Cross posting something I posted in the beginner's forum today for more points of view:

Premodern indifference theory taught us to build value ranges and then add bluffs as the price for our opponent to call got worse. The more expensive it was for our opponent to call, the more we bluffed. The selection of these bluffs was often debated, with one side saying to bluff from the bottom of your range while the other said to bluff with the best draws. After looking into this topic further, I decided that we needed to define some terms that had come into favor not just in the forums, but in the literature as well. Here's my attempt to define flop texture:

http://forumserver.twoplustwo.com/15...xture-1471300/

I learned a lot thinking and posting about this topic, and in that time I also came around to the idea that humans needed to play mixed strategies in order to compete effectively in the modern poker environment.

Then I participated in this thread:

http://forumserver.twoplustwo.com/15...whosnext+mixed

and I found a gem here:

What this means is that the old view of building value:bluff ratios is not going to give an optimal solution. Instead, as solvers have shown, in order to maximally exploit the best counter strategy, a mixed strategy must be used at nearly every decision point. That doesn't meant we mix with every combo in our range, but rather that there will be hands within our range that take two or more actions at frequencies because our opponent's strategy makes us indifferent to our options. The effect is that with draws, we see lots of mixing. With strong hands we see some slowplaying in addition to the many bets we will make with strong hands; slowplaying is extremely important to our defense of overbets. With some bluffcatchers that don't have any draws to go along with the showdown value, we will see mostly calling or folding with very little mixing.

Now, the next step at least for me, is to study asymmetrical range game theory more. I might come back later with some questions but I'm just gonna post this as is for now.

Interesting thread Bob. As you study, can you show some examples of threshold hands, that are the tipping point between pure solution and mixed strategy? Like, how strong does our value need to be, so that slowplaying at frequency becomes more optimal than betting for value.

Are you sure that so many decision points require mixed strat to be optimal?

And, lets make sure that we are all on the same page for vocabulary:

Mixed Strategy: sometimes take different action with the exact same holding in the exact same decision.

Pure strategy: always take the exact same action with a holding and a decision.

So, if I am reading correctly, the premodern system was:

Bluff this spot with this ratio of value and bluff combos.

And modern would be:

Sometimes bluff this spot at frequency PLUS sometimes slowplay value and sometimes bluff with merged ranges, and sometimes fold.

I agree with all of that except to say that the mixing of fold and call is a rarity that I'm not really concerned with at the moment.

Particularly with draws, yes I think this is true.

With slowplays it's about winning stacks in no limit, where in limit we rarely slowplay anything of value because the pot grows linearly as opposed to geometrically, as it does in no limit.

Question time:

In multiway pots, does the need for a mixed strategy dissolve as you add players?

I think yes, but it probably also has to do with the ranges involved and how they interact with the potsize. Contrast these two situations for example:

9 handed 100 big blind no limit holdem $1/$2

utg raises 3x, utg+1 3 bets to 10x, folds to button, button calls, small blind folds, big blind calls, utg calls. 80 bucks in the pot and everyone has about 180 bucks left in the stacks.

compare that with the same action but with shorter stacks. Seems like the need for a mixed strategy goes down to me.

or this action with 100 big blinds:

HJ raises 3x, co 3 bets to 10x, button calls, small blind folds, big blind calls, HJ calls. 80 bucks in the pot again and everyone has about 180 bucks left to play with. However, the wider ranges seem to allow much more room for draws to realize expectation both by checking or betting, thus there will be more mixing than the above situations.

Now the same action with 200 big blinds.

Seems to me that there should be more mixing as the number of players goes down and the stacks get deeper and the ranges get wider.

In real time, we thankfully get to play one hand at a time. So any long term ev loss is negligible from this hand in an individual sense, unless this is the last hand we're ever going to play. I think this means that we can play very straight forwardly with short stacks and strong ranges and many opponents.

Regarding family pots,

A value hand will have to be higher in range, near nutty, to be a bet, and bluffs will be harder to push through. However, draws to the nuts are just as strong and even more profitable multiways.

If a balanced player checks a nfd flop in a family pot, some percentage of the time, this can force balanced opponents to bluffcatch or pay off a flush with stacks.

As for wide ranges and deep stack NLHE family pots, I think that format is so far away from solved, that GTO may not be worthwhile (Even if a solution exists).

If you're going to use examples such as this:

Please back it up with some realistic preflop action so we can discuss it.

I'm inclined to disagree here. In 4+ way pots, my betting range is (value) and I don't think I'm missing much, if any long term expectation with such a betting range.

Well there are standard plays and profitable exploits that we should be making, but understanding why and when to make the standard play is, is pretty much the gto of multiway poker imo. Knowing that we're making an exploitive strike isn't necessarily the source of our ev with a particular exploitive play. However I think that having a good idea of where the margin lies is crucial to understanding the ev of such plays even on an intermediate level.

I have a loose theory regarding how far from the margin we can exploit, depending on which street we're on, the number of opponents, and how many bets have gone in.

On the most basic level, if everything is equal, there is the most room for exploitation on the river heads up. On a deeper level? I think we would have to break it down into categories:

exploitive folds: 0ev exploit to avoid a -ev call vs a value heavy opponent. A penny saved is a penny earned. Very valuable on the river when the pot is large.

exploitive calls: slightly + ev relative to folding, even on the river a bluffcatcher's ev will be zero or slightly positive with rare exception.

exploitive bets: many possibilities for exploitation here. Slightly profitable with low equity relative to folding, but very profitable with high equity relative to folding. Very valuable on the river with value hands, because bluffcatchers are drawing dead. Not so valuable on the river with bluffs.

Bolded to note that I was referring to no limit games, where the bets are much larger. In limit games, exploitive folds facing a single bet heads up on the end isn't the place to attempt exploits with the rare exception of a very value heavy opponent when holding a very marginal hand.

Here's an example of what I'm talking about:

https://forumserver.twoplustwo.com/5...=calling+minds

My response:

To add that now? I'm a mixed strategy guy, but I have this loose theory that vs players that deviate far from optimal lines, a pure maximally exploitive strategy is best. For example, never bluffing the weak draws vs the above opponent, but always bluffing the dominating draws vs that same opponent.

Do we need to, or benefit from having a separate branch of theory designed to maximize what a normal human can implement without software? It seems that some point we fool ourselves if we think we can randomize correctly with a mixed strategy in a live of tracking free situation. How do we know if the error from poor, but human, implementation of mixed strategies is not large enough to make indifference based strategies better in practice.

I think some mixing, even if done at the wrong frequency, is superior to playing a pure strategy.