Kohei's Poker GTO and Game Theory Notes

Range advantage occurs when a player's range of possible hands is stronger than their opponent's in a specific spot, often due to how the board matches their range or how the range is distributed. Range advantage is generally decomposed into two distinct elements:

1. Equity Advantage: When a player's range has a higher overall probability of winning the hand on average.
2. Nut Advantage: When a player's range contains more of the strongest possible hands (the 'nuts') compared to the opponent's range.

In game theory, poker is classified as a non-cooperative, extensive-form game with imperfect information.

• Non-cooperative: Players act in their own self-interest; gains for one player often result in losses for others, making formal agreements between players impossible.
• Extensive-form: The game is played in turns. Players can observe previous actions, allowing the game to be mapped as a game tree to compare payoffs across different paths.
• Imperfect Information: Not all information is disclosed to all players (e.g., hidden hole cards). Players may not know exactly which decision node they are at when future branches exist.

Strategies for responding to 3-bets or further raises:

• Raising Back: The range for raising against a 3-bet is very narrow and generally independent of position. The raising range is typically $\le 1/2$ of the frequency of the previous raise.
• Call vs. Raise/Fold: Players who are not the last aggressor (e.g., UTG who 2-bet and then faced a 3-bet from BTN) only have the option to call once it is guaranteed they will be heads-up. Until that point, the equilibrium strategy is strictly raise or fold.
• Calling Range Composition: When calling is an option, the range is slightly wider than the last aggressor's. It favors draw-heavy hands that can become polarized post-flop (e.g., A5s, 87s) and excludes condensed range hands that are likely dominated (e.g., AJo).

When moving from 6-max to 9-max (or other table sizes), ranges shift as follows:

• General Trend: Participation rates decrease across almost all positions; it is not a simple slide based on the number of players behind.
• Early Position Adjustments:
  • Offsuit hands, suited connectors, and low pocket pairs (previously 2-bet in 6-max) are the first to be removed or have their frequency significantly reduced.
  • Low-kicker Aces and Kings are removed next. The weakest are folded 100% (e.g., A2o in HJ), while mid-low strength hands see a decrease in 2-bet frequency (e.g., K8s-K5s in HJ).
• Late Position Adjustments: BTN, SB, and BB ranges remain mostly unchanged. However, because early position ranges are stronger, the overall frequency of calling and 3-betting decreases.

Using highly granular bet sizes allows an opponent to precisely identify which hand you hold (since a specific size might map to only one value hand and one bluff hand). To prevent this, players group bet sizes into a few average variations (typically 3-4 types).

Commonly used bet size groups include:

• Bets: 1/3 pot, 3/4 pot, or 3/2 pot.
• Raises: 1/2 pot.

When betting, the hands you expect to be called by are 'Value Targets,' and the hands you expect to fold are 'Bluff Targets.'

There is a direct correlation between the strength of the hand (or the strength being represented) and the bet size:

• Larger Bet Sizes: Force the opponent to call with a narrower, higher-equity range. Therefore, larger bets are typically used by ranges containing a high proportion of very strong hands (nuts) to target high-equity value/bluff targets.
• Bluff Selection: Hands with low equity when checking are the most effective bluff candidates, as the difference between their check-EV and bluff-EV is greatest.

Betting serves a purpose beyond increasing the pot: it denies the opponent's equity. By betting, you force hands with low-to-medium equity (like bottom pairs or weak draws) to fold, preventing them from realizing their equity. This gain is called 'Fold Equity.'

• Bluffing: Aims to lower the opponent's equity realization (EQR). Even if a single bluff is called, continuing to bet across streets can be profitable if it forces folds at a certain frequency.
• Marginal Hands: Betting with marginal hands can be valuable because it prevents the opponent from getting a 'free card' via a check, thereby increasing the EV of the marginal hand through equity denial.

EQR is heavily influenced by the player's ability to take aggressive actions:

• Position: Players In Position (IP) generally have higher EQR because they have more opportunities to bet and raise, which amplifies EQ through value bets or forces folds via bluffs. Players Out of Position (OoP) typically have lower EQR.
• Stack Depth:
  • Short Stacks: EQR variance is smaller because the limited number of bets before an all-in occurs caps the potential for EQR fluctuation.
  • Deep Stacks: EQR variance is larger as there are more betting rounds and opportunities to manipulate the pot size.

An Equilibrium Strategy (the goal of GTO) is a strategy that makes the opponent indifferent to their own choices, meaning the opponent's payoff remains the same regardless of the strategy they employ.

Key characteristics of Equilibrium Strategies:

• No Counter-Strategy: There is no single strategy that can consistently beat an equilibrium strategy.
• Bluff Protection: By playing an equilibrium strategy, you can defend against bluffs regardless of the opponent's bluffing frequency.
• Exploitation Baseline: Understanding the equilibrium allows a player to identify how an opponent's strategy deviates from it, enabling the player to apply a counter-strategy (exploit) in the opposite direction.

A Strategy is a predefined algorithm or set of guidelines for choosing actions within a game tree.

• Pure Strategy: An action taken with 100% frequency. Example: In a 'push or fold' scenario with very short stacks (e.g., 6BB), a specific hand is either always shoved or always folded.
• Mixed Strategy: Using multiple different actions based on a specific frequency to remain balanced and avoid being exploited. Example: Opening a hand like KT from UTG with a specific frequency (e.g., 14.25% open, 85.75% fold) rather than always folding, to prevent the opponent from easily countering the strategy.