Theory of the game of poker. Basics of poker - an introduction to the strategy of a successful game. It allows you to get rid of assumptions

For an avid card player, poker probabilities are one of the most exciting moments in a tournament.

For those who play poker regularly, it will not be difficult, as they say at school, to memorize such possible options development of events.

Those gamblers who have been familiar with the concept of probability theory since university lavas will be able to perfectly apply the acquired knowledge in practice in poker.

Calculations can be done both on your own and armed with special poker programs, which are offered today in a great variety. But one way or another, you need to think and reason, analyze and make a decision on your own, because no program will help the brain develop and improve.

Below will be the information that will help to calculate the probability in poker with the aim of winning. After the expiration of time, it is important to keep all the data presented in your head so as not to depend on tables on electronic, or, for example, paper.

Only in this way it will be possible to state the fact that success is guaranteed!

Probabilities in poker are measured from zero to one hundred percent. It shows how often this or that development of events can occur during a poker tournament.

Understanding this term and its meaning gives the poker player the opportunity to really assess the situation, analyze the perspective of each action, which can be performed in a specific scenario.

The poker odds table will be a useful hint from which you can get information about what the pot odds are in poker. It is these data that will help you make the right decision during the card competition.

Table variations

There is no single standard, described in one table, armed with which one could consider himself the "master" of poker and win uncontrollably. Everything would be too simple and boring.

Poker is a canvas of mathematical calculations. Which, at the exit, can answer the question of whether it makes sense to take risks or to fold. The calculation of probability in poker depends on how the hand went, and the table is formed based on this.

The following probabilities are known:

• Preflop ;
• with traditional preflop exposures;
• forming a combination with a pocket pair;
• with two card elements in the same suit;
• with 2 cards of different suits;
• with two unpaired cards on the flop in poker.

And this is not the whole list. There is also a table of probabilities in poker, which is called "flop textures". This information will be useful to the participant on the preflop. Here you can get acquainted with the possibility of dropping flops of a specific structure.

So, collect preflop:

• Three cards of the same rank have a probability of 0.24%;
• Combination with a pair in a set (for example, 7-7-2) - 17%;
• Three cards of the same suit - a little more than 5%;
• 2 suited cards - 55%;
• The combination "rainbow" (complete inconsistency) - 40%;
• 3 by increase (one by one) - 3.5%;
• 2 ascending - 40%;
• The absence of cards by seniority in order is more than 55%.

Based on the above data, which appear before the participant in the form of a table, you can independently, having realistically assessed what you see, understand that there is a high probability of hitting a paired flop, but at the same time, a flop with 3 cards of the same rank is more often an exception than a regularly repeating rule.

Armed with a table, you can study the probability of poker combinations in a particular hand and evaluate your own chances of success!

The prospect of improving your own situation?

There is an answer to the question, but it is difficult to call it unambiguous. It all depends on distribution. The theory of probability in poker regarding the improvement of the dropped hand also appears in the form of tabular data.

Below we present the prospects in percentage terms, which will answer the question, what is the probability of combinations in poker to improve the combination in poker flop to turn:

• poker set to Ful haus - 15%;
• Two pair to a Full House combination on the turn - 8.5%;
• combination flush in poker to Flash on the turn - 19%;
• open-ended straight draw to a straight on the turn - 17%;
• gutshot to a straight on the turn - 8.5%;
• a pair to trips on the turn - about 4.5%;
• pair to one of the 2 over cards on the turn - about 13%.

Calculating the probability in poker to strengthen and improve your own positions during the competition makes it possible to decide whether to leave the game or continue to fight for the pot, because the tabular information indicates the real prospects for winning.

More about probabilities

The table of probabilities in poker, based on which the prospect of improving the collection from the flop to the river, appears in the form of the following prospects, expressed as a percentage:

• Set - full house / river - 33%;
• 2 pair - Full House/River - 17%;
• Flush draw - flush / river - 35%;
• Runner-runner draw - flush to the river - a little more than 4%;
• Open ended straight draw - straight to the river - 17%;
• Pair to one of 2 overcards - river - 24%.

The above situations will come to the aid of a poker player when it is necessary to analyze post-flop variations.

The probability of combinations in poker, or rather their improvement from turn to river, is possible in the following percentage of data:

• Set to Full House or even higher - 22.7%;
• 2 pair to full house - 8.7%;
• Flesh-dro before flush - 19.6%;
• Two-way straight to straight - 17.4%;
• "leaky" straight to straight - 8.7%;
• Pocket pair to thrips - 4.3%;
• Pair to one of the over cards - 13%.

So, armed with the data above, you can evaluate the prospect of improving the set with the last river card. Analyzing information on different situations, it is worth focusing on the fact that the probability increases significantly when compared with a similar opportunity from the flop to the turn due to the card that has already been released.

One way or another, in order to lead a successful and exciting fight, the calculation of probability in poker must be carried out without fail. Being well versed in this matter, you can safely enter tournaments and play big.

The main thing is that excitement does not play a cruel joke and fails to push a sound mathematical miscalculation into the background.

True connoisseurs are well aware of the rule: the more time it takes to think and reason about card combinations, the better it will affect the professionalism and skill of the poker player.

Poker is a long game. Even a simple calculation, at times, will help to figure out the opponent and understand what cards he has in his hands. Such knowledge allows you to control the situation and correctly follow the right path to victory.

The theory of probability in poker is far from the last role. It allows you to adequately assess your own capabilities and the realities of the competition, its outcome. Possession of information about the probability is an excellent tip, which is designed to come to the rescue and save money if necessary, or will become a reliable support in obtaining a victory and winning a large cash prize.

What about finances? The tremendous pleasure from the process of a reasonable, logical, deliberate competition is incomparable to anything.

Poker has evolved a lot in recent years. It has changed so much that many books, videos and other related content are out of date.

The old school players made millions from the exploit, and the modern pros make a fortune based mainly on theory, while the exploit has faded into the background.

In this article, we will look at:

• Fundamentals of a theoretically competent poker game
• Why You Need to Use a Theory-Based Strategy (BOT)
• Examples from Doug Polk's game that demonstrate the importance of theory
• Four Obvious Benefits of a Theory-Based Game

So go ahead!

Fundamentals of a theoretically competent poker game

John Nash developed his game theory while at Princeton University around 1950. Since poker has gained incredible popularity over the past 15 years, the level of players has grown to such an extent that now it is not possible to win on a consistent basis without fundamental knowledge in the field of game theory.

Mathematically, every decision you make at the table affects your win rate, from the hand you decide to play in a particular position to a small check on the river in a modest pot. All this can be measured using the mathematical expectation (MO). If your decision is potentially profitable, then MO is positive (MO+), if not, it can be considered negative (MO-).

A very simple example of applying a theoretically sustained strategy is using an open-raising range. Below is an example of a typical open-raising range for a UTG player (first to decide preflop).

Obviously raising with strong hands in this position is a wise decision, but choosing exceptionally strong hands to raise will make your play predictable. If we add hands like 9s8s or 6h6c to the opening range - we balance him, and it will definitely strengthen our game. With this strategy, from time to time we will be able to hit even a weak flop like in the picture below.

Why it is necessary to build a game based on theory

You may be wondering: why put so much emphasis on theory when we will be making most of the profits from exploiting weak or inattentive players.

There are two main reasons:

• With this strategy, you will win in the long run no matter how well your opponents play.
• Make adjustments to own game it's easier if you already have a basic strategy to build on (more on that below).

From a COT point of view, reviewing and analyzing your own hands should take into account how the hands actually played - this will determine how balanced your strategy is. Moreover, from the point of view of the IOS, you must know how to act in any game situation and not reduce everything to two cards dealt to you. When analyzing a game, you should be thinking about how you should actually play a given hand.

If you are value betting in certain situations, then you should also include bluff-oriented hands in your range so that your opponent does not adapt to your play. If you only value bet a certain river, your opponent will be able to quickly spot the danger and fold. On the other hand, if in certain situations you resort to bluffing too often, your opponent will sooner or later understand everything and can easily get rich at your expense.

If you're still unsure if a theory-based strategy is the right way to make money, then these hypothetical examples from Doug Polk should help you figure things out:

Examples of using the theory

On the river, you bet $100 into a $100 pot, so your opponent must call to win $200. So your opponent's pot odds are 2 to 1 and he must win at least 33% of the time to break even.

This quick calculation shows the optimal proportion of bluffs in your river betting range: 33% (one bluff for every two value bets). This frequency is optimal because it allows you to steal the pot most often without the chance of running into resistance.

Let's test 4 different bluff-value betting scenarios so you can understand why a 33% bluff and 66% value betting range is the best option from the point of SOT, and why your opponent will not be able to oppose it.

(For the sake of simplicity, let's assume that we always win when Villain calls our value bet and always lose when he calls our bluff.)

Scenario #1 - 0% bluff, 100% value bet:

Your opponent can fold 100% of the time. With your betting range, you will win $100.

Scenario #2 - 100% bluff, 0% value bet

Your opponent can call 100% of the time. Now you will lose $100.

Scenario #3 - 50% bluff, 50% value bet:

If you call 100% of the time, you win $200 on value bets and lose $100 on bluffs. With your betting range, you will only win $50 if your opponent calls every time (50% * - $100 = - $50, 50% * $200 = $100, $100 - $50 = $50).

This scenario shows that the tactics complete failure from a bluff is more profitable compared to the one when the bluff is used 50% of the time.

Scenario #4 - 33% bluff, 67% value bet:

If your opponent calls every time, you again win $200 on value bets and lose $100 on bluffs. But this time you will only lose $100 33% of the time and win $200 67% of the time, so you make a profit of $100 (33% * $100 = $33, 67% * $200 = $133. $133 - $33 = $100).

The bluff to value bet ratio used in this scenario is optimal because:

• You win $100 if your opponent always calls
• You win $100 if your opponent always folds

You make a profit of $100 regardless of your opponent's decision. This win-win scenario is only possible with a perfectly balanced range.. Regardless of which option your opponent chooses, your range will bring the same profit.

Adjusting this ratio to exploit weak players can bring even more profit, but this requires careful and intelligent adjustments based on clear patterns in the opponent's game. If you want to develop and reach new heights, then using a strategy based on theory is a must.

Four Obvious Benefits of a Theory-Based Game

In conclusion, let's look at the four main benefits that COT offers.

It avoids loop thinking.

The outdated doctrine of poker in the 90s is based on the desire to understand what "level of thinking" the opponents are playing.

• At first you study only your own hand
• Then you try to figure out what the opponent might have
• Then you try to imagine what this opponent thinks of your hand.
• Then you analyze what your opponent thinks about what you think he has….
• And so on.

In theory, at one of these stages you should stop - that is, you should conditionally determine the level of thinking of the opponent, after which you adjust your own game to him. But the reality is that this scheme does not work well against weak players. And against more experienced players it, theoretically, can be repeated until the end of time, while both players will try to climb one level of thinking higher.

Patrik Antonius is the last person on earth I should be giving poker advice to. But still, we mere mortals can avoid getting into such a situation if we use a theoretically based bluff strategy. Then we don't have to "rethink" our opponent on the flop with zero equity.

It allows you to get rid of assumptions

Another benefit of COT is that it eliminates potentially false guesses about opponents' play. Of course, if you've been playing against a particular player for a long time, you can draw certain conclusions from his game, but in other cases, unreasonable general assumptions can cost you the pot.

For example, it's extremely unwise to say things like "there will NEVER be a bluff in this spot" or "he ALWAYS folds in this hand." Similarly, you shouldn't assume that an opponent you don't know can't have a certain hand in a range, or that he's only playing tight or loose in certain spots.

A well-thought-out strategy based on theory allows you to ignore these guesses and help build a strong game.

Objective analysis

Many players misjudge how they play a given hand based solely on the outcome of the hand. But the more a person plays poker, the more he realizes that this approach is fundamentally wrong.

Objective analysis is not easy, especially when the raffle ended in a huge success or a complete disaster. If you made a full house on the river and destroyed your opponent, this does not mean that this will happen every time.

Once you have developed the correct COT for a particular spot, you should apply it for the next session to see how well it performs over the long term over the entire range you choose, rather than just two specific cards.

Every successful poker player knows that admitting one's mistakes is a must. successful game. Game theory makes it easier to recognize these errors.

It makes it easier to adjust your own game

Why is theory so important for adjusting your own game strategy? To understand this, let's play a little game.

Let's say you've pocketed everything you know about poker, except for some of the outdated knowledge of the game, and you're about to play your first hand.

Live $1/$2.Effective stacks $200.

The player sits in the big blind with A♦ 9♦
btn is dropping. BTN raises to $7. sb drops. The player calls.

Flop($14) A♠ T♦ 3
The player checks. BTN bets $9. The player calls.

turn($32) J♣
The player checks. BTN bets $21. The player calls.

river ($74) 9♣
The player checks. BTN bets $50. The player calls.

BTN shows A2♣. The player wins $174 with two pair.

How to evaluate the aggression of a player on the button with his weak top pair? How can you exploit it in the future? Without a competent theoretical analysis of his particular hand, this will not be easy.

On the other hand, if you know how to theoretically play A2o in a given situation in the BU, you will know exactly how much he turned away from her. This knowledge will allow you to quickly determine how to exploit a given enemy.

Here are a number of adjustments we can make to crush his aggressive strategy:

• Small exploit: Call his barrels light (but not too much).
• Big exploit: Aggressively attack his check-back range (which seems to be very weak) with big bets for thin value combined with an appropriate amount of bluffs.

Very often, knowing theoretically based hand strategies makes it easier to exploit opponents, because in such a situation you know exactly how much their play deviates from optimal. When you don't know what to eat right almost impossible to understand what is wrong.

Conclusion

The desire to develop a theoretically perfect game strategy seems like a perfectly reasonable impulse, but, in truth, such a game does not yet exist. A human or robot has yet to finally "calculate" poker, so we still highly recommend using game theory to maximize your game strategy. This means that you have to work on your game both at the table and away from it.

Original name: "Theory of Poker" ("The Theory of Poker")

Year: 2005

Language: Russian

Chapter: Books about the mathematics of poker

Disciplines: no limit hold'em

Despite the title "Theory of Poker", this book is not written for specific beginners, but for those players who already know how to play and want to improve their skills. Sklansky has also written about the psychology of poker from a professional point of view.

He set himself the goal of introducing readers to the theory of poker so that each of them could overcome the dependence on luck and become a real master, relying only on experience.

The book contains a huge knowledge base, a lot of useful information and illustrative examples that help to assimilate the information as easily as possible.

Read Sklansky's The Theory of Poker poker book by downloading the book in PDF or Fb2 format, or listen to excerpts from the book online at our YouTube channel.

David Sklansky is a talented player and mathematician. He made a huge contribution to . Sklansky has 14 books to his credit, of which he is the author and co-author. Many now successful pros have learned from his books.

Not all great decisions are made from the pulpits, but it would be a mistake to assume that our decisions would be the same if there were no lecturers and book authors who accumulate and then transmit information to their audiences. Another thing is that it is the audiences of universities that become the vanguard of the interaction between science and the public, thereby acquiring the image of " open doors» into the world of science, however, what about those who do not have access to the classroom?

Now we are talking not so much about the benefits of higher education, but about the number of intermediaries between us and information itself. The concepts of "probability theory" and "game theory" are considered important in poker. I am more than sure that you have heard of them, but not everyone has discovered them while sitting in the classroom. On the Internet, reading books, maybe even just discussing them with friends - you got access to information that once came exclusively from the mouths of representatives of the scientific community.

We will try to consider the essence of these concepts, we will try to find moments for their application, and in addition, we will accompany them with examples from the game. For people who speak English, at the end of each of the paragraphs, we will attach links to the relevant online versions of the courses offered by Harvard and Yale universities as part of open educational programs.

Probability theory

The main content of the theory of probability lies in the development of methods for calculating the probabilities of some random events(relatively complex) with the help of the probabilities of other random events (simpler ones) that are somehow related to the first ones. The probabilities of the second, simpler, random events in the vast majority of real applications of probability theory are estimated based on experimental data, conducting mass homogeneous experiments. After that, using the formulas of probability theory, the probabilities of more complex events (the word "random" in probability theory is usually omitted) are calculated, associated with simpler events, without conducting any experiments.

However, when we talk about probability, we always mean the probability of an event occurring. The concept of an event is one of the basic concepts of both the general axiomatic probability theory and the naive elementary one. The term random event is used in probability theory only in relation to stochastic experiments, and the term "event" is used as an abbreviated form of the term "random event".

We cannot separately define the terms "random event" (in the sense of probability theory) and "probability". A probabilistic-random event is a random event that has a probability (which implies the possibility of unlimited repetition of the experiment under unchanged conditions), and only a probabilistic-random event has a probability (random events associated with unique experiments have no probability).

It is important to understand that if we are talking about an event associated with a unique experiment, then only one thing can be said: it will either happen or it will not happen. Unique experiments with a random result are not the subject of probability theory.

In the theory of probability are important: the concept of "event", the classical "definition" of probability; total probability formula; Bayes formula; concept independent events; concept of conditional probability.

In applications of probability theory, it is important to understand the following. For real problems, the stability of the frequencies of occurrence of certain events, i.e. the existence of probabilities of these events, and the values ​​of the probabilities are usually established in the course of experiments. This gives grounds to apply the theorems of the mathematical theory of probability to calculate the probabilities of more complex events associated with the experiment under study. However, since in reality the stability of the frequencies and the very values ​​of the probabilities of the initial events can only be established approximately, it cannot be guaranteed that the conclusions obtained using these theorems, as applied to the experiment under study, are correct at least approximately (it is better to say, with the degree of accuracy with which it is established frequency stability) - with the lengthening of the chain of logical conclusions and the increase in the number of operations performed with the initial probabilities (which in real problems are always known only approximately), the accuracy of the values ​​obtained and the reliability of the final conclusions decrease.

However, for poker, this concept has become a whole worldview. Every decision you make must have a mathematical basis based on the knowledge of odds and probabilities. Popular in the community are ready-made probability tables containing solutions for all typical situations. How useful can this be? If we try to summarize this in a few words, then the concept of "probability" in gambling has always existed, but the concept of "mathematical probability" is inextricably linked with poker as a "game of skill". In fact, examples of the use of probability theory are very widely represented in the life of any player. Some of them, more than others with the abilities of a "lecturer", are able to transfer this knowledge, and most importantly, understanding - to other players. Vivid examples include the works of Rounder, Moshman, Janda, and others. In addition to these books, as mentioned earlier, English-speaking users can familiarize themselves with the open course of lectures by Joe Blitzstein (personal website and twitter) link .

Game theory

The section of mathematics that studies the choice of optimal strategies in conflict situations, within which there is a struggle between participants, is called "Game Theory". Do not forget that each side pursues its own interests and seeks, first of all, the most profitable solution, possibly (but not necessarily) to the detriment of rivals. Game theory allows you to choose based on information about the participants in the interaction, resources, and also takes into account the possible consequences of their decisions.

Game theory has a tendency to popularize. In many ways, this is due to the names of John Harsanyi, John Nash and Reinhard Zeljen, as well as Robert Aumann and Thomas Schelling.

To determine the essence of game theory, one should refer to its basic definitions. Game - a mathematical model of the situation, characterized by the following characteristics: the presence of several participants; uncertainty of participants' behavior; mismatch of their interests; the interconnectedness of the behavior of participants (since the result obtained by each of them depends on the behavior of all participants); Finally, it is important to have some rules of conduct known to each of the participants. Strategy - a set of rules that determine the sequence of actions of the player in each specific situation that develops during the game. Party - each of the options for the implementation of the game. A move is a player's choice of one of the possible solutions. The outcome of the game is a payoff function, the value of which depends on the strategy used by the player.

The basis for the calculation procedure in game theory is the expression various characteristics in a quantitative way. In this sense, we turn to the "utility theory" of J. Von Neumann and O. Morgenstern, which states that decisions have a utility function.

Depending on the conditions that exist at the moment of decision-making, game theory qualifies the decision-making process for the following qualifications: First, decision-making under conditions of certainty; Second, decision making under risk; thirdly, she separately considers elections under conditions of uncertainty (which is exactly the case with poker); and, finally, fourthly, game theory especially considers decision-making under conditions of conflict situations or opposition from the enemy.

Why should game theory be remembered by poker players? The minimax theorem guarantees that each antagonistic game has optimal strategies. It gives existence, but does not determine how to search for these optimal strategies. In addition, it has a number of specific methods for each type of game and their features, but all of them, one way or another, rest on the methodology for determining utility. And now again remember the books of Rounder, Moshman, Janda - after all, this is what they all talk about. Determining the utility of decisions under uncertainty.

Fold: Folding EV is 0. Always, this is the first rule of the club (if you know what I mean).

Call: The EV of a call in this situation is -500$. I call this situation a bluff call - a product of our genius. In our case, the only time we don't lose money is when we share with others 23.

Raise: <1501$ поскольку после нашего рейза у соперника 2 варианта: он принимает нас, и мы теряем 1500$; фолдит, и мы забираем банк 1000$ + 500$ ставки соперника.

Let's call the raise as X and the fold as Y, and let the math (or rather, its deep micros) begin.

How to beat micro with one click?

The opponent must choose, so X+Y=1
Then, X=1-Y
EV of a raise 1500$ will be (1500)*(Y)+(-1500)*(1-Y) = 3000*(Y) – 1500
We if
3000Y-1500>0
3000Y>1500
Y=1/2 (for us, consider Y>51%) - fold probability, with which opponent must meet your raise so that it is

If you want to delve into this topic, but understand the very concept of game theory, without forcibly binding to only games in a state of uncertainty, we invite English-speaking users to listen to a course of open lectures by a Yale University professor