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 after eliminating a player (typically 50% for a PKO, and 100% for an SKO). The other (1 - Instant_Bounty%) added onto your own head.
Intuition behind this formula:
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" (in a differential sense), which naturally leads to the square root function, BPP% = Field% ^ 0.5
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.
