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Balancing ranges is necessary to obtain an unexploitable strategy, that much I think is a fact.

What I was wondering though is the pros and cons of two different balancing approaches:
1) Balance value raises with bluffs (pure or semi) and keep the medium strength hands in the calling range
2) Balance all hands across all actions.

Analysis:
1) Seems to be the popular approach among pros and it's probably easiest as you can just drop the bluffs if you are playing against a non-thinking player that won't adjust against you anyway. However, I wonder if the polarization of the range that happens because of this is actually a good thing.
2) To me this seems to be the true unexploitable strategy. If we balance our entire range across all actions we never reduce our range depending on our action so it becomes really really hard to play against us. On on hand, this adds more medium and low strength hands to our raising range so I wonder if we get bluffed more. On the other hand, this adds very strong hands to our calling range which reduces the amount we get bluffed in those spots.

Discuss.

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Cheers.

Assume the following spot: we are on the flop in a 3b pot and villain cbets. We have a little room to play, but not much, so our options are: call, raise/call and raise/fold.

The problem with approach 2 is: you take hands that are profitable to call, but unprofitable to call a shove (i.e. we raise/fold) and turn them into a bluff where we have to give up equity too often. In approach 1, the hands we fold wouldn't call anyway, so we didn't loose any equity. In general, you take the hands you have a value raise and raise them and balance with those hands that cannot call. From those hands, you just take the best ones, which might be a bottom pair, a gutshot or an overcard, just in case you get called.

So would you say the following should always be true in terms of hand strength/ranking for 1:
raise/call hands > call hands > raise/fold hands > fold hands?

_________________
Cheers.

no

take our example from above and a flop like As6s5h. Depending on the pot odds, we might call a hand like 7h4h but raise fold a hand like 33. Hand strength is a bad measure here: it assumes villain has a 100% range. Given that, 33 is actually pretty decent on this board. However, people don't have 100% in most spots and against any reasonable range here, 33 is just horrible as its either behind or will be behind a large % on the river, while other hands like 87 might not be strong handstrength-wise, but perform still pretty well against their range as 6 outs make us the nuts and thus beat his complete range and 2 other outs still are pretty decent versus his range.

Well let's assume we use a ranking against our opponents range in that spot, is the above true in that case?

For the fun of it I calculated a few scenarios:

Read: 33 has 47.37% of winning against a 100% range.

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Cheers.

This helps quite a bit because it solves one issue (static 100% range). Now it depends how you define the HS ranking. If we take traditional hand strength, i.e., 5 card hand ranking on the flop, I can still imagine cases where a hand like 98o has a higher ranking compared to 87o. EHS or EHS2 don't have these issues. In this case, I'd think that EHS is enough. EHS2 is imo just a heuristic to value implied odds hand higher to fix the symptom of a 100% range assumption.

Let's keep the "how to come up with the ranking" part to the other thread.

Given a perfect ranking, would this statement be true for a perfectly balanced strategy:
raise/call hands > call hands > raise/fold hands > fold hands? Can we assume a pure strategy for each of those ranges?

This obviously is completely unbalanced:
raise hands > call hands > fold hands.

But the way I imagine #2 to work is this:
raise & call hands > fold hands, and then come up with raise/call percentages for the entire range.

_________________
Cheers.

If I knew the answer to this question I wouldn't be spending my life try to find Nash Equilibria with gigantic numerical models

How's that coming along?

The problem with approach 2 is: if you have a hand which is in the call range, it means that its +EV to call but not +EV to raise/call. So we'll raise/fold it in approach 2, giving up equity, while in approach 1, we take hands which would fold anyway, but are profitable to raise/fold and so add to the overall EV.

I would assume a range should be balanced like this:
Sort your range against your opponent's range (from strongest to weakest).

slowplay-call
value-raise
call
bluff-raise
give-up

Or if we are first to act:

slowplay-check
value-bet
check
bluff-bet
give-up

But I'm just speculating tbh.

Polarizing your range is critical to a bots success imo. I have had to do this to keep the bot unpredictable both preflop and postflop. You have to be careful to balance out the entire range however. You have to check back (or check raise oop) strong, medium and air/weak hands equally. If you check back the medium hands all the time (which tends to be the norm for pot control) a perceptive villain will pot the turn putting yourself in a difficult decision. Keeping the "calling range" polarized is key. So I think balancing all hands across all actions is the best way by far but yeah its pointless against a non thinking player.

The other thing to think about is betting/raising on a dry board which complicates it further. Its hard to get much credit on a dry board as you just cannot rep much.

Have you seen my post? That's why I decided to put our best x hands in our calling range. We use our very best hands for this purpose because those hands are least likely to be outdrawn. And since we play NL (right?) we can get all our money in later anyways.

For more than one betsize, I'd expect a solution like this:
(Let's asume 2 different possible betsizes)

Slowplay-Call
Value-Raise Small (Pseudo Thin Value)
Value-Raise Big
Value-Raise Small
Call
Bluff-Raise Small
Bluff-Raise Big
Fold

Obv we could implement more betsizes the same way. I believe a pure-strategy-equilibrium would look like this. If I was true, we could solve games much faster, since we would only need to get the percentages right.
What do you guys think? Am I too optimistic?

If you choose to ignore the fact that ranking hands in only one dimension can't account for card removal, the nash-ranges would be: raise-raise, call, raise-fold, fold. That is from strong to weak hands with no range splitting across bet-sizes and only one round of re-raising remaining.

This is something that can be solved for a vast set of different rules. However, ranking in one dimension and selecting ranges based on that is bad. Even when using methods better than EHS and EHS^2 for ranking, what hands to fit into each bucket depends on the other hands in that bucket (nut-blockers etc.).

So, as long as these effects can't be accounted for, I would believe balancing all hands across all action to be superior when calculating strategies (sadly).

Let's assume I solved the hand ranking problem in a way that solves the card removal effect problem.

I was thinking about the slow playing idea but I am not convinced that is needed/the right way to go. I think I'd rather take the bottom part of my raise/calling range and add that to my calling range than taking the top part. You are giving up equity of those premium hands by not putting more money in the pot, whereas if you don't raise/call the bottom part of that range you are giving up less equity and still have a pretty hard range to play against for your villain.

_________________
Cheers.

I would always raise the same amount to avoid any bet sizing tells but you could randomize that I guess. Remember that the board is very important on whether a bluff raise could be effective so many of those air hands are going to folded unless you decide to float. This will mess up the balance a bit between value and bluffs. The float is another thing that should be added to a calling range %. I am starting to add in more floats on wet boards. Right now I only have two floats that I can easily rep..a flush draw and a broadway straight draw but I plan to add in more.

I like slowplaying on certain boards but calling vs raising on a wet board can be disaster. Raising is usually the better play almost always. I would not put your best X hands in with a calling range all the time as it just going to be board/hand dependent. Nut flush yes, str8 on two flush board..no. You get the idea...

Remember that we're speaking about an equilibrium here. Which means the opponent raises your "slow play" with enough hands to give your premium hands the same value check-raising/check-calling as bet-calling. With one bet-size however, I think you're correct.

Whether you can use the lower part of the bet-calling range for slow play (checking) or if you have to use a higher section in the case of more than one bet-size might depend on your and your opponent's distribution of hands. It will most likely not be the lower section at least, and it might vary between rivers (assuming that's what we're dealing with here). There's not one optimal generic structure, and I also believe it has been proven that splitting up the range in different bet-sizes is part of optimal play for certain situations.

If you want to read more about the stuff, this paper might prove interesting. It deals with finding the correct betting structures for rivers with different hand distributions, then solving the threshold values with linear programming.

Thinking about the whole discussion more, I think there is no optimal handranking. The problem is, that a GTO strategy looks very different for different stack sizes. For instance, given you play super deep, you will have weak hands in your calling range on the flop that will rarely make the nuts, because when you do you'll have tons of implieds. These hands vanish as the stacks decrease. Hence, your hand ranking needs to incorporate stack sizes too which makes it probably similar hard to compute.

Furthermore, I disagree with much that has been said about equilibrium solutions in this thread. For instance, calculating CFRM solutions with many different bet sizes definitely makes use of them instead of converging to a single one - I guess that its a paradim that has been teached as its very hard for human players to find the right betsize in the right frequency. Furthermore, it is true that there are spots where we put monsters in our slow playing (calling) range, but this is also very dependent of the SPRs.

Imo its imo wrong to try to incorporate expert knowledge that way. Expert knowledge can be used effectively for areas like exploitation or good abstractions, but not for guiding a CFRM algorithm to the best solution against the nemesis (this is a thesis of mine which I cannot back up by any math or something, feel free to prove me wrong )