The fundamentals of advanced poker theory should only be tackled after you have a firm grasp of basic theory. In fact, learning the basic fundamentals will get you halfway there and allow you to win against opponents who do not understand those fundamentals. The ultimate goal of this article is to give you insight on how to think as a professional poker player does.

Advanced poker theory articles. How Equity Realization Impacts Every Hand You’ll Ever Play This article covers a concept that impacts all of your poker decisions. 3 Spots You Should Actually Consider Minimum Defense Frequency (And 3 Spots You Shouldn’t) Minimum defense frequency is a concept that you always have to consider when you face a bet.

Once you have learned the basics and are knowledgeable about basic theory and understand how profit is generated in poker, as covered in part 2 of this basic poker strategy series, it’s time to start working our way toward converting that theory into a workable strategy. To that end, we will be discussing more advanced poker theory including these key concepts:

• Defining Profitable and Unprofitable Actions
• Strategy Versus Tactics
• Maximizing EV
• Thinking In Ranges
• Reading Flop Textures
• Understanding Commitment

• Course Description This course takes a broad-based look at poker theory and applications of poker analytics to investment management and trading. This course is offered during the Independent Activities Period (IAP), which is a special 4-week term at MIT in January.
• Poker Economics (PDF - 2.6MB) (Courtesy of Aaron Brown. Used with permission.) 7: Game Theory (PDF) (Courtesy of Bill Chen. Used with permission.) 8: Decision Making (PDF - 1.8MB) (Courtesy of Matt Hawrilenko. Used with permission.).
• Poker is any of a number of card games in which players wager over which hand is best according to that specific game's rules in ways similar to these rankings.Often using a standard deck, poker games vary in deck configuration, the number of cards in play, the number dealt face up or face down, and the number shared by all players, but all have rules which involve one or more rounds of betting.

Defining Profitable and Unprofitable Actions

The first step in bridging theoretical concepts into an actual game plan is to get to the heart of what actions actually cause a surplus, or deficit, in our bottom line. We already know that we win by our opponents playing worse than us along a multitude of decision points. However, how do we define those “leaks” in their game?

What is a Leak?

A leak is any negative expected value (-EV) play that is part of an overall game plan that loses money over the long term whenever the situation arises.

But what are the causes of leaks? In my view, it’s better to think of leaks as mistakes. Incidentally, this brings us to an area of misconception in poker.

Most players think that if they lose a pot one way or the other that they must have made a mistake. Additionally, when they hear a professional poker player talking about himself or his opponents making mistakes, the thought comes to mind of big pots where the player reraised all-in when their opponent had a monster. Or, perhaps they might imagine the times they failed to lay down AK and ran into AA. These are not examples of mistakes in poker.

What is a Mistake in Poker?

When a professional poker player talks about making a mistake, he is likely articulating a subtle and nuanced situation in which he failed to play optimally given the information available. A specific example of this is if a professional decides to check back a river and then when reviewing his play, realizes that there were many worse hands that might have called his bet. He will recognize that he made a mistake and cost himself some long-term earnings and that he would have been better off value betting in that spot. Conversely, he may make the opposite error and bet in a spot where he is never getting called by worse and once again realize he made a mistake.

By the same token, when a professional player is discussing the mistakes of his opponents, he is likely talking about any number of strategic or tactical errors in that player’s game that cause long-term losses. For example, he may observe an opponent cold calling time after time in the blinds with marginal hands. A professional will realize that his opponent is making tactical pre-flop mistakes due to a flawed strategy.

Strategy Versus Tactics

As you now know, to play profitable poker, you have to have a game that has fewer leaks than your opponent. The best way to build that game is to use the fundamentals as the foundation of your overall strategy. Then, once you understand what a profitable strategy looks like, all you have to do is make sure that your tactics are compatible with that overarching strategy.

Chess offers a great analogy as to how we can organize our thought processes in poker and translate poker theory into a practical plan of action. In chess, there are two games that are being played at all times. There is a strategic game and a tactical game. The strategic game is a player’s overall defensive strategy and his long-term future plans regarding offensive attacks. The tactical game is made up of specific short-term moves designed to carry out that long-term strategy. The strategic game is fairly rigid, while the tactical one is constantly reacting and evolving. Poker works the same way.

In poker, our strategy leads us to either carrying out superior or inferior tactics. Below are some examples of the strategic and tactical elements of poker. The first chart contains strategic weaknesses and the tactics that manifest that failed strategy. The second chart contains an example of optimal strategies and the tactics that make those strategies superior.

Fundamentals = Strategy

In my opinion, it helps to think of your strategy in poker as fundamentals and any action you do within the bounds of that overall strategy as a tactic. Therefore, as you learn and progress as a player, you should always focus on applying advanced theory in building a strong strategy with focused tactics that fit what your strategy is trying to achieve with precision. This will allow you to develop a systematic process that, in the heat of play, allows you to make quality decisions and avoid costly mistakes. Remember, if you are unsure what to do, always go back to the fundamentals and decide which tactical play most fits into your overall strategy.

Maximizing EV

To maximize means to make the most when you are winning the pot and lose the least when you are not. This is an important element of poker to understand, as it forces us to analyze every situation completely. In fact, sometimes an obvious play, while +EV, may not be the most profitable action available. In addition, we will occasionally find ourselves in what appears to be a hopeless spot and may tend to make a hasty decision in order to rush on to the next hand. Regrettably, over the long term, missing out on “concealed” superior plays will atrophy our win-rate.

The Road to Maximizing Profit

The first step toward maximizing begins with sound and disciplined fundamentals. Those fundamentals allow us to create profitable situations more often while avoiding -EV spots. By simplifying things and avoiding complicated decisions, the most profitable course of action will present itself much more frequently.

In other words, maximizing gets a whole heck of a lot easier when we actively create advantageous scenarios, instead of flying by the seat of our pants or following some cookie-cutter strategy which is, in fact, tantamount to bankroll suicide. Indeed, once we understand what the most profitable situations are, we can constructively place ourselves in those spots on a regular basis. We can actively seek out profit rather than passively sitting around waiting for it to come to us.

Poker is a People Game

In order to maximize profit, we must constantly adjust our tactics to fit the current situation, or “table dynamic”. To this end, we must consider what everyone else is doing, in order to optimize our decisions. All this information, based on how our opponents play, is what defines the table dynamic.

Pay close attention to this point, as it is one of the core concepts that you must master. In fact, the table dynamic drives everything that any poker player, who wants to maximize his or her profit, does. If you take nothing else away from this guide, keeping this concept at the forefront of how you plan every hand will be a much better player.

“Consider the table dynamic before any decision, even the trivial ones. “

I cannot stress this point strongly enough so I will put it another way. Anyone who does not implement this concept will never realize their full potential in poker.

The Importance of Counterstrategies

There is one advanced theoretical concept that may be the most important of all when it comes to exploiting opponents. And that is:

Whenever possible, pre-flop and post-flop decisions should be based on a counterstrategy.

Whether you are open-raising or limping, c-betting or 3-betting, each action you take should be based on a plan of exploiting one or more opponents on your table at that moment. All decisions should be based on a way of practically applying different tactics with the end of maximizing profit in every table dynamic.

However, there are times when you do not have specific “reads” on the players at the table and will need to have a strong “vacuum “strategy that is tailored for the typical nature of the game you are playing. In other words, sometimes you will have no info on how your opponents play and will need to rely on a base strategy that is tailored to work optimally in a vacuum where you know nothing about the other players. Then, as new information comes to light, you begin forming counterstrategies and adapting your play according to how your opponents are playing.

Balancing Theoretical Knowledge With Skill

As we have established, the first stage of an effective hand planning process is about assessing the dynamic of the table. Even so, beforehand, it seems only prudent that we discuss the engine of poker strategy by looking at the technical aspects of how one should plan hands. These elements include both learning and understanding how to employ all of the tactics which make up the decision-making process. To put it bluntly, it doesn’t matter how good you are reading situations, if you don’t have the skill to generate a plan that maximizes all of that good information.

To that end, before we get into the details of how to form strategies to systematically adjust and beat any table of opponents, you will need both a theory-based and practical understanding of what drives the poker tactics you will learn to employ. In fact, depending on your level of experience, this may require a complete overhaul of the way you thinking about poker hands.

Thinking in Ranges

You must learn to think beyond just the individual cards that everyone holds at the table and learn to consider their entire range of hands that they might hold. Indeed, the most key strategic decisions you will make in forming tactical plans for hands is based on perceived hand ranges for both you and your opponents. Furthermore, equity against those ranges clarifies present decisions based on future contingencies.

If you are not currently aware of how to form and use hand ranges in your play, then your poker life is about to change. Once you stop thinking only about your current holding, you will suddenly become illuminated to many aspects of poker that you probably do not currently know to exist. You will find avenues for profit far beyond just making hands and will begin to truly understand how to proceed in almost all situations in poker.

What is a Hand Range?

A hand range is defined as a group of holdings that a poker player possesses in any given situation. As an example, let’s say that we are under-the-gun and are deciding whether to open-raise or not. The choice is usually fairly straightforward since we already know what our typical range is for raising in that spot. Here is what it might look like.

How to Write in Ranges

Being able to communicate a hand range in writing is important once you start discussing hand ranges with other players in your study. The above example shows a hand range, expressed in yellow, on a grid of all possible Texas Hold’em hands. The traditional way to write these ranges would be 55+,ATs+,KTs+,QTs+,JTs,T9s,ATo+,KQo. Suited hands have the “s” behind them and off-suit hands have the “o.” However, there is a much simpler shorthand that has become popular in recent years that makes writing ranges much faster.

Poker Range Shorthand

A quicker way to write hand ranges is to combine both suited and offsuit hands into one shorthand. After designating the pair, you then continue with the highest unpaired grouping of hands (always Ace x), so on and so forth. The way it is done is to first take the highest card in the range, followed by the lowest unsuited kicker, finishing with the lowest suited kicker.

Let’s say a range includes A8o+ and A5s+. That range would be written as A85. If the kicker is the same for both suited and unsuited versions of a hand range, you just write the kicker alone by itself. Let’s say a range includes T7o+ and T7s+, which would be written as T7. To build the range, you just start with the pairs and work your way from highest top card down to the lowest top card in the range. Take the UTG range we talked about a bit ago. It would be written as 55+, AT,KQT, QTs, JTs, T9s.

In case you are wondering, I used a software program known as Pokerstove to build this range. I recommend practicing with the software (or similar software such as Equilab) and familiarizing yourself with how it works. We will also be using it in the future to do equity calculations.

How Can We Use Hand Ranges?

Whenever making any decision, or considering the decisions of your opponents, strong poker players think in ranges and not absolute poker hands. In other words, since we can never know the exact hand our opponent holds, we must contemplate all possible hands he could hold in order to make the best decision possible. Conversely, our opponents also don’t know our exact hand, so if we execute similar actions with a wide variety of hands in our range, our exact holding becomes camouflaged and we are much more difficult to play against.

Hand Strength Tiers

The easiest way to break down our range into different parts, to be played in various but sometimes similar ways, is to separate individual hands by texture and strength. Once you have compartmentalized your hands, forming betting strategies or “lines” becomes much easier and much more organized.

Typically, professionals have a repertoire of standard lines that have been shown to be profitable for playing various tiers and against particular opponent types. Of course, minor adjustments to those lines can be made to match the exact circumstances developing in the table dynamic.

Here is a system of hand strength tiers that have served me well for years:

• Tier 1– This is the post-flop “nut” range. Hands that you never fold and would like to get all-in with. Hand types in this range are two-pair using both cards, three of a kind, flushes, straights, full houses, and better.
• Tier 2– This is your strong non-nut range. It includes top-pair hands with a decent kicker and big draws.
• Tier 3– Showdown value hands. This includes any hand that has a chance of beating an opponent’s air or bluffing range but is seldom good against a value range.
• Tier 4– Air. Hands that have little or no showdown value and little chance of improving by the river.

Hand range tiers are relative based on the pre-flop action and the opponent(s) we are facing. The benefit of playing this way is that it simplifies matters for you and inherently balances your range. In other words, it makes you much more difficult to read, since you will sometimes play completely different holdings the same way and other times play hands that appear to be identical in another way.

What is Relative Hand Strength?

Assigning your hand to an appropriate tier is not always cut and dried. Situational factors can cause the strength of similar-looking hands to vary wildly. For example, top-pair might be the nuts against a particular loose and aggressive post-flop player, but against a tight player, it might be considered weak if you face a lot of action.

How you decide to group your hand is based on a number of elements: the pre-flop action, your opponent(s) in that particular hand, and your image at the time. Each of these variables factor into your decision regarding the relative strength of your hand.

Understanding your relative hand strength is a critical skill to develop since it is one of the keys to becoming an expert player. For now, I suggest keeping matters simple when it comes to categorizing your ranges and focus on mastering the concrete hand groupings that I have provided for you.

How Do We Put an Opponent on a Range?

In online poker, we often have the luxury of using tracking software that tells us precisely how often an opponent bets or raises. In anonymous games or in live play, you have to make educated guesses as to the frequency your opponent is likely performing a particular action.

Pre-Flop Ranges

Here are some common range percentages you will frequently encounter and potential hands that make up those ranges:

• 3%: JJ+, AK (a tight 3-betting range)
• 8%: 88+, AJ, KQ (a common 3-betting range)
• 14%: 55+, AT, KJT, QJT, JTs (a common early position range)
• 20%: 22+, AT2, KT, QT, JT (a common cutoff range)
• 35%: 22+, A2, K98, Q98, J98, T98, 65s+ (a common button range)
• 50%: 22+, A2, K82, Q82, J7, T76, 976, 875, 765, 54s+ (a common aggressive stealing range)

These ranges are not hard and fast. The exact makeup of an opponent’s range can vary wildly. What this list does illustrate is the large number of holdings that make up common hand ranges and the relative differences between tight and loose ranges. Through reads and experience, we can learn to narrow down ranges based on the thought process or level of thinking of our opponent. To familiarize yourself with the possibilities, I recommend spending time building various ranges in an equity calculator.

Post-Flop Ranges

Post-flop ranges are more complicated due to the community cards in play. On the flop, we can extrapolate how well our opponent’s pre-flop range connects, but in order to narrow it down into a post-flop range, we need information. We gain clues about a player’s hand by their post-flop actions. If we bet and an opponent calls, we can generally narrow his range a bit. If we bet and opponent raises, we can further narrow that range.

For now, don’t worry if this subject seems a bit overwhelming. Figuring out post-flop ranges on opponents takes months just to begin to understand and thousands of hands of experience to obtain a knack for. For the purposes of this guide, I will focus more on how to play your own ranges based on the potential for opponents to have connected to various board textures. Going into detail on all of the nuances of forming post-flop ranges would take countless hours and is well beyond the scope of these writings. For now, I want you to understand common pre-flop ranges and remain mindful of how they connect more solidly on particular flop textures.

Reading Flop Textures

On any given hand, once you are able to determine your relative hand strength, the next step is to decide how to proceed based on the texture of the flop.The cards that come on the board determine the possibility of your opponents connecting with it, which in turn impacts your betting strategy. You will need to learn how different hand ranges connect with different boards and how to form plans accordingly.

There are three different types of flops: dry flops, wet flops, and flops that fall somewhere in between. The first step to honing your board-reading ability is to be able to immediately categorize the various types at a glance.

Dry Flops

A dry board is one that is poorly coordinated and does not allow for the possibility of many made hands or draws. Generally, dry boards are unlikely to connect with the ranges of our opponents and will typically show a much higher success rate for c-betting. Since most players are only playing their cards, you should expect much stronger made hands to be turned up when all-in on dry boards.

Examples of Dry Flops:

• 2♦7♥2♠
• 9♣9♥4♦
• K♦8♠2♣
• Q♠5♥5♣
• A♣A♦9♠
• 2♦6♣J♠

Wet Flops

Wet Boards are highly coordinated and tend to contain multiple straight draws and typically include the possibility of a flush draw. These types of boards connect well with a variety of ranges, thus your c-bet success will tend to be lower.

Examples of Wet Flops:

• 9♥6♣J♥
• K♠Q♣T♦
• 4♦6♦8♣
• J♠7♥Q♣
• 9♦9♠T♦
• T♠8♠5♠

Neutral Flops:

Neutral flops fall somewhere in between dry and wet ones. How you approach hand planning on neutral flops is mainly determined by the number of players involved in the hand. A good rule of thumb is that if a flop is neutral, then you should err on the side of wet in multi-way pots and dry when heads up.

Truly dry boards will tend not to hit anyone, even multi-way, while neutral boards will usually in some way hit at least one of the multiple players. The development of your judgment in this area will come through experience.

Examples of Neutral Flops:

• 7♣6♥K♠
• A♠J♦7♥
• T♥6♦2♣
• 9♥2♦4♦

Once you understand how think in ranges, categorize your hand strength, and read board textures how do you decide how to play your hands? Much of your decision-making process will come down to “commitment.”

Understanding Commitment

In poker, commitment can be defined as your willingness to get all-in with your current holding on the current board against the current opponent. Basically, the stronger the relative strength of your hand, the more money you want to commit to the pot. Conversely, the strategy we employ with medium-strength to weak hands is also driven by commitment. In fact, every post-flop decision we make is somehow influenced by the relative strength of our hand (or range) as it relates to our opponent’s perceived range.

This really isn’t as complicated as it sounds. Simply put, we form our lines to get a lot money in the pot when we are strong and get less money in the pot when we are not so strong. Furthermore, we tailor our lines to “maximize” our hands based on the tendencies of our opponents. This is a bit oversimplified, but that is the gist of it. The tool that we use to decide how much money to put in the pot with different parts of our range is known as stack-to-pot-ratios.

Stack-To-Pot-Ratios

Stack-to-pot ratios, or SPR, refers to the ratio between the effective stack size and the size of the pot on the flop. The concept of SPR is based on commitment, meaning that the more of your stack you have invested in a pot, the less strong your hand needs to be in order to profitably get all-in. To figure out your SPR, divide the current pot into the effective stack. For example, if the pot is $5 and you have $28 behind as the effective stack, your SPR is 5.6.

As your stack increases, your commitment level with various holdings go down. Higher SPRs on the flop, say 10+, are usually not conducive to stacking off on the flop with one pair hands since most opponents won’t commit with a hand worse than that for so much money. In other words, when you flop top pair and get it in with a high SPR, you will often be beaten.

Stack Size and SPR

SPR is one of the main reasons that I play a small starting stack of between 30 and 40 big blinds instead of the traditional 100 big blind buy-in. The truth of the matter is that 100 big blinds is an awkward stack size to play, as it creates multiple uncomfortable situations every session. I would much rather have 250 big blinds than between 80 and 120. With the deeper stack, at least you have reasonable implied odds and can leverage your stack on later streets.

As a short stack or “CAP stacker” your decisions are almost always fairly obvious. Holdings that cannot typically get all-in on the flop profitably with 100 big blinds suddenly become viable to do so with a small stack. Take the following example:

You are playing in a .50/1.00 game and bring it in for your standard min-raise on the button after it folds to you. The small blind calls and the big blind folds. The pot is $5 entering the flop. Here is what your SPR would be with various pre-flop effective stack sizes:

• 250bbs: 49.6
• 200bbs: 39.6
• 150bbs: 29.6
• 100bbs: 19.6
• 50bbs: 9.6
• 40bbs: 7.6
• 30bbs: 5.6
• 20bbs: 3.6

Against opponents with an average commitment range, you can comfortably stack off on the flop with top pair hands with around 4.5 SPR and overpair hands with around 6 SPR. Against opponents who commit more loosely, those numbers rise to 7 and 10, respectively. As you can see by the example, any stack less than 40 big blinds land us really close to where we want to be after a pre-flop open min-raise. And considering many opponents will loosen up on the flop versus a small stack, we can certainly err on the higher side of the SPR envelope when making a decision on whether or not to stack off.

Now, I am not advocating that you have to play a short stack in order to succeed at poker. I am merely pointing out that a simplified decision-making process would be beneficial for beginning to intermediate players. Once more experience and skill is gained, you might find that the best starting stack size for you is higher. Remember, that doing well and maximizing our earn in poker is about mastering the fundamentals and then finding the style of play that best fits our personality and strengths.

What’s Next?

In part four, we will tie together all of the basic theory and fundamentals that you have learned so far and show you how to start building your own core strategy that is to be used as a baseline from which to make adjustments to your opponents. I call this core strategy a “vacuum” strategy, which is the way you will play when you have no reads or any information on your opponents. Once you have that down, you are ready to get started learning how to attack your tables by exploiting the weaknesses of your opponents. That’s where the real profit (and fun) comes from in poker.

November 17th, 2020

In this article I will lay out the foundational knowledge required to begin learning modern GTO (Game Theory Optimal) poker. At this point, the GTO vs Exploitative debate has essentially been settled and most reasonable people have acknowledged that if you want to be a profitable poker player you need to have a solid understanding of theory. Any exploitation should generally occur as a deliberate deviation from what you understand to be a theoretically correct play. But, if you do not have a foundational understanding of what theoretically correct play even is, your exploitative plays will be based on accurate hunches (at best) and complete delusion (at worst).

Mathematical Foundations

Before diving into the foundations of GTO we will cover some very basic poker math. These concepts have been discussed ad nauseam in a variety of poker books but are still wroth being briefly mentioned.

Hand Combinatorics

There are three categories of hands:

• Suited holdings
  • 4 combos per holding
  • eg. AKs - 4 combos, one of spades, clubs, diamonds, hearts
• Unsuited Holdings
  • 12 combos per holding
  • eg. AKo - 12 combos
• Pocket Pairs
  • 6 combos per holding
  • eg. 66 - 6 combos

Pocket Pair Combinatorics

Pocket pairs have six combos. So for the holding ‘66’ there would be be:

• 6 hearts 6 diamonds
• 6 hearts 6 spades
• 6 hearts 6 clubs
• 6 diamonds 6 clubs
• 6 diamonds 6 spades
• 6 spades 6 clubs

Removing just one 6 from the deck takes the amount of possible combos to 3.Removing another 6 from the deck brings the possible combos of 6 down to one.(Check for yourself by trying to exclude a suit and see how many are left.

Suited holdings have 4 combos. Let’s say we have AKs, removing the ace of hearts would take the number of combos down to 3. Removing the king of hearts would have no effect since the AhKh combo has already been removed. Thus, the number of a suited combos in an opponent’s range is easy to calculate.

Off-suit Holdings have 12 combos. The removal factor on off-suit holdings is slightly more nuanced. Take the example of AKo

AKo - 12 combos

Each Ace is mapped to exactly three king cards (every suit of king other than it’s own suit). Likewise, each King is mapped to exactly three Ace cards.

Removing an Ace from the deck will make the possible number of AKo combos 9. Removing an King subsequently will take the number of combos down to 6.

Outs and Equity:

The amount of outs a hand has is simply the number of possible cards left in the deck that can result in a player making a draw of a pair etc.

A common mental math calculation is to multiply the number of outs by 4 on the flop for the probability of making a hand such as a flop or a straight by the river and to multiply by 2 on the turn.

Hand Type	Outs
Flush Draw	9
Open Ended Straight Draw	8
Gut-shot Straight draw	4

Pot Odds:

When an opponent bets, pot odds can be calculated as

[frac{bet}{pot + bet}]

This will give you the frequency at which you would need to call to theoretically keep your opponent indifferent to bluffing you with any two cards (more on this later). But as we will later discover this approach to defense in the game is heavily outdated, however for players who are vastly under or over-defending vs raises and bets, this concept can be helpful.

Hand Odds:

This will give you the amount of equity against your opponent’s range required to defend against a bet. Again, this model is outdated and has been disproven by solvers but is still useful for beginners.

[frac{bet}{2*bet + pot}]

Range Morphology

A range is a set of hands. Preflop, you have a range for each action, whether you open-raise, 3-bet or flat an open defines that range. You range will interact with flops in a variety of ways. There are a few words we use to describe ranges and their states.

• Linear Range: A range that contains the best hands, the worst hands, and everything in-between
• Condensed Range: A range that is condensed contains mostly medium strength hands with very few if any strong and weak hands
• Polarized: A polarized range contains the strong hands and weak hands but very few if any medium strength hands
• Capped Range: A capped range is a range that does not contain the very strongest hands on a board runout
• Uncapped: An uncapped range contains the best possible holdings

Basic Game Theory Optimal Concepts

Now that I have given you a brief summary of the foundations of poker strategy we will now dive into the basics of game theory optimal play.

History of GTO Play

The idea of an optimal strategy to poker is as old as the field of game theory itself. Von Neumann cited poker as the inspiration of his early works on game theory and wanted to formalize the strategies behind bluffing. John Nash, and Harold W. Kuhn also developed poker toy games which they solved to equilibrium in order to further develop their ideas on games of hidden information.

Why Things Are Different Now

Poker Theory Books

The old way of conducting game theory research was to come up with a game, formalize the rules, and attempt to calculate an equilibrium and utility payoff structure. After deeply analyzing the game tree, mathematicians could calculate optimal solutions either by hand or with some form of computational assistance.

This was the technique used by Edward O. Thorpe when he first invented card counting for blackjack. Now, with very fast computers, we can simply run simulations of model poker games and observe their solutions.

Rather than theorizing about what might be optimal play, we can simply observe optimal play and work backwards in terms of developing heuristics to imitate software.

GTO Concept 1: Minimum Defense Frequency

In around 2013, Matthew Janda published his widely acclaimed book “Applications of No-Limit Hold’em.” This work popularized the concept of minimum defense frequencies and formed the back-bone of success behind many successful poker players such as Doug Polk, Ryan Fee and others.

Yet, the concept of MDF was not anywhere close to being representative of actual GTO play. It was however, so vastly superior to whatever techniques others were using in their calling/raising strategies at the time, that it made a great many players rich beyond their wildest dreams.

Let us suppose that you are facing a bet on the river. Your hand consists of a medium strength bluff catcher, where your opponent either holds the nuts, or total air. In other words; your opponent has a ‘perfectly polarized range’ consisting of hands with either 100% equity or 0% equity at an even distribution vs your bluff catcher.

How would be go about calculating the optimal frequency at which you need to call in order to keep villain indifferent to bluffing? In other words, how can we call in such a way that the EV of betting is effectively zero, and villain gains no additional value by bluffing?

[EV[Bluffing] = 0 = (Pot)*(1 - C) - (Bet)*C]

When villain attempts to bluff us and we fold, he wins whatever is in the pot. When villain attempts to bluff us and we call, he loses whatever it is that he has bet.By solving for the appropriate calling frequency we arrive at the conclusion that:

[C = frac{Pot}{Pot + Bet}]

The idea is that by calling at least C% of the time, we can assure that our opponent will not gain any additional utility by betting with a hand that would lose at showdown.

GTO Concept 2: Value to Bluff Ratios

• Let there be two players, an IP (in position) and an OOP (out of position) player.
• Let IP have a perfectly polarized range on the river (hands with either 100% equity or 0% equity) so that whenever OOP calls we effectively win.

Mit Poker Theory And Analytics

Question: What proportion of our river betting range should be value hands (hands that win at showdown) vs what proportion should be bluffs to make the EV of our opponent calling zero?

• Let X be our bluffing frequency on a river bet
• Let P be the original size of the pot
• Let B be the size of our bet
• Assume EV of Calling - EV of Folding = 0 (Our opponent is indifferent to either actions)
• EV of Folding is always zero, therefore we need to only solve EV[Call]

Our opponent will win the size of the pot + our bet when he calls our bluffs and will lose the size of our bet when he calls our value bets.

[EV[Call] = 0 = (P + B)x - B(1 - x)]

Solving for x

[X = B/(P + 2B)]

Now using our equation we can determine the frequency at which we should be bluffing for any given bet size where B is a fraction of P.

• Pot Sized Bet (B = P): 33% bluffs
• 3/4 Sized Bet (B = .75P): 30% bluffs
• 1/2 Sized Bet (B = .5P): 25% bluffs
• 1/3 Sized Bet (B = .33P): 20% bluffs

Let’s of course notice that the larger our bet-size the more bluff heavy our range is! To most players this seems counter-intuitive but let’s use some very simple reasoning to think through it.

When an opponent calls our value bet, we win what’s already in the pot + whatever we risked. When an opponent calls our bluff, we only lose what we risked. Larger bet sizes cause a rational opponent to fold at a higher frequency meaning that we must accordingly add more bluffs. Additionally, we can afford to lose at showdown often when we bluff since value bets will win us much larger pots. Large bet sizes are positively correlated with a more polarized range (separation between bluffs and value bets) whereas smaller bet sizes are more useful when ranges are too heavy to want to generate folds. This is evident in many solver simulations.

Conclusion

In this article we covered the bare bones fundamentals of poker. If you are a new player I hope you enjoyed this introduction. In future posts, we will dive deeper into solver based poker strategies!