To me it seems the micros players are out of balance in two big ways:
1) Too many of their big bets are for value so they aren't bluffing enough, and 2) They are defending too many marginal hands for too long. Another way to say it is that when putting money in passively they are too weak, and when putting money in aggressively they are too strong.

An optimal exploitative approach seems to be to make these adjustments: 1) Overfold to big bets and raises, particularly on turn and river, 2) Bet more often for thin value and increase bet sizing, 3) Bluff less often and only when there are multiple reasons to do so (opponent shows lack of interest, you have equity, blockers, and range vs range is favorable).

Yet when Doug Polk played his bankroll challenge he didn't make those adjustments and he still beat the game. I know he has said that he doesn't need to adjust to exploit poor players, they exploit themselves automatically. My question is this: Where does a GTO player's edge come from against micro players that are out of balance in the way I described in the first paragraph? It seems GTO means we'll be paying off a lot of value bets and failing with a lot of bluff attempts, so it must be getting compensated for somewhere. Do micro's players not win enough pots because they aren't bluffing enough? This seems like it should be obvious but I'm not seeing where their advantage comes from.

First of all, if you feel your opponents are exploitable and they will not adjust to you exploiting them, then playing an exploitative approach will make more money than a strictly GTO approach. So use your exploits if you're confident in them.

A player loses EV vs a GTO opponent - that does not adjust to exploit them - by making pure strategy EV mistakes. In other words, performing an action that is the opposite of what a GTO opponent considers a "pure strategy" or 100% action. 72o is folded UTG 100% of the time, and therefore opening or limping with that hand would be a pure strategy EV mistake.

Let's say the board is A7233r in a sb vs btn 100bb 3bet pot where the GTO player is OOP and has bet every street and the opponent is IP now facing a river jam.

The IP player is indifferent to calling with certain hands as the GTO opponent has a "perfectly" balanced river range. For example, if the IP player has JJ, the EV of call and the EV of fold are both roughly 0 and therefore JJ is folded and called both at some frequency. However, AQ is a pure call (100% and no folds)and calling is very +EV with AQ. If IP were to fold AQ on the river, that would be a pure strategy mistake, and he would be "losing" that EV. Similarly, if IP were to call with a hand that is a pure fold (such as a hand that cannot beat some bluffs of the GTO player) this would be a -EV call and the GTO player would "gain" that EV.

This simple river spot logic can be applied to earlier streets as well but it gets a lot more complicated.

Another example of a pure strategy EV mistake is incorrect hand selection preflop. Certain hands perform better in terms of their playability and raw equity than other hands. In an extreme example imagine an opponent raising a range from CO that folds suited aces, but opens offsuit connectors and 1 gappers like 67o/57o. This will influence the EV of his range not only preflop directly (in terms of having to fold too much to 3bets) but it will also impact the EV of his range on future streets due to the raw equity of his range being lowered, as well as the playability of those individual hands in that range. While a GTO opponent will not adjust his play, the overall EV of the opponents range will still be lowered.

There was a short video by David Alford that had the A7233r hand example in it:

https://www.youtube.com/watch?v=4eKdtqFMZe8

GTO CALLING: If they are only betting nutted hands the GTO strat will lose to those bets by calling the correct amount (which takes into account that they should be bluffing some percentage), but like you said, this means they are not bluffing enough so we win more pots overall. GTO also calls with less hands vs bigger sizes. If the bettor is not betting the optimal frequency matching the bet size then the GTO strat exploits their mistakes.

GTO BETTING: Vs a GTO strat the player calling a bet must do so with the correct range. Any deviation loses ev. Players at the micros who overcall will lose overall even if we are continuing to bluff the correct amount. If they fold too often they also lose. GTO forces villain to call the optimal amount and that is very hard to do for any player.

bolded. im curious how you got from point A --> B.

Doug Polk did not overtly exploit the randoms he was playing, but he absolutely did keep an eye on every single hand that was shown, and got an actionable read versus a few of them. This is why he always pulled up the history to see what they had, and would say, “ yea or nay” on the stream.

Seeing cards is just so much more valuable than seeing frequencies across lolsample size. If I am curious about an opponent, and the call is close, I will call just to see what their combo actually was. If it truly is a close call, then I am paying next to nothing to get valuable info.

If they never bluff, then a perfect GTO solution is winning money all the other times that they check-fold instead of bluff, but will pay off too often when they bet. Over a large sample, all the small and medium folded pots will absolutely outweigh the fewer times they actually have a hand to bet with. Non-bluffers are folding their stack away, bit by bit, and only survive against very bad players, such as low stakes casino patrons. My personal anecdotal experience is that non-bluffers will actually be bluffing some small portion of the time, and will bluff me more often than each other, since I also show my fair share of bluffs. This makes no sense, since I call way more frequently than they do, almost at MDF (usually tighter as standard, but sometimes way lighter vs overbluffers).

Vs underbluffers on the river, I fold a lot, but I call more liberally on early streets.

Vs overbluffers on the river, I call more on the river, but I fold more on early streets.

This is assuming that the opponents have decent value ranges. If not then we’re opening a new can of worms.

do the worms have heads?

In the academic sense a non-gto player loses to a nash equillibrium strategy for the same game by definition.

Only the opposing nash equillibrium strategy reaches the state of equillibrium where no player can do better by changing strategies.

This doesn't tell us where in the strategy the EV comes from, but there are only a few possibilities which are listed below. Note these categories are not mutually exclusive and one could argue some overlap in poker, but perhaps not in other games.

1. Payout structure

2. Player action sequence

3. Information assymmetry

4. Correct choice of strategic options


So basically EV simply comes from using the information given to choose the correct strategic decision with the correct frequency and bet sizing to produce the highest dollar EV outcome for your position.

This is true for all versions of poker, at all stakes.

So in a sense you can always "play the same way" in any version/stake of poker but that doesn't mean the strategy at each individual decision point is the same every time.

My guess is that Polk's strategy was probably slightly different at the lower stakes than at other stakes.

My understanding for limited research and reading on simulations is that we can't really say WHY the EV at any point in a simulated game is the best. We can only prove that our algorithm does infact converge or produce a correct solution.

This is not true.

Pleas explain? Perhaps I mixed terminology when I shouldn't have with GTO and nash equillibrium?

My understanding that in a game with a nash equillibrium, the game equillibrium occurs when both players cannot gain additional EV by deviating from his or her current strategy.

So by definition of the state of the equillibrium, if any player is playing a strategy from that set of equillbriums and their opponent is not then their opponent can do better by moving to the "counter strategy" in the nash equillibrium? Thus they would always be losing by not playing the counter equilibrium strategy.

If you want to argue say that any 1 decision point a particular strategy can do better than a nash equillibrium against an opponent who is also not at equilibrium then I have no objection to that.

But if one strategy is an equillibrium strategy and the other strategy isn't the non-equillibrium strategy will always lose EV by my understanding of the definition.

The loss of ev is greater than or equal to zero.

Yes it is.

Sigh, maybe it should be always reminded when we are talking about heads-up two player zero-sum no rake poker.

This is where a theoretical Nash solution would be unbeatable, only able to be equalled.

I think it’s misuse of the term “zero sum “ for ev distribution after blinds and antes are posted.

Nash equillibriums don't require only 2 players. They just require a finite number of players, a finite number of strategy options, and the condition of non-cooperative strategies.

I concede poker above 2 players provides no such guarantee of total non-cooperation even in a fair game.

Just pointing out the comment about nash being the best strategy can be applied to poker with more than 2 players, with some minor caveats that imho don't drastically change the message.

When all of your loss is all of your opponent’s gain, and visa versa, that is zero sum.

This does not mean the pot has to be exactly half yours.

Although many games are not “fair”, which means many games favor one of the players, while still being zero sum.

I agree.

JG, consider what might happen to your ev if you decline to bluff the river heads up out of position with the bottom of your range vs gto.

You won’t suffer a loss of value because your bluffs will be exactly breakeven.

Agreed. I had no arguments about your comment about the EV gain/loss being greater than or equal to 0. So that is certainly a counter example of a non nash equillibrium strategy that does just as well against a nash equillibrium strategy so I stand corrected.

Card removal will require you to bluff when your removal makes it +EV.

When both players are GTO, the only strat left to either player is card removal.

My current understanding of river betsize and bluffing ranges out of position heads up on the river looks like this:

if I check call the turn, I check river near 100%. Thus I rarely or never bluff there.

if the turn checks through, I bluff, varying my betsize from 1/3 pot with very weak hands up to a pot sized bet with good blocking hands.

with extremely unshowdownable hands from the very bottom of my range, I can bluff at 0ev, which would make the in position player's pure bluffcatchers indifferent to calling or folding. I can do this effectively with 1/3 to 1/2 pot sized bets.

with good blocking hands, I size my bets depending on my showdown value. As my showdown value increases, so does the size of my bets up to (pot).

Is there anything to stop us from bluffing these good blocking hands in addition to the hands from the bottom of our range?

Not vs gto, but perhaps vs an adjusting player.

Just because you cannot gain by deviating, does not imply that you must lose. You could stay equal. And there are many examples of games where this can in fact be the case.

Also, the argument you just put out assumes only 2 players (by your use of "both"). See below.

Firstly, Nash equilibria do not require non-cooperation, as far as I know.

The bolded is actually flat wrong. Once you go to games with more than 2 players, the properties of the Nash equilibrium can change dramatically. There are several reasons for this. One is that the definition of the Nash equilibrium involves all but one player playing the Nash equilibrium. In a 2-player game, "all but one player" means the one other player. But this falls apart with 3+ players.

By definition of the Nash equilibrium, if all but one player is playing NE strategy, the last player cannot gain by deviating from the NE. However, with 2 players, if the deviating player stays equal, it is necessarily the case that the other player will too, and if the deviating player loses, it is necessarily the case that the other must win (assuming a zero-sum game, and "winning" being relative to the NE). But in a multiway game, one player's deviations can actually shift utility around between the GTO players. This means that the entire idea of GTO being the "best strategy" is completely blown up. It is now theoretically possible to stick to GTO and lose if someone else's deviations pull you into losing territory. In fact, it is even theoretically possible for this to happen while the deviating player stays equal with the NE value.

Not in addition to, but these are a subset of your worst hands. They will never win as checks, no sdv, but are +EV since they block many combos that would call the bluff.

So, when turn checks through, the only reason not to bluff river is a read on a non-adjusting value heavy sticky reg. An adjusting exploit player you should be willing to bluff, and get called. You need him to pay off the majority of the time you are not bluffing.

I think what I am trying to say is that, because a real deck of cards produces hands that are dependent, can be equal, and do block each other, this is why an actual GTO solution can not be avoided by playing a strategy pair that features always folding, always calling, or always flipping a coin when faced with a final decision.

GTO doesn't exist yet except for rough approximations of certain situations that require significant guess work on the part of the person running the sims. And as stated above multi-player situations can cause a GTO player to lose.

I don't think the bold is necessarily true, but I agree with the sentiment.

I've won a lot of pots with very weak hands by checking the river. You might be surprised at how much showdown value you have vs players that don't bluff much on the river.


Ok I've thought about this for a few days and this is what I've come up with:

Asymmetric (0,1) distributions, with blocking effect in play on the river after the big blind calls preflop, then flop and turn both check through.

vs

Asymmetric (0,1) distributions, with blocking effect in play on the river after the big blind calls preflop and flop, but turn checks through.

These are probably the two most common bluffing spots for the big blind on the river, but there will be different sets of ranges in play depending on the runout. I would think that the case of (flop check bet call), we will see many more big bets with the good blocking hands than in the case of (flop check turn check through). Also, in both cases I would bet 1/2 pot or more with the bottom of my range, increasing my betsize as my showdown value increases* up to the point where check folding seems more profitable, however the bottom of my range is defined differently in either case because the action is different.

*I think this is strategically congruent with my river value bet sizing philosophy, the stronger my hand is, the more I bet.