A research idea: Skill-Adjusted Tournament Equity

A research idea: Skill-Adjusted Tournament Equity

I’ve been thinking a lot recently about whether we’re actually valuing tournament chips correctly. I know this isn’t a completely unexplored area. We already have concepts like ICM, Future Game Simulation (FGS), node-locking, population exploits and player modelling. What I am interested in is whether this can be formalised mathematically in a way that hasn’t really been done before.

The basic question I’m trying to answer is:

Are equal chip stacks really worth the same amount if the players holding them have vastly different abilities?
Example
Imagine a 9-man Sit & Go. Everyone starts with 1,000 chips. Standard ICM says every player owns exactly the same tournament equity because everyone owns the same percentage of the chips. But intuitively that doesn’t seem true. If one player is a proven long-term crusher and seven others are recreational players, I’d argue that the crusher’s stack has a higher future value than the recreational players’ stacks, even though they contain the same number of chips. Not because the chips themselves are different, but because the player holding them is expected to convert those chips into prize money more efficiently. That’s the part I’m trying to quantify.
A possible framework
At the moment I’m thinking about something like this:
Tournament Utility = Function(Chip Stack, Player Skill, Remaining Field, Payout Structure)
Where:
Tournament Utility = the player’s “true” tournament equity or expected value in the tournament.
Chip Stack = the player’s current stack.
Player Skill = a measure of that player’s overall ability.
Remaining Field = the skill level of every other player still in the tournament.
Payout Structure = how the prize pool is distributed.
The idea is that tournament utility shouldn’t depend only on chip counts, but also on who is holding those chips and who remains in the field.

One possible adjustment could look something like:
Adjusted Tournament Equity = ICM Value × (1 + Calibration Factor × (Player Skill − Average Field Skill))
Where:
Adjusted Tournament Equity = the model I’m trying to estimate.
ICM Value = the player’s tournament equity under standard ICM.
Calibration Factor = a value determined through simulation that measures how much skill influences tournament equity.
Player Skill = the player’s skill rating.
Average Field Skill = the average skill level of the remaining players.
I don’t think this is the final equation. It’s simply a starting point for trying to formalise the idea mathematically.

A thought on ROI
One way I’ve been thinking about this is through ROI.
Suppose we have a 9-man $10 Sit & Go with no rake, so the prize pool is $90. Everyone starts with 1,000 chips, and under standard ICM each player’s starting stack is worth $10 in tournament equity.
Now imagine one player has a proven long-term ROI of 20% in this exact field.
Their buy-in is still $10, but the expected value of entering the tournament is:
Expected Return = Buy-in × (1 + ROI)
So in this example:

Expected Return = $10 × (1 + 0.20) = $12

That extra $2 isn’t created from nowhere. In a zero-sum tournament (ignoring rake), it ultimately comes from the players with negative ROI. In other words, stronger players consistently convert the same starting stack into a larger share of the prize pool by exploiting weaker players.This is one of the main motivations behind the idea. Standard ICM assumes equal chip stacks imply equal tournament equity, but in practice the future value of a stack may also depend on who owns it and who remains in the field.I’m not suggesting the buy-in itself changes. Rather, the expected value of holding that starting stack changes because some players consistently convert chips into prize money more effectively than others.

Why I’m interested

One thing that has always interested me is that tournament poker isn’t just about maximising chip EV. It’s about maximising long-term tournament utility. Suppose two very strong regs are playing against one another. Even if one of them has a small positive chip EV decision, is taking that spot always optimal if it risks doubling one of the few players capable of exploiting the rest of the field? Likewise, if I’m playing against a recreational player who has significant postflop leaks, shouldn’t that increase the future value of preserving a stack that can continue playing pots against them? My intuition is that the answer is yes. The difficult part is turning that intuition into mathematics.

Measuring “skill”

I’m still unsure exactly how skill should be represented. My initial thought was to use ROI because it’s an observable measure of long-term performance. However, ROI itself is influenced by factors like field strength, rake and variance, so it may not be sufficient on its own. It’s possible that “skill” should instead be a weighted combination of things like:
Long-run ROI
Postflop performance
ICM accuracy
Exploitative ability
Overall decision quality
Exactly how those should be weighted is something I haven’t figured out yet.

The biggest challenge

The plan I’m considering is building a Monte Carlo tournament simulator in Python. The idea would be to simulate hundreds of thousands (or even millions) of tournaments using players with different skill ratings and then calibrate the model from the results rather than inventing constants arbitrarily. Unfortunately, that’s also my biggest roadblock. I’m comfortable thinking about the poker theory and the mathematics, but I’m not a programmer. So part of this project is probably going to involve learning enough Python to build the simulations or working with someone who can help implement them.

Final thoughts

This is very much a work in progress, and I’m sure there are holes in it that I haven’t spotted yet. I’m posting it because I’d genuinely like feedback from people who have thought about these ideas before. In particular, I’m interested in whether anyone knows of research that attempts to mathematically adjust tournament equity based on player skill rather than just stack sizes and payouts.

If nothing else, I think it’s an interesting problem:

How much is a tournament chip actually worth once you account for who owns it?