Bounty Models Explained: Solving Knockout Tournaments

We’ve just rolled out a revolutionary update! You can now find optimal postflop ICM strategies tailored for any tournament format, including classic freezeouts, satellites, progressive knockouts (PKO), regular knockouts (KO), and mystery bounties. Learn more about the update here!

To achieve this, we developed new methods for accurately modeling the value of bounties. This involved creating algorithms to calculate the expected value (EV) of knocking out players both in the current hand and in future hands. In this article, we’ll unveil the engineering solutions that make solving these bounty formats possible!

The Scope of Bounty Tournaments

Let’s start with a brief summary of what bounty tournaments are before we dive into the details.

You can read more about knockout formats and how to strategically approach each of them in this article.

How GTO Wizard AI Models Bounties

The latest update allows you to plug in any stack sizes, assign bounties, set arbitrary payout structures. One of the major engineering challenges we faced was finding a way to model the value of a bounty in knockout tournaments.

At first, this seems like a simple problem. If one player stacks another, they get some portion of that player’s bounty. The solution looks straightforward. Just add the value of a bounty to a player’s EV whenever they knock someone out.

Not so fast! There are two hurdles:

To calculate the expected value (EV) of anything, you need two pieces of information:

Traditional CFRCounterfactual Regret Minimization (CFR)
Counterfactual Regret Minimization is an algorithm that is commonly used to solve large imperfect-information games. solvers calculate all the way to the river. That means players’ strategies on every runout and betting line are known and well-defined. This information makes it easy to calculate how often one player will knock out another, and therefore the EV of knockouts is trivial to determine (for a computer). However, GTO Wizard AI only solves one street at a time.

However, the work involved for traditional solvers can take minutes or hours to complete for one flop; it’s not feasible to do this on a web application like GTO Wizard. We use a shortcut to solve spots thousands of times faster than traditional solvers. Instead of calculating the entire strategy to the river, GTO Wizard AI calculates:

Hours worth of solving compressed down to seconds!

While this speedup is great, it also means that if you solve a turn spot, it doesn’t calculate the river strategy. Future streets are abstracted, so there’s no way to directly calculate the knockout probabilities. We needed a new method!

Let’s walk through a simple example to demonstrate how all this works.

It’s the first hand of a PKO. Some maniac player open-shoves.

You call the shove with AA and stack them. Congrats on winning their bounty! But how much value did you gain? Can we quantify this? Let’s compare your value before and after stacking them to make it simple.

Before KO

Before the hand began, you invested $200 to play this PKO, and have an equal share of the chips to every other player. Ignoring any skill edge, you’d expect to win $100 in prizes and $100 in bounties. In other words, before the first hand, your stack was just worth the $200 you paid to enter the MTT.

After KO

After stacking the maniac, you immediately win $50 from their bounty.

You’ve doubled your stack, so you have 200bb out of 10,000bb or 2% of all the chips. There’s $9950 remaining in bounty prizes, and you expect to win 2% of those, so your future bounty EV is $199. This is a straightforward Chip EV calculation.

Lastly, we calculate the ICM value of your stack. With a Chip EV approximation, you’d expect to win 2% of the untouched $10,000 prize pool. However, the independent chip model doesn’t value chips linearly. Doubling your stack won’t quite double its value. After all, knocking out that maniac didn’t just benefit you; it also slightly benefits everyone else who remains since they are one step closer to the money. Therefore, the ICM value of your stack is closer to $195. (This varies by a few percent depending on the payout structure).

All in all, you’ve gained about $244 total!

Summary

Modeling the value of bounties is crucial for solving optimal tournament strategy in today’s games. To recap, we break the problem down into three parts:

To be precise, our neural network predicts current and future bounty EV in one step.

With these three pieces of the puzzle, we have a complete picture that accurately predicts chip utility in tournaments, allowing MTT players to solve optimal postflop strategies for all formats and take their game to the next level.