Notes - Modern Poker Theory

August 11, 2025

Chapter 1: Poker Fundamentals

This chapter introduces the core concepts and metrics necessary for understanding advanced poker strategies.

The Basic Concepts

The chapter defines fundamental poker terms, including positional definitions such as Early Position (EP), Middle Position (MP), and Late Position (LP), as well as the Blinds (Small Blind and Big Blind). It also clarifies general poker terminology like Hero (the player from whose perspective the hand is played) and Villains (opponents), Pre-flop and Post-flop actions, In Position (IP) and Out of Position (OOP), and Relative Position. More specific terms like Stack Size and Effective Stack (measured in big blinds), Bet-size (in big blinds or pot fractions), and the concept of "The Nuts" (the strongest possible hand at a given moment) and "Effective Nuts" are also explained.

Other key poker terms elaborated include:

• Speculative Hand: Unlikely to be best now, but can improve later.
• Air: Hand with no showdown value or draw potential, wins only by bluffing.
• Bluff Catcher: Hand that can only beat a bluff.
• Pure Bluff: Betting/raising with a hand unlikely to improve, to make a better hand fold.
• Semi-Bluff: Betting/raising with a hand that has a decent chance to improve later, to make a better hand fold.
• Showdown Value: Hand can realistically win at showdown against Villain's range.
• Nut Advantage/Nut Threshold: A player has this if effective nut hands form a larger part of their range than the Villain's.
• Player Types: Passive, Aggressive, and Regular.
• Hand Types: Pocket Pair, Draw (Gutshot, Open-Ended Straight Draw - OESD), Broadway, and Any Two Cards (ATC).
• Situational Terms: Coin Flip/Race, Cooler, Versus (vs), Toy Game (simplified game model), Game Abstraction (Information Abstraction, Action Abstractions), and Rake (commission fee).
• Calling Station: A player who never folds a made hand.

Player Actions

A comprehensive list of player actions is provided:

• Call (c): Matching the current bet.
• Limp (l): Calling the minimum bet (1bb) to enter the pot.
• Raise (r): Increasing the price (also called a 2-bet).
• Check (x): Passing one's turn.
• Voluntary Put Money in the Pot (VPIP): Entering the pot by calling or raising.
• Raise First In (RFI): First player to enter the pot by raising (also open raising).
• Steal: RFI from BN, CO, or SB.
• Isolate: Raising after someone entered the pot, to play heads-up.
• Minraise/Minbet: Raising/betting the minimum allowed.
• Overbet: Betting larger than the pot size.
• Three Bet (3-bet or 3b): A re-raise.
• Resteal: 3-betting after someone steals.
• All-in/Push: Betting all chips. This includes Open Shove/Open Jam (first to go all-in) and 3-bet-Jam/Shove/Reshove/Rejam (all-in after a raise).
• Four-bet (4-bet or 4b), 5-bet, etc.: Refers to the number of bets in a round.
• Cold 4-bet: 4-betting without being the initial raiser.
• Cold Calling (cc): Calling a raise, 3-bet, or 4-bet without previously putting money in voluntarily.
• Squeeze (sqz): 3-betting after a raise and a call.
• Continuation Bet (C-bet): Post-flop bet by the last aggressor.
• Donk Bet/Lead Out (DK): OOP player bets into the aggressor from previous round.
• Slow Play/Trap: Playing a strong hand passively to induce action.

Action Lines

Common sequences of actions are defined:

• Limp/Fold (l/f): Limping, then folding to a raise.
• Limp/Raise (l/r): Limping, then re-raising a raise.
• Check/Fold (x/f): Checking, then folding to a bet.
• Check/Call (x/c): Checking, then calling a bet.
• Check/Raise (x/r): Checking, then raising a bet.
• Bet/Bet (b/b): Betting turn after flop call (also double-barreling).
• Bet/Bet/Bet (b/b/b): Betting flop, turn, and river (also triple-barreling).

Hand Range

Advanced players think in terms of opponent's range of possible holdings, narrowing it based on actions (hand reading). The best players also consider their own range and its equity against the opponent's, along with concepts like range polarization, balance, and nut advantage. The 13x13 Hand Grid is introduced as a graphical tool to represent all 169 possible pre-flop hands. It's crucial not to give away information by splitting ranges (e.g., raising only strong hands and limping medium ones), as this makes a player easily exploitable. Professional players construct well-thought-out, unexploitable ranges.

Combinatorics

This section explains how to count individual combinations (combos) of hands within a range. While initially difficult, thinking in terms of combos is essential for progress, as all top players use this approach. Software tools like flopzilla.com or power-equilab.com can effortlessly provide range statistics (e.g., percentage of sets, two pairs, draws).

Key Metrics

Pot Odds and Outs

Pot odds represent the reward-to-risk ratio and, when compared to the probability of winning (hand odds), help determine profitability. The chapter provides methods for approximating odds using "outs" (e.g., multiply outs by 4 for turn/river percentage) and tables for common scenarios. It also discusses Backdoor Draws (needing two cards to complete, roughly equivalent to one extra out) and Dead Outs (outs that also improve the opponent's hand).

Equity (Eq)

Equity is defined as your share of the pot based on your current chance of winning or splitting if all cards were dealt without further betting. Free online tools like pokerstrategy.com/poker-software-tools/equilab-holdem can calculate equity.

• Hand vs Hand Equity: Emphasizes that hand strength is relative to the opponent's hand or range, not absolute.
• Hand vs Range Equity: Calculating your hand's equity against a range of possible opponent hands.
• Range Versus Range Equity: Crucial for post-flop scenarios to see how each player's range interacts with community cards and how equities shift on different board textures. Understanding equity is key for poker success and requires practice with calculators.

Expected Value (EV)

EV is the average amount of money you expect to win or lose in the long run in a given situation. A positive EV (+EV) play is profitable. Poker relies on consistently taking the highest EV action. Small, consistent EV edges accumulate into significant long-term winnings.

• Win Rate: EV can be expressed as big blinds won or lost per 100 hands (bb/100).
• Fold Equity (FE): The chance of making an opponent fold. The chapter provides formulas and examples for calculating the minimum fold equity needed for a bluff to be profitable.

Hand Playability / Equity Realization (EqR)

In real poker, players give up equity by folding, which transfers to others. Equity Realization (EqR) is the fraction of a hand's raw equity that actually materializes in EV. Hands can over-realize (capture more than their equity share) or under-realize (capture less). EqR is affected by:

• Position: IP players (acting last) have a significant advantage in realizing equity, as they can check behind for free cards.
• Hand Strength: High-equity hands are easier to play and realize equity well. Medium-strength hands are the most challenging due to pressure.
• Hand Suitedness: Suited hands realize more equity (avg. 16% more) due to flush draws.
• Hand Connectedness: Disconnected offsuit hands have low EqR and are difficult.
• Range Advantage: A strong range advantage allows weaker hands within that range to realize equity because opponents cannot bet aggressively.

Stack to Pot Ratio (SPR)

SPR is the ratio of effective stack size to the pot size, crucial for planning future betting and tailoring bet sizes.

• Low SPR (0-5): Favors made hands (top pairs, overpairs).
• Medium SPR (6-11): Value shifts from top pairs to speculative hands; EqR relies more on suitedness/connectedness.
• High SPR (11+): Hand value comes primarily from "nuttiness" (potential to make the nuts), favoring sets, nut draws, high flushes/straights. Single pairs struggle to reach showdown.

Range Morphology

This classifies hand ranges based on their equity distribution:

• Linear Range: Composed of the highest equity hands without gaps.
• Polarized Range: Consists of high-equity value hands and low-equity bluffing hands. A perfectly polarized range has only nuts and bluffs.
• Depolarized/Condensed Range: The opposite of a polar range, it lacks the top and bottom hands, consisting mainly of middle-equity hands.
• Capped/Uncapped Range: A range is capped if it's missing the strongest hands, and uncapped if it contains all strong hands. A condensed range is always capped, but a capped range isn't necessarily condensed. The author notes that real poker ranges are typically a mix of these types.

Chapter 2: The Elements of Game Theory

This chapter delves into the fundamental principles of Game Theory as applied to poker, stressing its importance for understanding optimal play.

The Core Concepts

• Game Theory: A mathematical and scientific field that studies strategic interactions between rational decision-makers. It's widely applicable, including to poker.
• Game: Any interaction where players' payoffs are affected by others' decisions.
• Utility: The measure of happiness players derive from outcomes.
• Zero-Sum Game: A situation where one player's gain perfectly balances another's loss (poker is, neglecting rake and ICM effects in tournaments).
• Strategy: A complete plan specifying a player's action at every possible decision point.
  • Pure Strategy: Always taking the same action in a given situation.
  • Mixed Strategy: Playing more than one pure strategy with specific frequencies.
• Dominant Strategy: A strategy that yields a higher payoff regardless of what other players do.
  • Strictly Dominates: Always a better outcome.
  • Weakly Dominates: At least one better outcome, others the same.
  • Strictly Dominated: Always a worse outcome.
  • Weakly Dominated: At least one worse outcome, others the same.
  • Intransitive: Neither dominates nor is dominated; the better choice depends on opponent's actions.

Maximally Exploitative Strategy (MES)

An MES is the most profitable set of actions against an opponent's fixed strategy. To calculate it, you must know the opponent's full strategy. Each hand is played in its most profitable way, and if a mixed strategy is used, it means all chosen actions have the same Expected Value (EV). An example demonstrates how a Big Blind (BB) can maximally exploit a Button (BN) who shoves 100% of hands by calling all hands with at least 45% equity.

Counter Exploitation

This concept highlights that if you adopt an MES, an adaptive opponent can in turn modify their strategy to exploit your MES, thereby improving their own expectation. The example continues by showing how the BN, if aware of the BB's MES, can adjust their pushing range to exploit the BB.

Nash Equilibrium (GTO Strategy)

The Nash Equilibrium, also known as a Game Theory Optimal (GTO) strategy, is a state where no player can unilaterally change their strategy to improve their own expectation.

• Strictly dominated strategies are never part of a Nash Equilibrium, as they are irrational.
• If a hand is played with a mixed strategy at equilibrium, it means all strategic choices for that hand have the same EV. The chapter illustrates Nash Equilibrium solutions for the push/fold game previously discussed. The key insight here is that playing a GTO strategy guarantees a certain EV regardless of the opponent's play, whereas an MES is vulnerable to counter-exploitation.

The Indifference Principle

This principle states that if a player plays a mixed strategy at equilibrium, then all actions they take with a non-zero frequency must have the exact same EV. This implies that the opponent is playing in such a way as to make the player indifferent to their choices. It is deeply connected to the concept of balance; for instance, an opponent's calling frequency should make the worst hand in your mixed range break even.

The Clairvoyance Toy Game

This simplified game is used to model and understand complex poker dynamics.

• Setup: Player 1 (P1) has a polarized range (AA or QQ), Player 2 (P2) has a condensed range (KK). P1 knows P2 has KK, but P2 doesn't know if P1 has AA or QQ. Both start with 50% equity.
• Eliminating Dominated Strategies: Actions that are always worse are removed. For example, folding AA, calling with QQ (as KK always wins), and betting KK (as P1 won't fold AA or call QQ) are all dominated strategies and should never be used. Checking with AA is also suboptimal because P2 never bets KK, so P1 misses value.
• Optimal Play: To prevent P2 from exploiting, P1 must find an optimal bluffing frequency. On the river, the bluffing frequency should equal P2's pot odds (how often P2's call needs to be good to break even). P1's optimal strategy involves a 1 bluff to 2 value combo ratio in their betting range (50% bluff frequency for QQ).
• Minimum Defense Frequency (MDF): This is how often a player must defend against a bluff with 0% equity to make the bluff break even. It's calculated as 1 - Alpha, where Alpha is the frequency a bluff needs to work.
• Toy Game Summary: P1's optimal play results in an EV of $75, while P2's is $25, despite both starting with 50% equity. This highlights the power of P1's informational advantage and optimal betting strategy. For a perfectly polarized range against a bluff-catcher on the river, the optimal bet-size is always all-in to maximize EV by allowing more bluffs.

GTO in Multi-way Situations

• While Nash Equilibrium guarantees unbeatability in heads-up (two-player) subgames, it's different in situations with three or more active players.
• In multi-way pots, EV loss from one player's mistakes is not equally distributed among other players. Some players might even lose EV despite playing equilibrium strategies.
• Conclusion: A sound GTO core strategy and deep understanding of GTO principles are critical for modern poker, especially in tough games, but GTO isn't a replacement for critical thinking. While GTO offers near-unexploitable strategies, particularly for heads-up play, they are a starting point for multi-way situations and shouldn't be blindly followed. The goal is to improve your strategy and exploit opponents' weaknesses.

Poker Math Everyone Should Know

This section presents a table relating bet-sizing to key GTO concepts like Alpha (how often a bluff needs to work) and Minimum Defense Frequency (MDF). For example, a half-pot bet gives an opponent 25% pot odds, requiring the bluff to work 33% of the time, and the opponent must defend 67% of their range.

Chapter 3: Modern Poker Software

This chapter explores the essential software tools used by poker players to analyze and improve their game in the evolving poker landscape.

Equity Calculators

These are considered the most essential tools for any serious poker player. They are used to understand hand vs hand, hand vs range, and range vs range equities. Examples include Power Equilab.

• Pros: Intuitive, easy to use, supports multi-way equity calculations, allows custom playability profiles, and provides fast results.
• Cons: Limited to basic equity calculations, does not provide EV or strategic advice, and has limited post-flop range analysis capabilities.

Range Analysis Tools

Tools like Flopzilla allow players to input hand ranges and analyze how they interact with specific flops, breaking down the range into various hand categories (e.g., top pair, sets, draws).

• Pros: Reports equity by hand combinations, can analyze card removal effects, offers post-flop filtering, fast calculations, and is simple to use.
• Cons: Does not offer EV or strategic advice, cannot create complex decision trees, primarily useful for heads-up analysis, and has limited pre-flop utility.

EV Decision Trees

Tools such as Card Runners EV enable users to build decision trees to represent player actions and calculate the Expected Value (EV) of every decision point.

• Pros: Reports both Equity (EQ) and EV for different actions, includes ICM calculations vital for SNGs and MTT final tables, supports complex tree creation and scripting, multi-way calculations, comprehensive range filtering, calculates MES (Maximally Exploitative Strategy), and can incorporate bounties and rake. It also accounts for card removal and bunching effects.
• Cons: Requires significant manual input for ranges, pot sizes, and bet sizes; can be time-consuming; necessitates knowledge of the opponent's full strategy or strong assumptions; and has a steep learning curve.

Pre-flop Nash Calculators

Historically, these were the preferred tools for MTT and SNG players to approximate pre-flop push/fold strategies, especially for shallow stacks. Holdem Resources Calculator (HRC) is a key example.

• Limitations: They do not support post-flop play, which leads to inaccurate pre-flop strategies for deeper stacks due to necessary "action abstractions" (e.g., assuming no betting after flop or removing non-all-in options). This makes them suitable primarily for push/fold scenarios.

GTO Solvers and Artificial Intelligence (AI)

Poker is an imperfect information game, making it significantly more challenging to solve computationally than perfect information games like chess. While AI like Libratus has mastered Heads-Up No-Limit Hold'em (HUNLHE), other variants like tournaments and 6-max cash games are not yet fully solved.

• Approximations: Modern solvers, powerful computers, and strategic abstractions allow for good approximations of No-Limit Hold'em (NLH) GTO play by breaking down complex games into smaller, solvable parts.
• Nash Distance (Ɛ-Nash or Epsilon Equilibrium): This term describes a strategy that is near Nash Equilibrium, meaning it loses only a small value (Ɛ) to its worst-case adversary. It's used to quantify how close a strategy is to true equilibrium.
• PioSOLVER: A prominent GTO solver.
  • Pros: Provides highly accurate results for EQ, EV, EqR, and full strategies; capable of MES and MinES calculations; offers extensive frequency analysis and reporting; supports Limit and No-Limit Hold'em; high compatibility with other poker tools; advanced scripting; ICM calculations; and has a heads-up pre-flop solver. Known for outstanding customer support.
  • Cons: User interface can be initially overwhelming; GTO calculations can be time-consuming, especially for deep stack simulations; requires a steep learning curve and high hardware requirements. It only solves for heads-up situations.
• MonkerSolver: Considered the most versatile solver, capable of solving Hold'em and Omaha from any street with any number of players, using various abstraction techniques to manage game size.
  • Pros: Solves complex multi-way NLH and PLO situations; finalized simulations are easy to browse; includes ICM calculations and rake adjustments; offers equity graphs and manual range export.
  • Cons: Has a rough graphical user interface that can be overwhelming; a very steep learning curve; and requires high-capacity hardware, often necessitating supercomputers or rented servers for accurate solutions.

GTO Poker Training Web Apps

These platforms provide precomputed GTO solutions.

• Pros: No installation or sophisticated computer hardware needed; offer carefully crafted, high-accuracy GTO strategies developed by experts; feature sleek and intuitive user interfaces; provide real-time advice; have a large and growing library of solutions; include multi-way pre-flop and post-flop solutions; require minimal user input; offer rake-adjusted cash game solutions; and advanced post-flop analytics.
• Cons: Use fixed bet-sizes; cannot calculate exploitative strategies against weak players (only provide GTO core strategy); and limit users to precomputed solutions, meaning no custom calculations.

Chapter 4: The Theory of Pre-flop Play

This chapter delves into the foundational mathematical and strategic concepts crucial for pre-flop poker. It highlights that understanding these elements is vital to becoming proficient in advanced poker strategies.

Blinds and Antes

Blinds and antes serve a critical purpose in poker by incentivizing players to engage with more hands rather than simply waiting for premium holdings like AA. The presence of this money in the pot encourages active participation.

Win Rate

A player's skill can be quantified by their win rate, measured in big blinds won or lost per 100 hands (bb/100). This metric is directly influenced by the skill level of opponents, emphasizing the importance of game selection for success.

• Positional Impact: Win rates vary significantly by position. The Big Blind (BB) and Small Blind (SB) inherently start with negative win rates (-100bb/100 and -50bb/100, respectively), making them the least profitable positions to play from. Conversely, the Button (BN) typically boasts the highest win rate because it allows players to profitably play the most hands. Win rates generally decrease as positions move further away from the BN.
• Rake's Influence: House rake further reduces overall win rates, meaning players must be highly skilled just to break even. High rake, especially in online micro-stakes, can make winning very challenging even for strong players.
• Maximizing Profit: To maximize win rates, players must consistently take the highest Expected Value (EV) action with every hand at every decision point. The EV of folding is always 0, so players should only engage in hands where their chosen action has a non-negative win rate.
• Example: In a 9-max MTT with a 20bb effective stack, if the action folds to the SB, various actions (all-in, raise, limp, fold) have different win rates for specific hands. For instance, raising 2.5bb with AA is the highest EV play, even though going all-in with AA is still +EV, it's significantly less profitable. This highlights that a +EV play isn't always the correct play; it must be the highest EV play compared to all other options.
• Cash Games vs. Tournaments: While win rates directly correlate with profitability in cash games, this is not always true in tournaments due to the differing monetary value of chips. Nonetheless, win rates remain useful for evaluating skill and identifying leaks in tournament play.

Complex Strategies

This section compares the profitability of simple push/fold strategies versus more complex strategies (involving limping and various raise sizes) across different stack depths and positions.

• Increased EV: Employing a complex strategy allows a player to capture a larger portion of the pot, yielding an extra 4.62% EV (7bb/100) when out of position (OOP) and 14.6% EV (22bb/100) when in position (IP), compared to a push/fold only approach.
• Lower Fold Frequency: Complex strategies enable players to profitably play more hands, especially as stack depths increase, thus reducing overall folding frequency.
• Pushing Frequency: Open pushing is most dominant with very shallow stacks (under 10bb) but becomes rare for the Button (BN) with 17bb or more.
• Limping Frequency: Limping is used significantly more often (42.51% more) when IP (BN) compared to OOP (SB), becoming a dominant strategy for the BN. For the SB, limping's importance increases with deeper stacks.
• Bet-sizing (Small Blind): When playing OOP in the SB, larger open raise sizes are beneficial as stacks get deeper, starting at 2.5x at 15bb and increasing to 3.5x at 30bb.
• Conclusion: While offering more options increases expectation, overly complex strategies can hurt overall EV if the minimal gain is outweighed by the difficulty of implementation. For shallow stacks (10bb or less) or against superior opponents, a simpler push/fold strategy can be beneficial to reduce mistakes and minimize the opponent's edge.

Playing First In

Knowing how to act when the action folds to you is paramount for poker players. Developing sound pre-flop ranges beforehand is key to making better decisions and avoiding fundamental mistakes that cost money in the long run. Prepared players have a plan and know when to adapt their ranges to exploit opponents, rather than second-guessing themselves.

Main Variables that Affect Pre-flop Hand Ranges

Several factors influence the optimal pre-flop hand ranges:

• Absolute Hand Strength: This refers to a hand's strength in a vacuum. Higher card values and pocket pairs are stronger. Suitedness improves raw equity and playability, for example, A♥5♥ is preferable to A♥5♦ due to flush potential. Connectedness (e.g., 87s) can compensate for lower high card value.
• Position: The earlier a player's position, the stronger their absolute hand strength must be to be playable. Weaker hands can be played profitably from later positions.
• The Size of the Pot: Larger pots, often due to antes, allow more hands to be played profitably. For example, in a 9-max MTT with 12.5% antes, the BN can shove 10bb with 46% of hands, but only 33% without antes.
• Bet-size: Larger bet-sizes generally mean fewer hands can be profitably played, as the risk of losing more when facing a re-raise and folding increases.
• Likelihood of Being Raised/Re-raised: If opponents don't defend often enough, players can exploit them by opening a wider range. Conversely, against good opponents, there's a limit to profitable hands due to frequent re-raises.
• Rake: Rake has a significant impact, causing players to increase raising frequency, reduce limping and calling, and fold more.
• Bunching Effect: This phenomenon describes how folding ranges impact the hand distribution of players yet to act. For example, a BN's Raise First In (RFI) range must be tighter at a 3-max table compared to a 9-max because more weak hands have already folded, increasing the likelihood of blinds having strong hands. The earlier the position, the lower the bunching effect.

Limping

• Open Limp: Limping allows players to enter the pot with a wider range for the minimum cost. However, it risks giving away information about hand strength and encourages multi-way pots, which can be undesirable. Strong players generally avoid open limping from positions other than the SB.
• Facing Limpers: If an opponent open limps (especially outside the SB), it often indicates they are a weak player. The strategy to attack them depends on their type:
  • Aggressive Limper: Isolate with very strong hands that can withstand check-raising or leading.
  • Trappy Limper: Over-limp with suited/connected hands and small pairs, or raise a polarized range (hands that can withstand 3-bets and hands with good blockers for raise-folding).

Open Push

Going all-in pre-flop is a powerful action that guarantees 100% equity realization but involves high risk.

• General Rule: The fewer chips a player has and the fewer opponents remaining, the more inclined they should be to shove.
• Examples: An UTG player with 10bb can only push 14.4% of hands, while the SB can push 74.7% due to fewer players to act. Similarly, a BN with 12bb can shove 41.5% of hands, but with 15bb, only 37.2%.
• Maximum Stack Depths: A table provides maximum stack depths by position for profitable open pushes in MTTs (e.g., UTG max 10bb, SB max 17bb).
• Range Types:
  • Linear Range: Used when shoving the entire range, consisting of the top hands with the best equity.
  • Condensed Range: Used when splitting the range (shove/raise/fold), comprising hands too good to raise/fold but not strong enough to raise/call.
  • Polarized Range: Used for non-all-in raises, including hands with good blocker value (e.g., Axo) for raise-folding, and premium hands (e.g., KK) for raise-calling.

General Guidelines for Pre-flop Bet-sizing

• Positional Sizing: The earlier a player's position, the smaller their bet-sizing should be, due to the increased threat of opponents re-raising.
• Range Width Sizing: Wider ranges should use bigger bet-sizes, as more of the opponent's hands will have the correct equity to call a smaller size.
• Defense vs. Bet-size: A larger bet-size faced means fewer hands need to be defended to remain unexploitable.
• Consistency: Multiple bet-sizes should be used for different situations (e.g., position, stack depth), not for different hand strengths, as this gives away valuable information about the player's range.
• Min-raising: This is optimal when opponents have good rejamming stack depths (under 25bb) because it offers them the worst possible price on their all-in re-raise.
• Larger Bets (Deeper Stacks): With deeper stacks, bigger bet-sizings are better to discourage calls from speculative hands that have better implied odds.
• Positional Bet-sizing: OOP favors larger bet-sizes to reduce the SPR and positional disadvantage. IP favors smaller bet-sizes as they retain their positional advantage even if called.
• Optimal Bet-size: Solvers often show indifference to similar bet-sizes (e.g., 2.25bb vs 2.3bb). The best bet-size is often one that takes opponents to an unfamiliar branch of the game tree, forcing them to make mistakes. Bet-sizing is more of an art form than a science.

General Pre-flop Play Heuristics

• Highest EV Action: Always take the highest EV action with each hand.
• Relative Strength: Play a hand distribution that is strong relative to the random hands yet to act, or relative to the hand distributions of players who have already entered the pot.
• Mixed Strategies: Hands played with mixed strategies (e.g., sometimes raising, sometimes folding) are threshold hands. These are the first hands to become profitable against opponent leaks.
• Exploitative Adjustments: When exploitatively opening a wider range, it's acceptable to over-fold to 3-bets, as you're leveraging the fact that you won't be 3-bet often enough.
• Key Opponents: The BN, SB, and BB are the most important players to consider when opening marginal hands, as they are most likely to provide action.
• Thinking Ahead: Always plan several steps ahead to avoid difficult spots. For example, before 3-betting a medium hand, consider your response if your opponent re-raises.
• Identifying Leaks:
  • Wrong Action Frequencies: Easier to spot, especially online with HUDs. If your RFI frequency is off, cut down or add hands to correct it.
  • Wrong Hands in Ranges: Harder to spot, as frequencies might be correct but the composition of hands within those frequencies is flawed. This is a common reason pros struggle to move up in stakes.

Chapter 5: 6-max Cash Game Equilibrium Strategies (100bb)

This chapter presents baseline GTO strategies for typical 6-max online cash games (100bb, 5% rake, $3 cap). The author emphasizes that the goal is not memorization, but to understand why the solver plays hands in a certain way, incorporating core GTO principles like pot odds, equity, and range polarization. Charts are guides, not rigid rules.

Recommended Bet-sizings

Specific bet-sizings are recommended based on extensive testing and expert discussions:

• Heads-up: BN RFI: 2.5bb; BB 3-bet: 10bb; BN 4-bet: 23bb; BB 5-bet: All-in (100bb).
• 6-Max: RFI (LJ, HJ, CO, BN): 2.5bb; SB: 3bb. 3-bet (IP): 8.5bb; 3-bet (OOP): 10bb (add 2-3bb for squeeze). 4-bet: 23bb; 5-bet: All-in (100bb).
• Blind vs. Blind (BvB):
  • SB Limp: BB Raise vs SB Limp: 3.5bb; SB 3-bet: 13bb; BB 4-bet: 28bb; SB 5-bet: All-in (100bb).
  • SB Raise: SB: 3bb; BB 3-bet: 9bb; SB 4-bet: 24bb; BB 5-bet: All-in (100bb).

GTO Raise First In Strategies

Detailed RFI strategies are presented for each position, starting with blinds and moving to earlier positions.

Small Blind

The SB is challenging because it's OOP, reducing equity realization, but offers a price discount to enter the pot.

• Split Range: The optimal approach is to split the range into limping and raising. This allows playing more hands due to the discount while keeping overall raise frequency low, which makes BB 3-bets less effective.
• SB Limp vs. BB Iso Raise: SB re-raises a linear range of high-equity hands. Against a 4-bet, AA is slowplayed (98%), while KK, QQ, AK, JJ, TT, 99, and A5s are 5-bet pushed.
• SB 3x Raise vs. BB 3-bet: The 4-bet range is polarized, consisting of hands with good blockers (e.g., A5o). Hands like 66+, AKo, and AQs+ are used for stacking off against a 5-bet.

Button

When the action folds to the BN, the strategy is typically raise or fold.

• No Open Limp: Open limping is generally unprofitable in cash games due to rake and the increased likelihood of multi-way pots or being raised. A raise or fold strategy protects the range and makes it easier to defend against 3-bets.
• BN RFI Range: The equilibrium strategy is to open a maximum of 43.4% of hands, including most pairs, suited aces, and various suited/offsuit connectors. In soft games, this can be expanded to 55-60%+ if blinds overfold.
• BN Defense vs. Blinds: BN defense against both SB and BB 3-bets is similar, involving a polarized 4-betting range with good blockers and board coverage. Stacking off range against a 5-bet is 66+, AK+.

Cutoff

The CO plays differently from the BN due to the presence of the BN (who has position) and the blinds.

• Tighter Range: The total playable hands from CO are reduced to 27.8%. Weaker suited/offsuit hands are typically folded.
• Defense vs. 3-bets: When IP, the CO primarily defends by calling, leading to a very polarized 4-betting range. However, against the BN, the CO is more incentivized to 4-bet to either take the pot pre-flop or reduce the SPR, mitigating the BN's positional advantage. Playing OOP against the BN post-flop makes equity realization difficult, so the CO folds more and calls less.

Hijack

Ranges from middle position, like HJ, must be tighter than late position due to more players to act and playing OOP more often.

• HJ RFI: A typical HJ RFI range is 21% hands. Detailed tables show HJ's equity against various 3-bet ranges and appropriate responses (4-bet, call, fold).

Lojack

The LJ position requires even more caution, with an overall range shrinking to 17.1% hands. This is due to facing five other players, three of whom have position. LJ plays high equity hands with strong blockers, and a small portion of suited connectors and small pairs for board coverage.

Playing vs 3-bets: General Heuristics

Well-constructed ranges are designed to maximize profitable hands while remaining unexploitable.

• Defense Requirement: Players must defend enough of their range against 3-bets to make opponents indifferent to 3-betting the bottom of their range.
• Range Type vs. 3-betting:
  • If a player has a flat calling range (e.g., BN or IP), their 3-bet range should be polarized (best hands plus non-calling high equity hands with good blockers).
  • If a player doesn't have a flat calling range (e.g., HJ, CO, SB), their 3-betting range must be linear (only best hands, folding everything else).
• Exploitative 3-betting:
  • If an opponent is likely to 4-bet or fold, increase 3-bet range to include hands with good blockers (e.g., A5s) (polarized 3-betting).
  • If an opponent is likely to call 3-bets, 3-bet a linear range of high-equity hands with good post-flop playability.
• Bet-size Impact: A smaller 3-bet size means a player needs to defend more hands to remain unexploitable.
• 5-betting: Generally, the "getting in" range for a 5-bet is TT+ and AK.
• Cold 4-betting: This is generally preferred over cold calling, as cold calling gives away information and allows active players to realize equity. Cold 4-bets typically range from ~3% of hands (vs. tight ranges) to ~6% (vs. wide ranges).
• Positional Variation: The response to a 3-bet varies significantly depending on the position of the 3-bettor (HJ, CO, BN, SB, BB), influencing 4-betting, calling, and folding frequencies. For example, against the SB (who is OOP), players need to 4-bet less often and can call more due to having position.

Specific Ranges Playing Versus Open Raises

This section provides granular details on GTO responses to open raises from various positions, including 3-betting, calling, and 4-betting frequencies and hand compositions. Key factors include:

• Board Coverage and Blockers: The solver often includes a small weight of weaker pocket pairs or suited connectors in 3-betting ranges to ensure good post-flop board coverage and to make it harder for opponents to use blockers effectively.
• Equity Realization: Hands that struggle to realize equity OOP (e.g., offsuit disconnected hands) are often folded, even if they appear strong in a vacuum.
• Slowplaying: Premium hands like AA and KK are sometimes slowplayed (flat called or checked behind) to protect a player's calling range or to induce more action.
• Polarization and Linearity: The type of 3-betting range (polarized vs. linear) depends heavily on whether the player has a flat-calling range (polarized) or is restricted to 3-bet or fold (linear).

Chapter 6: The Theory of Tournament Play

This chapter highlights the fundamental differences between cash games and multi-table tournaments (MTTs), focusing on concepts like variance, bankroll management, and the Independent Chip Model (ICM).

Variance

Variance refers to the natural fluctuations of results around a player's true win rate. It is the square of the standard deviation of results.

• Factors Affecting Variance:
  • Volume: The more hands or games played, the smaller the total variance. High volume players experience less variance.
  • Spread of Win Rates: A wider spread in win rates (e.g., in chunked samples of hands) leads to higher variance. Different game formats have different inherent standard deviations.
• Importance: Understanding variance is crucial for interpreting poker results, estimating potential downswings, practicing smart game selection, and determining proper bankroll management. Short-term results can be highly misleading due to variance.

Poker Tournament Metrics

Traditional cash game metrics don't fully capture tournament skill.

• Total Cashes/Earnings: While popular, this metric alone is insufficient as it doesn't account for buy-ins, expenses, or taxes.
• Profit: More valuable than raw earnings, but can still be misleading if a lucky big win skews lifetime results for a losing player.
• Return on Investment (ROI): Calculated as Net Profit / Total Buy-ins, ROI is a more accurate measure of a tournament player's skill. However, ROI generally decreases at higher stakes due to tougher games. Fast/turbo tournaments have lower ROI due to less time to exploit weak opponents. Bounty tournaments reduce variance but also lower EV for strong players as bounties are distributed to those who knock out opponents.

Decreasing Variance and Increasing Profitability in MTTs

• Increasing Volume: Playing online MTTs is a good way to increase volume, as players can multi-table. Recommended play-to-study ratios are 50% play/50% study for new players (focus on in-game experience) and 80% play/20% study for winning players (to maintain skill and prevent stagnation).
• Increasing ROI: The easiest way is to play lower stakes against weaker opponents. The toughest is to become a better player.
• Exploitation over Balance: The author emphasizes that GTO poker is fundamentally about exploitation, not just balance. Equilibrium strategies arise when players are maximally exploiting each other. Successful tournament players like Daniel Negreanu and Phil Hellmuth are cited as examples of effective exploitative play, even if not always "theoretically perfect". The advice is to study theory but focus on practical application and adapting to opponents, not fearing "bad" or "stupid" plays if they are exploitative.

MTT Bankroll Management

Players should determine how much of their net worth they are willing to allocate as a bankroll. A professional's bankroll should be a large percentage of their net worth, while recreational players should use an amount they can afford to lose.

Game Selection

Similar to cash games, tournament players should choose games where their ROI will be higher and variance lower. Playing a wide spread of buy-ins significantly increases the probability of experiencing large downswings, as illustrated by the author's personal experience of losing $120,000 in high-stakes MTTs despite being a winner in mid-stakes.

The Mental Game

Mental game is highlighted as absolutely key to poker success, especially in MTTs. The author emphasizes the importance of resources like books and coaching for overcoming mental limitations. Key mindset traits of winning players include a positive attitude, responsibility, strong work ethic, emotional control, openness to criticism, long-term results orientation, mindful multi-tabling, and attention to health and routines. Losing players typically exhibit the opposite traits.

Tournament Life

This concept refers to the value of still having chips in a poker tournament. Tournaments are about survival and outlasting opponents, not just accumulating chips.

• EV of Future Hands: Staying in the tournament has value because of the EV of future hands that can be played.
• EV of Gaining a Higher Payout: There's value in folding to stay alive and potentially secure a higher payout if other players are eliminated before you. This differs from cash games where folding's EV is always zero.
• ICM and FGS: Mathematical models like the Independent Chip Model (ICM) and Future Game Simulation (FGS) were developed to quantify these non-chip EV factors.

The Independent Chip Model (ICM)

ICM addresses the Principle of Diminishing Chip Value in tournaments: chips risked are worth more than chips won, and the more chips a player has, the less each individual chip is worth monetarily.

• Function: ICM assigns a monetary value to chip stacks based on their equity share of the remaining prize pool. It calculates the probability of each player finishing in a specific place and multiplies it by the payout for that position.
• cEV vs. $EV: In cash games, chip EV (cEV) and monetary EV ($EV) are the same, but in MTTs, decisions should always be based on $EV. ICM often dictates tighter play around bubbles to preserve $EV.
• Limitations: ICM overvalues short stacks. It does not consider skill edge, future game play, or position (e.g., a 4bb stack on the BN is more valuable than UTG).
• General Guidelines: ICM analysis leads to general guidelines like calling tighter in tournaments than cash games, avoiding marginal +cEV spots, and recognizing that medium stacks are forced to play tightly around short stacks and the money bubble.

Risk Premium

Risk premium is the extra equity required for a play to be +$EV in tournaments compared to cash games, due to the principle of diminishing chip value. It is higher when calling large bets, especially all-ins, as tournament life is at stake. Even for small bets, it influences BB defending ranges. Risk premium also applies to making bets.

Tournament Life / Gambler's Ruin

The Gambler's Ruin problem mathematically models the probability of a player losing all their chips.

• Fair Game: In a fair game, a player's probability of winning increases linearly with their chip stack.
• Skill Edge: If a player has a skill edge, their probability of winning does not increase linearly; the curve is concave. This implies that the marginal value of each additional chip decreases. Therefore, players with a skill edge should be more risk-averse and avoid marginal gambles. Taking high-variance gambles benefits the weaker player.
• Skill Deficit: Conversely, if a player has a skill deficit, the probability curve is convex, meaning the marginal value of chips increases. Such players should be more willing to gamble for big chunks of their stack.

Conclusion

The complexity of tournaments (varied table sizes, stack depths, blind structures, payment structures, player skill levels) makes a single unified model for optimal play impossible. There is no "definite answer" to optimal play, and blindly copying successful players' strategies can be detrimental. The ultimate goal is to develop personal judgment based on GTO principles to decide when to follow equilibrium strategies and when to deviate for maximum profit.

Chapter 7: MTT Equilibrium Strategies: Playing First In

This chapter focuses on cEV-based equilibrium strategies for playing first-in (PFI) in MTTs, specifically for 9-max tables with a 12.5% ante, across key stack depths (15bb, 25bb, 40bb, 60bb).

General Guidelines

• Deep Stacks (no antes): Ranges are tighter, speculative hands gain value, and calling becomes more attractive.
• Pre-flop Decision-Making: It's vitally important to think ahead before making pre-flop decisions, considering your own stack, opponents' stacks, and their likely responses (re-raising or calling). Players should know whether their hand is a raise-fold or a raise-call, and if it's better as an open shove or a min-raise. Avoiding autopiloting and staying mindful is crucial to prevent blunders.

Bet-sizing

• RFI Bet-size: The specific RFI bet-size doesn't have a significant EV impact as long as it's reasonable and paired with reasonable ranges. The optimal range is typically 2bb-2.5bb from BN to UTG, and 2.5bb-3.5bb in BvB situations.
• Min-raising (<25bb): Min-raising is recommended when stack depths are in the "rejam region" (less than 25bb). This is because:
  • Raise-folding is expensive with short stacks; min-raising allows for folding the worst hands.
  • It gives opponents the worst possible price on their all-in re-raises.
• Deeper Stacks (30-35bb+): The threat of being rejammed is lower, so raise sizes can be increased to 2.3x to cut down opponents' implied odds and discourage calls from weaker hands. At 100bb+, 2.3x-2.5x is common.
• Positional Bet-sizing: Generally, bigger bet-sizes are preferred when deep-stacked and out of position (OOP). A smaller 3-bet size from the SB is also common due to the threat of the BB.
• Pot-Commitment: If a 3-bet size requires investing over 1/3 of your stack, you'll be pot-committed (unless your hand is trash). In such cases, it's often better to go all-in to reduce opponent's strategic options.
• 4-bets and 5-bets: For stacks under 40bb, all 4-bets are typically all-in. For 50bb-100bb, non-all-in 4-bets (2.25x-2.5x IP, 2.5x-3x OOP) are seen. All 5-bets are always all-in.
• Simplification: While solvers may show complex mixing, using a simpler strategy with fewer bet-sizes is generally recommended for human players.

Short Stack Push/Fold Charts

These charts, generated with HRC using FGS, indicate the maximum big blind stack depth at which a hand can be profitably pushed from various positions. Hands in black can be pushed with 10bb or less. The principle is that with fewer than 10bb, open pushing is often the correct play from any position, with tighter ranges for earlier positions and larger stacks.

Small Blind PFI Strategy

The SB has the highest VPIP (Voluntarily Put Money In Pot) frequency (average 83%) when action folds to them, due to the discount and only one player left to act.

• 10-15bb: Play a push/limp/fold strategy.
  • Pushing Range: Hands with good blockers (Ax, Kx), hands with high equity but poor post-flop equity realization (small pairs), or high equity hands that struggle to call a jam (suited connectors).
  • Limping Range: High equity hands that can call an all-in (mid-to-high pairs, broadways, suited Ax/Kx) and hands that play well in limped pots.
• 17bb+: Non-all-in raise sizes are introduced, increasing from 2.5x at 17bb to 3.5x at 30bb.
• 25bb: Open pushing largely disappears, shifting to a raise/limp/fold strategy. The limping range includes traps (strong Ax, mid pairs) and hands that don't want to raise and fold to a jam. The raising range is polarized, with the strongest hands and those with good blockers.
• 40bb: No all-in range. Most hands are played as a mixed strategy, with stronger hands raised more and weaker hands limped more. The limp/raise range is composed of strong hands and bluffs with blockers/board coverage.
• 60bb: Limping frequency increases, raising frequency decreases. No hands are played as a pure raising strategy, but many are pure limps. Limp/reraising hands are the best hands for stacking off pre-flop, along with blockers and board coverage hands.

Button PFI Strategy

The BN has a unique advantage: no other player can force them to play OOP.

• 10-30bb: The solver favors a limping range, often preferring limping over min-raising at 10-12bb.
• 15-30bb: Min-raising becomes important to allow for IP post-flop play.
• 30bb+: BN no longer open shoves, as stacks are too deep.
• 40bb+: BN stops limping and favors a simpler raise/fold strategy. This is because the BN can defend very well against 3-bets by calling IP and realizing a lot of equity.
• Drawbacks of BN Limping: Compared to the SB, limping from the BN doesn't offer the same EV gains and increases variance due to more multi-way pots.
• Non-Limping Strategy: Without limping, BN's VPIP increases with stack depth (38.9% at 10bb to 55.5% at 80bb), as deeper stacks favor BN's positional advantage. Open pushing frequency decreases significantly with deeper stacks (3.5% at 20bb, 0% at 25bb).
• 15bb: BN splits between all-in and min-raising (almost 50/50). Pushes hands with good equity but poor post-flop playability (small pairs, offsuit broadways).
• 25bb: No all-in range. High card value and raw equity are more important. Small pairs (33, 22) are folded because they perform poorly as min-raises (reverse blockers, difficult post-flop play, low implied odds).

Cutoff PFI Strategy

For CO and earlier positions, limping strategies are generally not recommended due to increased complexity and the higher risk of being over-limped or raised by multiple players. The focus is on a raise/fold strategy.

• VPIP: CO VPIP increases with stack depth (31.4% at 10bb to 37.5% at 80bb).
• Shoving Frequency: Push/fold at 10bb and below, min-raises at 12bb, and all-in frequency drops to 0% at 20bb.
• 25bb: No all-in range. Small pairs (33, 22) are folded in this position when first in.
• 40bb/60bb: CO opens slightly wider, incorporating hands that play well with deeper stacks, such as small pairs and suited hands.

Hijack PFI Strategy

From middle position like HJ, ranges must be tighter than late position due to more players and more frequent OOP play.

• VPIP: HJ VPIP peaks at 25-30bb and then stagnates or decreases, as the positional disadvantage with deeper stacks becomes more relevant. Overall average VPIP is 26.89%.
• Shoving Frequency: Push/fold at 10bb and below. Min-raising becomes dominant at 15bb. All-in option disappears at 20bb and deeper.

Lojack PFI Strategy

LJ play is similar to HJ, but the overall range shrinks to 22.54% VPIP due to a slight positional disadvantage.

• Shoving Frequency: LJ has a raise-folding range at 10bb stack depths. There are open shoves at 15bb (4.9% with 77-55, AQo, AJo, A9s, A8s, JTs+), but none at 17bb or deeper.
• 25bb: Significant increase in VPIP compared to 15bb. Hands that flop good top pairs, straight draws, and flush draws (e.g., J8s, T8s, 98s) start opening, but 44 and 33 are folded.

UTG+2 PFI Strategy

In early position (EP) like UTG+2, the average VPIP drops below 20% because players must contend with at least six opponents.

• VPIP: Peaks at 21% (25-30bb), averaging 19.63%.
• Shoving Frequency: Overall drop in open jam frequencies due to the increased chance of opponents having premium holdings.
• 15bb: Tiny all-in range (2.2%), but raise/fold is preferred as jammed hands are indifferent to min-raise. All-in range disappears at 17bb.
• Deeper Stacks: Offsuit broadways lose value, and small suited connectors become more relevant. At 60bb, the range shifts towards stronger, less dominated hands and small pairs for board coverage.

UTG+1 PFI Strategy

This position has an average VPIP of 16.75%, peaking at 18.60% with 25bb.

• 12bb: Opening range is split almost 50/50 between a push and a min-raise.
• 15bb: Stacks are too deep for jamming, so only a raise/fold strategy is used.
• Deeper Stacks: The range shrinks somewhat compared to 25bb, focusing on hands with good playability that can flop nuts or nut draws. Offsuit broadways decrease in value. At 60bb, the range tightens further, with offsuit broadways and suited connectors becoming almost unplayable, favoring high raw equity and good blockers.

UTG PFI Strategy

UTG holds the tightest range of all positions, with an average of 15.42% VPIP.

• VPIP: Doesn't increase much with deeper stacks; instead, the type of hands played differs.
• 10-14bb: Mixed strategy (min-raising and open jamming).
• 15bb+: All-in range disappears, becoming raise/fold only.
• Deeper Stacks: Decrease in RFI for offsuit broadways, increase in suited connectors and small pairs for board coverage.

Chapter 8: MTT Equilibrium Strategies: Defense

This chapter provides a comprehensive guide to defending against open raises from various positions (BB, SB, BN, MP, EP) in MTTs, across different stack depths.

Theoretical Considerations

• Average Action Frequencies: Tables provide general frequencies for responding to open raises based on Hero's and Villain's positions.
• Facing an Open Shove:
  • If the call is for more than 1/3 of your stack, you should rejam all-in over the top, except in very heavy ICM/bubble situations.
  • If the call is for less than 1/3 of your stack, call with your continuing range or min-re-raise to ensure heads-up play.
  • The calling range varies with pot odds, position, and the shover's position and stack depth. Practicing with push/fold apps is recommended.
• Overcalling (after a raise and a call): Players should use a tighter range than when facing only a raise. While there's more dead money and the caller typically doesn't have a premium hand, their range is still strong enough to withstand 3-bets. 3-betting is often better than cold-calling. Hands that play well multi-way include suited Ax, suited connectors, and medium/small pocket pairs, while offsuit high/low cards are poor.

Big Blind Versus Small Blind (BvB)

This is unique because the BB is IP throughout the entire hand, leading to significantly higher equity realization.

• BB vs. SB Push: The BB's calling range depends on pot odds. It's tighter than what traditional push/fold apps suggest due to the bunching effect and other SB strategic options. At 12bb, BB calls broadly; at 15bb, it tightens slightly; at 25bb, it shrinks further. For <30bb, BB's only 3-bet size is all-in, trapping with bigger pairs and rejaming with high equity/poor post-flop EqR hands.
• BB vs. SB Raise:
  • 40bb: No 3-bet jams. 3-betting strategy becomes polarized (TT+, AJ+ for value; offsuit Ax/Kx, etc., for bluffs). BB calls 4-bet with 66+, AJs+, ATo+.
  • 60bb: Similar 3-bet frequency, but bluffing portion changes to include a wider variety of combos for board coverage.
• BB vs. SB Limp: BB either checks back (56% average) or raises (44%). The raise size increases with stack depth (2x at 10bb to 3.5x at 30bb+). Even smaller bet-sizes can be used exploitatively against weak limpers. Ranges are detailed for various stack depths, with pushing, raising, and checking back decisions influenced by blockers, equity, and post-flop playability.

Defending the Big Blind Versus IP Player

• 15bb: Average BB fold vs. min-raise is 22.56%. BB is too shallow for non-all-in 3-bets. All-in frequency decreases as opener's range strengthens, but overall BB VPIP remains constant due to increased calling frequency. Solver likes all-in with hands ahead of opener's range but with bad post-flop equity realization.
• 25bb: Average BB fold vs. min-raise is 22.54%. All-in frequency drops as opener's range strengthens, but non-all-in 3-bet frequency remains fairly constant (~5%). Folding range increases drastically against stronger opener ranges because weak hands struggle to realize equity OOP at this depth.
• 40bb: BB is too deep to 3-bet all-in against most positions. Folding frequency increases significantly against stronger Villain ranges (24.2% vs. BN to 45% vs. UTG). Against polarized ranges from EP, BB should defend tighter and use stronger hands as bluffs, with lower 3-betting frequency.
• 60bb: Slight increase in BB's defense frequency compared to 40bb, possibly due to weaker hands having better implied odds. However, it's recommended to cut down on weaker hands against tough opponents who overbet.

Defending the Small Blind

• 15bb: Average SB VPIP is 22%. SB plays 3-bet or jam vs. LP opens. Against MP/EP, the opener's range is strong enough that rejams don't get enough folds, so the SB calls more and 3-betting frequency decreases. Suited connectors, small suited Ax, and broadway hands are good calls in multi-way pots. AA is slowplayed 25-33%.
• 25bb: Similar trends of VPIP decreasing as opener's range strengthens. Offsuit hands mostly folded, except for Ax for rejams vs. LP. Small pocket pairs are good rejams vs. LP but better calls vs. EP.
• 40bb: Tiny 3-bet all-in range, often practically ignored. SB calling range remains constant (~12.21%) vs. all positions.
• 60bb: Average SB VPIP 24%. Deeper stacks favor 3-betting hands with better post-flop playability, as play is likely to go to the flop, forcing SB to play OOP. SB will call many hands vs. a 4-bet since IP is too deep to 4-bet jam.

Defending the Button

• 15bb: Average BN VPIP 15%. Calling range is constant at 5%, mostly middle to small pocket pairs, suited broadways, suited Ax, and slowplayed AA.
• 25bb: BN VPIP 21%. Aggressively rejams vs. CO (5.9%) but rarely vs. UTG (1.2%). Non-all-in 3-betting range is constant (~3.32%) and calling range (~14.22%) includes strong equity hands benefiting from position.
• 40bb & 60bb: BN VPIP increases as the opener's range weakens, maintaining consistent calling frequencies while 3-betting frequency lowers against stronger openers.

Defending the Cutoff

• 15bb: No flatting range due to concern about BN calling or squeezing. The CO's strategy is to rejam the entire playable range (8.3% vs. UTG to 13.3% vs. HJ).
• 25bb: CO rejams a narrow range, so a 3-bet/call/fold strategy is preferred. The flatting range is capped (mid-to-small pairs, suited broadways, ATo+, KJo+).
• 40bb: CO plays a 3-bet/call/fold strategy. 3-betting range is polarized. Flat-calling includes mid-to-small pocket pairs, suited Ax, suited broadways, and premium suited connectors, sometimes slowplaying AA and AKs.

Defending the Hijack

• 15bb: HJ defends against open raises by rejaming a linear range (average 9.3%).
• 25bb: GTO strategy rejams at a very low frequency (~1.45%), so in practice, a 3-bet/call/fold strategy is better, converting mixed call/rejam hands to pure calls and mixed 3-bet/rejam hands to pure non-all-in 3-bets.
• 40bb & 60bb: HJ 3-bets a polarized range (5.48% average at 40bb, 6.13% at 60bb). The 3-bet/fold range consists of high equity hands with good blockers and decent post-flop playability.

Defending UTG+1

• 15bb: UTG+1 vs UTG rejams a linear range of 7.1%.
• 25bb: No rejaming range. Incorporates a 3.6% flat-calling range (JJ-66, AQo+, suited broadways) and a 3-betting range (5.1%) that is less polarized than from LP/MP.
• 40bb: 3-betting range shrinks slightly (4.8%), changing to include more suited Ax and fewer suited broadways. Flatting range increases slightly (4%), incorporating more suited broadways.
• 60bb: UTG+1 3-bets more often and calls less than at 40bb, benefiting from UTG defending 3-bets by calling OOP. UTG+1 realizes more equity and 3-bets a wider range.

Chapter 9: MTT Equilibrium Strategies: Playing Versus 3-Bets

This chapter details strategies for defending against 3-bets after open-raising, emphasizing the importance of understanding opponent tendencies and adjusting baseline strategies.

The Key Factors

When facing a 3-bet after opening, key factors to consider are:

• Villain Bluffing Tendency: Is the opponent likely to be over-bluffing or under-bluffing in that specific spot?
• Profitability of Calling: If calling is close to 0 cEV, it's likely a losing play in terms of $EV. The author stresses that the presented baseline ranges are merely guides; players should actively analyze these variables to make in-game adjustments. The goal is to understand the solver's logic rather than memorizing exact frequencies. Simplifications and rounding are acceptable in practice, as human opponents rarely play perfectly.

Short Stack 2x Open Versus Rejam (10-25bb)

• Action Frequencies: The average fold frequency after a min-raise vs. a rejam with short stacks (10-25bb) is 55.04%, with an average call of 44.96%.
• Defense Frequency: The deeper the effective stack that is being rejammed, the less frequently the opener has to defend. The earlier the position of the player going all-in, the tighter the opener should defend (exception: SB is called lighter than BB).
• Calling Range: When facing a rejam, the calling range should consist of the top X% of hands that have the highest equity against the rejamming range. This section provides detailed calling and folding ranges for various positions (BN, CO, LJ, UTG) against all-ins from different opponent positions at both 15bb and 25bb stack depths.

Mid Stack Open vs Non-all-in 3-bet (25-40bb)

• 4-betting: The solver's only 4-bet size in this situation is all-in.
• Positional Differences: When IP, the opener should 4-bet more with short stacks and call more with deep stacks. When OOP, the opener should fold more with deep stacks and call more with short stacks.
• Hand Types for 4-bets: Generally, hands with good equity but poor playability are preferred for 4-betting all-in (e.g., mid-small pocket pairs, big offsuit Ax, small suited Ax).
• Hand Types for Calls: AA, KK, suited connectors, suited broadways, and big suited Ax tend to perform better as calls.
• Detailed action frequencies and hand ranges are provided for different opener positions (BN, CO, LJ, UTG) against non-all-in 3-bets from various opponent positions at both 25bb and 40bb stack depths.

Big Stack Open vs Non-all-in 3-bet (60-80bb)

• 4-betting: With deeper stacks (60-80bb), all-in 4-bets become less common and are replaced by non-all-in 4-bets. The overall 4-betting frequency decreases (especially IP).
• Calling Frequency: The calling frequency increases with deeper stacks, as playing post-flop becomes more enticing.
• Blinds' 3-bet Sizing: Blinds should use larger 3-bet sizes when stacks are deeper to prevent the opener from calling with a very wide range.
• Playing vs. 5-bets: Generally, call any premium hand and fold blocker-type bluffs.
• Detailed action frequencies and hand ranges are provided for different opener positions (BN, CO, LJ) against non-all-in 3-bets from various opponent positions at 60bb stack depth.

Chapter 10: The Theory of Post-flop Play

This chapter initiates the exploration of post-flop strategies, emphasizing that a deep understanding of why we bet is crucial. It shifts the focus from subjective reasons for betting to a more scientific, game theory-based approach.

Theory of Betting

The primary reasons for betting in poker are:

• Leveraging the advantage of knowing your own cards: Betting allows a player to exert informational advantage over opponents who only know a player's range. By incorporating diverse hand types (value and bluffs) into betting ranges, players make it harder for opponents to consistently make correct decisions, forcing them into mistakes and causing them to lose EV. This concept extends to board coverage, meaning a range can make strong hands on a wide variety of runouts, especially when playing balanced strategies.
• Realizing equity or preventing your opponent's equity realization: Betting helps realize equity by denying the opponent's equity and can set up the Stack to Pot Ratio (SPR) favorably. Range composition is key, as demonstrated in the clairvoyance toy game where a player with a polarized range can over-realize equity against a depolarized range, especially with deeper stacks. Conversely, with shallower stacks, it becomes harder to deny opponent's equity, leading to a more even pot distribution. Well-balanced ranges are essential for multi-street play with deeper stacks.

Strategic Actions

• Folding: Folding a hand when checking is free is a strictly dominated strategy and should never be used. Even with a bad hand, it always retains some equity or backdoor equity that could be realized if opponents check. Proactive folding also reveals information about your range, which can be exploited by aware opponents. Folding is only valid when facing a bet and all other options have negative Expected Value (EV).
• Checking: When checking, a player can either check/fold, check/call, or check/raise.
• Calling: Calling a bet in a heads-up pot realizes equity, either by reaching showdown or seeing another card.
• Raising: Raising applies the same principles as betting: leveraging informational advantage, realizing equity, or denying opponent's equity realization. It's crucial to be aware of your own ranges to ensure they remain well-balanced, as actions convey information.

Equity Buckets (EQB)

This new concept categorizes hands within a given range based on their hand vs. range equity, providing a more detailed understanding of post-flop equity distribution than traditional range morphology (linear, polarized, condensed, capped/uncapped).

• Strong Hands: Hand vs. range equity ≥ 75%.
• Good Hands: Hand vs. range equity ≥ 50% but < 75%.
• Weak Hands: Hand vs. range equity ≥ 33% but < 50%.
• Trash Hands: Hand vs. range equity < 33%. Hand value is dynamic and relative to ranges and board type.

Post-flop Bet-sizing

No-Limit Hold'em allows flexible bet-sizing. This section explores the importance and optimal number of bet-sizes.

• The Pot-size Raise: A bet laying exactly 2-to-1 (33% pot odds). Formulas are provided for calculating bet-size as a fraction of the pot.
• Alpha and Minimum Defense Frequency (MDF) Revisited: While useful as a rough guide, Alpha and MDF are misleading for post-flop play because they assume zero equity for bluffs and zero EV for checking back. On the flop, most hands have some equity, and checking back is not always 0 EV. EV calculations must compare all actions, not just bet/fold.
• Splitting Your Range Into Multiple Bet-sizes: The Toy Game demonstrates that in symmetric ranges, an infinite number of bet-sizes can be optimal.
  • Example A (No x/r allowed): Hero can split range across three bet-sizes (Full, 2/3, 1/3 pot), capturing 56.11% of the pot.
  • Example B (OOP x/r vs 1/3-bet allowed): Hero stops using the 1/3-pot size, EV reduces. This highlights how opponent's ability to check/raise (x/r) impacts optimal bet-sizing.
  • Example C (OOP x/r vs all bet-sizes allowed): Hero reverts to a single, full pot-size bet, checking back weaker hands.
  • Example D (All-in allowed, SPR 2x pot): Hero splits range into all-in and pot-size, gaining a small EV advantage. All-in limits villain's options. However, with SPR 3+, Hero reverts to only full pot-size.
• Summary of Toy Game Outcomes: Shows that using multiple bet-sizes is dangerous, especially in early betting rounds, as it gives away information for exploitation. Choosing the wrong bet-size can significantly reduce EV.
• SPR Effect on Optimal Bet Sizes:
  • SPR ≤ 1: Optimal bet-size is all-in.
  • SPR 1-2: Range split between pot and all-in.
  • SPR 3: All-in bet stops, range splits between 75% and 125% pot.
  • SPR 5-10: Range splits between 75% and 150% pot.
  • Simplifying to a single bet-size (removing least frequent) results in minimal EV loss (around 0.21% of pot across all SPRs).
• Conclusion on Bet-sizing: While multiple bet-sizes might offer marginal EV gains against GTO opponents, their complexity can lead to more human mistakes. It's more profitable to adapt bet-sizing exploitatively against weaker opponents. There's no single "optimal" bet-size; it's an art form. Most online MTT regulars use a 1/3-pot size c-bet, which is simple and exploits population over-folding. The author recommends proficiency with at least two bet-sizes (big and small) to adapt to opponents and situations. Proper simulation setup requires considering all players' proper bet-sizes to avoid skewed results.

Chapter 11: The Theory of Flop Play

This chapter delves into how different flop types interact with players' ranges and influence post-flop betting strategies.

Suit Isomorphism

Suits are strategically equivalent pre-flop. Post-flop, suit isomorphism allows for reducing the 22,100 possible flops to 1,755 strategically different ones, simplifying study and solver calculations by avoiding redundant computations.

Dynamic and Static Boards

• Volatile (Dynamic) Boards: Future cards are very likely to change hand values, e.g., low and connected boards with possible flush and straight draws.
• Static Boards: Good hands are likely to stay good, and bad hands likely to stay bad, e.g., offsuit disconnected flops with high cards.

Flop Classification Scheme

This scheme simplifies the vast number of flop situations by grouping them based on common characteristics that influence play.

• Flop Structure:
  • Trips: All three cards of the same rank (e.g., A♥A♣A♠).
  • Paired: Two cards of the same rank (e.g., K♥K♣5♠). Can be rainbow or two-tone.
  • Unpaired: All three cards of different ranks (e.g., Q♥7♦4♣).
• Flop Textures:
  • Monotone: All three cards of a single suit (e.g., A♥K♥T♥).
  • Two-tone: Two cards of one suit, one of another (e.g., A♥K♥Q♠). Subtypes: high-mid, mid-low, high-low.
  • Rainbow: All three cards of different suits (e.g., Q♥9♠7♦).
• Flop Rank: Determined by the highest card (e.g., Kxx, Axx). Higher ranks are more common (Axx is 21.74% of all flops). 85% of all flops are rank 9 or higher.
• Flopped Straights: Categorized by the number of possible flopped straights (e.g., 0, 1, 2, or 3).
• Flop Families: Groupings by card rank (High, Mid, Low, Ace), e.g., A54, A53, etc., as "Ace with two low cards".
• Flop Subsets: Attempts to represent the entire game with a smaller number of flops.
  • Will Tipton's (103 flops): Had some issues in fully representing the game.
  • PioSOLVER's (25-184 flops): Optimized for approximating equities and EV for the pre-flop solver, but not guaranteed for post-flop strategies. The author instead used super-computers to solve all 1,755 flops to analyze data.

The Flop Donk Bet (DK)

Donk betting (when OOP bets into the previous street aggressor) goes against traditional poker norms.

• GTO Perspective: GTO does not recognize "betting lead." Betting is about leveraging informational advantage, range composition, and SPR.
• Frequency: Overall BB donk bet frequency is only 2% (for 1/4 and 2/3 bet-sizes), while average IP c-betting is 84%. However, the solver does donk bet at high frequencies in specific spots.
• Conditions for Donk Betting:
  • Occurs on flops where BB has the equity advantage (e.g., 654r, where BB is 51% vs IP's 49%), making BB's bets more profitable.
  • Crucially, when BB's range is more polarized (more strong/good/trash hands) than IP's (more weak hands), incentivizing BB to take the betting lead. This is similar to the Clairvoyance Toy Game.
  • Denies IP Equity Realization (EQR) and helps BB realize equity by leading with a well-balanced range.
  • High donk bet frequency flops are typically ranks 7-x-x and 6-x-x with 1-3 possible straights. Rainbow flops are donked more than two-tone, and monotone least.
  • On high donk frequency flops, BB over-realizes equity (103% EQR), capturing 52% of the pot.
  • Bet-size for Donk Bets: Smallest bet-size (1/4-pot) is preferred on high donk bet boards when stacks are deeper (30-40bb), allowing BB to call a re-raise without going all-in. At 20bb, 2/3-pot size is preferred for higher equity hands to protect equity.
  • More frequent against UTG (67%) than BN (53%) because UTG ranges miss these boards more often.
• Value of Donk Betting: While donk betting is not widespread, on specific low-frequency flops, it can be a valuable tool, especially if opponents are not used to facing them. However, removing the donk bet option typically results in minimal EV loss for OOP (e.g., 0.4% donk on A76r, 0 EV loss when removed).

The Power of Position

Position is a massive advantage due to acting last, allowing free cards and better equity realization.

• IP vs OOP:
  • IP (In Position): Benefits from deeper stacks, greater informational advantage, and higher EQR (e.g., 110% EQR for IP vs 90% for OOP with symmetric ranges, resulting in 5% more pot captured by IP). IP's strategy often involves betting multiple streets.
  • OOP (Out of Position): Struggles to realize equity, especially with weak hands, and is forced to check/fold or use larger bet-sizes to reduce SPR and positional disadvantage. OOP often checks strong hands to protect their checking range and induce bluffs. OOP also avoids splitting their range on boards where they have a large range disadvantage by checking 100%.
• Flipping Positions: When ranges are flipped, IP captures a significantly larger portion of the pot (e.g., 12.8% EV increase on A76r for BB when in position).

Chapter 12: The Flop Continuation-bet (c-bet)

This chapter analyzes IP's continuation betting (c-bet) strategies.

Overall Flop Metrics

• Importance of C-betting: IP playing a 100% check-back strategy results in a 26bb/100 EV loss, highlighting the massive importance of c-betting for IP.
• IP vs BB Metrics: Both BN and UTG (IP) over-realize equity by 15%. UTG, with a stronger range (tighter open), captures a larger portion of the pot than BN. IP clearly has the overall range advantage.
• Hand Breakdown in C-betting:
  • Strong hands: Incentivized to bet big to grow the pot.
  • Good hands: Often checked back for pot control or to realize equity.
  • Weak hands: Benefit from checking back against strong ranges; good as semi-bluffs against weak ranges.
  • Trash hands: Generally c-bet more often than weak hands for fold equity (pure bluff).
• Bet-size and Stack Depth: Bigger bet-sizes (and overbets) are more relevant with deeper stacks, while all-in bets are more frequent with shallower stacks for guaranteed equity realization.
• C-bet Frequency Trends:
  • Flopped Straights: C-bet frequency decreases as more straights are possible (favors BB). Flops with zero straights are highest c-bet ones.
  • Open-Ended Straight Draws (OESDs): Flops with three OESDs have lowest c-bet frequency, favoring larger bet-sizes (IP needs protection).
  • Flop Rank: Lower ranks (e.g., 6xx) are c-bet at lowest frequencies (62%). Axx (96%) and 2xx (100%) are highest.
  • Flop Texture: Paired boards are often min-bet.

Flop C-betting by Structure

• Trips: Absolute best for IP (81% pot capture). Bigger bet-sizes on lower ranks, 1/3-pot most frequent.
• Paired: BB gets many strong hands, polarizing their range. Small bets are optimal to force BB to reveal information and get folds from trash/weak hands. Min-betting is preferred.
• Unpaired: Least c-bet flop families are LLL, MMM, MML, HMM.

Flop C-betting by Texture (Unpaired Flops)

Unpaired flops make up 82.82% of all flops.

• Monotone: Most c-bet but lowest EV for IP. IP's strong hands diminish, BB's increase due to flushes/draws. Small bet-sizes are preferred to isolate strong ranges and avoid over-committing with depolarized hands. Lower ranks and more connected boards are c-bet less.
• Two-tone: Higher c-bet frequency with bigger bet-sizes compared to monotone. IP has 24% strong hands vs BB's 5%.
• Rainbow: IP's range advantage is even greater than two-tone, leading to a higher preference for larger bet-sizes. Absence of flush draws means IP can triple barrel more effectively.

Developing IP C-betting Strategies

Categorizes bet-sizes into "Big" (1/2, 2/3 pot) and "Small" (min-bet, 1/3 pot). C-bet frequency: High (80%+), Mid (60-80%), Low (<60%).

• Big Bet-size Strategy: Used when BB has a more depolarized range (4-5% strong hands, many good hands). IP sizes up when OOP has many potential calling hands.
• Small Bet-size Strategy: Used when BB has more strong hands (7-9%). IP sizes down when OOP potentially has more raising hands.
• Mid-frequency C-bet Flops: IP's range becomes more depolarized, splitting range (checking back good hands, c-betting polarized range) for bigger bet-sizes.
• "Information Bets" are a mistake: Betting weak hands just for information is -EV if it leads to being check/raised or folding later.
• General IP C-bet Guidelines:
  • Understand opponent's equity buckets.
  • High c-bet frequency/big bet-size if opponent has lots of trash/few strong hands.
  • Low c-bet frequency/small bet-size if opponent's range hits well.
  • Check back strong hands if having a checking range to keep it protected.
  • Bet-folding high equity hands is a disaster.
  • Bet small at high frequency if expecting folds/raises; size up if expecting calls.
  • Play passively in multi-way pots unless nut draws.
  • If hand can get value across three streets, start with flop bet.
• Way Ahead or Way Behind: On boards like A♠T♥T♦ where OOP is either monster or trash, IP checks back good hands for pot control and strong hands as traps, while c-betting trash/weak hands balanced with remaining strong hands.
• IP C-bet Examples (A♥Q♦3♠, Q♥J♥T♥, 9♥8♥4♦, J♠6♥6♦, 8♥6♦2♠, 5♥5♦4♥): Detailed breakdowns of c-betting ranges (value/bluff), frequencies, and reactions to check/raises for various flops and stack depths, showing how solver mixes strategies to maintain balance.

C-betting in 3-bet Pots

Ranges are narrower and more polarized for aggressor, more condensed for caller.

• IP 3-betting (Hero Aggressor): Solver creates uniform post-flop situations. Hero has polarization advantage (2:1 strong hand advantage). Small bet-sizes generally preferred, often 1/4-pot. IP checks back ~38% of the time, often with strong hands as traps. Trips and rainbow paired flops are most c-bet. Low and Kxx boards are most c-bet ranks. Axx boards are c-bet 53% by IP, often checked back to get information later.
• OOP 3-betting (Hero Aggressor): Hero realizes less equity OOP. Checking range must be more balanced. Uses bigger bet-sizes to reduce SPR and positional disadvantage. SB c-bets slightly more frequently than BB due to stronger range. Higher c-bet frequency on low boards, checks more on middle/high boards that connect with IP's range. Axx boards c-bet 77% (OOP) vs 53% (IP) to avoid giving IP free turns.

C-bet Defense

• MDF is Detrimental: Relying solely on MDF is a mistake as it ignores equity and range distribution. Folding can be the highest EV play if calling/raising are -EV.
• Adjusting to Villain: If Villain c-bets with a skewed range (always strong/weak), exploit by over-folding or bluff-raising accordingly.
• BB vs IP Defense (40bb): BB's range is weaker vs UTG than BN, leading to more folds vs UTG c-bets.
• Strategy Overview (BB vs IP):
  • X/R Sizing: Larger x/r vs small bets, smaller x/r vs large bets (except all-ins) to make IP indifferent.
  • X/R Frequency: Higher overall x/r frequency when IP's bet-size is smaller (more depolarized betting range).
  • Folding: Mostly trash hands.
  • Hand Categories: Strong hands mostly x/r (or x/c to x/r later), good hands mostly call (some x/r for protection/unblocking), weak hands mostly x/c (some fold, some x/r), trash hands mostly fold (some x/c, few x/r bluffs).
• General BB C-bet Defense Guidelines:
  • Avoid splitting range early unless many strong hands.
  • X/R ranges are polarized (made hands + raise/folding bluffs).
  • Play aggressively with high equity, low showdown value hands.
  • Deeper stacks mean stronger x/r hands.
  • Consider blocker effects for value and bluffs.
  • Don't give up on A-high/K-high too quickly.
  • Fold trash regardless of MDF.
  • River scenarios influence earlier street play.
• BB C-bet Defense Examples: Detailed analysis of BB's response to various IP c-bets on specific flops, including slowplaying nuts, value-betting, and bluffing based on range composition and blocker effects.

Chapter 13: GTO Turn Strategies

Turn play is challenging but similar to flop play, focusing on pot size, effective stacks, ranges, and board. GTO solvers require these inputs but don't care about prior history or who was aggressor; humans use history to infer ranges. Focus on key strategic lines instead of all possible permutations.

Key Strategy Lines

• Flop Line: IP Checks back (x/x): Ranges are somewhat reversed from flop, with OOP's range more polarized and IP's more condensed, leading to OOP developing a turn betting strategy. If IP checks a non-GTO range (e.g., no strong hands), OOP can exploit by betting aggressively.
  • OOP First Action: OOP has various bet-sizes (1.2x, 2/3, 1/3 pot) and strategies vary across different flop textures and turn cards.
  • IP vs Bet: IP's typical response to OOP's turn lead.
  • IP vs Check: Happens frequently. IP's strategy depends on OOP's perceived range; often uses smaller bet-sizes on static boards where OOP's range is polarized (more folds).
  • OOP vs Bet: OOP folds more to larger IP bets; x/r more to smaller IP bets, especially on polarized boards.
• Flop Line: IP C-bets & OOP Calls (x/b/c): IP's c-betting range is often polarized (strong & trash), while OOP's calling range is more condensed.
  • OOP First Action: Generally checks to IP (more depolarized range) but will donk bet on turns that polarize their range and condense IP's. Smaller donk bet sizes preferred compared to flop x/x line.
  • IP vs Donk Bet: Rare line. IP usually raises at high frequency due to position and uncapped range.
  • IP vs Check: Happens frequently. IP has substantial polarization advantage, often betting with strong hands and checking back good hands to see a free river. Often uses 2/3-pot bet-size. IP does well on high cards (especially offsuit).
  • OOP vs Turn C-bet: OOP in tough spot, reasonable equity disadvantage, depolarized range. IP's range even more polarized. OOP often x/r all-in as a smaller raise would over-commit.

Turn Categories

Turn cards are grouped by impact on ranges.

• Paired Board: Turn cards that pair the board.
• Flush: Turn cards that complete a flush.
• Straight: Turn cards that complete an OESD.
• Ace: Significant effect.
• Overcard: Turn cards higher than top pair.
• Brick/Blank: Doesn't connect meaningfully. These can be further subdivided by backdoor draws.
• Examples of Turn Play (9♥8♥4♦): Demonstrates how OOP's EV varies dramatically with turn cards (e.g., low cards/straights/paired boards good for OOP, aces/overcards good for IP) and how range morphology shifts with runouts.

Chapter 14: GTO River Strategies

The river is crucial, often where most money is won or lost, despite being the easiest street conceptually.

Setting up a River Abstract Model

River characteristics simplify calculations:

• Fixed Hand Values: No more cards, so each hand is 100% or 0% equity.
• Linear Ordering: Hand strength ranking is history independent.
• No Future Betting: Calls or IP checks end the hand. Model setup involves listing hands, assigning weights, establishing blocker effects, and setting up a game tree. Identifying river situations as "polar vs. bluff-catcher" helps apply GTO strategies.

River Abstract Models

Explores typical river situations using game setup (Board: 2♠2♣2♥2♦3♣, Pot: 100 chips, Stack: 500 chips initially).

• Perfectly Polarized vs. Bluff-catcher: P1 (nuts/air) vs P2 (bluff-catchers). P1 always goes all-in with nut hands and Alpha % of bluffs. P2 calls 1-Alpha of the time. Position doesn't matter, P2 never bets.
• All of P1's Range is Stronger Than P2's Range: P1's EV is the entire pot. P1 can exploitatively bet when IP, or check/bet OOP, to get P2 to put more money in.
• P1's Range is Polarized but Has More Nuts Than Air (e.g., 90% nuts, 10% air): P1 can bet big enough with all hands to force P2 to fold, winning 100% of the pot. The minimum bet-size to force a fold is calculated. If P1 can't make this minimum bet, they must give up some bluffs, betting nuts + optimal bluff frequency.
• Role of Traps (P2 has some nut hands): If P2 has traps (slowplayed nuts), P1's optimal bet-size decreases and EV can diminish. If too many traps, P1 is forced to check 100%. Having 10-20% traps is usually enough to prevent thin value-bets and high bluffing frequencies from opponents.

Linear Distributions

Applies to situations where players' ranges contain a mixture of nuts, air, bluff-catchers, and traps.

• Strategy: Betting player bets value hands (top), bluffs (bottom), and checks middle (checking range has some value hands if OOP, as traps). Defending player calls hands above a threshold and folds the rest.
• Example (Symmetric Ranges): Formulas provided for threshold calling and betting hands when checking hands have 0% equity.
• Complex Situations: Calculating thresholds by hand is impossible; best done with a GTO solver.
• Examples (P1 IP, P2 OOP / P1 OOP, P2 IP): Demonstrates varying optimal bet-sizes (all-in, 125% pot, 25% pot) and range compositions for betting, checking, and bluffing, based on SPR and positional advantage.

Blocker Effects

Blockers influence optimal bluffing and value betting.

• Example 1 (Nut Flush vs. Nut Blocker): If Hero has nut flush + nut blocker, best bluffs are nut flush blockers. Betting non-nut blockers loses EV.
• Example 2 (No Nut Blocker): If Hero lacks the nut blocker, cannot remove traps from Villain's range. Overbetting loses EV. Hero bets pot-size and prefers to bluff with low spade combos rather than non-nut flush blockers to avoid exploitation. If Hero has some nut blocker combos, they split value range between all-in and full pot, using low spade combos as bluffs.

Practical Applications

Summary of river characteristics and how to apply them.

• OOP River Betting:
  • Favored when OOP has traps or nutted hands, and IP tends to bet at high frequency.
  • After IP checks back turn, OOP's range is more polarized, leading to OOP betting 20-30% of the time on average.
  • After x/c, x/c, OOP donk bets when river (or turn+river) significantly shifts range polarization in their favor (e.g., four-flush/straight boards, runner-runner pair).
• IP River Betting:
  • Simpler than OOP. After OOP checks, their range is usually weak/bluff-catchers.
  • If IP's range is polarized, bet big with value and enough bluffs to make OOP indifferent.
  • If IP's range is condensed, check back middle strength hands, bet top, bluff weakest.
  • If range not very nutted, use small bet-size, especially if OOP can trap.
  • Always add small frequency of nutted hands to small bet-size to protect against x/r.
• IP Play vs. River Bets:
  • Exploit over-bluffing Villains by folding less, over-folding to under-bluffing Villains.
  • IP must check back turn with hands capable of raising river bets, otherwise OOP can exploit.
  • Defending close to 1-Alpha is a good approximation.
• Overall Strategy: Understanding river play helps improve pre-flop, flop, and turn play by focusing on arriving at beneficial river situations with the correct range composition.

The book concludes by emphasizing that while poker theory and software are vital tools, they are not oracles and should not be followed blindly. Understanding fundamentals and applying knowledge to exploit opponents' weaknesses is key to long-term success.