Model (ICM)

ICM simply represents your current chip stack in terms of 'dollar equity' in the prize pool of the tournament. It is most commonly used in Sit and Go play as the numbers are easier to calculate (due to fewer paying places)but also has applications at MTT final tables. When you or your opponents have short stacks in comparison to the blinds (less than 10BB is usually quoted) then it is possible to calculate a mathematically ‘correct’ range of hands to push all-in with or call an opponents all-in bet. Playing using ICM is an Unexploitable strategy, as long as you can put opponents on correct hand ranges for calling or pushing then you will profit from ICM based decisions over time. Even if you choose not to use ICM yourself, it is worth learning about this - since many of your opponents will be basing their decisions on this math, once you understand what is driving their decision making it will be easier to put them on a range of hands.

As with any Unexploitable poker strategy there is a counter. Adjusting play based on stack sizes (especially the presence of very small stacks) or making mathematically ‘bad’ calls against regular opponents to prevent them correctly using this strategy in future are just 2 examples of poker game theory in action. For more on ICM see my SNG Strategy Articles section.

Planet Mark's Rec : If you are thinking about Poker Game Theory, then you are already more advanced than the majority of players. What you need to do next is stop playing against other experienced players on the big sites, and check out sites where the majority of players are amateurs, recreational poker fans or just plain fish! Your returns will be much bigger at 888 Poker - I strongly recommend you check out the tables at 888 Poker now and see just how soft their games are for yourself!

Poker Game Theory Application #3 – Stack Sizes and The Gap Concept

The gap concept states that you need a stronger hand to call an opponent's bet than you need to raise yourself. In a no-limit Holdem tournament the ‘gap’ is usually small when stacks are shallow compared to the blinds (since implied odds are far higher) and shrinks considerably as the game approaches the bubble.

Poker Game Theory suggests that opponents understanding of the gap concept can be exploited. This happens when you strongly re-raise a player who has raised a pot ahead of you when stacks are getting shallow. If your opponent understands the gap concept he may be open-raising with a wide range of hands. The second ‘gap’ – that between the hands he will open with and those he will call a re-raise with – may be very wide. If your opponent raises with 20% of hands but will only call with 5% he will fold to your re-raise 75% of the time.

Using mathematics to exploit this concept will gain chips in many tournament situations. This is particularly true when stack sizes are balanced in such a way as to commit your opponent to the pot. The danger of using such a strategy is that yet another opponent picks up a monster hand behind you – so this is best used from late position!

More recently the 'exploitative' strategy of re-raising light is well known by regular tournament players. You might find your re-raised raised once more (4-bet) simply because your opponent knows that you will fold a good percentage of your hands (this could continue to 5-betting etc if stacks are deep enough). Remember, a novice player is unlikely to be bluffing when they make the 4th bet - since they are not considering game theory, just looking at their own cards.

Poker Game Theory Application #4 – S.A.G.E. Poker System / Nash Equilibrium

The SAGE poker system uses Nash Equilibrium math to create an unexploitable system for heads-up play at the end of a poker tournament. ICM does not work heads-up, since you are effectively playing for a single prize (the gap between the 1st and 2nd place payouts). Sage has a points system based on the number of big blinds in your stack, and tells you whether to shove all-in or call your opponents all-in with certain hands.

SAGE is best used when you are playing against an opponent who is more skilled or experienced than you are. While this system is 'unexploitable' it is not flexible enough to adjust to errors that your opponent is making. To take advantage of errors you need to step outside of the nash equilbrium 'game-theory' system, and call more or bet more depending on how you perceive the weaknesses of your opponent. For example, if your opponent folds too often (according to game theory) you can exploit this by betting more frequently.

Remember, applying Game Theory is even more effective against weak opposition – find out which is the best place to find those easy to exploit opponents in our Best Poker Tournament Sites section.

Planet Mark's Rec : If you are thinking about Poker Game Theory, then you are already more advanced than the majority of players. What you need to do next is stop playing against other experienced players on the big sites, and check out sites where the majority of players are amateurs, recreational poker fans or just plain fish! Your returns will be much bigger at 888 Poker - I strongly recommend you check out the tables at 888 Poker now and see just how soft their games are for yourself!

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The movie A Beautiful Mind (Full Screen Awards Edition) beautifully describes the life of John Nash, who is very famous for his works in the area of game theory. I suggest you read more about him (Biography of John Nash) and especially understand what a “Nash Equilibrium” is (Wikipedia Article about Nash Equilibrium). Not only is the topic and the person John Nash extremely interesting, but it also widens ones horizon and – after all – will dramatically increase your understanding of poker. I believe this is crucial to improving. I don´t have a very high opinion on learning tricky lines, how to play 2nd pair in reraised pots. You´ll learn some easy guidelines that will do the job for a certain time, but following rules without a deep understanding of them (besides some poker idol saying “that’s fine”) will not get you anywhere long term.

I cannot stress often enough the importance of understanding game theory. I´m no math expert, nor any sort of “feel bs player”, I think my biggest strength is “thinking like game theory”. It’s not about exact numbers (unless you play vs Bryce Paradis), it’s about knowing what you´re doing. What if, then, but if this, then what, but if he does this against, how do i, considering that…
Almost every reader has probably heard about game theory, and more specifically in poker about “game theoretic optimal” (=GTO) play. It has been discussed and talked about a lot. Whenever the term is used, we like to think we´re talking about an extremely advanced concept.
I don´t agree with this. Game theory is not only extremely simple to understand, but also very intuitive. With “simple” I do not mean that everybody can understand it within seconds, but more that every step to understanding it is comprehensible – at least in regards to poker. Before we get deeper into the topic, let us define some terms that will be frequently used. The definition is rather simple and if you´re looking for something to use in your doctoral thesis, I´m not going to be a great help – your search slave Google can do this for you.

Game theoretic optimal play (= GTO):

Does NOT mean most profitable play in a certain situation. It simply means to make a decision that will show a profit – no matter what your opponent does (profit here can also mean to lose as little as possible, i.e. when playing in the Blinds). The beauty of GTO play is that – if exercised perfectly – your opponents can´t do anything about it. If you bluff with a GTO frequently, they will usually either fold too much or call too much, both way netting you with a profit.

Exploitative play:

You´re taking advantage of a weaknesses of your opponents, despite doing a GTO mistake yourself. One common leak used to be that players would cbet 100%. This was exploiting the fact that people folded too much to cbets. The trend changed to people checkraising 80% of the time, because players would fold everything that couldn’t beat 2 pair. This is exploiting another tendency. Exploitive play is NOT optimal.

Maximizing profit:

This is what we want to do while playing poker. Maximizing profit means, making the best decision that will profit the most. In this decision every current and factor is already included. If we can exploit a player, we will play exploitative. If we play against a very good player, we play GTO (practically speaking, as close to GTO as possible).

Practical Examples:

You have a soccer/basketball game and the opposite team always attacks you by using a key player. Now you can exploit this by covering him up with 4 people at once. Since the team will still try to involve him in their attacks every time, you will have no problem beating this team. The “flaws” of this strategy is pretty obvious, since you open up yourself for a deadly attack, because 4 players can´t defend where they usually defend.
This is EXACTLY what exploitative play is. You´re opening up yourself, because your opponent has a big leak and will not take advantage of you. This strategy works, until your opponent finds out.
The two described “methods” only worked because the opponents were rather unaware. What if your opponents start to counter-attack you? Are you prepared to fight back with counter-exploitation and with the ultimate weapon – GTO ?! Or does your understanding stop after “being checkraised”. This situation occurs every time you “don´t know what to do”. This happens exactly when you learn to “not call checkraises with one pair”. It’s the reason why it is so dangerous to implement “rules” without understanding WHY.

When to err on the side of exploitative play:

Your opponent is clearly very bad. Bad means, he knows exactly what he is doing. Lets say we are only raising on the flop with 2pair or better, but at the same time a huge calling station.

No-Limit Hold’em, $20.00 BB (2 handed) – Hold’em Manager Hand Converter from HandHistoryConverter.com

BB ($2887)

Hero (= Weaktight Station) (SB) ($3184)

Preflop: WEAKTIGHT STATION is SB with 3, 3

Hero bets $30, BB raises $120, Hero calls $100

Flop: ($280) 3, 10, 9 (2 players)

BB bets $180, Hero raises $680

Let say BB has AA:

If BB plays GTO his play is either to reraise all in or to call (more likely reraise all in). Folding AA in a reraised pot is criminal. If BB is folding AA to our raise, he is folding approximately 80% of his range. This again means that we *could* be raising with any hand and exploit him. For that reason, he is simply never folding those AA, no matter what.

If BB plays exploitative, he will have made his decision very quickly. As we described, we are only raising 2pair or better on the flop, but call a lot. His bet with AA is very good, but after we raise, it’s time to release. He is exploiting us extremely by folding. If we knew that he is folding AA, we would be bluffing any two cards (=ATC).

If we try to maximize our value, we will have to call in this spot. We know that our opponent will fold way too much and he will continue to put in more money with a lot of worse hands

Game Theory Applied To Poker

When to err on the side of GTO (Game Theoretic Optimal) play:

Game Theory Applied To Poker Tournaments

In the same example, if our opponent knew that we can be raising any Top pair, flush draw, straight draw and gutshots or total air, he of course should not even think about folding AA. Also, he should be going all in with a wide variety of hands.

Game Theory Applied To Poker Chips

More examples and articles about this topic will follow. For today Ii´ll keep it to that.

Bill Chen has wrote an amazing poker book. It probably is the most “scientific” book to this day. I have to warn you though, it is not easy material. Nevertheless invest the couple bucks, and be patient with yourself. I am the owner of this book for over 1,5 years now, and I will openly admit that I haven´t finished it. I treat this book like an exercise book. Every time I want to know more, I´ll devote some hours and dig deeper. It will increase your understanding. Again, it’s pretty dry and if you´re not completely, seriously interested in this topic, save yourself guilty feelings and the money. You shouldn’t feel guilty at all. The book is there to help you, and you choose when. If you don´t at least play NL200, I also advise you to save the money for later.

Game Theory Applied To Poker

The Mathematics of Poker by Bill Chen and Jerod Ankenman