Research Publications
Independent mathematical research, closed-form derivations, and empirical probability studies for quantitative poker game theory.
Pot Odds, Equity Fundamentals & The Rule of 2 and 4
Mathematical foundations of poker profitability. Explores the relationship between pot odds ratios, required equity calculations (Risk / (Risk + Reward)), and probabilistic outs estimation using hypergeometric distributions.
Pot Odds = Call / (Pot + Call) - • Implied Odds: When to Call with Drawing Hands in Poker 18 min →
- • Reverse Implied Odds: Hidden Costs of Dominated Hands in Poker 18 min →
- • Outs Counting and Probability: The Exact Combinatorics of Draws in Poker 19 min →
- • Break-Even Equity Thresholds: Mathematical Foundations of Poker Decisions 18 min →
Expected Value (EV) Modeling & Decision Trees in Poker
Formal application of Expected Value: EV = (P(Win) × Win Amount) - (P(Loss) × Loss Amount). Demonstrates multi-street EV calculations, fold equity dynamics, and the exact mathematical thresholds for profitable bluffs and calls.
EV = (Win% × Pot Won) - (Lose% × Amount Risked) - • PokerStars vs GGPoker Rake: NL10 and NL50 on Identical Pots 6 min →
- • GGPoker Rake at NL10 and NL50: Caps and Effective Cashback 7 min →
- • Fold Equity and Semi-Bluffing Mathematics: The Dual-Engine EV Equation 18 min →
- • ICM Tournament Equity: Why Chips ≠ Dollars 22 min →
- • Multi-Street Expected Value Calculations: Decision Trees, River Planning, and Geometric Sizing 20 min →
- • Rake Impact on Expected Value: The Hidden Win Rate Leak 19 min →
- • Minimum Defense Frequency (MDF) & Optimal Bluffing Ratios (Alpha): The Mathematical Limits of Game Theoretic Defense 20 min →
Kelly Criterion, Variance Analysis & Poker Bankroll Risk Models
Advanced application of the Kelly Criterion to poker bankroll management. Features risk-of-ruin equations, standard deviation modeling for win rates (bb/100), and probabilistic analysis of downswing duration.
f* = μ / σ² - • Risk of Ruin in Poker: Variance, Downswings, and Bankroll Survival Math 20 min →
- • Kelly Criterion in Poker: Cash Games vs Tournaments (MTT) Mathematical Formulation 22 min →
- • Shot Taking and Moving Up in Stakes: Quantitative Bankroll Growth and De-Staking Rules 18 min →
- • Downswing Probability Analysis: Extreme Value Theory and Drawdown Duration in Poker 22 min →
Combinatorics, Card Removal & Range Morphology
Quantitative breakdown of the 1,326 hole card combinations and 19,600 flop textures. Explores blocker effects on opponent frequencies and range-vs-range equity distribution mapping.
C(n,k) = n! / (k!(n-k)!) Choose a platform
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All Mathematical Publications
Browse and search our full curriculum of open-source derivations, bankroll proofs, and range models.
PokerStars vs GGPoker Rake: NL10 and NL50 on Identical Pots
Compare documented regular-table USD rake on matched NL10 and NL50 pots, keeping player brackets, rewards and game formats separate.
GGPoker Rake at NL10 and NL50: Caps and Effective Cashback
Read GGPoker's USD rake caps by format, calculate a pot's fee, and measure cashback from redeemed rewards instead of an advertised maximum.
Pot Odds & Equity Fundamentals: The Mathematical Core of Poker
Master the foundational mathematics of poker. Learn how to calculate pot odds, determine required equity, and use the Rule of 2 and 4 to make profitable decisions at the table.
Expected Value in Poker Decisions
How to calculate Expected Value (EV) for calls, bets and bluffs in poker, with worked examples, break-even bluff frequencies and the role of variance.
Minimum Defense Frequency (MDF) & Optimal Bluffing Ratios (Alpha): The Mathematical Limits of Game Theoretic Defense
Axiomatic derivation of Minimum Defense Frequency (MDF), bluffing profitability threshold (Alpha), optimal river bluff-to-value geometry, and a mathematical proof of why blindly following MDF on dynamic textures causes catastrophic EV loss.
Bankroll Management & Kelly Criterion for Poker
Learn the rigorous mathematical foundation of poker bankroll management, including the Kelly Criterion adaptation for poker, Risk of Ruin formulas, and practical buy-in rules.
Combinatorics & Range Analysis in Poker
Advanced poker combinatorics framework for precise range construction, blocker effect quantification, and Nash equilibrium hand analysis.
ICM Tournament Equity: Why Chips ≠ Dollars
Formal analysis of the Independent Chip Model (ICM) demonstrating why tournament chips have diminishing marginal value, and how ICM equity calculations fundamentally alter optimal decision-making compared to cash game strategy.
Range vs Range Equity: Beyond Hand vs Hand
Advanced framework for calculating and applying range-versus-range equity in poker decisions, covering equity advantage, nut advantage, equity distribution curves, and solver-derived strategic implications.
1,326 Hand Combinations & Blocker Effects in Poker
A rigorous mathematical exploration of the 1,326 starting hand combinations in Texas Hold’em, the structural asymmetry between pocket pairs, suited, and offsuit holdings, and the powerful card removal effects (blockers) that shape modern GTO strategy.
Flop Texture Probability: Wet, Dry, and Monotone Boards
Quantitative analysis of all 19,600 possible flops categorized by texture type. Covers exact probabilities for dry, wet, monotone, paired, and high-card boards with solver-derived strategic implications for each category.
Board Runout Distributions: Turn and River Card Probabilities
Exact street-by-street probability analysis for turn and river card distributions. Covers draw completion rates, board pairing frequencies, brick runout probabilities, and their impact on optimal bet sizing and pot geometry.
Implied Odds: When to Call with Drawing Hands in Poker
Rigorous mathematical framework for evaluating implied odds, stack-to-pot ratios, future bet extraction formulas, and deception coefficients in No-Limit Texas Hold'em.
Reverse Implied Odds: Hidden Costs of Dominated Hands in Poker
Deep mathematical analysis of reverse implied odds, domination penalties, stack-to-pot ratio risks, and solver defense rules in No-Limit Texas Hold'em.
Outs Counting and Probability: The Exact Combinatorics of Draws in Poker
Rigorous combinatorics of poker draws, derivation of the Rule of 2 and 4, clean vs dirty outs discounting, backdoor equity, and blocker modeling.
Break-Even Equity Thresholds: Mathematical Foundations of Poker Decisions
Axiomatic derivation of break-even equity, pot odds geometry, bluffing alpha, Minimum Defense Frequency (MDF), semi-bluffs, and ICM risk premiums.
Fold Equity and Semi-Bluffing Mathematics: The Dual-Engine EV Equation
Rigorous mathematical derivation of semi-bluffing expected value, fold equity isolation, blocker removal effects, multi-street barrel planning, and GTO ratios.
Multi-Street Expected Value Calculations: Decision Trees, River Planning, and Geometric Sizing
Advanced mathematical derivation of multi-street poker decision trees, backward induction from river to flop, geometric bet sizing formulas, and equity realization dynamics.
Rake Impact on Expected Value: The Hidden Win Rate Leak
Rigorous microeconomic analysis of poker rake drag, mathematical proof of pot odds distortion, preflop range adaptations, and network benchmark audits.
Risk of Ruin in Poker: Variance, Downswings, and Bankroll Survival Math
Stochastic derivation of poker risk of ruin equations, Brownian motion modeling, required buy-in matrices, and dynamic de-staking protocols.
Kelly Criterion in Poker: Cash Games vs Tournaments (MTT) Mathematical Formulation
Rigorous information-theoretic derivation of the Kelly Criterion, discrete vs continuous formulations, heavy-tailed MTT distributions, and fractional bankroll sizing.
Shot Taking and Moving Up in Stakes: Quantitative Bankroll Growth and De-Staking Rules
Statistical confidence intervals for stake progression, mathematical gambler ruin success modeling, win rate degradation factors, and dynamic de-staking rules.
Downswing Probability Analysis: Extreme Value Theory and Drawdown Duration in Poker
Rigorous stochastic modeling of poker downswings, Brownian maximum drawdown analytics, Gumbel extreme value theory, and the CUSUM structural break test.
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