Poker Ruin Probability Calculation and Risk Management Model

Context: STRATEGY article: poker-ruin-probability-risk-management

Tool Purpose

The ruin probability calculation and risk management model is a core tool for poker players to manage their bankroll and control risk. It helps players quantify the probability of going bankrupt given a certain bankroll size, win rate, and variance, thereby formulating a sound bankroll strategy. Common models include the Kelly criterion and the Risk of Ruin formula.

Formula Principles

Risk of Ruin Formula

The classic approximate formula for the probability of ruin (RoR) is:

RoR = e^(-2 * B * µ / σ²)

Where:

• B is the current bankroll (in big blinds or buy-ins)
• µ is the expected profit per hand (in the same unit)
• σ is the standard deviation of profit per hand (in the same unit)
• e is the natural constant (approximately 2.718)

This formula assumes that profit follows a normal distribution and is only applicable to long-term, fixed-strategy scenarios.

Kelly Criterion

The Kelly criterion is used to determine the optimal bet size to maximize long-term growth:

Kelly% = ([Win rate](/term/win-rate) - Loss rate) / Odds

In poker, it can be simplified as:

Kelly% = (Win probability - (1 - Win probability)) / Odds

But more commonly it is based on expectation and variance:

Kelly% = µ / σ²

Where µ is the expected profit per hand and σ² is the variance. Due to the high variance in poker, mainstream advice recommends using half Kelly or even more conservative percentages, such as 1-2% of total bankroll.

Usage Steps

1. Determine parameter units: Unify bankroll, profit, and standard deviation into the same unit (e.g., big blinds bb). For example, profit per 100 hands (bb/100) and standard deviation (bb/100).
2. Collect data: Use hand-tracking software (such as Hold'em Manager or PokerTracker) to obtain your historical data and calculate the average profit per 100 hands and standard deviation.
3. Calculate ruin probability: Plug into the Risk of Ruin formula. For example, if B = 1000bb, µ = 10bb/100 hands, σ = 100bb/100 hands, then calculate RoR.
4. Set acceptable risk: Professional players generally keep RoR below 5%, while recreational players may accept 10-20%.
5. Adjust bankroll: If the current RoR is too high, increase the bankroll or move down in stakes.

Practical Example

Example: Player A is profitable at NL10 (blinds $0.05/$0.10). Data shows a profit of 5bb per 100 hands (µ = 5bb/100), standard deviation 70bb/100 (σ = 70bb/100), current bankroll 400bb ($40).

Calculate RoR:

RoR = e^(-2 * 400 * 5 / 70²) = e^(-4000 / 4900) = e^(-0.8163) ≈ 0.442 = [44](/term/44).2%

The risk of ruin is as high as 44%, which is too risky. Doubling the bankroll to 800bb ($80) reduces RoR to:

RoR = e^(-2 * 800 * 5 / 4900) = e^(-1.6327) ≈ 0.195 = 19.5%

Still relatively high; further increase bankroll or use a half Kelly strategy.

Frequently Asked Questions

Q: Should I use full Kelly or half Kelly?
A: Due to the high variance in poker, full Kelly often leads to excessive risk. Half Kelly (i.e., half the calculated value) is safer, with a minimal loss in long-term growth but a significant reduction in risk.

Q: How much bankroll do I need to be safe?
A: Depends on your profit variance. A general recommendation for cash games is at least 20-40 buy-ins (assuming a standard deviation around 70bb/100 and a win rate of 5bb/100, 40 buy-ins corresponds to approximately 5% ruin probability). Tournaments require even more.

Q: How long should the data sample be to be reliable?
A: At least 100,000 hands to reasonably estimate win rate and standard deviation.

Q: Does the formula apply to tournaments?
A: It requires adjustments because tournament payout structures are complex. The ICM model combined with simulations can be used.

Further Learning

• Deep dive into the Kelly criterion: Kelly, J.L. (1956). "A New Interpretation of Information Rate"
• Book on Bankroll Management: "The Mathematics of Poker" by Bill Chen and Jerrod Ankenman
• Use specialized software like "Winamax Bankroll Manager" or Excel templates to calculate RoR
• Learn more advanced models: Gambler's Ruin, multi-table bankroll management, etc.

Remember, risk management is the foundation of long-term profitability. Never underestimate the risk of ruin caused by variance.