Poker Bankruptcy Probability Calculation and Risk Management Model

Purpose of the Tool

The ruin probability calculation model is used to assess a poker player's risk of losing their entire bankroll given a specific bankroll size, win rate, and variance. Proper application of this model helps players determine suitable buy-in levels, stop-loss limits, and bankroll growth strategies, thereby avoiding bankruptcy due to short-term fluctuations.

Principle of the Calculation Formula

The most common ruin probability formula is based on random walk theory. Assuming the expected win per hand is μ, the [standard deviation] is σ, and the bankroll is B, then the ruin probability over an infinite time horizon (ignoring rake) is approximately:

P(ruin) ≈ exp(-2μB / σ²) (when μ>0)

• μ: average win per hand (in units of buy-in)
• σ: standard deviation of win per hand (in units of buy-in)
• B: bankroll (in units of buy-in)

This formula assumes independent and identically distributed (i.i.d.) wins that approximately follow a normal distribution. In actual poker, win distributions have fat tails, but the formula still provides a good estimate.

For a finite time horizon, more precise numerical methods can be used, such as simulation or exact formulas (e.g., the risk rate function).

Usage Steps

1. Estimate μ and σ:

   • From historical data, calculate the win rate per 100 hands ([bb/100]) and its standard deviation ([bb/100]).
   • Convert big blinds to buy-ins. For example, assuming a buy-in of 100bb, then μ (buy-in/hand) = (bb/100)/100 /100? Note: bb/100 represents the number of big blinds won per 100 hands. Converting to bb per hand: bb/hand = bb/100 /100 = bb/10000. Then divide by the buy-in (100bb) to get μ in units of buy-in per hand. Similarly for σ.

2. Set an acceptable ruin probability: Typically 1% or 5%.

3. Calculate the minimum required bankroll: B_min = - (σ² / (2μ)) * ln(P_accept) (P_accept is the acceptable ruin probability)

4. Adjust dynamically: Recalculate when bankroll or win rate changes.

Practical Example

Assume an online cash game player has a profit per hand μ = 0.01 buy-in (i.e., 1% of a buy-in per hand), and a standard deviation σ = 0.5 buy-in. They want the ruin probability to be below 1%. Calculate the minimum required bankroll:

B_min = - (0.5² / (2*0.01)) * ln(0.01) = - (0.25 / 0.02) * (-4.605) = 12.5 * 4.605 ≈ 57.56 buy-ins

Thus, they need at least about 58 buy-ins. If their bankroll is only 30 buy-ins, the ruin probability will be much higher than 1%.

If their bankroll is 30 buy-ins, the actual ruin probability is: P = exp(-20.0130 / 0.25) = exp(-0.6/0.25) = exp(-2.4) ≈ 0.0907 = 9.07%, exceeding the acceptable level.

Common Questions

Q: The formula assumes μ>0. What if I am a losing player? A: For a losing player, the ruin probability is 1 (infinite time horizon), so the formula does not apply. Losing players should first improve their skills.

Q: How to accurately estimate the standard deviation σ? A: A large sample of hands (at least tens of thousands) is needed. Use the standard deviation function on the win per hand data. Online platforms provide data that can be exported for analysis.

Q: How to calculate when playing multiple tables or sessions simultaneously? A: The win per hand should be adjusted for the number of tables. Typically, the overall standard deviation increases, but the formula still applies using the combined μ and σ.

Further Learning

• Study more precise ruin models such as the "risk rate function" ([Risk of Ruin] formula with time horizon).
• Research bankroll management strategies like the "Kelly criterion" to maximize long-term growth.
• Use poker tracking software (e.g., Hold'em Manager) to export data and calculate manually with Excel or Python.