ICM Power Law Theorem: Shortcut to Approximate Placement Probabilities

As you guys may know, I like to mess around with poker math, and every now and again a new piece of poker wisdom pops out. Here's my latest discovery!

I've found a neat shortcut for approximating the ICM placement probabilities in MTTs:

P(bottom x %) = (x) ^ StackMultiple

Where:

• x = % of field to be eliminated
  
• StackMultiple = Hero Stack ÷ Average Stack
  

For example: 1,000 players remain, top 150 get paid, and hero has 1.5× the average stack. The probability of finishing in the bottom 85% (busting), is simply:

85% ^ 1.5 ≈ 78.3%

So, the probability of finishing in the top 15% of the field (finishing in the money) would be:

100% − 78.3% = 21.7%

Generalizing this example to a range of StackMultiples gives us this table:

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This formula is accurate for flat fields, where every other player but hero has exactly the same stack. The ICM math simplifies neatly in this special case, which is how I derived the formula.

But how well does it perform in real tournaments where stacks aren't perfectly even? Actually, quite decently! For realistic stack distributions, this formula predicts how often different stack sizes will cash within about 1% accuracy of the full ICM calculation. It's more accurate in medium-to-large fields and less accurate in smaller fields. I wouldn't recommend using it at a final table, but it's great for quickly estimating your chances of cashing deep in an MTT.