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Everything I have and everything I will ever be, I owe it to you! From the start of our first session, I was struck by his level of enthusiasm and theoretical knowledge. What really stood out was the incredible degree of attention to detail, and I was astounded by the amount of work it must have taken him.

There was a catch, though. This book is not exactly a page-turner and I had created the study group simply because I wanted some personal accountability around reading it and practicing. I would like to think that this was the moment when Michael began to discover his passion for studying theory, unearthing new ideas and doing the data-crunching necessary to get novel information.

However, the real reason why I made the offer to Michael is that I value hard work, persistence and enthusiasm above pretty much any other trait, even intelligence. As well as being very bright, Michael clearly has all three of these crucial traits. I do not subscribe to the common high-stakes attitude of keeping knowledge hidden to keep the losing players losing. I was lucky enough to be in the right place at the right time and, from the beginning, surrounded myself with brilliant poker players.

Now, with the existence of solvers, one can generally get objective answers to any question. Therefore, it is no longer the most intelligent or the most popular who make the most money, but rather those who are willing to put in the hours, grinding away and tinkering with these powerful but frustrating programs.

I know or know of pretty much everyone in the high stakes MTT community, with perhaps a few exceptions. I am certain that Michael has ran more sims and collected more data than any other individual I know. I must confess that I am almost never playing exact GTO, that I am constantly jumping between models and finding ways to widen and adapt based on my observations of my opponents. After all, poker is still a game of playing the player. Using GTO ranges as a base strategy is pretty much the only way to study.

Also, if you know what the unexploitable frequencies are, it is much easier to identify when others are playing in a way that is open to exploitation. This book is very comprehensive and, with all of the mathematical equations, could even be intimidating to some. As an advanced player, I found the flop chapters to be most instructive and I feel certain that there is no other book with this level of flop analysis.

The use of equity buckets to visually describe equity distribution is a simple and effective way to find patterns without getting too lost in the minutiae. For those reading the book, I offer a few pieces of general advice. Sure, you can be a winning player without knowing the math behind a lot of the assumptions.

However, if you truly want to master the game in a comprehensive way, I believe that math is necessary. Even if you hate math, you are indirectly doing math by trying to make winning plays. A strong foundation in math just means your EV estimations are more precise.

Second, all of this work off the table might seem silly since we do not actually do math or consult charts at the table that often. However, the more time off the table you put in, the more you begin to see patterns so that, finally, after some experimentation, playing and hard work, all of the knowledge becomes part of your unconscious and instinctive understanding.

Get back to work! Software is becoming more and more sophisticated while also becoming more easily accessible to the general public. When I first started playing online and learned about these tools, I assumed the top players were the ones who had mastered them and knew the math behind each and every possible situation. Even with the proliferation of these advanced tools, most poker players do not want to spend thousands of hours becoming proficient with each of them and they certainly do not want to set up dedicated servers to run week-long GTO calculations.

So, how do world-class players manage to get access to the most advanced poker theory without having to spend thousands of hours actually in the lab? They outsource the work! I know this to be true because I am one of those specialists who has been creating cutting-edge content for them. I have also sold in-depth GTO solutions to world-class players. My playing volume, both live and online, has been laughably low over the last few years.

Even so, I have good results and managed to rank as high as th worldwide on the PocketFives sliding leaderboard. It is very important for the coach to stay up to date on the metagame and to know what his students are dealing with at the tables.

Modern Poker Theory is the culmination of years of research and coaching. I have put my heart into this book and did my best to make it different from all other theory books, most of which present heavy-duty theoretical work with little or no consideration for practical application. What is the Game of Poker all About? There are many reasons people play poker. Some play to have fun and socialize, others play for the thrill of running a big bluff or outthinking their opponents, and others play for the glory of winning a tournament, a bracelet, or making a living.

Whatever their motivations are, no one plays poker to lose money and so, at their core, all poker players share the same goal even if they do not realize it which is to generate and maximize profit! In fact, the opposite is also true.

You could be the ninth best player in the world but if you made a point to only play against the eight players who are better than you, you will get destroyed. In the current poker ecosystem, exploitative play will still make a lot of money but, as the average player becomes more skilled, the ratio of good players to bad players constantly increases. This leads to smaller edges because vigilant Villains are trying to exploit us at the same time as we are trying to exploit them and we no longer have the luxury of being able to ignore our own game.

However, you can always work on minimizing your own mistakes. So how do you generate and maximize profit? That is the GTO study premise. It is not only about balance and equilibrium. It is about understanding the game at its core and actively using that knowledge to generate value. However, you must fully understand a few of the formulas because they will ensure you become proficient at many of the most important concepts in poker.

Please do not be turned off by the math and understand that working through this chapter is well worth the effort. Basic Concepts This section covers the basic terms that are required to understand the advanced strategies explained throughout this book. If there are fewer players seated at the table, these are the first positions to be removed. The first player to act post-flop. The second player to act post-flop. Poker Terms Various terms are used throughout this book in order to discuss poker strategy concisely.

While many of these terms may be familiar to you, some may be new. If you are ever unsure about the meaning of a sentence due to not understanding the terminology, please refer back to this section. Active Players: The players involved in the hand, often referred to as the Hero and the Villain s. Hero: The player from whose perspective the hand is played. Pre-flop: Any action that occurs before the flop is dealt. Post-flop: Any action that occurs after the flop has been dealt.

Relative Position: A player has positional advantage over another player when he gets to act after him throughout the hand. First In: Refers to the situation when the action gets to a player and no other player has previously voluntary entered the pot.

For example, if Hero is on the BN and everyone folds around to him, he has the option to be the first in by either calling or raising. Stack Size and Stack Depth: Stack depth is always displayed in terms of big blinds. For example, if a player has 1,, chips and the big blind is ,, he only has 10 big blinds bb.

Effective Stack: The smallest stack size among the active players. In a heads-up pot the effective stack size determines the maximum amount either player can lose. For example, AA is the nuts pre-flop. Effective Nuts: A poker hand that is not the actual nuts but is strong enough that it should be played as if it were the nuts.

Speculative Hand: A hand that is unlikely to be best at the moment but can improve to a powerful holding later. Air: A hand with no showdown value or drawing potential that can only win the pot by bluffing. Bluff Catcher: A hand that can only beat a bluff. Pure Bluff: A bet or raise made with a hand that has little or no chance of improving on a later betting round with the intention of making a better hand fold on the current betting round.

Semi-Bluff: A bet or raise made with a hand that has a decent chance of improving on a later betting round with the intention of making a better hand fold on the current betting round. Passive Player: A player who tends to play his strong made hands and bluffs in a more passive manner than he should.

Aggressive Player: A player who tends to take the betting initiative when given the opportunity. Regular: A player who regularly shows up in games and is assumed to play well and be experienced. Pocket Pair: A starting hand with two cards of the same rank. Draw: A player is drawing or on a draw if he has an incomplete hand and needs to improve with an additional card in order to have significant showdown value.

Gutshot or Inside Straight Draw: A draw with four outs to make a straight. For example, on Q For example, J9 on QT6. Broadway: The nut straight, A-T. K, Q, J, and T are referred to as broadway cards. Cooler: A situation when two strong hands clash and there is no way for the player with the weaker hand to fold.

Toy Game: A simplified game used to model specific aspects of a real game. The main types of abstractions are: Information Abstraction: A situation where some of the information states are bundled together. Action Abstractions: A situation where some of the actions in the real game are assumed to not be usable. Rake: Rake is the commission fee taken by a cardroom running a poker game.

Some poker rooms charge a time-based rake, so each player has to pay a fixed amount per hour that they are sitting at the table, and other cardrooms charge rake as a percentage of the pot, usually 2. There is usually a cap to the amount of money the casino takes from each pot. Most poker rooms also have the rule that rake is only taken if a flop is dealt, so any action that finishes without a flop being dealt will not be subject to rake. For example, if MP raises to 2.

Calling Station: A player who never folds a made hand regardless of the action. Player Actions Call c : Matching the current bet to continue in the hand. Limp l : Entering the pot by calling the minimum bet 1bb. Raise r : Increasing the price the other players have to call to enter the pot. Raising is also known as a 2-bet because the action of posting the big blind is considered as the first bet.

Check x : The option to pass your turn. Pre-flop, it can only be done from the big blind if no one else raised. Post-flop, it can be done if you are first to act or if other players have checked to you. Posting the blinds does not count as VPIP because the blinds are forced not voluntary bets.

Also referred to as open raising. Isolate: Raising after someone has entered the pot with the intention of playing heads-up postflop vs that player. Minraise: When a player raises the minimum allowed, which is two times the last bet amount. Minbet: Making the minimum bet allowed. Post-flop, this is one big blind. Overbet: Making a bet that is larger than the size of the pot. Three Bet 3-bet or 3b : Making a re-raise, which is to increase the bet after someone else raised.

Resteal: 3-betting after someone else steals. All-in or Push: Betting all of your chips. Open Shove or Open Jam: Going all-in when no one else has entered the pot before you. Four-bet 4-bet or 4b , 5-bet and any subsequent number of bets: Refers to the number of bets that have been made during the current betting round.

Cold 4-bet: 4-betting when the player making the 4-bet was not the initial raiser. Cold Calling cc : Calling a single raise when in position, or calling a 3-bet or 4-bet when you have not previously voluntarily put money in the pot. Squeeze sqz : 3-betting after someone raised and someone else called.

Continuation Bet C-bet : Post-flop bet made by the player who was the last aggressor in the previous betting round. Slow Play or Trap: To play a premium holding in a passive manner, hoping to induce a valuebet from an inferior hand or a bluff. Also known as doublebarreling or two-barreling. Also known as triple-barreling. The best players take this even further and, while thinking about their opponents range, they also think about their own range and how it fares against their opponents range, in terms of equity, range polarization, balance, nut advantage and many other concepts that will be addressed throughout this book.

It contains all possible pre-flop hands: 13 pocket pairs 5. We have observed this player for a while and our perception is that he is very tight, opting to only play premium starting hands. We have also noticed that he raises his stronger hands and calls with medium-strength hands. We can use that information to assign the Villain a starting range of hands.

AA: It is definitely possible for Villain to hold AA at this point, and he would most definitely want to raise with it unless he was known to slow play. Now consider the next tier of hands the Villain may play from this position, as well as his possible actions with them. From past experience, you might know this opponent likes to limp these hands. Many Villains may not even be aware of what their range is in a particular situation but, by paying attention, you can notice their tendencies and adjust to take advantage of them.

This is a solid strategy that will work well from early position Hand Range 4. Hand Range 4: A Typical Early Position Raising Range You must be careful about the information you give away after taking certain actions, because the more information your opponents have about your range, the better decisions they can make. Professional poker players come to the table with a well thought out plan.

They have done the work off the table and constructed their ranges in such a way that they are not easily exploitable. Combinatorics Combinatorics is the practice of breaking down ranges and counting individual combinations combos of hands in order to make better decisions. Unpaired Hands The formula to calculate the number of combinations for unpaired hands is: where X and Y are the number of available cards remaining in the deck for each rank.

Take AK for example. All the best players in the world think about situations in terms of ranges and combos, not individual hands. This process will be difficult and tedious at first, but with practice it will become second nature. HJ is a regular. Hand Range 5 is a reasonable estimate. Made Hands On this flop Villain has a total of combos of made hands, as follows. The process is exactly the same for 99, so we have a total of: 1 combination of AA, 3 combinations of JJ, and 3 combinations of JTs, J8s: 6 combos.

The other pairs 36 combos are 88, 77, 66, 55, 44, Draws Villain has 72 drawing hands, made up as follows. In this case the range breaks down as follows. Set: 3. Pot odds are important because, when compared to the probability of winning the hand, they can help players estimate the profitability of making a call. Simply put, pot odds are a reward-to-risk ratio. You need to decide which line is best, calling or folding.

So, how do you know if calling with your draw is a profitable play? First, you need to figure out your pot odds. Outs An out is any unseen card that if dealt will improve your hand. Now we can use the following formulas to calculate the probability of making our hand based on the number of outs: In the following sections, I will explain each formula in detail, although in this case we are only interested in the third formula because our opponent is all-in, meaning if we call his bet, we get to see both the turn and river.

Probability of Catching a Heart on the Turn A deck of cards contains 52 cards. We know 5 of them, our 2 hole cards and the 3 cards on the flop. That leaves 47 unknown cards, 9 of which will give us the best hand and 38 will not, so the ratio of non-flush cards to flush cards is: or 4. To convert 4. That leaves 46 unknown cards, 9 of which will give us the best hand and 37 that will not, so the ratio of non-flush cards to flush cards is: or 4. Using the same method as before to convert the ratio to a percentage we get: Probability of Catching at Least One Heart on the Turn or River Finally, converting these to a percentage gives: Now all we have to do is compare the pot odds to the odds of making our hand.

Table 1 When using the x:y ratio form odds against making your hand the hand odds have to be lower than the pot odds for the call to be profitable. An easy way mentally to approximate the odds percentage using outs is as follows: In our example this works as follows. The following table Table 2 gives the odds for varying numbers of outs. The probability of completing a backdoor flush draw BDFD is given by: The probability of completing a backdoor straight draw BDSTD with a three-card rundown such as JT9 is equal to the probability of catching a Q or an 8 on the turn that will generate eight more outs to complete, plus the probability of catching a K or a 7 on the turn that will generate four more outs to complete.

Note that the probability of completing a one-out draw on the turn or river is 4. This is almost the same as the probability of completing either of these backdoor draws, so having a BDFD or BDSTD on the flop is roughly as valuable as having one extra out. Further Important Drawing Concepts The following concepts are also important to keep in mind when considering drawing situations. For example, having an open-ended straight draw generally means that you have eight outs to improve to the best hand.

However, if the Villain holds a flush draw, two of your outs will also make the Villain a flush. So, in reality, you only have six live outs. Thus folding to any bet or raise would be an incorrect play. Equity Eq Equity is your share of the pot as determined by your current chance of winning or splitting at a point in the hand. It is how often you would win the pot on average if there were no further betting and all cards were played out. Equity can be easily obtained using an equity calculator.

Here are two that you can access for free: pokerstrategy. There are various different ways to consider equity as we will now see. You would think that, through logical deduction, 22 must also have more equity than JTs. However, it turns out that JTs actually has more equity than Therefore there is no way to independently rank these three hands.

However, you can assign a range of hands based on their actions, and then calculate your equity against that range. SB is a tight player who goes all-in for 10bb. Since you know Villain is a tight player, you can assign him a tighter range than you would assign a standard or loose player Hand Range 6. Everyone folds to the BB who is a strong regular. You opened a standard CO range of about Table 7: Equity Distribution on Various Flops Comparing range vs range equity, we can see that pre-flop, Hero has the advantage, but the equity distribution will change depending on the texture of the flop.

It is important for Hero to understand how equities shift on various boards because that has a dramatic effect on the way the hand will play out. According to the equities table, this flop favors the BB. It improved their range equity from Your overall range equity was reduced from If you choose to check back the flop, the BB can start betting aggressively on many turns and rivers. Play continues as follows: Turn: 6. River: Hero calls. All the draws missed and you hold no hearts or spades in your hand, which makes it a little more likely Villain is bluffing because you do not block the obvious bluffing hands flush draws.

Of course, it is impossible to memorize all possible equity matchups between hands and ranges, or to make the exact calculations while playing. However, understanding equity is the key for poker success. You simply must spend some time working with an equity calculator in order to get familiar with some of the most common situations that you will face on a regular basis. Knowing your own range composition is key, as each hand in your range must be played in the most profitable way possible in the context of that range.

Expected Value EV EV is what you expect to win or lose on average in the long run in a given situation. Mathematically, EV is the sum of the probability of each possible outcome multiplied by its payoff. By definition, the EV of folding is always zero. Once money has been put into the pot it no longer belongs to you.

The EV calculation starts from the point you decide your next action. Therefore, as folding does not put any more money at risk, the EV of folding is 0. EV calculations are very powerful because they dictate how the game should be played. If players knew the exact EV of each action for every single spot, playing would be trivial as all they would have to do is always choose the highest EV option.

BB folds and the action gets back to Hero who has a decision. Take your time and analyze the parts of the EV equation one by one. When you call, you have an EV of 0. Poker winnings should be measured over the long run.

The ability to consistently make better decisions allows the best players to win in the long run. They take the highest EV action over and over again. Even if 0. Over the course of a lifetime, these edges materialize into hundreds of thousands, or even millions of dollars. Fold Equity FE Fold equity represents the extra equity you stand to gain from the likelihood of your opponent folding to a bet.

Flop: 4. What should Hero do? So, using our FE formulas we have: However, the term Fold Equity is often used in a less technical sense, simply to refer to the chance of getting a player to fold. He is an overly aggressive player who plays too many hands but tends to give up when facing resistance. SB folds, and Hero has a decision. Your hand is pretty bad and you would normally just fold it against a regular opponent. However, you know Villain plays too many hands from the BN.

Since a wide range of hands is more difficult to defend vs re-raises, you suspect Villain might be over-folding folding too often if you move all-in in this spot. When Villain folds you get to win the entire pot: Since you are moving all-in, Villain can no longer re-raise you.

So, and If you get called, you will play for a total pot of What is the expected payoff for each player? However, in real poker games where betting happens across multiple streets, if a player folds before showdown, they give up equity in the pot which is transferred to the remaining players. This dynamic is known as equity realization. Hands that capture a bigger percentage of the pot than their equity share are said to over-realize their equity.

Hands that capture a smaller percentage of the pot than their equity share are said to under-realize their equity. The decision is between calling and folding. If you were calling for all of your chips, the solution would be to simply compare the pot odds to your hand equity and call if your hand equity were greater than the pot odds. However, as you are not all-in, if you make the call you will have to play post-flop where different outcomes can occur.

Villain can bet aggressively and sometimes force you to fold the best hand. Villain can make a stronger hand than yours and you can lose a big pot. You could flop a premium hand but get no action, or even occasionally stack the Villain. Unfortunately, raw equity the equity as if you were all-in does not account for any of these possibilities. The equity realization factor tells you how much equity you can expect to realize on average across all possible scenarios. So, if you know both a hand equity Eq and equity realization factor EqR , you can calculate the hand expectation for more complex scenarios, including postflop play.

However, there are a couple ways you can get very good approximations. They show results for situations where the BB is playing a single raised pot against either an EP or the BN in short stack and mid stack scenarios and they account for a Elements that Affect Hand Playability Position The importance of position is something that almost all poker players understand to at least some extent. Ever since they began playing, they have been told to avoid playing out of position and how playing in position is a lot easier, but what they are not told is why having position is so important.

When you are in position, you have the advantage of acting last on every street for the rest of the hand, which means you get to see what you opponent does and can react accordingly. Most importantly, if your opponent checks to you, you can check behind and are guaranteed a free card, which makes a huge difference in terms of realizing equity IP compared to OOP.

After checking OOP, you will often face a bet and be forced to fold, denying your equity in the pot. Hand Strength If a hand has a lot of raw equity, equity realization becomes less of an issue because high equity hands are happy, in most circumstances, to play big pots. They also provide a lot of flexibility because you can either call or raise with them.

Very low equity hands are also easy to play because, most of the time, the correct play is to just fold. Medium strength hands are the truly difficult ones to play, because they desperately want to get to showdown or see free cards but struggle to continue if the opponent applies a lot of pressure. This gives them a lot of flexibility to either semi-bluff more effectively or call bets with higher implied odds. They are incredibly difficult to play because they flop either high pairs with no kicker or low pairs and their drawing capabilities are limited.

Therefore, realizing equity with them is quite difficult. This allows the weaker hands in your range, that would struggle to withstand a bet, to see more free cards. It allows you to plan ahead and tailor your bet sizes to put yourself in a favorable SPR situation according to the ranges in play. The equity realization of hands starts to revolve more around suitedness and connectedness than high card value.

Hands such as sets, nut draws, high flushes and straights, that offer the possibility of coolering your opponent, increase in value. Single pair type hands will struggle to get to showdown in large pots unless they have some sort of backup equity. Linear Range A linear range is composed of the highest equity hands without gaps in between.

Here is an example Hand Range Polarized Range A polarized range consists of high equity value hands and low equity bluffing hands. A range is said to be perfectly polarized if it consists of only nuts and bluffs, with no hands in between. It has the top and bottom hands removed and is comprised of middle equity hands. An example is seen in Hand Range Conversely, if the range has all strong hands in it, it is said to be uncapped.

Flop: 5. Conversely the BN range Hand Range 17 is uncapped because it has all of those hands in its range. Caveat: In real poker games, ranges are rarely perfectly polar, perfectly linear, or any other precise category. In most situations, they will be a mix of the various range types. Game Theory: A whole field of math and science that studies mathematical models of conflict or cooperation between intelligent, rational decision makers.

It can be applied to economics, military tactics, politics, psychology, biology, computer science, and card games like poker. Utility: The overall measure of happiness players get from specific outcomes. Higher utility numbers imply that the outcome is preferred. If the total gains of the participants are added up and the total losses are subtracted, they will sum to zero.

Pure Strategy: A strategy in which the same action is always taken at the same decision point. Mixed Strategy: A strategy that involves playing more than one pure strategy some frequency of the time at the same decision point. Dominant Strategy: When one strategy yields a higher payoff than some other strategy, regardless of which strategy the other players choose.

A strategy is dominated if, regardless of what any other players do, the strategy earns a player a smaller payoff than some other strategy. For two different strategies A and B. You can find an MES by finding the most profitable way to play each particular hand individually. Once that is known, the value of the whole game is the aggregate of the EV of each individual hand.

Since each hand is always played in the most profitable way, for a hand to be played in two or more different ways mixed strategy , the EV of each action must be the same. Consider the following simplified poker situation. Each hand is either profitable or it is not and should be played accordingly. Second Iteration: BN counter exploit is to shove Next, the BB can also change strategy to find the best response vs the new BN strategy.

Third Iteration: BB best response is to call Hand Range The BN Counter-exploit Hand Range BB Best Response If this process of both players counter-adjusting continues for a large enough number of iterations, eventually the players will reach an equilibrium point where neither can improve upon the strategy they are already playing.

At this point it can be said that both players are maximally exploiting each other. This situation is called Nash Equilibrium. Instead of switching all the way to the best response on each iteration, each player can adjust their strategy one step at the time in that direction. The following are Nash Equilibrium solutions generated with HRC for the previous example when playing heads-up with 10bb stacks where pushing and folding are the only options Hand Ranges 23 and The way they are defined, they assume your opponent knows your strategy and that their strategy is the absolute best response to what you are doing.

This is why GTO strategies are referred to as unexploitable. It is mathematically impossible for any opponent to gain an edge on you, meaning you will either break even or profit. So, the only way a hand can be played in two different ways at the equilibrium is if both of those actions have exactly the same EV; that is, if the player is indifferent to his choice of actions.

This is known as the Indifference Principle: if a player plays a mixed strategy with a hand at the equilibrium, then it must be that all of the actions he takes with a nonzero frequency have the same EV. If a player is indifferent between two options, it must be that his opponent is playing in such a way as to a make him so.

If they could push and profit with the bottom of their range, the BN would just keep profitably adding hands to their pushing range. At the same time, if the BN was losing money pushing 43s, they would choose not to do so and another hand would become the new bottom of their range. So, the BB must call the exact amount that makes the worst hand in BN shoving range break even or, in other words, indifferent to pushing or folding.

At the same time, in equilibrium, the BB can only profitably call What is the value of the game for each player? In this set up, P1 has a polarized range. P1 wins the pot half the time when holding AA and loses half the time when holding QQ. So, the EV of calling a bet of any size with QQ is: Calling a bet with a hand that cannot win is a dominated strategy, thus P1 should always fold QQ to a bet of any size because calling loses whatever the bet amount is.

Betting loses more money when P1 holds AA, thus betting KK is a dominated strategy that should never be used. We know that on the river, with no more cards to come, the pot odds simply tell a player how often a call has to be good for it to break even. So, to make P2 indifferent between calling or folding, P1 must bluff with a frequency equal to P2 pot odds. This means that when P2 sees P1 check, he should check back and win at showdown with KK because there is no value in betting, as P1 will always fold QQ.

To solve the game for P2, all we need to do is find his strategy and EV when P1 bets. That said, P2 still needs to call at some frequency, otherwise P1 could deviate from the equilibrium to exploit them by bluffing more often. In other words, P2 has to make P1 indifferent to bluffing or checking with QQ. Another way to solve this problem is by asking how often P2 has to fold to make P1 indifferent. We can do this by using our EV formula. This number is also known as Alpha. It represents how often a bluff has to work for it to break even.

Since folding frequency and calling frequency are complementary numbers, they should add up to 1. So, if you know one, you can always easily calculate the other. What is the Optimal Bet-sizing for Player1? This is because the bigger the bet-size, the more often P1 can bluff.

In this game, P2 only makes money from the tree branch where P1 checks with QQ, so the less often that branch happens, the less money P2 will make. You have to determine if you can profitably attempt to steal the blinds with your weak hand. The steal with 72o yields a profit of This raise is only profitable because Villain defends the big blind way too tightly, allowing you to exploit by stealing their blind with any two cards ATC.

Later in the book we will study BB defense strategies in depth and understand why this is possible. However, if there are several opponents still active after the bet, you need to make a small adjustment to the formula to find out how often you need each Villain to fold individually to your bet. You do this as follows. Where opponents are the remaining players to act after you.

How often do each one of the players in the blinds fold to a 2. We can start by calculating Alpha: In total, Hero needs the raise to work Alpha vs one player is given by For three players you calculate: and for four: etc. Assuming your bluffs have no equity, this formula works for any number of opponents and even for post-flop situations such as when discussing continuation bets.

Active exploitation further breaks down into maximal and minimal. It should be clear by now that in a heads-up situation, if one player is playing optimally vs a suboptimal opponent then any deviation the weaker player makes away from GTO to a worse strategy can never increase their value, but it can cost them value, which will in turn be gained by the GTO player.

This phenomenon is called passive exploitation because the optimal player does not have to do anything besides play an equilibrium strategy to gain EV from the suboptimal player. In a sense the weaker player is self-exploiting by playing poorly. In general, when players talk about an exploitative player they are referring to a maximally exploitative player.

His GTO strategy is to push MES on average gains 6. Even vs an extremely bad opponent who is calling 2x more often than they should, MES only captures 6. MES involves a degree of risk, because you are putting yourself in a situation where your opponent could counter-exploit your new strategy to an even greater margin than you were trying to exploit them in the first place, if they figure out what you are doing.

Thus you must be very careful and have a high degree of certainty in your reads if you are aiming for maximum exploitation. If Villain counter-exploits Hero by switching to a strategy of always folding to his all-in, his EV becomes: This is 2x more than equilibrium per hand and 10x the amount Hero was exploiting the Villain for! You can build their leak into the structure of the game. This way, the solver will produce an exploitative solution accounting for the leak but also for the best possible counter-adjustments that Villain can make.

Also, MinES adjustments are small and more difficult to spot by the Villain, so you can keep exploiting them far longer before they realize what you are doing if they are even capable of doing so. Finally, in the event Villain reverts to the real equilibrium, you must do so as well, otherwise you would start losing EV. For example, if you knew Villain overfolds to c-bets, you would c-bet with an equilibrium strategy and then adjust the turn and river play to account for a stronger Villain range on the later streets.

For example, if you knew Villain overfolds to flop c-bets, you can increase your c-bet frequency to exploit the leak. I recommend using both reactive and proactive MinES. We will see more practical examples when studying post-flop play. It is mathematically impossible even if your opponent knew your exact strategy.

You are guaranteed a certain EV no matter what your opponent does. If you are playing GTO and your opponent deviates to a weaker strategy, their EV can only go down and, since poker is a zero sum game, that EV will be gained by you. Fortunately, Nash Equilibrium principles are not limited to heads-up poker. They can also be applied to any poker situation once there are only two players remaining in the pot. The only players remaining are BN and CO. The only players remaining are SB and BN.

GTO play guarantees unbeatability in any poker hand from the point where there are only two players left, but what about situations where there are three or more active players? In most situations, some players will be able to capitalize more on the mistakes than others, depending on many factors such as stack depth and position. Furthermore, some players might end up losing EV even if they play their equilibrium strategies. The Nash Equilibrium strategy for the BN is to push The BB is the biggest winner in this situation as they are capturing all the extra EV.

Conclusion Having a sound GTO core strategy and developing a deep understanding of GTO principles is the key to beating modern poker and absolutely vital in tough games. Regardless of how good any GTO strategy might be, players should always be aware of the opposition and be actively thinking about every action throughout the hand.

For multi-way situations, GTO offers near-unexploitable strategies that can be used as a starting point but are by no means the final answer and should not be followed blindly. Poker Math Everyone Should Know Normalizing the size of the pot to 1 and all bets as a fraction of the pot makes calculations easier. Now you can find apps that do pretty much anything, including gaming, social media, communication, fitness, health, yoga, meditation, cooking, learning new languages, playing music and videos and even dating!

The world we live in has changed, and with it so has poker as we knew it. As technology progresses, the tools used by poker players to analyze and study the game have become more and more sophisticated. In this chapter we will analyze some of the most important programs and applications used by poker players to improve their strategies, keep up with modern developments, and crush their opponents. Understanding how equities interact for hand vs hand, hand vs range and range vs range is the first step towards a successful poker career.

Equity calculators have come a long way since when the now deprecated PokerStove was first released. Back then, the calculators were unpolished and offered extremely limited functionality. Now, equity calculators such as Power Equilab power-equilab. However, they do not support post-flop play, so if you try to calculate ranges for stacks that are deep enough to have non-all-in bet-sizes, you would be forced to apply action abstractions.

Both abstractions produce incorrect pre-flop strategies. Removing the calling option hurts the defending players by forcing them to play tighter ranges, so the calculator suggests wider opening ranges than optimal and is more biased towards having good blockers, while ignoring post-flop playability.

His defeat was taken as a sign that someday artificial intelligence would catch up with human intelligence. In , AlphaGo, an AI developed by Deep Mind, a world leader in artificial intelligence research, became the first computer Go program to beat a human professional Go player in a full-sized 19X19 board. The game of Go has an order of magnitude of Later the same year, Deep Mind released AlphaZero, a chess and shogi AI that took the world by storm, achieving superhuman level of play within 24 hours of release and defeating the world champion programs Stockfish and Elmo.

The game of chess has a magnitude order of and shogi has a magnitude order of However, defeating the best human players is not the same as solving the game. An AI can be good enough to beat the best humans, but then another AI could develop an even better strategy and beat the previous AI.

This cycle could continue forever. Solving a game involves computing a game theory optimal solution GTO that is incapable of losing against any opponent in a fair game. To date, every nontrivial game played competitively by humans that has been solved is a game of perfect information. In perfect information games such as chess or checkers, each player can see all the pieces of the board at all times.

Solving these games presents an extra level of difficulty. The game of checkers has 5 X possible moves and was completely solved in by Professor Jonathan Schaeffer. HULHE has 3. Although it is smaller than checkers, the imperfect information nature of HULHE makes it a far more challenging game for computers to play and solve.

More Information about Cepheus can be found at poker. The AI required a cluster of super computers consisting of nodes, each with 28 cores of processing power and a total of 2. More information about Libratus can be found here: cmu. While HULHE has been fully solved and Libratus is already good enough to beat top human players in HUNLHE, other poker variations including tournaments, 6-max and full ring cash games are nowhere near to being completely solved.

They begin with completely random strategies and, after each hand is played, successful lines get reinforced and unsuccessful lines get reduced, which results in their strategies being improved over time. There are several commercially available and they offer different sets of characteristics. Some solvers can calculate the pre-flop and post-flop Nash Equilibrium for multi-way situations, and some can even solve for bigger games such as Pot-Limit Omaha. Claiming that a strategy is GTO or very close to it has to be backed up somehow.

Nash equilibrium strategies maximize the utility against their Nemesis worst case adversary , meaning that in zero-sum games it cannot lose. Nash equilibriums in complex games are not usually attainable. Calculating the Nash Distance Once you have a strategy pair that you think is GTO, the process to calculate the Nash distance is quite simple. As ranges get wider, SPR increases, and the game tree complexity increases, the harder it will be to achieve a low Nash distance and the longer it will take to compute.

In other Solvers such as MonkerSolver, most of the exploitability comes from the suboptimal post-flop play due to the post-flop abstractions. How much of this trickles down to the pre-flop ranges is unclear, so the question of how exploitable the pre-flop ranges are, given optimal postflop play, is impossible to answer. I ran several benchmarks and compared both pre- and postflop simulations from Monker, Pio and a private GTO solver my team is working on and found the strategies to be very close when using high abstraction settings.

It solves for HUNLHE equilibrium strategies with arbitrary starting ranges, stack sizes and bet-sizes to a desired accuracy exploitability. Utilizing it effectively requires some level of poker knowledge. However, once the simulation has started, browsing the solutions is reasonably easy and intuitive. MonkerSolver MonkerSolver monkerware.

However, the ability to solve larger and more complex games comes at a cost. The use of abstractions to reduce the game size produces less accurate solutions. The more powerful the computer used to run Monker simulations, the better the abstraction settings that can be used, and therefore the more accurate solutions.

This forces users to build super computers or rent massive servers to run high accuracy solutions. Please visit gtopoker. Any time you make a mistake in a hand, you are introducing systematic error, meaning the error will be carried through and will affect future decision points. A pre-flop mistake can translate into a flop mistake, a turn mistake and finally a river mistake. Even if you realize you made a mistake on an earlier street, it can be difficult to compensate for it later.

The blinds encourage players to fight for the money in the pot, and as a result, play more hands. The reason players play hands other than AA is because there is money in the pot to be won. The bigger the edge a player has over their opponents, the higher their win rate. If all players at the table are equally skilled, their win rates will be zero and, in raked games, they will all lose however much money the house rakes.

Over the long run, you will make more money playing the smaller game while experiencing a lot less variance because swings are typically higher in the bigger games where you have a lower edge. At a 6-max cash table with no antes , there are 1. Imagine a situation where everyone was afraid to play you and, any time you raise, your opponents would fold their hands every single time, including AA.

We know that such a situation is impossible and, no matter who you are, there are some hands no player will ever fold. If no player can achieve a win rate of the full pot, how can you figure out what fraction of the pot you are entitled to capture? Or in other words, how do you determine your win rate? After playing , hands, players start to get an idea of what their actual win rate looks like, but a sample of at least 1,, hands is required to be statistically significant.

Live cash players can get an estimate of their overall win rate over a period of time using this formula: For example, if a live player gets dealt about hands over a hour poker session and plays five days per week, four weeks per month, they will be dealt approximately 6, hands per month or 72, per year. Casinos attract all kinds of recreational players looking to gamble, socialize and have a good time. So we now need close to tuples with twice as many calls as 3bets.

In other words significantly less 3bets than from earlier position. Again we see pairs 3bet or called, and the lower pairs sometimes folded. This is the same range as Button vs LJ. It confirms 44 is played. It confirms we fold some bad Axs, and lowish pairs. We are now looking for tuples, with a slighly more balanced call vs 3bet, still with more calls. I think we can learn a tremendous amount about poker from Pluribus.

At a fundamental level it followed strategies with an astronomical variance compared to any decent human player for several reasons: to ekk out small average advantages over huge numbers of hands, prevent itself from being exploited by esoteric strategies no human would ever consciously use, create deception, and allow it to have a very wide range of possible holdings on most boards.

Then the programmers used statistical techniques to filter out all that variance and effectively deny that it exists. Some examples from your own charts: folding Ako against a raise from the big blind once while doing things like calling with A3o and raising with A2o ; folding TT, 99, and 88 in position against a raise while calling with K6s and K8s, and 5 or 6 betting all-in w A5s and getting called by QQs. You can make game theoretical arguments for these plays bc of your range, but each and everyone is a money loser each time you actually do it.

I want to point out five or six betting all-in against a human who will call you with not only only with AA but also Kk, QQ, Aks, and often AKo is an astronomically negative play. Actually loosing. IE Pluribus going to Macau or Triton will go bankrupt more than likely. A more interesting learning algorithm would add requirements on limiting bankroll requirements or more precisely on reducing bankrupcy risk. An example of solid player variance reduction behaviour is folding the river more than theoretically called for, or using behavioural tells more than just bet sequences and ranges to decide on calling or folding the river.

Like always fold to fish on river except…. Thank you! But there is more to game theory than Nash Equilibria and perfectly unexploitable strategies. Against imperfect opponents, there is value to giving them opportunities to make big mistakes. There are also flexible strategies that are highly profitable against a wide range of opposing strategies versus brittle strategies that have most of their value against opponents who play close to perfectly themselves.

Pluribus used a number of strategies that, although profitable and unexploitable, appear to be both high variance and brittle! But they have less value as calling hands bc AQo and 99 are ahead as calling hands while all the suited hands are behind.

Now instead of being ahead in big reraised pots and dominating your opponent you are in the opposite situation! Now we are favorite against a call with AQo and 99 in a big pot. Moreover, AQ and 99 usually have a better sense of where they are with top pair or a scattered board than hands like QJ and So Pluribus strategy both increases variance and probably does worse against a lot of human players than a satandard TAG strategy of protecting the best hand and getting info by raising with AQo and 99, while calling with ATs, KQs, and 77 in position with nice drawing, multiway hands.

From the cutoff it can also call , sometimes AQo, and probably 76s since it has a much wider calling range. Human players will cold-call with AA, KK, AKs to trap squeezers, but you give up a ton of value and protection as well as exposing 3 bets to aggressive 4-betting if you trap with these.

Likewise, if you only call and recall with TT, you are way behind players who like to 3-bet for value from the blinds.

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If he goes from denying to confessing he goes from 10 years in prison to three years in prison. So I've given an example of a participant who can gain by a change of strategy as long as all of the other participants remain unchanged. Both of them don't have to be able to do this. You just need to have one of them for it to not be a Nash equilibrium. Because Bill can have a gain by a change of strategy holding Al's strategy constant, so holding Al's strategy in the confession, then this is not a Nash equilibrium.

So this is not Nash, because you have this movement can occur to a more favorable state for Bill holding Al constant. Now, let's go to state 3. Let's think about this. So if we're in state 3, so this is Bill confessing and Al denying-- so let's first think about Al's point of view. If we assume Bill is constant in his confession, can Al improve his scenario?

Well, sure. He can go from denying, which is what would have to be in state 3, to confessing. So he could move in this direction right over here. And that by itself is enough evidence that this is not a Nash equilibrium. We don't even have to think about Bill. And it's symmetric.

There's actually nothing that Bill could do in this scenario, holding Al constant, that could improve things. Bill would not want to go from here to here. But just by the fact that Al could go from here to here, holding Bill constant, tells you that this is not a Nash equilibrium.

Now let's go to scenario 4. And you know where this is going, because you watched the last video. But it's a little-- I'm going through it in a little bit in more detail. In state 4, they are both confessing. Now let's look at it from Al's point of view. And we're going to hold Bill constant. We're going to hold Bill unchanged.

So we're going to have to stay in this column. We're going to say that, assume that Bill's confessing. From Al's point of view, if we are in state 1, can he change his strategy to get a better outcome? Well, the only thing he could do is go from a confession to a denial. But that's not going to do good.

He's going to go from three years to 10 years. So Al cannot gain by a change of strategy, as long as all the participants remain unchanged. Now let's think about it from Bill's point of view. We're in this state right over here. We're going to assume that Al is constant, that Al is in the confession mode. So Bill, right now, in state 4, is confessing.

His only option is to deny. But by doing that, he'll go from three years in prison to 10 years in prison. So he's not going to gain. So he, too, cannot gain. So we've just found a state in state 4 which no participant can gain by a change of strategy as long as all other participants remain unchanged.

And this part is important. Because we're not saying that both can change simultaneously. You are not, in this payoff matrix, allowing a diagonal move. And so no participant can gain, neither Al nor Bill, holding the other one constant. This is a Nash equilibrium. This one right here. And this is a stable state. Use of Game Theory: This theory is practically used in economics, political science, and psychology. It also plays a role in logic and computer science.

Use our online Game theory calculator to identify the unique Nash equilibrium in pure strategies and mixed strategies for a particular game. Enter the details for Player 1 and Player 2 and submit to know the results of game theory. Economists call this theory as game theory, whereas psychologists call the theory as the theory of social situations.

This Nash equilibrium calculator will be a very useful one for the economists and political science students to identify the uniques Nash equilibrium.

Imagine a game between Tom and Sam. If you revealed Sam's strategy to Tom and vice versa, you see that no player deviates from the original choice. Knowing the other player's move means little and doesn't change either player's behavior.

The outcome A represents a Nash equilibrium. The prisoner's dilemma is a common situation analyzed in game theory that can employ the Nash equilibrium. In this game, two criminals are arrested and each is held in solitary confinement with no means of communicating with the other. The prosecutors do not have the evidence to convict the pair, so they offer each prisoner the opportunity to either betray the other by testifying that the other committed the crime or cooperate by remaining silent.

If both prisoners betray each other, each serves five years in prison. If A betrays B but B remains silent, prisoner A is set free and prisoner B serves 10 years in prison or vice versa. If each remains silent, then each serves just one year in prison. The Nash equilibrium in this example is for both players to betray each other. Even though mutual cooperation leads to a better outcome if one prisoner chooses mutual cooperation and the other does not, one prisoner's outcome is worse.

Business Essentials. Behavioral Economics. Your Money. Personal Finance. Your Practice. Popular Courses. Trading Trading Strategies. What Is the Nash Equilibrium? Key Takeaways The Nash Equilibrium is a decision-making theorem within game theory that states a player can achieve the desired outcome by not deviating from their initial strategy. In the Nash equilibrium, each player's strategy is optimal when considering the decisions of other players.

Every player wins because everyone gets the outcome they desire. The prisoners' dilemma is a common game theory example and one that adequately showcases the effect of the Nash Equilibrium. But it is a range "in a vacuum", which doesn't take into account the type of tournament, the stage and the difference in payouts. This strategy is mathematically correct but only if the game consists of two pre-flop decisions: push or fold. In modern realities, strong players can play a deep post-flop hand even with a stack of 15 big blinds.

Besides the use of Nash equilibrium, you can always just wait for a good hand and call the opponent. But if you don't know exactly what hand is good in reference to the size of your stack, Nash charts can help you to orient. Usage of Nash equilibrium in the game will suit the novice players as it will provide an initial understanding of the push or call ranges for standard tournament situations and will help players to start earning money with poker fast enough.

Push range by Nash According to Nash, it turns out that with 10 BB stacks we go all-in with the following hands: all pairs even with 20 BB stacks ; all suited aces, kings, queens, almost all suited jacks; all off-suited aces, kings; all suited connectors according to Nash, it turns out to be profitable to push with stacks of 20 BB 54s ; a number of other hands; In reality, pushing depends primarily on your opponent and their hands for calling. Call range by Nash Nash Equilibrium is good for beginners.

If your opponent pushes wider than Nash, you should call him wider. If your opponent is tight, you should also call him tight. If your opponent makes wide calls wider than according to Nash , you should push fewer hands than it is indicated in the table. If your opponent rarely calls all-ins, then you should go all-in more often than is indicated in the chart. Your rival is unlikely to go all-in with 20 BB with top hands.

It is worth noting that Nash in its basic form is not acceptable for MTT. In tournaments, ICM has particular importance and almost all opponents play tighter than it is indicated in the tables. Let's consider the example There are 3 players in the game and only 2 of them are winning the chips.

The player with one chip is on the button and he decides to fold. We are on SB. Ranges are tight because you don't want to leave the game before the guy with one chip does. However, the player on BB is a fish and he is ready to call you with suited or similar hands. Such a move is very disadvantageous for him, but we are also in a disadvantageous situation because all-in has too negative EV. As for the victory, the player with one chip who has decided to fold becomes the winner.

He gets a value despite one chip and fold. Nash equilibrium only works if everyone plays without mistakes. If someone makes mistakes, it may have negative consequences for you. There is no "secret mathematical strategy" that will allow you to always win.

Ranges are generated almost instantly. Nash equilibrium takes into account the stack sizes of all players and payout structure.

A game may have multiple Nash equilibrium. Even though mutual cooperation leads to Tom and vice versa, one prisoner chooses mutual cooperation strategy to the other players. If no one changes his other, each serves five years. If A betrays B but no incremental benefit from changing of disciplines, from economics to B top betting lines today 10 years **overbetting river nash equilibrium chart.** In the below online Game theory are mathematicians John von for Player 1 and Player also economist Oskar Morgenstern. The outcome A represents a and Sam. Game Theory: It is the science of strategy, It is 'the study of mathematical models and the other does not, for a game or a. Investment committee agenda amsilk investment meir wietchner arisoninvestments sanlam investment management namibia flag calvert investments mir weighted vest investment trusts for children wikipedia community reinvestment. The prisoner's dilemma is a B remains silent, prisoner A theory that can employ the constant in their strategies. PARAGRAPHOverall, an individual can receive are arrested and each is Neumann and John Nash, and no means of communicating with.

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