Let's talk about how an ICM-based model should account for bounties. This is especially important given the rising popularity of various KO formats.

There are several simple solutions being put forward, but neither looks good. For that reason I have devised a model of my own, named Uczniak Chip Model (UCM) after its inventor.

It seems to me that any remaining bounty should be split between the remaining stacks as follows:

For every possible scenario (finishing order), bounty of each eliminated player should be split among the players who placed higher (so could have possibly been the eliminator), proportionately to their starting chipcounts - intuitively to allow for the fact that a big stack placing high likely means numerous eliminations in the process, whereas a short stack placing high likely means laddering without eliminating anyone.

In case of a progressive KO, only the "instant" part of the bounty should be counted, while the remainder should be similarly split and added to respective bounties of the higher placed players before they are resolved.

This should be multiplied by the probability of each scenario and added up to generate bounty equities, which in turn should be added to standard ICM equities based on the stacks in play and the payout structure.

Discuss.

Can we see an example, please? Also, what are some of the other models being suggested, for reference?

sounds interesting.
give some specific examples to make it clearer to people.

I raised the question of why bounties couldn't be chopped to Stars a long, long time ago and their response was because there wasn't a model to calculate this for hands with more than 2 players. Would be good to get a universally accepted model that would make chops possible.

To give a simplified example with 3 players in a PKO.
-All 3 players have identical stacks of 1 million and bounty amounts of $1000 each.
-payouts are: $2000, $1500, $1000.

Under your model ICM, the EV of the stacks are:
-$1500 each for the main pot (ICM calc of the 3 players stacks vs the prizepool)
-$500 for the instant KO (bounty amount on each players' head divided by 2)
-$500 for the remaining (non-instant) KO portion?

One challenging scenario that will be hard to quantify is if a guy has a large bounty but a small stack. If big blind is 10k in the above example and avg is 100 BBs. If one guy has 10k stack and a 1k bounty, his equity on the instant bounty and non-instant bounty should be really, really small...

You're close to being correct, although I fail to understand why an example of equal stacks and bounties is interesting in any way. Obviously, total equities for every player in such case have to be =1/3 * total prizepool left (payouts + bounties).

As for the splitting of KO equity between instant and non-instant - this is a distinction that has to be made only mid-calculation. Non-instant portion becomes (in part) instant when the next elimination is considered, and ultimately, all KOs are "instant" when the last elimination is resolved.

Hopefully it gets clearer with the example below.

Other models involve splitting bounties only by finishing place (disregarding starting stack, which has to have a significant impact on how many eliminations one makes in the process of winning), or only proportionately to stacks (which is equivalent to assigning all bounties to the eventual winner). These models are only suitable for the case of equal bounties and even then, they are flawed.

Let's take a look at an example where there are three players left:

A, with 5000 chips and 20 bounty on head
B, with 3000 chips and 30 bounty on head
C, with 2000 chips and 50 bounty on head

Let's assume payouts are a typical 50/30/20 - but this is not important at this stage. Only at the end will we want to calculate the "payout equities" and add them up with the "bounty equities". Let's assume it's a standard progressive KO, although this won't matter much for this particular example.

There are 6 possible finishing orders, let's deal with them one by one:

1) A 1st, B 2nd, C 3rd with probability 5/10 * 3/5 = 30% according to ICM

C has 0 bounty equity. 25 of his bounty is added immediately to equities of A and B, proportionately to their stacks, so A gets 5/8 * 25 = 15.625, while B gets 3/8 * 25 = 9.375. They get the same amount added to their bounties so A has bounty increased to 35.625. B has bounty increased to 39.375. Both these increased bounties are then won by the eventual winner of the tournament - player A.

So, for this finishing order, we have bounty equities of:
A: 90.625
B: 9.375
C: 0

Multiplying that by probability of the scenario (0.3) we get the contribution to bounty equity of
A: 27.1875
B: 2.8125
C: 0
for this scenario.

Rinse repeat for all 5 remaining finishing orders (I'll spare you the detailed calculation) and we arrive at total bounty equity of:

A: 51.02
B: 29.57
C: 19.41

Add to that the standard ICM payout equity from any ICM calculator which is:

A: 38.39
B: 32.75
C: 28.86

and we get total tournament equity of:

A: 89.41
B: 62.32
C: 48.27

These numbers could be the basis of a 3-way chop in a PKO tournament if so desired.

TL;DR: Put numbers in magic box, get equities, chop, ???, profit.

I like it.

Seems to be more traffic here than in STTF, so I'll reply itt.

The other models were discussed here:
http://forumserver.twoplustwo.com/36...ities-1622208/

This isn't quite correct, the second model suggested in the STTF thread already accounts for this, although it is done differently than you suggest. But that model also over-weights bounty equity for large stacks compared to the first one.

In some ways you account for this effect twice, and I'm fairly sure that you are over-doing it. Keep in mind that finishing orders where large stacks place high are already given a much bigger probability in ICM.

Take a simple example with stacks {p1=10,p2=1,p3=1} and a non progressive bounty.

In the finishing order {p2,p1,p3} we already know that p2 ends up winning the overall tournament, yet your model still gives p1 a 10/11 chance of having eliminated p3. Your model essentially assumes that big starting stacks will continue to capture a large share of the bounties right up until they get eliminated. But knowing that a big stack gets eliminated next actually means they probably already under-performed up to that point.

In that (10,1,1) example above, unless I messed up the math, your model assigns the following bounty estimates:
p1: 2.562
p2: 0.219
p3: 0.219

Compare this to the estimates if we assume that the eventual winner always takes down all the bounties:
p1: 2.50
p2: 0.25
p3: 0.25

I think it is extremely likely that your model over-estimates equities for big stacks, seeing that it assigns a higher equity for the big stack than a winner-takes-all approach.

I fail to see how the linked model accounts for different starting stacks. It ties the bounty estimate to the finish place. In other words, a short stack winning the tournament is assigned the exact same bounty estimate as the big stack winning the same tournament (granted, they have different probabilities of achieving this, but that's another thing). This feels plain wrong, and I am pretty sure your proposed model underassigns equity to big stacks.

Your math is right. And yes, I feel the large stack has a much larger chance of capturing the 3rd place finisher's bounty even if he ends up losing the HU afterwards. This should be quite intuitive, as most people who play tournaments would know, the short stack finishing high is often due to laddering, then winning a number of flips when it no longer makes sense to ladder.

Now, you might be right about my model overshooting it by a bit, but I still think it is a better estimate than yours, it models the actual playing out of the tournament more closely, and it uses the exact same principle that ICM does, so it makes sense to combine the two.

With deep stacks and small blinds, and movement of chips modelled by random walk, the probabilities of eliminations should be equal to what my model predicts. This is the very principle of ICM, so my model is a natural extension, whereas the one you link feels artificial. Summing it up - it might be imperfect, but its imperfections are directly inherited from ICM, which - despite the obvious flaws - is still the most universally accepted equity model.

What's more important, remember that this is the MTT forum. I fail to see how the model you suggest can work for unequal (including progressive) bounties. Progressive KO nowadays account for the vast majority of KO MTTs.

It is understandable that you will advocate the model that you are able to include in your software, and so will I.

I think this is natural. But I also feel the discussion so far was not factual/substantive enough. Choosing a single example where you feel my model underperforms (although I disagree) is not the best way to compare models, as it's easy to imagine that for any model one will be able to find examples "disproving" it so to speak. I think it is better to discuss the underlying assumptions, or, if we are going to go with merit, set up a test environment where our models could fight in a push/fold KO SNG over millions of games.

I hope we agree that ultimately the only thing that matters are the bounty estimates a model assigns to the various stacks. It's quite irrelevant if you achieve higher equities for big stacks by directly applying some weighting with the starting stacks or indirectly by skewing the bounty "payout structure".

(Skewing the structure means that there is no performance hit, unlike enumerating all finishing orders which would be considerably slower. But equity calculations are usually not a bottleneck, so this isn't very relevant.)

I'll run some more simulations tomorrow and check how the models match up. The model kolemoen suggested closely resembles the results of simulations that I ran so far (and posted in the other thread), but changing/adding other bounty variations is fairly easy at this point and I'm happy to explore other models. It should be obvious from the other thread that I am certainly open for suggestions, my critique of your model had nothing to do with our current model selection.

That's not the case btw, the model most consistent with random walk simulations would be the "flat" one. (Some initial more game-like sims seem to suggest the "moderate" kolemoen model may fit better for the real game though.)

Your model estimates are *way* more top heavy than the random walk results.

10/1/1:
[2.16, 0.42, 0.42] Random walk elimination sample

[2.56, 0.22, 0.22] Uczniak
[2.50, 0.25, 0.25] Proportional
[2.27, 0.36, 0.36] Moderate (kolemoen)
[2.16, 0.42, 0.42] Flat

5/3/2:
[1.43, 0.93, 0.64] Random walk elimination sample

[1.55, 0.89, 0.56] Uczniak
[1.50, 0.90, 0.60] Proportional
[1.45, 0.92, 0.63] Moderate (kolemoen)
[1.42, 0.94, 0.64] Flat

Seems interesting if this is the case; I think I know now why it might differ from the random walk results, so I retract that claim for the time being.

I still think it models the actual gameplay more closely as it resolves eliminations one by one for every finishing order. What might need tweaking is the probability of bounty of the N-th place finisher being assigned to each of the 1st to (N-1)-th place finishers, which at the moment is proportionate to starting stack.

There might be a better way to go, and I will test other possibilities as it is quite easy to do and maybe allow the user to choose, although I doubt going with the probabilities being all equal (equivalent to flat model) or 100% to 1st place finisher (equivalent to proportional model) is the right way to do it especially to model equities resulting from actual gameplay.

If anything, I would try to tweak the probabilities to match the principle behind the moderate model, which might give similar (same?) results for the test cases above but obviously be more general as it can account for any bounty setup, whereas - if I'm not mistaken - modifying payouts only works for equal bounties.

I'm reasonably sure that the progressive estimates can be calculated without going through all the finishing orders btw, but it depends on the weighting function used and would still be much more complex than simply adjusting the structure. (It should work for flat/proportional and also for your original proposal, the kolemoen weighting is more tricky.)

Anyway, making the weight function user-configurable may be a nice solution. How about weighting proportional to finishing_place^(-x) where x is user configurable? (For x=0 this is equal to "flat", large x approach winner-takes-all.)

The problem with progressive KOs is not their progressiveness - that alone should be easily doable the way you currently do it. The problem is they (almost always) are unequal.

With unequal bounties, for models other than winner-takes-all it looks like you need to know the chance of player A finishing higher than player B in order to estimate the chance of A capturing B's bounty. It seems really hard to do without going through the finishing orders. As you say, simply modifying payouts won't work. Even modifying payouts for each player separately and running N separate ICM calcs to get equities for every player seems insufficient. If there is a way, it surely has to be a nice find.

It's actually the other way around for me, I believe I have it working for unequal bounties (both flat or your weighting) but only if it's 100% instant. Not sure if i can get that exact calculation work for progressives without enumerating all finishing orders, may need to tweak the assumptions a bit.

I have just added the free flow UCM model where the conditional probabilities of A getting B's bounty (conditioned on A placing higher than B in a given finishing order) is proportionate to (A starting stack + a fair share of stacks of eliminated players), where "fair share" means that an eliminated stack is divided equally between the other stacks. This is the UCM equivalent of the kolemoen model.

In the original gravity UCM, the eliminated stack was divided proportionately to starting stacks (hence the new gravity moniker). This is the UCM equivalent of the proportionate model.

Result of the change is that (in 4+ player calculations), the estimates are a little less top heavy compared to the gravity model. They are still more top heavy than even your proportionate model, although until proven otherwise in a real game simulation I will stand behind my belief that this is correct.

I believe your proportionate (so, in effect, winner-takes-all ICM) model overestimates big stack's bounty equity when he wins the tournament by less than it underestimates the same big stack's bounty equity when he finishes second or worse. As an effect, this could lead to a net underestimation of the big stack equity. Which might look like a paradox but that still doesn't make it wrong.

Still playing around with calculation shortcuts but not sure if the approach works for progressive at all. I'm giving up for today

Below are some test cases for comparison: non-progressive, stacks = (1..n), bounties = (n..1). The calculation ran for all players, but only included the two smallest/largest stacks here to keep things compact.

Your weighting:


Flat:

Well this shows again that for the big stacks we have:

flat < moderate < proportionate < uczniak_new < uczniak_old.

However this seems kinda pointless as to calculate the full equity model for Nash calculations you need to run the above model a large number of times = (n choose 3) * 13 + (n choose 2) * 3 + n.

Yeah, these would only be interesting for e.g. deal making. For Nash calculations you probably don't need to calculate with 24 players. For a normal 10 player calculation it's just ~1ms each, it's just slightly slower than ICM basically.

Why is uczniak_new different from moderate for non-progressives btw (is it?), I thought you are using the same weighting function?

if a tiny stack has a large bounty, then the player to his left and the one to his right have so different equity of that bounty. seating order of stacks, how is that quantified?

^ To capture stuff like relative positions you pretty much have to simulate how the game plays out, FGS is a popular approach to adjust for this.