Poker Bankruptcy Probability Calculation and Risk Management Model

Tool Purpose

The bankruptcy probability model is the core tool for poker bankroll management, used to evaluate the likelihood of a player losing all their funds given a certain win rate, standard deviation, and bankroll size. Understanding this model helps players set safe and effective bankroll levels to avoid going broke due to short-term variance.

Formula Principle

The most commonly used Risk of Ruin (RoR) formula is based on normal approximation or random walk theory. The basic formula is:

RoR = e^(-2 * B * WR / V)

Where:

• B: Bankroll size (in big blinds or buy-ins)
• WR: Win rate (in big blinds per hand or bb/100 hands)
• V: Variance (in big blinds^2 per hand, usually approximated by the square of standard deviation SD)

Typically, the standard deviation SD is around 80-100 bb/100 hands (for cash games) or much higher for tournaments. The approximation for variance is V ≈ SD².

Note: This formula assumes infinite time, fixed win rate and variance, and ignores opponent adjustments. In practice, a safety margin should be accounted for.

Usage Steps

1. Estimate your win rate (WR): Use data from at least the last 100,000 hands to determine your average profit per 100 hands (bb/100).
2. Calculate standard deviation (SD): Obtain your standard deviation from hand history data (usually provided by tracking software). For cash games, SD is typically 80-100 bb/100.
3. Set a target risk of ruin: A common safe value is RoR ≤ 5%, while conservative players may aim for ≤ 1%.
4. Solve for bankroll B using the formula: Rearranged as B = -ln(RoR) * V / (2 * WR). If RoR = 5%, then -ln(0.05) ≈ 2.996.
5. Adjust based on results: If the required bankroll is too high, you can increase WR by improving your skills or reduce variance (e.g., by playing tighter or fewer tables).

Practical Example

Example: A cash game player has WR = 5 bb/100 and SD = 90 bb/100, and wants a risk of ruin below 1%.

Step 1: Calculate V = SD² = 90² = 8100 (bb²/100 hands). Note that the units for V must match WR: WR is 5 bb/100, V is 8100 (bb/100)².

Step 2: Target RoR = 1% → -ln(0.01) = 4.605.

Step 3: B = -ln(RoR) * V / (2 * WR) = 4.605 * 8100 / (2 * 5) = 4.605 * 8100 / 10 = 4.605 * 810 = 3730.05 bb. This is approximately 37.3 buy-ins (assuming a 100bb buy-in).

Conclusion: At least 37 buy-ins are needed to keep the risk of ruin below 1%. If you only have 20 buy-ins, the risk of ruin would be approximately e^(-2205/8100) = e^(-0.0247) ≈ 0.9756, i.e., a staggering 97.6%! This indicates an insufficient bankroll.

Frequently Asked Questions

Q: How do I accurately obtain the variance in the formula?
A: Standard deviation can be obtained directly from poker tracking software (e.g., Hold'em Manager). For cash games, the standard deviation is typically 80-100 bb/100; multi-tabling or high-variance styles may result in higher values. If you have no data, conservatively assume SD = 100.

Q: What if my win rate is negative?
A: With a negative win rate, the risk of ruin is inevitably 100%, and the formula breaks down. You should first improve your skills to achieve a positive win rate.

Q: Does the risk of ruin model apply to tournaments?
A: It is partially applicable, but tournament variance is much higher (standard deviation is often 200-400% of buy-in per event), and more complex models like ICM should be used. The simple bankroll formula can serve as a rough reference, but a more conservative approach is needed.

Further Learning

• Kelly Criterion: Used to determine optimal bet sizing to avoid excessive risk.
• Confidence Intervals and Variance Simulation: Use Monte Carlo simulations for a more accurate risk assessment.
• Multi-Table Bankroll Management: Does playing more tables reduce hourly variance? In reality, standard deviation does not decrease linearly with the number of tables; adjustments are needed.

Author's recommendation: In addition to mathematical calculations, maintain a psychological buffer (e.g., an extra 10-20 buy-ins) to handle downswings, taxes, etc.