Pearls in Graph Theory: A Comprehensive Introduction by Nora Hartsfield and Gerhard Ringel. I used this book to teach a course this semester, the students liked it and it is a very good book indeed. The book includes number of quasiindependent topics; each introduce a brach of graph theory. It avoids tecchnicalities at all costs. AS YOU MIGHT HAVE GUESSED, “GAME THEORY” IS THE MATHEMATICAL STUDY OF GAMES. Game Theory and Poker. John Nash developed game theory as a branch of mathematics at Princeton University around 1950. As poker has become more popular over the last 15 years or so, players have improved dramatically, to the point where it’s very difficult to consistently to beat the game without game theory knowledge in your corner. Popularized by movies such as 'A Beautiful Mind,' game theory is the mathematical modeling of strategic interaction among rational (and irrational) agents. Beyond what we call `games' in common language, such as chess, poker, soccer, etc., it includes the modeling of conflict among nations, political campaigns, competition among firms,.

1. Game Theory Applied To Poker
2. Game Theory Optimal Poker Strategy
3. Game Theory Applied To Poker Tournaments

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

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

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.

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

Game Theory Applied To Poker

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.

The Mathematics of Poker by Bill Chen and Jerod Ankenman

Game Theory with Poker

Poker is a wonderful game that involves both luck and skill. What separates poker from the rest of the games at a casino is that, if done correctly, a player can maximize his expected outcome over time if played enough. This of course, involves knowing the game quite well, knowing the probabilities of certain events, both where you win and lose, and simply put knowing when to bluff and fold. A key part about maximizing your profit from your play style is to know what your opponent is doing/thinking. In this sense, Game theory is a tool that can be justifiably applied to playing poker.

In this blog post, I would like to focus on Texas holdem, a game with each player dealt 2 cards, and eventually a five card board shared between players. There are three rounds of betting open (not including pref-lop) open to all players at the table. The trick to poker is to use the every signal/action from other players and use it to your advantage. Based on preflop action of players, you should immediately assign a range to said player. For example, If a player 3-bets (or re-raises) pre-flop, you know he is strong and should put him at the top of his range. This will provide you with more information on what to do when certain cards come out on the flop. If they hit match his range, you may want to stay away from this hand, however if it does not, you can pursue this hand aggressively, regardless of if you have a good hand or not (this is when bluffing comes in as a key factor).

The other part in game theory other than knowing what the other player is thinking, is them knowing if you will make the optimal decision. Believe it or not, according to a pro in an article on game theory (see link at bottom).

“According to the pro:

• everybody knows who I am
• everybody knows how I play
• there’s no reason to get out of line”

This follows a concept referred to as “Game Theory Optimal” or GTO. The best online player last year was said to followed this strategy, and the thing about online poker is it can be broken down purely to probabilites and signals of players (you take out the option of live reads at a live table). This theory helps you decide when to bet 80% of the time and to call 20% of the time, and then when to call a bet 20% of the time and fold 80% of the time. Unfortunately for humans, bots (which are supposedly being implemented into online gaming sites unknowingly) are much better at learning probabilities, applying GTO with perfect optimal, and taking our money. So unless you took diligent notes in your Game Theory class, make sure to stay away from online poker once bots are introduced.

https://www.pokernews.com/strategy/game-theory-optimal-solutions-and-poker-a-few-thoughts-22654.htm#targetText=GTO%20stands%20for%20%E2%80%9Cgame%20theory,to%20being%20strictly%20results%20oriented.

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