whiteshark's avatar


262 points

1. Problem Definition
I'm going back to the study of some underlying GTO principles currently and there is a question that has been bugging me for a while now. So in heads up play, a Nash equilibrium between two players is defined as a set of two strategies from which neither player can deviate in order to increase their EV. Often times, equilibrium play results in certain hands being indifferent between two actions (say betting and checking), which manifests in a mixed strategy for these hands. Let's say Player A has two different hands that are both supposed to mix between exactly 67% betting and 33% checking.

Scenario 1: In the first scenario, Player B is sticking to the equilibrium response no matter what Player A does. Player A can now freely choose any betting and checking frequency for the mixing hands without losing EV. Since Player B is not adapting and as betting/checking for these hands are equal in EV, there are infinitely many alternative strategies for this player that generate the exact same EV vs. the equilibrium response. Player A can choose any deviation from the optimal mixing frequencies.

Scenario 2: In the second scenario, Player B is allowed to adjust his strategy if Player A deviates. Now, let's say Player A chooses to bet one of the hands that are supposed to mix 80% and check it 20% of the time. This would be an asymmetrical deviation meaning that Player A deviated from optimal frequencies and the overall betting frequency of his whole range changed. This play should lose EV. Now there exists an exploit that Player A can choose as Player B was not sticking to the optimal mixing frequencies. This simply means that one player deviates from GTO and loses to the other player's exploit.

Scenario 3: Essentially, here comes my question: What if Player A deviates from the equilibrium by choosing different bet frequencies but deviates symmetrically? This means that if one hand bets 80% rather than 67% of the time (13% more betting) and the other hand bets 54% rather than 67% of the time (13% less betting), Player A deviates from the optimal mixing frequencies of each individual hand but keeps the overall betting frequency of the whole range fixed. Does an exploit exist now for Player A? My personal take is that there shouldn't be one. We often get taught that deviations from game theory optimal play allow for exploitation (like in Scenario 2). But wouldn't we need to refine this to saying that this relates only to asymmetrical deviations, whereas symmetrical deviations are no problem at all?

In other words: Do symmetrical deviations from mixed betting frequencies lose EV in GTO?

2. Experiments in GTO+

So I've played around a bit in GTO+ to investigate this and for simple toy games the intuition above indeed seems to be supported. The game is AQJ vs. KK (one street), so IP has a perfectly polarized range composed of nuts and air and OOP has a perfectly condensed range consisting of only bluffcatchers.

Equilibrium Solution
For the parameters that I've set, this is IP's range construction in equilibrium. The EV of the game for IP is 0.5555. The overall betting frequency is 55.6%. The value region always bets (A) and the required bluffing frequency is equally distributed among the two bluffing candidates Q and J (no removal effects). IP has a total of 4 bluffing combos (12 combos, each betting 33% of the time, 12 * 0.33 = 4).

Experiment for Scenario 1: OOP never adjusts
Here, IP commits mixing errors but OOP's response is node-locked to the initial equilibrium strategy. The EV of the game for IP does not change, still 0.5555.

Experiment for Scenario 2: Asymmetrical Deviations from Mixed Frequencies
Here, IP commits a slight mixing error (betting 63% of the time globally) and OOP is responding freely. As IP is bluffing slightly more than he should, OOP is calling all of his bluffcatchers (100% calling, 0% folding). The EV of the game for IP now now reduced to
0.4074, OOP is exploiting.

Experiment for Scenario 3: Symmetrical Deviations from Mixed Frequencies
Finally, IP is again deviating from the equilibrium but does so in a way that keeps the overall betting frequency exactly at 55.6%! J is now never betting, and Q is betting 4 out of 6 times. So as in equilibrium play, IP has exactly 4 bluffing combos again, but these arise as a consequence of symmetrically deviating from equilibrium play. The EV of the game for IP is now back to 0.5555. The symmetrical deviation (betting one hand more than he should and simultaneously betting the other hand less) did not lead to any EV loss.

3. Conclusion
So in this toy game, symmetrical deviations do not seem to lead to EV loss. I'm sure that in more complex games across several streets, they will do. My question essentially is when this will be the case and why... Any thoughts??

Dec. 27, 2022 | 5:09 p.m.

Nov. 10, 2022 | 8:30 a.m.

Hi all, father of a 3 1/2 year old here.

My little daughter recently has put an eye on the poker chips in our shelf and asked me what they are about. I was explaining it to her a bit and we started playing this little poker-alike card game where we simply get a handful of chips and we iteratively get dealt two cards. The better hand wins. So of course for a three year old a huge portion of this game is to simply explore the different picture cards (king, queen, jack), counting the number of symbols printed on the lower cards to derive whether a card represents a 7 or a deuce (or whatever number) and she didn't even pick up on the suits yet :-D

Now I do have to say that she directly got very much overly excited about this game in order to see who "wins" each flip. A good friend of mine (no poker background) now confronted me with this game fostering gambling behavior and there being a reason why games like poker have an age restriction, essentially arguing it would be way too early to introduce a game with clear gambling elements.

Does anyone with older kids have experiences and an opinion on this? How did this go for you?

Nov. 9, 2022 | 1:40 p.m.

This commentary is spot-on, thank you very much for this.

Feb. 1, 2022 | 5:09 a.m.

Nice video! Which country are you from and why aren't you able to play on Stars currently?

Jan. 28, 2022 | 4:52 a.m.

Not sure if I would approach this spot with a solver since our opponent is so clearly out of line that I would just try to determine whether his range is weak enough to call. If the opponent is unknown, I'd go with pool tendencies. Also, Snowie is not a solver ;-)

Against unknown I'm blindfolding myself, crying a little (but very manly), tanking until I have 3 seconds on the time bank, and then I call.

Jan. 27, 2022 | 7:41 p.m.

no it's you

Jan. 24, 2022 | 6:05 a.m.

I agree that calling river would be burning money against a pool that is strongly unblanced towards value when overbetting river and your holding not beating any value hands that would elect to overbet on this river.

A note on your flop play that you hopefully find handy:

You c-bet 50% pot OTF. On paired boards (especially the rainbow variety), small bets (around 1/3 pot, maybe even 1/4) are almost exclusively used. This is mainly because due to the pairing the board is so dry that BB will have a hard time connecting to the flop, as way fewer pairs, two pairs, flush draws, straight draws and backdoors are possible. So BB fails to equalize the range advantage that you carried to the flop.

With a small bet, you force him to call relatively wide in theory. Out of the reasons outlined above, he can't do that since he's simply lacking suitable hands. The next category of holdings he would defend are high card hands, which he largely has 3-bet pre. Those high card hands that remain are dominated by your top region, hence these are also unprofitable defends for him. With a 1/3 bet, you hence force him to vastly overfold, which is a dynamic you should take advantage of as large parts of your range will profit from getting to fold out much EQ for only a small bet. With a 1/2 bet, you're actually making it easier for him, he can now be more selective in his continuing range.

Once you go 1/3, just be aware that this will in turn imply overbet sizes OTT. Your strong hands from the flop could have gone bigger but went into the small size for deception purposes. Hence they now need to get their value on later streets. So 1/3 flop, 150% turn, 150% river will be sizing choices that you see on many runouts on this kind of board.

Jan. 24, 2022 | 5:55 a.m.

Alright, I mean I would have never called extra loose with hands like SCs or one gappers but I always wondered whether setmining specifically goes up in value vs. tight ranges. But I definitely see that overall it's not worth it since we're loosing a lot of EV that we should have with these PPs when bluffcatching or checking down to showdown.

Jan. 24, 2022 | 5:34 a.m.

Thanks for the feedback. Just gonna fold these medium PPs from now on ;-)

So you open for 2.2 so I assume you have some small pairs in your EP rfi. Those are auto folds vs any 3bets.

Next are the mid pairs like 77-99, which are close to 0 ev to slightly positive calls vs optimal 3bet ranges and sizes.

Not sure about this statement. If you look closely into the ranges then the small pairs 44-22 call pure vs. the 3-bet (+EV) while medium PPs are mixing call/fold (0EV). Not 100% sure why this is but this is something that I have noticed in many positional configurations already where we're the OOP defender.

Jan. 24, 2022 | 5:31 a.m.

RunItTw1ce thanks for your advice, will do.

Jan. 24, 2022 | 5:23 a.m.


So this is a question that I have been asking myself pretty often in a session as I can't really get my head around it. Consider this spot where we open with a PP and are confronted with a 3-bet.

Now in theory, many PPs are mixing when facing a 3-bet and being IP (let's just ignore that the 3-bet size is too large for the moment). For instance, this is Wizard's response vs. an optimal opponent. TT-55 are mixing, and 44-22 are pure calls unblocking the bluffing region.


However: What if we're facing an opponent that's obviously too tight? In this hand, MP played 15/12 with 3% 3bet over 178 hands. I can't quite get my head around whether

a) PPs overperform here since the 3-bettor's range is too strong. Hence, when we hit a set, we stack MP way more often than we should since he's lacking all kinds of folds once we hit a set. If we don't hit a set, we can just XF against a range that's too strong.

=> Adjustment: Medium PPs turn into a pure call

b) PPs underperform since a lot of their EV does not come from hitting sets, but bluffcatching on static runouts and checking down vs. broadway cards. This will obviously work poorly against a 3-betting range that's only comprised of the top 3% of hands.

=> Adjustment: Medium PPs turn into a pure fold

Which one of the two is it?

Jan. 23, 2022 | 6:38 a.m.

First of all thanks a lot for reviewing my footage again and the couple of kind words throughout the session. I would have loved a take on my "funky opening sizes", but I get that the EVs between different opening strategies are close and it's probably not worth spending too much time on it. That said, while I'm aware that the EVs are close versus the solver, I do think that there is room for quite some extra EV to be generated when playing versus erring humans.

I agree with almost all of your takes throughout the video. My main take aways from this video (also considering RunItTw1ce's comments) are:

  1. I c-bet too much. Probably best shown by the AcTc hand at 29:45 that you pointed out. I do try to take these broadways that have overcards but will unblock all (backdoor) flush draw folds on later streets into my c-betting range as our range needs these low equity bluffs on all kinds of runouts. But this should be for a low frequency OTF and in the video I overdid it.

  2. I don't call enough vs. small bets in spots where ranges are wide. My rationale in the JcTd hand at 10:40 was that the Jc blocks the missed flush draws and I'd rather call hands without a club, but probably this is not enough to justify a (pure) fold. The fold with 77 at 34:55 was definitely too much.

  3. I put in too much money versus strong ranges with mediocre hands. This showed in the 4-bet pot with AQo at 5:00 both pre and OTF as well as at 38:15 where I'm also not sure where the XR with 87o on AT5r vs. a recreational in UTG came from haha.

Thanks again and I'm looking forward to your review in January 2023 on (at least) NL50! ;-)

Jan. 23, 2022 | 6:02 a.m.

Comment | whiteshark commented on Delayed C Bets

Thank you, again, for this superb video. Your theoretical rigor is out of question and this is why you have rapidly developed into one of my favorite coaches on the site. At times I find myself checking regularly whether the new Josh Lessner video is up already.

That said I do think that more can be done in terms of adding to your videos. The shorter formats in which you reduce your message to a couple of major points are great, but the theory part could definitely be accompanied by some more applications in which you drive these points home some more. Going through one or two boards in Pio is nice, but there is definitely room for hand reviews in PT4/HM3 or drilling the spot in Pio for example.

Jan. 23, 2022 | 5:44 a.m.

I would very much appreciate such a video. I heard Matt Berkey say in a podcast once that the deeper you get, the more room there is for detrimental mistakes (often in overplaying hands) that you can capitalize on if you know the spots to look out for. Would make a great video.

Jan. 20, 2022 | 8:18 p.m.

I enjoyed this video through and through and I have to say that I really like the way that you are talking about poker! It feels a bit like putting some type of sunglasses on that lets me see the at times chaotically seeming output of a solver in a new light that I can grasp and understand.

Very much looking forward to your content in 2022 and may you be even more amazed by the game than you were in 2021. Cheers!

Jan. 12, 2022 | 4:25 a.m.

Update 1:

Poker Results:
One week into the year 2022, here comes the first update that I'm writing. My results at NL25z have sadly been pretty devastating since I started this blog, as I lost another 8 stacks in the last 3k hands. It currently feels like most things are just going against me at the tables and I'm not sure about the diagnosis yet. I'm sure I'm making tons of mistakes, but so do my opponents and I'm just not convinced that my current results represent the EV that I should have at NL25z.

Let's focus on the positive things:

  1. I'm pretty happy about my redline, as it used to be hanging pretty drastically in the past and I now achieve to finish more and more sessions with break-even non-showdown winnings.
  2. I'm very happy with my mindset. I can authentically say that I manage to play my A-game almost every session and have not noticed any kind of tilt yet even in this losing strech. On the other hand, this rules out tilt as a potential explanation for my current negative results. So either it's just a stretch of negative variance, or I don't have a skill advantage over my opponents and the rake.

Here are two exemplary hands of today's session that are representative of the last week:

In the AKo-hand I expect to lose at showdown the majority of the time. Against the 40% potsize bet I only need to be good around 20% of the time though. In game I thought that unblocking all the the AsXs flushdraws this is a call, whereas holding the As myself I'd opt for a fold. Also, I think there is at least the slight possiblity of the BU value betting some AK himself against AQ/AJ which take my line as well.

In the QQ-hand I have no idea what the minraise on the turn is supposed to mean, but since BB coldcalled my 3-bet preflop I suppose he's a weaker player and I'm unblocking all FDs. From a theoretical standpoint, I'm way to high up in my range to fold, but even when we're just playing our individual hand against a recreational, I don't think we can get away here.

Poker Study:
My study has been focused around MMAsherdogs's ultimate cash game course's section on 3bet pots IP. Ultimately I was able to resolve a problem in my bluffing range construction that I often referred to as the "fundamental dilemma of bluffing range construction" in front of myself. The "dilemma" just points to the fact that in a polarized c-betting approach, we prefer to c-bet non-made hands with equity (FDs, BDFDs) but on static runouts, these are exactly the holdings that block our opponents' folds. Since I had constructed my polarized c-betting ranges like this in the past, I was often lacking suitable bluffs on turns/rivers.

The solution that I have prepared now revolves around "low equity bluffs" merely consisting of overcards that
- block the opponent's top pairs but
- unblock his folds (FD and BDFD floats) on later streets

These low equity bluffs only get into our c-betting range on low RNG numbers (indicating aggression) and predominantly barrell through on static runouts. This morning's sesssion provided a nice example of this line (RNG <30 on both flop and turn):

Career and Family: This was my last week of vacation and we're currently preparing the birth of our second child. My wife is planning to give birth at home after she had a Caesarean section with our first daughter. I'm strongly rooting for her that she will manage to leave the experience of her first birth behind and I think she assembled the right team of people around her to get this through. I'm looking forward to supporting her, and can't wait to get to know our second daughter.

Jan. 8, 2022 | 7:13 a.m.

Thanks a lot for the video!

2:50 CO vs. BU 3BP with 55, what is your reasoning to fold with 44 but call with 66 against most opponents? If I check GTO Wizard then all of these medium to small pocket pairs are mixing vs. 3bets, whereas 33/22 are pure calling as they unblock all of the BU's bluffs. Thanks!

Jan. 7, 2022 | 5:01 a.m.

Thanks a lot! Update 1 coming soon.

Jan. 5, 2022 | 6:41 a.m.

Regarding meditation: What I'm doing is basically the Vipassana practice as inspired by several Western scholars/authors centered around "secular buddhism". It mainly centers around neutrally observing the mind (thoughts, emotions, triggers, physical pain) rather than working with specific mantras. I try to do this every day, though often it will really only be like 5 minutes.

Who I find most inspiring is the figure of Joseph Goldstein, co-founder of the Insight Meditation Society. These are two videos of him that I regularly re-watch :-D


Jan. 5, 2022 | 6:39 a.m.

Very much looking forward to the Kanu course though, I have heard quite some good things about it.

Jan. 5, 2022 | 6:29 a.m.

So I haven't started the Kanu course yet. Together with a study partner, I am currently still finishing up MMAsherdogs ultimate cash game course. We are meeting twice a week and we're basically taking the video lectures as inputs for our studying. MMAsherdog's course is solid, though you definitely have to get used to him just talking at Pio output for an hour straight and I think it definitely needs to be accompanied with a lot of follow-up self-study.

Once we're through with this course, which should happen in February, Kanu is next on the list.

Jan. 5, 2022 | 6:29 a.m.

Thanks a lot, man! I already thought that you would spot this journal haha ;-) Likewise much success in all your endeavors

Jan. 5, 2022 | 6:26 a.m.

Background/Personal Life

It has been quite a while since I wanted to start this journal but I feel like with the beginning of the new year 2022, the time has finally come to get this going ;-) I'm 27 years old and from Germany. I have been playing poker since my early adolescence with my friends, although I never felt that I had a particular talent for it or was particularly good at figuring out its strategic elements on my own. What I remember is just my 14 year old me spending hour after hour in front of my computer being absolutely mind-blown by what I saw on shows like High Stakes Poker, Poker After Dark, or the WSOP coverage. The figures, the personalities, the noise of the chips, the way people moved… Once I encountered this game, I was immediately sure that I wanted to be part of this community and world.

Several attempts in my adolescence at studying and succeeding at poker (mainly through pokerstrategy.com back then) remained unsuccessful. I always listened to the huge success stories of the few youngsters that immediately started crushing their way up to high stakes and couldn’t cope with the realization that I was not one of them. Today I know that the high school version of me just had never learned how to systematically approach a complex game such as poker, how to break it down into a bunch of smaller pieces, how to focus on long-term goals rather than rapid results, how to emotionally handle bad swings etc. I needed to grow significantly older than many other people need to in order to taste (at least parts of) these skills. It’s funny that back then, I mentally “put poker aside”, although I always knew that the moment will come for me to work through and succeed at this game.

Enter 2021. Eight years at university (where I’m currently acquiring my PhD) have taught me much of the things that I (at least think I) needed to learn in order to study and endure in a game like poker. I’m not sure what happened last year, but when the pandemic hit, I changed a lot of things in my life: I decided to quit academia after my doctorate and start working in the private sector in data science, a field that I’m dedicating large parts of my professional endeavors to. I started working out several times a week, stopped drinking alcohol (I definitely had a drinking problem in the years prior to that), started meditating almost daily, and stopped eating animals. Also, the moment had come for poker to return.

Poker Ambitions
I started out almost exactly one year ago at NL2. Since then I had an absolutely inspiring time grinding and studying this game, playing around 100k hands in 2021, moving from NL2 to NL25, and maintaining a study-to-play ratio close to 1:1 throughout the last year. From a passive poker consumer, I finally turned into an active player, playing short high-focus sessions almost every day and studying around 4-5 times a week. This is the graph of my year 2021 on Pokerstars. As you can see, NL25 is going rather meh, but I’m working on it.

Results for 2021:

It is not my goal to one day cancel my professional career for the sake of poker. I want to keep stable finances around my day-to-day job and excel in poker as much as I can next to my career and my family, soon to be enlarged by our second child. This means to rise to mid stakes online and playing as high as I can live. The game is NLHE Cash. My way to achieve this are daily grinds and daily study sessions of around 1-1.5 hours each. I’m a firm believer that it’s consistency that will get me where I want to be, not throwing in 10 hour sessions when I’m passionate and then leaving everything behind when I’m running badly in poker or life.

Focus Areas and Goals

- Beat NL50z by the end of 2022
- Rigorously work through the Kanu High Stakes course by the end of 2022 while throwing in lots of tangents and own investigations in the solver
- Connect to more people in order to expand my poker network

Personal Life
- Finish my dissertation by the end of 2022
- Continue my lifestyle in terms of sports, meditation, and diet
- Live with love and respect for my wife and children and be there for my family whenever they need me

Why this Journal and what to expect?

What to expect? This is not going to be about daily results and graphs. I’m going to post results regularly, but posts will often revolve around reflections on where I stand in terms of the above outlined goals. Also, I will map out theoretical ideas that come up in my study sessions.

Why this journal? My main goal is to connect to people, learn new things, and expand my poker network. I have no idea where this journal will lead to but I’m absolutely sure that good things will stem from it.

Let’s fucking go!!!

Jan. 2, 2022 | 9:13 p.m.

RunItTw1ce I think the confusion about ranges stems from the fact that you treat this as BvB 3BP, whereas the ranges in the video are for SB vs. BU 3BP :-) In the screenshot from Wizard you can see that you modeled SB vs. BB. In the video, OOP range has more AA-JJ/AK since OOP is the 3-bettor, not the defender.

Dec. 8, 2021 | 8:32 p.m.

Thumbs up, great video! I really like the long dedication to your slides where you are explaining the theory with only few illustrations in Pio. You can really tell that you understand the dynamics of the spot, which stands in contrast to quite some other videos where coaches basically just spend the entire time talking at Pio output, often being as surprised as the viewer about what they see.

Seeing that river jam with a medium pair like 77 63229 was definitely new to me and something I will need to incorporate into my game.

One question: At 2:30 you mention that the polarity of the IP range will drive the IP barreling frequency up by a good margin (polarity -> frequency). I'm usually applying the rule that the EQ of our range drives our betting frequency and the polarity of our range drives our sizing. Am I misapplying this heuristic?

Dec. 8, 2021 | 8:19 p.m.

Josh Lessner RunItTw1ce I would very much appreciate a video on drilling this spot as a follow-up

Dec. 5, 2021 | 3:10 p.m.

Something I stumbled upon recently is that when I'm looking through GTO wizard's preflop ranges, it seems like the BB is actually defending a wider range of hands vs. early position raises than when facing a raise out of LP. Below you some of the ranges and defence frequencies. As you can see, BB is defending more vs. MP than vs. CO/BU opens.

I'm wondering why this is, as intuitively you would expect BB to defend wider vs. a weaker opening range. My hunch is that it's a consequence of MP using a smaller opening size.

  1. This simply offers better odds for a call and BB needs to win the pot less often postflop to call profitably.

  2. Smaller sizes lead to lower SPRs which improves equity realization for all the medium/low suited hands that find a call vs. a smaller sizing. Although this hand class should in general be underrealizing oop, with an SPR implied by a 2bb open they can realize their EQ better than with an SPR as implied by a 2.3/2.5bb open.

Any thoughts on this?

BB defence vs 2bb MP open (continuing 43.5%)

BB defence vs 2.3bb CO open (continuing 36.3%)

BB defence vs 2.5bb BU open (continuing 39.1%)

Dec. 3, 2021 | 4:19 p.m.

I already liked the Pio vs. Population series, very much looking forward to how Exploretation will go! And of course, the French guy was nuts ;-)

Dec. 3, 2021 | 2:36 p.m.

Thanks a lot, this definitely cleared things up for me.

I think the key to understanding this dynamic on paired boards is the last T22 example that you gave. As I understand now, small sizes are not a result of the PFC outflopping a lot of the PFR's top range pre-flop through making trips (which I wrongly thought before). Rather, small sizes are a reaction to both players missing paired boards a lot, which is why PFR can now force PFC to overfold. The degree of overfolding that takes place increases if a small sizing is used, as PFC needs to defend more the smaller the sizing, but just won't have enough hands to defend the appropriate frequency implied by 1-alpha since PFC (i) misses the board a lot (ii) and his high card hands are dominated.

Dec. 1, 2021 | 8:38 a.m.

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