In optimal play does anyone have any idea about the likely ranges of certain statistics, for example went-to-showdown% (WTSD) would say quite a lot about the correct amount of bluffing overall. I think that most ‘winning’ players have a WTSD of say 27%-35%, and a won-at-showdown of 47%-54% [ranges from looking in my database/trawling the web/talking to people], do you agree with these and do you think that two hypothetical optimal players would have similar values?

Part of me thinks that it might be possible to bound this theoretically [perhaps within a rather wide range] because there must be simple counter-strategies that would crush any strategy that got to showdown above some proportion of the time – e.g. the optimal strategy cannot have so many bluffs that it doesnt win against a ‘calling station’ strategy etc [I wonder what its edge in bb/100 would be versus a calling station – this would measure the difference between the bluffy component and the value component].

I know this is possible in some ‘toy’ games like the ones in Chen and Anhkenman, with the so-called ‘golden-mean’ of poker giving the bluff to value bet ratio

I’m sure Teppo and Al have thought about this and I’d be fascinated to hear any comments they’d like to share

Thanks for those thoughts. Do you think that games of top players versus other top players (i dont know…say durrrr versus Phil Ivey) would begin to approach equilibrium? The war of adaptation must surely lead to ‘small oscillations’ about something that must be reasonably close to equilibrium. In the simpler case of HU limit holdem Polaris is thought to be pretty close to equilibrium and clearly the top human players don’t seem to have any edge, however neither are they crushed  by it, so in this instance they must be adapting to play near equilibrium themselves.

One of the reasons I’m asking these rather vague/naive questions is really just the frustration of having read a few books/articles (there are only a few!) on HU and watched a few videos etc many of which contain good advice, tips etc. The trouble is they often can be reduced to something like

‘if your opponent is doing X too much/too little in this type of situation then you can exploit this by doing Y less/more etc’

and this usually seems to be pretty straightforward to understand this in general – except for the fact that I often find it pretty hard to pin down the boundary between too much and too little! So unless I’m facing some sort of charicature player who exhibits some rather extreme traits, exploitation can be tricky – luckily though these people do seem to exist 

Thanks for those thoughts. Do you think that games of top players versus other top players (i dont know…say durrrr versus Phil Ivey) would begin to approach equilibrium? The war of adaptation must surely lead to ‘small oscillations’ about something that must be reasonably close to equilibrium. In the simpler case of HU limit holdem Polaris is thought to be pretty close to equilibrium and clearly the top human players don’t seem to have any edge, however neither are they crushed  by it, so in this instance they must be adapting to play near equilibrium themselves.

I have a couple of quick points on this part.

When some of the best players play against each other, for example durrr vs. Ivery as you mentioned, I believe it must lead to something that is relatively close to equilibrium. If not per hand or even per sessions basis (for example if one player is on tilt and the other is exploiting that), at least in the long run over the course of high enough number of hands and sessions.

Even players like Ivey, who is said to have lower theoretical understanding of poker than other top pros, but who is still the best player in the world due to his ability to exploit other players’ mistakes better than anyone else, should on average play pretty close to equilibrium in such situation due to the resemblance between the game and a process known as "fictitious play", which is a method that should lead to equilibrium when two players take turns in finding the best response against his opponent’s current strategy.

Another theoretical point I wanted to make about top pros not being crushed by Polaris, is that in order to break even they don’t necessarily have to play as close to equilibrium as Polaris does, as long as they can avoid making dominated mistakes that are the ones costing money against equilbirium players.

…The trouble is they often can be reduced to something like
‘if your opponent is doing X too much/too little in this type of situation then you can exploit this by doing Y less/more etc’
and this usually seems to be pretty straightforward to understand this in general – except for the fact that I often find it pretty hard to pin down the boundary between too much and too little! So unless I’m facing some sort of charicature player who exhibits some rather extreme traits, exploitation can be tricky – luckily though these people do seem to exist 

I’m with you 100% here. I think a lot of such advice, despite being true in a generic sense, is written that way because the authors don’t actually know how the game should be played when no information is available about the opponents’ tendencies. If there is no baseline given for you to know what is too mcuh and what is too little, such advice is usually trivially correct in theory, but worthless in practice.

I personally use very little exploitation in my own game, and only focus on that aspect if I face truly extreme players who for example are way too aggressive, or who are overbetting the pot way too much. Otherwise I don’t care about small differences in players’ styles, but simply follow my own poker philosophy where I don’t try to beat anyone but I let my opponents do the losing. And as long as my opponents make more mistakes than I do, this approach works both in theory and in practice.

(I do admit it though that there are many players on this site who make less mistakes than I do, and who are also probably outplaying me without me even knowing it.)

I’ve never studied the theory of fictitious play very thoroughly, so please correct me if I’m wrong. But I though you could construct it in a way that the players take turns at executing the best reponse strategy while the other player does not adapt until it is his turn to use best reponse, and the average of all the strategies used over time would (hopefully) converge towards equilibrium? Maybe there are alternative ways to construct the process that you are referring to?

Notice that I didn’t mean that the latest strategy would necessarily approach equilibrium, only the average strategy as observed over all trials. And in this sense I claimed that two "master-exploiters" playing each other should on average play relatively close to equilibrium, even if individual hands were far from it due to one always exploiting the other.

In practice of course there’s always some limit how close to equilibrium the players’ average strategies would end up. For example in durrr’s challenge against Antonius it seems that they are very even, but have somewhat different styles that can be seen by durrr winning more pots but Antonius winning bigger pots, and that kind of tendency might never change no matter how long they played. Or if one player was significantly better than the other, then there would be no reason for the game to have any resemblance with actual equilibrium.

I think it’s more likely they say "best response is Y so I’ll do Y until it appears opponent starts doing Z" which leads to undamped oscillations a little like you get with 3 player equilibrium solutions.

I probably expressed myself poorly again because I do agree with this 100%.

What I was trying to say is that if we averaged all their hands with some tracking software for example, then their average stats should not be too far from what equilibrium strategy suggests for the same situations. This doesn’t require the players to actually play close to the equilbrium in any given hand, but are free to oscillate around it while trying to exploit each other.

Disclaimer: I am offering opinion off the top of my head rather than a theoretical analysis. I am as likely to be wrong about any/all of this as I am right…..

I just don’t think you can look at the raw numbers without the context of who the opponents were. I’m not sure that there can ever really be a true equilibrium strategy due to the significant differences in the styles of play that will be encountered. The numbers, analyzed after the fact, might give some insight but I’d be very wary of trying to do that. What I’m ultimately saying is that the amount of $$$ won/lost is more a reflection of the key hands being played between players where a disproportionate amount of money is being wagered, IMHO.

I keep notes on all my opponents (and now thanks to Teppo I can import all those hands into PT and analyze them there, too). I rely on my notes from each opponent to help me come up with a plan for how to play each one of them. My numbers vary significantly from opponent to opponent, as they probably do with everyone who does the work to understand how a specific opponent plays. So, my overall stats are more a function of whom I play against and what strategy I am using to play against them. Successful (profitable) HE strategy relies on exploitation. As such, it causes the numbers to swing in the direction of what is profitable against a given opponent. A given pool of players is not likely to produce numbers that would correlate to a different pool of players unless the pools were huge and of approximately even ability.

In the end I just don’t think it’s possible to estimate a good range as it’s too dependent on the circumstances of each matchup.

Hi VAroadstter, I know what you mean, most successful (real money) HU players I beleive do the same (i.e. they go for what game theorists would call exploitative strategies – detecting and exploiting the flaws in opponent strategy).

However there is such a thing as the equilibrium strategy (for HU NL holdem) its the strategy where (when both players play it) neither player can gain any advantage by deviating. So its not the most profitable strategy against any specific opponent, however it does make it impossible to loose (in the long run) against anyone at all – and thats why its so interesting to try and figure it out. Its existance can be proved mathematically (but in a non-constructive way, i.e. the proof does not reveal what it is). I’m interested because (for example) I know ways I can exploit someone who cbets the flop too often, but unless they either do it WAY too often, I’m never sure where exactly these boundaries between too often/not often enough are – same goes for just about any other statistic you can think of. When I started the thread I was interested in a ‘simple’ (perhaps not simple!) statistic that we might be able to come up with some estimates for. For example just looking at the bet sizes relative to the pot gives us some bound of ‘not folding’ due to giving our opponents risk free profits, but this is a low estimate. I wondered if anyone had any ideas along these lines. 

I dont know if you’ve seen any of ‘Dog is Heads’ HU training videos on Deuces Cracked. In one he has a powerpoint slide which says

GTO: Just Say No!

(GTO = Game Theoretical Optimum = the equilibrium we are talking about)

his point being that everyone he’s ever played is so far away from it that its not the best weapon, you are better served by trying to exploit. I think that knowing more about the equilibrium would help us play better exploitative poker too, because we’d know what we were deviating from and by how much etc, you could better exploit your opponents because given a choice of two exploititive strategies against them you could choose the one which itself lost least versus equilibrium etc.

My thought FWIW – I think Teppo and Al are much more knowledgable than I am because they have both done serious analysis themselves – at least more serious than my simple stuff!