The Power Law of PKOs: Predicting Bounty Pool Decay
I've been simulating knockout tournaments to track how the bounty prize pool (the total sum of all unclaimed bounties) changes over time. I started with a 50% progressive knockout (PKO), where eliminating a player awards you half their bounty, while the other half is added to your own head.
After running a ton of simulations, I uncovered a simple yet powerful relationship:
BPP% ≈ √Field%
Generalizing this to any KO format:
BPP% ≈ Field% ^ Instant_Bounty%
Where:
- BPP% = The percentage of the bounty prize pool remaining
- Field% = The percentage of runners remaining
- Instant_Bounty% = The percentage of a bounty you claim upon eliminating a player (typically 50% for a PKO, and 100% for an SKO). The other (1 - Instant_Bounty%) goes onto your own head.
In a 50% PKO, each elimination removes half of the busted player's bounty from the prize pool. This means the BPP shrinks half as fast as the field, which naturally leads to the square root function, BPP% = Field% ^ 0.5
If I had realized that from the start, I probably wouldn’t have spent as much time on this. But the insight only hit me at the end while writing everything up!
Why does this matter?
Your assumptions about the bounty prize pool affect how you value bounties relative to chips (bounty power). It also impacts future bounty EV, and any solver-based strategy depends heavily on your assumptions about the remaining bounty prize pool. In short, BPP assumptions fundamentally shape KO strategy.
Mathematical derivation
Spoiler
Variable definitions:
- F = Field% (Starts at 100%)
- B = BPP%
- P = Instant bounty%
Since each remaining player has an average bounty of B/F, when a player busts, the BPP decreases by P times that amount:
ΔB = -P * B/F
Eliminations are discrete events, but we can treat it continuously. We want the change in B as F changes:
∂B/∂F ≈ (P * B) / F
Solving this differential equation yields:
B = F ^ P
Simulations:
Spoiler
I wrote (or rather, told an LLM to write) a Python script that simulates PKOs: two players go all-in at random, the winner is chosen by a coin flip, and stacks/bounties adjust following PKO rules. Repeat until one player remains. Track the BPP along the way, then average 100k PKOs for statistical significance.
Below is a graph comparing simulated BPP% to the theoretical BPP%: (so close you can't tell them apart)
If you zoom in on the top finishing positions, you'll see slight deviations. The log-log graph clarifies that it's a very close fit, though the top placements don't perfectly align.
I ran this for 50% PKOs ranging from 30 to 10k runners, and consistently got an R^2 of at least 99.99%.
Comparing to upper and lower bounds of PKOs:
Bounty theorist Dr. K. wrote an article for GTO Wizard outlining rough upper and lower BPP bounds. The lower bound assumes the biggest bounty always gets busted next, while the upper bound assumes the smallest bounty always goes next. Neither scenario is realistic, but the true BPP should lie between these extremes. My simulations confirm this:
Limitations and Final Thoughts:
It remains to be seen how well my random-shove model predicts BPP evolution in real games. On one hand, it does capture the idea that bigger stacks generally outlast shorter stacks and simulates overall chip movement—and there’s something very fundamental about the intuition that in a 50% PKO, the bounty pool should decrease “half as fast” as the field. But real tournaments feature strategic elements and ICM considerations that might create deviations. The best way to test this is by comparing the predictions to actual PKO data.
Additionally, I’m not accounting for rebuys. Many PKOs allow late registration, but my simulation treats the event as a freezeout. Presumably, this wouldn’t matter if you model everything after late reg closes, but it’s still a caveat.
Anyway, I’m happy to share this simple, elegant relationship, since it doesn’t appear to have been published before.
7 Replies
I wonder how this changes the common wisdom of always playing PKOs from the start. If the bounty pool only decays by ~20% with max late reg then perhaps the inherent ICM advantage of late reg makes up for it.
Very cool!
This looks very accurate for larger fields, but I think in practice it's not going to reflect actual prize pool distributions well deep in smaller fields. Which makes sense. The larger the field size, the more the distribution will align with the averages.
This should be quite useful for things like setting up sims when there is significant dislocation between the bounties on the table and the average bounty of the whole tournament.
I've been simulating knockout tournaments to track how the bounty prize pool (the total sum of all unclaimed bounties) changes over time. I started with a 50% progressive knockout (PKO), where eliminating a player awards you half their bounty, while the other half is added to your own head.
After running a ton of simulations, I uncovered a simple yet powerful relationship:
BPP% ≈ √Field%
Generalizing this to any KO format:
BPP% ≈ Field% ^ Instant_Bounty%
Where:
- BPP% = The percentage of the bounty prize pool remaining
- Field% = The percentage of runners remaining
- Instant_Bounty% = The percentage of a bounty you claim upon eliminating a player (typically 50% for a PKO, and 100% for an SKO). The other (1 - Instant_Bounty%) goes onto your own head.
In a 50% PKO, each eli
I wonder how this changes the common wisdom of always playing PKOs from the start. If the bounty pool only decays by ~20% with max late reg then perhaps the inherent ICM advantage of late reg makes up for it.
The bounty prize pool decays much more quickly than the ICM value of stacks increase because it is chip EV. I have a few articles in the pipeline on late reg that demonstrates the math, but for example I found a -16% instant ROI upon max late registering an ACR PKO that I analyzed. That reflected a +17% ROI boost for the regular prize pool, but a -50% decrease in the value of the late reg stacksize share of the bounty prize pool
I don't recommend late registering bounty tournaments where the bounties are in play from the start, and operators should stop the predatory practice of long late reg for such bounty tournaments.
Interesting. Maybe you can clear something up for me. It would seem that from an individual's perspective much of the bounty prize pool is always inaccessible. Players on all those other tables are going to bust and their bounties will be taken out of the prize pool, regardless if I play from the start or I max late reg.
Is it then really fair to say that I missed out on "x%" of the prize pool by late registering, when in reality if I did play from the start I would only interact with a small fraction of the overall bounty pool?
Interesting. Maybe you can clear something up for me. It would seem that from an individual's perspective much of the bounty prize pool is always inaccessible. Players on all those other tables are going to bust and their bounties will be taken out of the prize pool, regardless if I play from the start or I max late reg.
Is it then really fair to say that I missed out on "x%" of the prize pool by late registering, when in reality if I did play from the start I would only interact with a small fr
A couple of problems with that idea come to mind:
1) We have to cover other players to be able to access the bounty prize pool at all
2) Bounties are more valuable to win earlier, both in terms of their negative risk premium pressure and in terms of our overall EV in the tournament.
Regarding #2, think about it this way. In a $100 + $100 PKO:
Scenario 1 - We eliminate 2 different players in the early levels, winning two $50 instant bounties for a total of $100
Scenario 2 - We late reg, double up, and eliminate a player who has a $100 instant bounty after having eliminated 2 other players
In scenario 1, we took $100 out of the bounty prize pool, and $100 is "locked up" in our bounty, which we can win if we win the tournament.
In scenario 2, we took $100 out of the BPP, and the player we eliminated took $100 out of the BPP.
We win the same $amount in absolute terms, but a higher share of the BPP in the first example.
Tl;dr But here is a theorem:
Knockout pressure roughly cancels ICM.
So Cev is closer to $ev in PKO tournaments than in regular KO tournaments (where knockout pressure diminishes over time), where they are in turn closer than in regular MTTs.
And a corollary
The player who assumes Cev = $ev in PKO tournaments will have higher $EV as they will have more resources to think about plain poker strategy. While any cEV=/= considerations have only very marginal value.
[QUOTE=LoveThee;58899085]Tl;dr But here is a theorem:
Sometimes equity drops even out ICM, but a lot of times they don't. And the assumptions about differences between PKOs and knockouts are also incorrect.
