Ok that's what I thought. I find kind of situation incredibly difficult to play with any hand. I would rather call blindly in the BB vs an MP raiser with 4 callers behind than this spot with QJs. I have relative position to the preflop raiser.

It's almost impossible to find a hand that would rather see a flop 4 way OOP rather than 3bet and go HU with 2bb extra dead money in the pot.

They way you punish unbalanced preflop ranges is by 3betting aggressively with a strong playable linear range and not allowing them to see flops and realize equity.

In the BB it's a bit different because of the odds, but even then you should 3bet very aggressively with all your good connected suited hands. The notion that these hands should be played as calls is outdated and wrong, especially OOP.

QJs would be a mandatory 3bet both in the SB and BB in the situation you describe.

What is your guess as to what limit Snow could beat online and live?

If you have relative position on the preflop raiser it is worth it. Not in this spot being in the middle or worse.

Hello Mathew,

First of all I would like to thank you for your enormous contribution to poker, particularly theory. The example section of applications I read a couple of years back had quite a transformative effect; your recent book has just been completed for the first of hopefully several times. I look forward to giving applications further attention and exploring more of your work which will undoubtedly challenge and improve my thought processes.

As is often the way questions and comments on your material will typically centre on misunderstandings and differing opinions rather than affirming every fresh insight. So with that in mind I've raised a couple of thoughts persisting after the first read, naturally, though I intend to read it again and may add further comments.



Quote on mixed strategies:

A mixed strategy occurs when the line of two EVs is exactly equal. In theory, this is an extremely common occurrence because if a player always takes the same line with the same hand in certain situation he can be exploited.

This statement left me confused and in some disagreement. If the existence of lines of equal EV is a necessary condition to avoid exploitative situations then equivalent lines of EV would certainly be pretty handy to have; however, it is unclear as to why the presence of exploitable situations would guarantee the existence of equal lines of EV. Or put it another way, the claim appears to assert the existence of exploitable situations (almost) guarantees the existence of an unexploitable counter-strategy realised through the merging of equal EV lines into a mixed strategy.

I would claim, though, that optimal mixed strategies in all likelihood will contain lines with distinct EVs, except I feel in fairly trivial examples.

It follows implicitly from the exploitable property of some pure strategies that superior mixed strategies with distinct EVs will frequently be generated:

If a line yields the same EV regardless of frequency then the opponent is responding with a consistent (or equivalent) strategy: the strategy can't or isn't being exploited.

However, if a line is exploitable then there exists at least one frequency where line-EV alters in response to a change in the opponent's strategy.

Consider the following:

Assume the expected value of LineA when utilised 100% is X {~ EV(Line A | 100%) = X}. We define Line A to be exploitable when used > 80% of the time. So at 80% utilisation there is a change in the opponent's strategy resulting in an improved EV for LineA (since it is less exploitable) which we will set at 1.1X:

i.e. EV(Line A | 80%) = 1.1X

The other 20% is taken up by LineB. We define the EV of LineB when used 20% of the time as Y:

i.e. EV(LineB | 20%)= Y

Now for the mixed strategy to be superior than the pure strategy we must have:

0.8(1.1X) + 0.2(Y) > X

i.e Y > 0.6X

So the reduction in exploitability creates a slack in EV allowing the possibility of weaker lines to combine to provide an overall mixed strategy better than the pure strategy.

Intuitively, we'd find it quite reasonable to expect the EV of betting-the-flush-draw line on the flop to be markedly higher when bet at 5% rather than 95%. But what we gain in price we lose in volume: it is unlikely our vast number of checked actions will be sufficiently compensated by the rare but generously paid-off flop-bet flushes. But a say 80:20 distribution in favour of bet-flops, as Mathew suggests in the book, might well outperform the pure strategy. But as I've hopefully demonstrated, we can in theory do so through a mixed strategy with differing EVs.



Why we bet and raise:

The caution I and possibly a few others experience with the reasons to bet or raise is the fear of developing a faulty heuristic. With heuristics we're looking for effective methods to establish criteria for specific decisions. The effects of denying equity and creating a larger pot if we win will, naturally, often exist in many losing actions too. The danger (I feel) is that we may instantly pattern match in some charged situation and use it as an excuse to act aggressively - suiting our emotional state or viewpoint and falling into the confirmation bias trap as we tick the reasons-to-raise boxes.

Certainly though, I think you're right, these are the very insightful and distilled effects we're interested in.


My attempt to establish criteria for the two reasons to raise bet/raise you defined:

When bet or raise we reduce our opponent's range in order to achieve one or a combination of the following effects:

To gain more EV through the reduction of our opponent's equity than is lost to the surviving equity (much of which will benefit from our action);

To make more profit from our opponent's narrowed range than from the broader range surviving the passive strategy.

It probably isn't an especially tight definition. The first definition highlights the trade-off we're interested in when we're contemplating reducing our opponent's equity; the second addresses our desire to creating bigger pots when we're in particularly strong positions (although not always strong hands of course).



Hand Discussion:

Finally, in Hand 7 of one set of examples you mention the intuitively surprising preference poker snowie holds nut flush runner runner draws where just Ace of the suit is held over the concealed Ax(s) backdoor flush draws. I found it counter-intuitive too. One observation I made though was that when the flush draw arrives on the turn there appears some notable advantages for the open flush draw over the concealed form. With so many nut hands in our range it is much easier to deny our opponents equity and additionally run a very limited risk of being denied our own (or charged more to realise it) through a turn check-raise than when our flush draw is hidden.

Many thanks


ps there is a 'witch' somewhere in the text of the kindle edition :-)

If villain is constantly adjusting to you, he isn't playing optimally. So the example you're using is basically a watered down version of this...

"Imagine you a playing an opponent at NL$200 where you know if you 3-bet 72o pre-flop, bluff him off the superior hand post-flop, and then flip over your cards to show the table while simultaneously saying 'that's how we do it in the pros' while winking at his girlfriend, he'll go into a nerd rage and start buying in for the max and going all-in every single hand until he bust his $10k roll. Should you 3-bet your 72o pre-flop even if you know it's -EV for that isolated hand?"

Meh... sure, but this isn't the kind of content I care to produce. GTO villains don't respond to how we change our gameplan, so we should always take the most +EV against them. Exploitatively we can justify ****ty lines when villain adjusts poorly and this is very easy to prove mathematically as you've demonstrated.

No, you haven't and you can't, because you're assuming GTO robots will adjust their strategies to ours. All you've done is proven "exploitative poker can justify taking a -EV line now for a more +EV spot later," which is true.





For the rest of your post, if you find a better explanation of why to bet or raise that works for you, then you should definitely stick with that one. There is a trade off, and the more precise you want to make a guideline, the more complex it will be and often difficult to use. Do what works best for you.

Hope that helps and best of luck.

I'm not sure if this is directed at me, but I have no idea. I'd rather not guess on something I'm so uninformed on.

Thanks for your speedy reply Matthew,


In the example I mentioned from the book you discussed the problems of betting flopped flush draws in position every time: we risk being exploited.


Now in effect your statement on mixed strategies is claiming that we’re screwed unless we can find a line of equal EV to formulate a mixed strategy - if we can’t, we’re stuck with an exploitable pure strategy.


So what’s the problem with being exploitable? Well, in the example on betting flush draws you state that our opponent will know when we check back on the turn we can never hold a flush if we always bet the draw: we’re exploitable. In other words, he makes money from our predictability through adapting his strategy. So we’re being exploited, it is costing us money.


So the solution is to deploy a mixed strategy. But you argue we cannot do this unless we find a line of equal EV - we’re condemned to hemorrhage money with our pure strategy unless one exists. This is not true. Suppose we can stop this exploitation by introducing a line which is marginally less EV (in this case checking) around 30% of the time. Now we’re not exploitable, we’re not giving him a huge edge when the flush arrives and we don’t have it, all for a tiny price. So clearly the second line doesn’t need to be of equal EV to be of enormous benefit: so we’re just haggling over the price of the inferior line.


The statement that lines of equal EV must exist in theory to avoid exploitation is a strong one - it seems to imply that there is some inherent property in the construct of the game that determines these exist. I cannot see how this can be demonstrated or why it must be the case. If there were no lines of equal EV, would there be basis for mixed strategies? This seems unlikely.


I read a bit about PokerSnowie my recollection is that the strategies are formed through playing effectively against bots endless times, learning, making tonnes of mistakes but eventually creating a GTO strategy.


My guess when reading your comments at one point in the book about mixed strategies was that you reasoned that if PokerSnowie experimented with a mixed strategy of unequal EVs it could easily improve its strategy by eliminating the inferior and replacing it 100% with the superior line (you didn’t state this). That would be the case if the opponent never adapts. But PokerSnowie has arrived at its GTO strategy through playing an opponent which constantly adapts and exploits (as does Snowie) as Snowie takes inferior lines.


If you subtly limit Snowies opponent in a few areas to pure strategies and play trillions of times it will provide a different solution since its opponent can’t counter and so Snowie doesn’t need to be unexploitable in those situations.


Snowie is going to experiment with mixed strategies and it isn’t going to turn its nose up at mixing it with inferior lines. Suppose it experiments 80:20 with lines of distinct EVs. Snowie’s opponent adjusts its strategy and Snowie eventually settles on some EV for this mixed strategy. Now, Snowie decides to experiment with 100:0, a pure strategy (the line with greater EV). Snowie’s opponent over time moves to adapt, possibly to exploit him.


So Snowie’s opponent here will deploy distinct strategies against Snowie’s distinct lines. Which is Snowie better off with? Who knows? If Snowie’s opponent applied the same strategy in each scenario then we can state that the pure strategy dominates the mixed line but since it applies separate strategies we can’t say. As such if the 80:20 scenario with distinct lines of EV is more profitable than the 100:0 then Snowie will obviously reject the pure strategy in favour of the mixed line with distinct EVs. Now Snowie is only going to swap out this strategy for another mixed strategy. So through this method of learning Snowie can easily end up with mixed strategies with distinct EVs.


In the formation of its GTO Snowie, like people, is going to experiment, exploit and compare. The lines it arrives at will be as a result of its attempts to simultaneously exploit and avoid exploitation. Not to purposefully arrive at some approximation of an unexploitable strategy, but that’s where it will end up if it were playing itself. In the formation of this strategy if some line Snowie takes has a weakness to some possible counter-strategy, then it will be found out, its opponent will apply a different strategy, it will try to exploit. It is through the endless deployment of strategy and counter strategy that Snowie arrives at its balanced solutions. As such it is very likely to have experimented with mixed lines with distinct EVs and kept some because it knows its pure strategy was exploited and less profitable.


When you stated that if our opponent is constantly adapting to our strategy then he is not playing optimally, I’m not sure what you mean. Obviously playing GTO is not playing optimally, just unexploitably. Certainly he is not playing GTO if he is adapting. But generally if we are altering our strategy and our opponent isn’t adapting then he isn’t playing optimally.


regards

Slightly random question but did you create a new username solely to post these questions?

One other point. I'm not assuming GTO bots will adapt their strategy to ours, but I am assuming their strategies will allow for the presence of mixed strategies with distinct EVs. These will have been developed because as their strategies formed many pure lines will have been exploited and these*mixed strategies will emerge.

Another very simple way of looking at it: if it is the case that EVs must be equivalent in mixed strategies then even a line which is only inferior at the 10th decimal place, would in theory be rejected. We and Snowie produce mixed strategies for profit, what could we hope to gain from mixing it up that cannot compensate for an infinitesimal difference in line EVs within a mixed strategy?

Once we accept the premise that lines can be different in mixed strategies, then it is simply a matter of how much different can they be (as per my initial example). It's the only way I could query Mr Janda :-)

@Husker, I did create the account because of Matthew's book. I had an account many years ago with an old email address and a forgotten password.

correction:

If there were no lines of equal EV, would there be no basis for mixed strategies? This seems unlikely.

Snowie isn't GTO.

Thanks Husker,

From what I've read on the software, it's not critical to the points I've made that Snowie is GTO - it is essentially looking to exploit and avoid exploitation and thus will inevitably try out mixed strategies with distinct EVs and only reject if there are better pure strategies.

I'm planning on buying Snowie soon to try it out. I'm guessing there are discussion threads here on it.

ps It's the only way I could query Mr Janda :-) was an editing mix up: it was directed at you, not Matthew :-)

Also apologies for some the inconsistency with terminology in the long post, such as casually interchanging 'lines' with 'strategies'. And confusing descriptions such as 'mixed strategies with distinct EVs' where I mean 'mixed strategies containing lines with distinct EVs'. The editing feature here is short lived sadly :-( If it is too confusing, I'll repost in edited form.

I just checked out this thread and this jumped out. I haven't even read this post yet, but I will if I need to.

Many terms, especially mathematical terms, have very specific meanings. You cannot use the word "optimally" to mean "the maximally exploitative strategy," because that's not what it means. You are not the only person I've seen do this, and unfortunately we cannot have a fruitful conversation if you're re-defining words for what you think they should mean, instead of what they actually mean.

Let me know if you'd like me to read your post as it stands or if you want to re-write anything or be a bit more direct with your questions. Honestly, your previous post was quite hard to read, so if this one is as difficult as the last one and you're using words based on what you think they should mean (rather than what they actually mean) then responding is going to be too much work for too little gain for any of us.

So yeah, let me know if you want me to read this post and the ones that follow, or if you'd like to re-write anything or give me a heads up about anything I should be aware of before I jump in.

I understand what is meant by optimally in mathematics. If I'm not exploiting when possible then I'm not optimising my strategy against my opponent: that is a very basic definition consistent with the mathematical definition of maximising a function, which in this case is our strategy against an opponent. GTO strategies are designed to be inexploitable, like the classic Nash push-folds, but are clearly only optimal against opponents with the same strategy. But what I have stated isn’t predicated on us agreeing on a definition for optimally.

You don't have to read it all, you are free to choose. I will reduce to four points with some overlap, you can respond to if you wish:

1. The statement that lines of equal EV must exist in theory to avoid exploitation is a strong one - it seems to imply that there is some inherent property in the construct or nature of the game that determines these exist. I cannot see how this can be demonstrated in theory or why it must be the case.

2. Let's say PokerSnowie chooses a mix strategy containing lines with distinct EVs in some situation (Snowie will try anything, right?). For simplicity let's also assume there are no lines with equivalent EV. Now your statement on mixed strategies assumes that a pure strategy must be superior to any mixed strategy available (since there are no EV equivalent lines to create what you have defined to be a viable mixed strategy). Snowie's opponent eventually develops a strategy to play against its mixed strategy. Snowie moves to a pure strategy, Snowie opponent moves to counter. Snowie's opponent deploys one strategy against the pure strategy and another against the mixed. Snowie has to choose which strategy of the two it prefers. Since different strategies are deployed against Snowie in each case, why should Snowie automatically reject its current mixed strategy with lines containing distinct EVs for the pure one?

3. A very simple way of looking at it: if it is the case that EVs must be equivalent in mixed strategies then even a line which is only inferior at the 10th decimal place would, in theory, be rejected. We and Snowie produce mixed strategies for profit, what could we hope to gain from mixing it up that cannot compensate for an infinitesimal difference in line EVs within a mixed strategy?

4. Finally, in the example from the book you mention the problem with always betting flush draws on the flop: if we never check back on the turn then our opponent will know we don't have a flush and we're potentially exploitable. In other words our opponent can adapt his strategy to take advantage of our pure strategy of always betting flush draws. So very clearly you are stating our opponent will use a different strategy against the pure strategy than against the mixed one. Since we’re facing different strategies in each case, the mixed strategy can contain lines with distinct EVs and be superior to the pure strategy. Why? Because we can’t simply eliminate the inferior line in the mixed strategy and guarantee to generate a better pure strategy because the pure strategy faces a different examination to the mixed one. If they faced the same strategy it would be trivial to improve on the mixed strategy with distinct EVs - just take the line with the best EV and create a pure strategy, but when we do that our opponent adapts.

Hopefully, that's a little clearer.

regards.

bruh no offense but you need to take a break from adderall and ask concise questions instead of writing essays

I can see why you've posted so many messages in such a short time: no offence ;-)

I've asked why mixed strategies must theoretically contain lines of equivalent EVs. That's pretty short. The points I've written in the previous post (not short) assert that they do not have to be.

I can't do a better job of backing up why I believe what I do than is already in the book.

Perhaps a math or game theory PhD could show a hard proof for mixed strategies in a GTO strategy, but that's not something I'm capable of doing.

When we say a given line has the highest EV in theory, no adjustment will take place to lower that lines EV. So none of this adjustment back and forth takes place.

If two robots are playing against each other and not optimal, then trying new strategies and slowly adjusting to one another's game plan will happen. This may or may not result in the final unexploitable strategy containing mixed strategies. But again, the final mixed strategy will only exist if all lines contain the same EV, otherwise we're not at a GTO solution yet (Snowie for example used to recommend mixed strategies when the EV wasn't the same.... so we know by definition we're still pretty far from optimal here, since an optimal player would never do this).

Again, you are assuming a GTO villain will change their strategy based on ours. This does not happen. A GTO villain does not adjust to us in any way, so we will never take anything but the highest EV line. You can go all-in pre-flop 1000 times in a row and a GTO opponent is not changing its strategy when you go all-in pre-flop for the 1001th time.

Yes, because real opponents aren't optimal players.

Against a GTO opponent I would not worry at all about mixing up my play. I'd just take whatever line I think is easiest to implement that's part of a mixed strategy. So, if betting and checking a flush draw have the same EV against an optimal opponent, it does not matter if you take a pure strategy or mixed strategy. The maximum theoretical EV will be the same of either, and you should take the strategy that's easiest to implement.

This was clearer so thank you for that.

Best of luck!

Hi Matthew,

Thanks for the response. This has become quite protracted so I’ll try and wind this up - it shouldn’t need much further comment.

I’m certainly not assuming a GTO villain will adjust. Clearly we don’t need mixed strategies against a GTO, as you mentioned. But the reason we and GTO have mixed strategies is to deploy them against adaptive players, so the question I asked myself: if all we have available are lines with distinct EVs why would we reject these type of mixed strategy solutions when they should in theory beat pure strategies on occasion? Or why would the bot/program reject them as it forms its GTO (when in theory such scenarios could exist)?

I’ve not ventured into any of the software yet, but I will do shortly. From what you’ve said the mixed strategies solutions in GTO always turn out to hold equivalent EVs. So struggling for to find why that is always going to be the case (without being able to look at any of the EV-functions of the lines or performing some sensitivity analysis around the solutions) I’ve searched for some simplistic intuition as to why this arises: ****

It seems reasonable to assume the EV of a line will often be a function of the frequency used against an adaptive player - we expect to see the EV of betting a flush draw @ 5% to be much higher than at @95% similarly for checking it back. Perhaps the mixed strategy function will always tend to maximise when the line-EVs are equivalent*. Eg say betting the flush draw @95% has an EV of 2BB, but @5% the bet-line EV is 4BB. For the check-line @95% the line-EV is 1BB but @5% it is 2.2BB. So assuming the EV functions behave well, at some ratio/mix the EVs of each line will be equivalent. In this case it might be around (90:10) bet:check ratio, if the plots are fairly linear. So perhaps this will approximately correspond to the maximum EV of the mixed strategy. Although it’s certainly not clear that it will always be the case - that the function is maximised at equivalent EVs.

So I would read from this that mixed strategies with distinct EVs can certainly be an improvement upon pure strategies but they themselves tend to always be improved upon (assuming the functions behave well and there is not some step-sized change in the opponent’s strategy).

Anyhow I’ll get hold of the Snowie and the other program and perhaps look into it further.

Thanks for the comments.

* we can obviously create line-EV functions where the EV of the mixed strategy function is not maximised where lines have equivalent EVs, but perhaps those class of functions don’t crop up. *

correction: we don’t need mixed strategies against a GTO, as you mentioned.

you didn't state this.

I think you'll get a better understanding of mixed strategies once you have a play around with Snowie and can see for yourself how often they occur. I think it's quite hard to explain why/how it is the case that the EV of two potential lines is identical (the topic comes up quite often in the Poker Theory forum, if you want to have a look at that; there's also a massive thread about Snowie there), but it's very clear from solvers and Snowie that mixed strategies are very common, and that utilizing them prevents your range from being face up and therefore exploitable.

As a quick example, if the BTN opens 2.25bb at 100NL, then the BB should always 3-bet AK. That option is more profitable than flatting, so it should be done at 100% frequency. In short, for AK in the BB vs BTN, there is a pure strat: always 3-bet. With AQo, however, the 3-bet only happens 38% of the time. Mixing is used, and based on other hands also being mixed, it happens to be the case that 3-betting AQ has exactly the same EV as flatting it. It's a great hand to 3-bet and it's a great hand to flat. (Unlike AA, which clearly maximises EV by always 3-betting. It's too good not to!)
If Snowie chose instead to always 3-bet AQ, and never flat it, it could never flop TPTK on Qxx (or top two on AQx) if it calls pre. This inability apparently makes it slightly exploitable (or at least costs it some EV), so it chooses to flat it 62% of the time and 3-bet 38%.
It's often the case that many hands that aren't "slam-dunk 3-bets" or "obvious" flat calls utilize mixing. By putting hands into the flatting range at some frequency and the 3-betting range at some frequency, the total EV of each range is maximised. At equilibrium, the individual EV of each option where mixed strats are used is identical. Because, after all, if 3-betting AQ had a higher EV than flatting, it should always 3-bet.

EDIT, to add a little bit about post-flop. When it comes to flop action, it's even more common to see mixed strategies being used, and it's often with mid-strength hands. With the top of a range, it's usually the case that you have a "slam dunk value-bet", so you c-bet 100% of the time with the nuts, for example. You might also c-bet 100% of the time with a particular OESD combo (an "obvious" bluff). With something like TPGK, however, there are benefits to betting for value/protection, but there are also benefits to checking back to protect your checking range, keep the pot-size under control, induce bluffs etc. With hands where it's not an "obvious" value-bet, or an "obvious" bluff, mixed frequencies are often used, where you'll sometimes bet and sometimes check. The GTO strategy would require getting the frequency of the bets and checks correct. But in the real world, no one plays a large enough sample size where they can 'exploit' the fact that you bet or check one particular combo slightly more often than the GTO frequency. One combo out of several hundred is only a tiny part of your range.

I unfortunately don't really have anything to add other than what I've already said. I think if I were to respond I'll just be sounding like a broken record, and it's possible we may just not agree on the topic. I've been wrong many times before, so perhaps you're right and I'm wrong and I'll come around to understanding at a later time, but for the time being I'm confident with my thoughts as they're presented in the book. That GTO poker involves many, many mixed strats and every line that's part of a mixed strat will have the exact same EV for that single isolated hand against a GTO villai.

Best of luck!

Great post Arty, thanks I'll have another read tomorrow.

Because, after all, if 3-betting AQ had a higher EV than flatting, it should always 3-bet.

Now this is a point of confusion to me because I would expect the EV of 3 betting AQ to be a function of the frequency it is raised. Similarly for how often we call with AQ. So in the example where you mentioned the cost of our inability to represent the AQ on AQx if we never flat call with AQ, we must surely expect our flat-calling EV of AQ to be a function of how frequently we call with it: it is going to be tough to rep AQ on AQx if we only flat call 1% but if we flat call 20% then we have more options - so we should expect more EV from the flat-calling line if we call 20% rather than !%. This implicitly suggests that our flatting-calling EV with AQ is not fixed but a function of frequency.

Instinctively I feel that AQ can have identical EVs if we create some specific mixed strategy (as I tried to express in my previous example) which GTO-software finds, rather than having identical EVs in some absolute sense which we set into some mixed range. So GTO-software can calibrate lines within strategies holding equal EVs because the EVs of hands are functions of both ranges and frequencies of other lines - it has a lot of parameters to work with.

As you say I am rather blind at the moment so I need to play with Snowie. Anyhow, thanks again it is best I speak more informatively after playing around with the software.

Matthew, that's fine and thanks for your responses.

Wouldn't it be best to move all this discussion to the Poker Theory forum?

Will do. I was on the case but irritatingly was once again thrown out by 2+2 all was lost (bar an incomplete draft!). But I will post again somewhen soon.