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 Post subject: Headsup NL Short Stack Nash Equilibrium
PostPosted: Wed Aug 19, 2009 3:40 pm 
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Whom of you guys has worked on that, and managed to successfully find this? I would be interested in (possibly) private discussion on that with you.

Note: This is not about push(jam)/fold equilibrium which you can find e.g. in Mathematics of Poker and which (for 20bb stacks) is exploitable by 60bb/100h.

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 Post subject: Re: Headsup Short Stack Nash Equilibrium
PostPosted: Wed Aug 19, 2009 8:00 pm 
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Specifically you mean for NL I think...


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 Post subject: Re: Headsup NL Short Stack Nash Equilibrium
PostPosted: Wed Aug 19, 2009 9:18 pm 
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Yes, thx for the clarification, I corrected the title. (thou I haven't seen push/fold limit bots :)

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 Post subject: Re: Headsup NL Short Stack Nash Equilibrium
PostPosted: Mon Aug 24, 2009 9:09 pm 
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OK after a massive interest and 6 pages long thread ;)

I will do some more clarifications, and put this in hypotetical scenario. That wouldn't guarantee more responses, but if one is interested, he would at least have some more beef to work with:

Say we are on the SB, facing a BB, and both have initial stack of 20bb size (before posting the blinds).

I have fairly often seen people advising push/fold play on this spot (E.g. in headups-endgame SNG situations), with ranges varying based on how loose/tight the BB is. But some simple game theoretical calculations shows that this is far away not being the case.

So let's say we are facing an optimal push/fold player on the BB, in this spot the EV of our optimal play, if we play push/fold, as advised, is -36 bb/100h, or -0.3664 per hand.

Now, let's assume we have one more action in our disposal: Minraise. If we only add that action (while the BB keep playing push/fold), our EV becomes 23 bb/100h. So if you play against optimal push/fold/min raise strategy this is a difference of over 60 bb/100h.

Now I know that adding min raise to our arsenal is not something groundbreaking. The right amound of push/minraise & call/minraise & fold/fold, however, is not something that I have seen being discussed.

So my questions are:
1) Does anyone of you did similar calculations? If yes, can you confirm the above numbers?
2) Am I way off by missing important discussion on this topic - Are you aware of such elaborated discussion on the minraise range %, etc.
3) Have anyone of you worked further in this direction, adding more actions to SB or BB, and eventually including postflop play in the picture?

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 Post subject: Re: Headsup NL Short Stack Nash Equilibrium
PostPosted: Tue Aug 25, 2009 9:28 am 
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Your calculation is a bit stretched because you are giving the SB the option to {fold, raise all in, min-raise} while only allowing the BB to {fold, re-raise all in}. The big part that is missing is the ability of the BB to call the SB's min-raise and play postflop. Once you add in the minraise option then you must consider postflop play (and thus things get much more complicated).

As a HU player myself, I think that resorting to a strict push-fold equilibrium strategy when effectively 20bb deep is rather absurd against many opponents. For example, depending on the opponent, min-raising your big hands and shoving any flop could be more +EV than jamming them in preflop. By min-raising your big hands instead of jamming, you are giving your opponent an opportunity to re-raise you preflop with a (weaker) hand that he would not have called your shove with. On the same note, you are also giving your opponent an opportunity to call your min raise and fold postflop with a hand that he may have called a preflop shove with.

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 Post subject: Re: Headsup NL Short Stack Nash Equilibrium
PostPosted: Tue Aug 25, 2009 12:35 pm 
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The above I posted merely to validate if my calculations are correct, and as example. If one did similar calculations, we can directly compare results.

I agree with you - that I gave less options in this example to the BB, and normally he would call minraise, which would lead to a postflop play. The example is not without value, as if we know we are facing push/fold opponent, we can just implement the minraise/push/fold strategy and have guaranteed EV of 36bb/100. But even if this case, if we play exploitive, we might achieve better results.

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 Post subject: Re: Headsup NL Short Stack Nash Equilibrium
PostPosted: Wed Aug 26, 2009 10:39 am 
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I originally forgot to mention that I'm currently in the process of researching various equilibrium-finding methods and am concentrating on a HU NLHE equilibrium for various effective stack sizes (shortstacked 20bb being one of them).

Before getting into any calculations of my own, could you make some clarifications about your calculations as I might not be understanding something?

indiana wrote:
So let's say we are facing an optimal push/fold player on the BB, in this spot the EV of our optimal play, if we play push/fold, as advised, is -36 bb/100h, or -0.3664 per hand.


To me this makes no sense. If you are the SB using an optimal push-fold strategy (you are shoving with the optimal range of hands, folding everything else) against a BB who is also using an optimal push-fold strategy (calling with the optimal range of hands, folding everything else) then both players are in equilibrium each with an EV of 0bb/hand.

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 Post subject: Re: Headsup NL Short Stack Nash Equilibrium
PostPosted: Wed Aug 26, 2009 4:12 pm 
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BluffCatcher wrote:
To me this makes no sense. If you are the SB using an optimal push-fold strategy (you are shoving with the optimal range of hands, folding everything else) against a BB who is also using an optimal push-fold strategy (calling with the optimal range of hands, folding everything else) then both players are in equilibrium each with an EV of 0bb/hand.


Of course not. They have posted different blinds, therefore each EV from the hand is not 0bb/hand.

If you take EV of player1 at SB + EV of player1 at BB, then it becomes symetric, and EV will be 0bb/hand (if they use the same, optimal, strategy).

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 Post subject: Re: Headsup NL Short Stack Nash Equilibrium
PostPosted: Thu Aug 27, 2009 12:53 am 
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Yea of course you are right. I don't know what I was thinking (or drinking) at the time :drink.

Anyways, I've spent some time going through the EV calculations of a Nash pusher (SB) vs. Nash defender (BB) at 10bb and 20bb effective stacks and did not arrive at -0.36bb/hand for the small blind. It might be that we have used different pushing/calling ranges. Let me know if I made an error somewhere:

Nash pusher (SB) vs. Nash defender (BB) @ 10bb effective
SB pushing range: {22+,A2s+,K2s+,Q2s+,J3s+,T5s+,95s+,85s+,74s+,64s+,53s+,A2o+,K2o+,Q7o+,J8o+,T8o+,97o+,87o}
BB defending range: {22+,A2s+,K2s+,Q6s+,J8s+,T9s,A2o+,K6o+,Q8o+,J9o+}

SB pushes 56.6% of the time (750/1326)
BB calls a push 38.6% of the time (473/1225)
SB pushing range has 45.786% equity against BB calling range

EV_sb_pushes_and_bb_calls = EV_sb_wins + EV_sb_loses = (0.45786*10.5 + (1 - 0.45786)*-9.5)) = -0.3428bb
EV_sb_pushes_and_bb_folds = (1 - 473/1225)*1.5 = 0.9208bb
EV_sb_pushes = EV_sb_push_and_bb_folds + (473/1225)*EV_sb_push_and_bb_calls = 0.9208 + (473/1225)*-0.3428 = 0.7884bb/push
EV_sb = -0.5 + (750/1326)*0.7884 = -0.054bb/hand

-----

Nash pusher (SB) vs. Nash defender (BB) @ 20bb effective stacks
SB pushing range: {22+,A2s+,K3s+,Q5s+,J6s+,T6s+,96s+,86s+,75s+,65s,54s,A2o+,K9o+,Q9o+,J9o+,T9o,98o}
BB defending range: {33+,A2s+,A5o+,K9s+,KTo+,QTs+}

SB pushes 40.87% of the time (542/1326)
BB calls a push 23.51% of the time (288/1225)
SB pushing range has 43.823% equity against BB calling range

EV_sb_pushes_and_bb_calls = EV_sb_wins + EV_sb_loses = (0.43823*20.5 + (1 - 0.43823)*-19.5) = -1.9708bb
EV_sb_pushes_and_bb_folds = (1 - (288/1225))*1.5) = 1.147bb
EV_sb_pushes = EV_sb_push_and_bb_folds + (288/1225)*EV_sb_push_and_bb_calls = 1.147 + (288/1225)*-1.9708 = 0.684bb/push
EV_sb = -0.5 + (542/1326)*EV_sb_push = -0.22bb/hand

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 Post subject: Re: Headsup NL Short Stack Nash Equilibrium
PostPosted: Thu Aug 27, 2009 10:19 am 
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Bluff, I will come back to you with that.

In the meantime, someone else gave me this hint:
Quote:
There is a solution in the Kill Everyone book with more pre-flop actions allowed, but post-flop it just assumed that both players check it down if not all-in by the flop.


Does any one has these few relevant pages scanned or are they published somewhere in the net?

If it is really the optimal solution that they computed in the worst case this might be useful to compare results for correctness.

Otherwise optimal solution with more actions preflop and checking it down to the river when called is trivial to compute, but not very indicative of reallity.

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 Post subject: Re: Headsup NL Short Stack Nash Equilibrium
PostPosted: Fri Aug 28, 2009 8:25 pm 
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@BluffCatcher - You are right, that above is money/100hands, as the BB is 2.

Which makes it -0.18 bb/hand, which is different than yours -0.22, so I will investigate this. In fact, I have several implementations right now, and some are actually at -0.22 :)

As my EV is better for the SB, I can imagine that it still may be the correct one.

Also I see that your stack is 20.5 bb when all in, mine is 20. So both players have 20bb at hand start, after posting the blinds they have 19.5bb and 19bbs.

Otherwise the ranges are similar. 41% / 23% looks OK.

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 Post subject: Re: Headsup NL Short Stack Nash Equilibrium
PostPosted: Sun Aug 30, 2009 12:02 am 
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I found a video on stoxpoker.com that talks about Heads-up Push Folding, and they have a slide that says at 20bb effective stacks the SB loses 0.18bb/hand. Click here to see a list of videos, and then click "Stream 3 min Demo" for the "Heads Up: Push Fold" video (#1057). The slides will be skipping ahead very quickly, but if you pause the video at 2:01 you will see the slide that deals with push-folding @ 20bb effective stacks.

Furthermore, the video states that the SB loses 0.050bb/hand on average against a nash defender @ 10bb effective (1:53 in the video), and loses 0.27bb/hand @ 30bb effective (2:10 in the video). In the 10bb scenario, I arrive at roughly the same -0.050bb/hand result using the calculations I previously posted, but I don't have -0.18bb/hand in the 20bb scenario. Do you mind posting your calculations, as it does seem that -0.18bb/hand is the correct answer. Also, could you try calculating the SB bb/hand with 10bb starting stacks to see if you arrive at roughly -0.05bb/hand?

indiana wrote:
Also I see that your stack is 20.5 bb when all in, mine is 20. So both players have 20bb at hand start, after posting the blinds they have 19.5bb and 19bbs.

My reasoning was as follows:

I started with $EV_sb = -0.5bb + SB_push%*$EV_push + SB_fold%*$EV_fold. Every hand the SB posts the forced -0.5bb, and since we are restricting the SB to pushing or folding, some percentage of the time the SB will push ($EV_push) and some percentage of the time the SB will fold ($EV_fold).

Note though, that when the SB folds, his $EV is 0, so the formula is simplified to:
$EV_sb = -0.5bb + SB_push%*$EV_push

Assuming 20bb effective stacks, after posting the blinds the SB will have 19.5bb and the BB will have 19bb. When the SB pushes, there are these possible outcomes:
  • BB folds
  • BB calls and SB wins
  • BB calls and SB loses

When the SB pushes and the BB folds, the SB's $EV is (1 - BB_call%)*1.5bb. The SB gains the blinds.
When the SB pushes and is called by the BB, the SB's $EV is ((BB_call%*BB_equity%*-19.5bb) + (BB_call%*SB_equity%*20.5bb)). The 1.5bbs are already in the pot--they do not belong to the SB or BB anymore. So when the SB pushes, he risks 19.5bbs to win 20.5bbs (BB's 19bb stack + 1.5bbs in the blinds).

Thus, $EV_push = [(1 - BB_call%)*1.5bb] + [((BB_call%*BB_equity%*-19.5bb) + (BB_call%*SB_equity%*20.5bb))].


If anyone sees a glaring mistake in my calculations or reasoning please let me know.

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 Post subject: Re: Headsup NL Short Stack Nash Equilibrium
PostPosted: Sun Aug 30, 2009 11:48 am 
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Didn't know about this Stox video. But they don't consider raise?

What kind of tool they used to come up with their calculations? During the video they mention StoxEV, for situations when you face a raise. I'm gonna look into that.

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 Post subject: Re: Headsup NL Short Stack Nash Equilibrium
PostPosted: Sun Aug 30, 2009 8:05 pm 
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I'm pretty sure they do not consider raises in the video.

I'd be interested to know how you are considering raises. You are assuming the BB is playing strictly push-fold, but what kind of hand ranges do you have the SB min-raising? the SB min-raising then calling a BB push? the SB pushing all in? Unless you computed the optimal (equilibrium) ranges, then how can you guarantee the $EV for the SB to be 23bb/100 against a push-folding BB?

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 Post subject: Re: Headsup NL Short Stack Nash Equilibrium
PostPosted: Sun Aug 30, 2009 8:26 pm 
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Yes, I have computed the NEQ for this case (SB: push/fold/raiseCall/RaiseFold, BB: push/fold). The EV is not 23 bb/100h, but rather 12.5 bb/100 (same issue that I mistakenly posted the EV in $, not in big blinds).

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 Post subject: Re: Headsup NL Short Stack Nash Equilibrium
PostPosted: Sun Aug 30, 2009 8:47 pm 
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And here are the numbers:

blinds 1.0/2.0, stacks 40.0

EVsb: 0.23108

Small blind:
Jam%: 0.7%
MinRaise&CallJam%: 30.4%
MinRaise&FoldToJamJ%: 49.4%
Fold: rest

Big blind:
CallJam & FoldToRaise: 0%
FoldToJam & JamToRaise: 15.3%
CallJam & JamRaise: 32.4%
Fold to any: rest

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 Post subject: Headsup NL Short Stack Nash Equilibrium
PostPosted: Wed Sep 02, 2009 1:20 am 
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indiana wrote:
In the meantime, someone else gave me this hint:
Quote:
There is a solution in the Kill Everyone book with more pre-flop actions allowed, but post-flop it just assumed that both players check it down if not all-in by the flop.


Does any one has these few relevant pages scanned or are they published somewhere in the net?


Hi guys. I'm the author of Kill Everyone. I've been a lurker on these forums for the last several months - I'm getting more into bot development and post-flop analysis. I'm currently working on more equilibrium&exploitive stuff for my new project, which will at least be a new book, maybe a bot as well. I just want to say that these forums have been a great resource for me.

That being said, please don't publish any of our material where it's publicly available. I have no problem working with people individually on projects and I'll be happy to chat and discuss about any work that I've done or new potential avenues to investigate. Hope you understand.

And now that I've finally delurked, maybe I can start contributing more as well. :)

Thanks,
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 Post subject: Re: Headsup NL Short Stack Nash Equilibrium
PostPosted: Wed Sep 02, 2009 2:07 am 
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Hi Tysen,

Welcome to the forums, I selected some avatar for you so now you are completely delurked. :)

We don't have a practice to make publicly available any content that otherwise isn't. I was primarily interested in HU NEQ that is more than push fold, to simply compare results.

I have calcualated e.g. NEQ for NL and various stack sizes where everyone can raise two times the last raise, or he can jam (he can do that at any point), or he can call the last raise (When we have a call, I take the all-in equity, so there is no postflop play). Agh, and forgot to say - my program works ligthning fast, so basically, you can run these calculations (e.g. with exact stack sizes) realtime.

While push/fold is trivial, and I know it works correctly, I was interested to compare my calculations for the more complex cases with someone else that has computed the same. Hence the interest in any resouce (in particular your book) if these numbers were published already. I didn't just bought the book as I've seen that there is a new edition (that just came?) and I wasn't sure if this was in the new, or the old one (and amazon.de is still not selling the new one).

Furthermore, I would be interested to approaches for postflop play, and again practical comparison of results. E.g. how far away is the usage of allin equity from the true equilibrium? If not allin equity, to what extend we can separate the calculation of preflop play and flop play? Etc.

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 Post subject: Re: Headsup NL Short Stack Nash Equilibrium
PostPosted: Wed Sep 02, 2009 9:36 pm 
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indiana wrote:
We don't have a practice to make publicly available any content that otherwise isn't. I was primarily interested in HU NEQ that is more than push fold, to simply compare results.

I wasn't worried, I just felt obligated to put that out there... ;)

Quote:
While push/fold is trivial, and I know it works correctly, I was interested to compare my calculations for the more complex cases with someone else that has computed the same. Hence the interest in any resouce (in particular your book) if these numbers were published already. I didn't just bought the book as I've seen that there is a new edition (that just came?) and I wasn't sure if this was in the new, or the old one (and amazon.de is still not selling the new one).

There is a "revised and expanded" edition that just came out. The study we're talking about is in both versions. There's nothing that was taken out of the first edition - only additions.

Quote:
Furthermore, I would be interested to approaches for postflop play, and again practical comparison of results. E.g. how far away is the usage of allin equity from the true equilibrium? If not allin equity, to what extend we can separate the calculation of preflop play and flop play? Etc.

Yup, that's what I'm interested in as well. Assuming that there is no post-flop play is probably an okay assumption for 10BB and a very poor assumption for 100BB. How bad a job does it do at 20BB? 30BB? I'm not sure yet and I hope to figure it out.

A very related topic that I've always been interested in is the post-flop "playability" aspect of certain pre-flop hands. Hands like low pairs, suited connectors, and Ax might have similar equities against an opponent's range, but they play very differently post-flop and have different "comfort" levels. And that comfort level also depends on stack size...

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 Post subject: Re: Headsup NL Short Stack Nash Equilibrium
PostPosted: Wed Sep 02, 2009 10:31 pm 
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Do you have the software with you - can you calculate equity for various preflop setups?

E.g. preflop NEQ with 10bb (before posting the blinds), and options:
- For the SB: push, fold, call the blind
- For the BB: push, check

In case of call/check, we take the allin equity (as if there is no postflop play, but cards are directly dealt).

Can you calculate and post your results for that? We can then compare the equity and the ranges.

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