This is a spin-off from this thread, I figured it would make sense to post this separately.

Basically, support for bounty/knockouts for beta version of HRC breaks down to two parts:
1) Actually awarding any bounties for eliminations that occur within the calculation scope.
2) Estimating how bounties paid out in the remaining tournament affect the equities.

The first part is very straightforward conceptually, it's just an implementation of the bounty rules. This is completely defined by the rules of the game, no need for any estimates or approximations. If player A busts player B then you award him an additional $ amount, easy. (Actually implementing this took the lion's share of the development time though.)

The second part is more interesting, we have to come up with an approximation how the remaining bounties will be awarded. We currently assume that all players who eventually finish better than n'th place have an equal chance of having eliminated the player who finished in n'th.

So the 1st player always gets the bounties for himself and the player finishing 2nd, plus 50% he eliminates the player in 3rd place, 33% he eliminates the player in 4th, etc.

So let's assume we have 5 players left:
avg_knockouts(1st) = 1/4 + 1/3 + 1/2 + 2
avg_knockouts(2nd) = 1/4 + 1/3 + 1/2
avg_knockouts(3rd) = 1/4 + 1/3
avg_knockouts(4th) = 1/4
avg_knockouts(5th) = 0

We get the following results from the calculation above [1st, 2nd, 3rd, 4th, 5th]:
[3.0833, 1.0833, 0.5833, 0.2500, 0.0000]

If we have stacks of [10, 20, 30, 40, 50] and we do an ICM calculation with the structure above we get the following results:
[0.4143, 0.7684, 1.0475, 1.2822, 1.4874]

This means that we estimate the stack of 50 would on average get 1.487 bounties in the remaining tournament, while the stack of 20 would only get 0.768 bounties, etc.

In the overall calculation the bounty-value for these estimated future knockouts is added to the ICM estimates for the regular prize pool to get the resulting overall equity estimate including bounties.

Hi Plexiq,

Thank you for taking the time to come up with this model. It seems like an elegant way to implement bounties into icm calculations.

However I feel like distributing the bounty of an eliminated player evenly among all players finishing ahead of him is somewhat inaccurate and places too much equity on the lower finishes. For example, in my experience the player who knocks out 3rd place will finish first way more often than second. Also it seems unrealistic that the player who knocks out 6th place will have the same probability to finish 1st as 5th. That is because a knockout will usually go along with a big increase of your chip stack.

Have you thought about giving the player who scores a knockout a bigger than average stack for the calculations? For example, one could assume that the player who knocks out 6th place will have 1k chips, with the other 4 players having 500 each. Then the chance of that player finishing first will be 1/3; 4/15 to finish 2nd etc. (according to icm probabilities). Now for 5 players one could assume the player who knocks out 5th will have 2/5 of the chips, with the other players all having 1/5 and calculate probabilities based on those stack sizes.

We would end up with the following distribution of the 4 bounties that don't automatically go to first :
avg_knockouts(1st) = 5/15 + 4/10 + 3/6 + 2/3
avg_knockouts(2nd) = 4/15 + 3/10 + 2/6 + 1/3
avg_knockouts(3rd) = 3/15 + 2/10 + 1/6
avg_knockouts(4th) = 2/15 + 1/10
avg_knockouts(5th) = 1/15
avg_knockouts(6th) = 0

Summing it up we get [1.900, 1.2333, 0.5667, 0.2333, 0.0667, 0]
for [1st, 2nd, 3rd, 4th, 5th, 6th]
While your model for 6 players is at [1.2833, 1.2833, 0.7833, 0.4500, 0.2000, 0]

So the main difference is that there is a lot more equity on first place and less on 3rd to 5th
Now obviously the model I used also has it's flaws. Mainly that the player who gets knocked out will probably have a less that average chip stack most of the time. Maybe there is some kind of middle ground where one could come up with a model that includes a chip increase for the player with the knockout but makes it smaller than in the model I used.

For what it's worth I also compared with my actualy stats from pokertracker (around 1k tournaments) and there the distribution is the following (normalized the bounties so that they add up to 4)
[1.8013, 1.4116, 0.5707, 0.1560, 0.0604]
They also have more equity placed on 1st and less on 3rd to 5th.
Now those might be due to my specific playstyle and the sample size for the 4th and 5th place finishes is very small since you will get bounties very rarely on those places. So it would be nice to get more stats from other players for comparisons.

Thanks for the feedback.

I think our current version works quite well, especially when used with FGS, but there is certainly room for improvement. I agree that the knockout estimates from our current model are likely too flat. Your approach is an interesting alternative, but I'd guess you are right that it may be a bit too top heavy. Maybe a linear combination of the two could work.

We'll definitely do more testing and fine tuning on this. As I mentioned in the OP, implementing the first part (awarding bounties that occur within calculation scope) has taken up the vast majority of the development time so far, but we can now easily plug in different models for estimating the remaining bounties.

This is great as a reference, thanks. Can you elaborate on the bold part please, does the raw stat include bounties for knocking out 2nd place?

I'm glad to hear you are still open to fine tuning. I also think the program works quite well overall.

About the stats, I only included the bounties for knocking out 3rd to 6th place.
What I meant with normalizing is I multiplied the average bounty for each finish position with 4/x where x is the sum of the average bounties for each finish position. All stats are taken from 10$,25$ or 50$ 6 max hypers on Pokerstars. I can send you the raw stats per PM.

Edit: Guess, I can't write PMs yet. You'd probably have to write me so I can reply.

Will probably rely mainly on simulations for the fine tuning, but actual player stats would certainly be a nice sanity check.

Please PM if you have the info. Thanks!

Very interesting, 2nd point especially. Now IMO fgs issue is still quite complicated, because basing on my experience in that games fgs really affect many decisions

@kolemoen:
I get the following knockout averages from simulations, excluding top 2 bounties:
5 handed: [1.635 0.970 0.319 0.074 0.000]
4 handed: [1.172 0.679 0.148 0.000]
3 handed: [0.659 0.340 0.000]

The strategies used in those simulations were from an unrelated project w/o bounties, I'm not sure if the knockout statistics would change much if the strategies are bounty-specific, I'll look into that later.

Anyway, your suggestion is much closer to the simulation values than our original model, very nice! I've updated the bounty model to use your suggestion for the time being, we may do some further tweaking though.

Looks good! I'm surprised the numbers are actually so close since in theory my approach should have been a bit too top-heavy as you said. I guess your simulation will take into account that bigger stacks have better chances to win bounties due to being able to play more agressively, and that might even it out somehow.

I've added a configuration option to allow switching between the following models:
1) Flat (first post, compatible with Icmizer)
2) Medium (recommended, based on kolemoen's suggestion)
3) Proportional (remaining bounties are distributed proportional to stacks)

The proportional model was added upon user request. Note that this option certainly under-estimates equities for smaller stacks, but I believe it may be useful for comparison.

You can see my approach in this thread. Posted in MTTc as it can be used for any KO tournament, not only STT.

I replied in the MTTc thread, but essentially your model appears to result in estimates that are even more bigstack-biased than the proportional winner-takes-all approach, that really doesn't seem right imo.

hello!
could you please explain how to input pricestructuere?
for example for 5 $ (2.38+2.38+0.24) 6max hyper superknockout i can enter 714 for 1st and 2nd and 238 for KO? (just correspondent prizes in cents)

and also, do you know how programm simple nash calculates knockouts?
it uses the ICM Malmuth-Weitzman to determine the probability of each player being knocked out next and the knockout is spread evenly among who is left.
as i understand it should be close to "flat" right?

Yes, you can simply enter the cent amounts like you said.

And the simple Nash calculation should correspond to the flat model if it works like you described, yes.

Thank you!
Tested for a little bit. simple nash is close to flat model though its not 100%the same.
i found an interesting thing: in SN overcall (3d guy to allin) is way tighter then in any of HRC models though pushes and calls correlate well for "flat". for example i now witnessed a hand where in HRC overcall was 20% wider then in SN though push and call were mirroring one another. It should be a mistake in taking into account the 3KOs in the hand in one of the programms right? or how could it be possible

Wait, are they using Malmuth-Weitzman or Malmuth-Harville (ICM)?

We use Malmuth-Harville, so if they actually use Weitzman that may explain the difference.

as far as i understand they use weitzman ONLY to determine the probabilities of players being knocked out next. the explanation is not thorough and its in russian so its hard for me to translate
this caught my eye because SN correlates with HRC"flat" very good so i cannot see why such a difference for overcall in 3way. but i guess it cannot be helped since SN is no longer supported.
well, anyway its cool that HRC finally has knockout calculations, its about time! thank you for the work!

sorry, one more question, the sums of EQ% is not 100, its because i entered prizes in cents? how to enter them properly then?

Nope, that's intentionally. The equities still use the total regular prize pool as 100%, so including bounty equity you get a total of 100% + players x bounty%.

well thats not good, because it screws up all the EVdiffs of the hands. i look at EVdiff of the hand to judge how bad or good the push is, using positive/negative edges for big/short stacks, how am i to do that now if the numbers are completely different from 5year expierience 0_0

It's fairly trivial to convert back though.

e.g. for (2.38+2.38+0.24) 6max you mentioned earlier you will have an additional 16.6% for each remaining player, so with 3 left you have 100% + 3x16.6% = 150% total; simply divide the displayed equities by 1.5 (-33%) and you get the values in comparison to the total remaining prize pool.

The alternative would be to use 100% for the total prizes still to be awarded, but that will screw with users who want to easily convert to $.

why ucan convert to $ when the sum is not 100% and cannot when its 100? O_O when its 100 u just multiplay diff of the hand to prize pool

also its not obvious why would you need to see $ anyway. u play mix of differenet buyins for exapmle i play 5 / 10 /25 why would u need $?? 0_0 its buyin dependant

I didn't mean "screw" as in "not possible". Of course you can still convert to $, it's just slightly more complicated than multiplying with a fixed $ amount since you need to take care of the # players remaining. It's still quite trivial to convert to $, and it's also trivial to convert to 100% eqdiffs in the current version.

Regardless of the choice of display some users end up with some slightly increased complexity to get the numbers they prefer. I believe the group of users relying on "constant eq% sum" is considerably smaller than the $-conversion crowd, but I'm happy to change the display or make this configurable if it turns out to be a problem.

PS: And I completely agree that $-conversions are not a great idea in general, that doesn't change the fact that loads of users do them anyway
PPS: We can further discuss this via pm or email, it really doesn't add anything to the KO model discussion.

del