Let’s do this together as an exercise in poker game theory. This is one complete street of hold’em poker, on the river, two players in the hand, with a stack-to-pot ratio of 4. This is for starters, once the methodology becomes clear, then all the variables can be altered later for lower SPR, different boards, etc.

I have done most of this before, but not with the level of detail in combinatorics and blockers relative to SPR that I hope to get to in this thread.

All that means is that I know the method, but am always capable of mistakes and call me out when you think you see one.

All I need is for you all to participate, a little or a lot, whatever is interesting to you.

1.) I will set up the board and the previous betting sequence in the hand.

2.) Then I need input on what reasonable ranges are to assign to each player.

3.) Then, step by step we walk through the optimal max EV GTO strategy for both players and decide the value of the game.

4.) Then, the hard part, we break up the ranges of both players according to every combo in range and including blocker effects.

5.) Then we compare our results to solver simulations and discuss the differences, if any.

6.) Lastly, we discuss possible errors in our model and ways to improve the model, such as moving the whole analysis back a street to the turn.

Opinions and feedback before we begin?

1.) Setting up the board and the action.

Effective starting stacks are 130bb.

Table is 6-max no limit hold'em.

Action folds to the CO, who opens for a bet of 2.5bb.

Folds around to Big Blind who 3bets to 7.5bb and CO calls.

Flop (15.5bb) is Jc 7h 6d

BB bets 7bb, CO calls.

Turn (29.5bb) is Jc 7h 6d 2d

BB checks, CO checks...

River (29.5bb) is Jc 7h 6d 2d 7s

So, given this board and the action, what are some reasonable ranges for both players to have? There is no wrong answer, the ranges are just necessary for meaningful results. They can always be altered later.

Here is a link to a Discord server for poker chat and discussion, The server is called Degenaments and this is on a channel called robertsrubix. If this sort of thing is for you, feel free to join.

https://discord.gg/fXdjwp

Feedback so far is that BB should be raising larger, otherwise the ranges will stay too wide. So now the BB 3bets to 9, which the CO calls.


Updated action:

Action folds to the CO, who opens for a bet of 2.5bb.

Folds around to Big Blind who 3bets to 9bb and CO calls.

Flop (18.5bb) is Jc 7h 6d

BB bets 8bb, CO calls.

Turn (34.5bb) is Jc 7h 6d 2d

BB checks, CO checks...

River (34.5bb) is Jc 7h 6d 2d 7s

Players' effective stacks remaining are 113bb.






Also, feedback is that to get to the final ranges requires each action in the hand has to have ranges, which is true of course. So, start at the CO open. Upswing CO opening range is:

22+, ATo+, A2s+, 45s+, JTo+, K8s+, Q8s+, J8s+, T8s+, 97s, 86s

This looks like this:



This is 330 combos (24.89% of hands)

Really interesting idea. I'm going to check back in a bit later but one suggestion I'd make is making the situation a bit more normal in terms of stack size and 3bet size, although maybe that was your intention to make the situation harder

There is probably going to be some debate about the range that BB uses for its 3-bets. My own strat has a fair bit of mixing, and (looking at the board) the post-flop strat will be affected somewhat significantly by including (or not including) A7s, 77 and 76s in it.

I'm happy to go with Upswing's suggested 3b range if you want to post it.

Lets use yours Arty. I don’t have a paid membership at Upswing. The process is the fun part. Everyone’s ranges will be different.

OK. This is what Snowie's online pre-flop advisor (which is slightly out of date, but still reasonable) recommends for the BB's strategy:



If we want to use pure strategies (to make things clearer), a simple way to do it is to just 3-bet the hands that Snowie goes with >50% of the time. That leads to a 3-betting range that looks slightly odd:



It's curious that Snowie more frequently 3-bets 77/66, but calls 99/88. Go figure.

OK. So do you want to use this 9.2% range? TT+, 77-66, A2s+, KJs+, QTs+, JTs, AKo

What do you mean by out of date?

I would replace JTs with AQo as a standard play, what do you guys think about that?

Yeah I would just call pre with ATs A9s A8s
3! AQo, KQo for value.
And add some weaker suited 1 gappers to the 3! Bluff range: 75s, 86s, 97s etc

Here is Arty's pure strategy approximation of Snowie, plus a few suited connector steals, added AQo, and minus two of the weaker suited aces.



Now, what does CO flat call with?

How about this below. Imo, it should be mostly linear, any holes would be for 4b resteals (I was thinking CO could be using 55 and A5s for that purpose).

I believe the online and app advisor uses ranges that the full Snowie program was recommending about 2-3 years ago. When the artificial neural network did some more "learning", with the addition of the quarter pot sizing (for both pre and post), the pre-flop ranges changed slightly.
e.g. In 2013 (and on the online pre-flop advisor), Snowie would open J9s UTG in 6-max with an EV of 0.01bb. After the additional learning period, it reduced the EV of J9s and it became a fold. (An even earlier version of the AI even had QJo as an open UTG).
It doesn't make much difference to me. I think a lot of players mix with those hands, while others are adamant that one is a 3-bet and the other is a call. The same goes for the weaker SCs. All the edge cases work fine as 3-bets or calls, so it's pointless arguing about it.

We could have similar discussions about the flatting range. I don't think KJo and QJo should be in it and would rather call with 87s-65s. I don't think 55 is a good 4-bet. It's a call for me. I actually flat queens in that spot too, but I'm on board with 4-betting it if you want.

It's kind of interesting that this exercise goes to show we shouldn't trust someone that says "The solver says XX hand is a call on the turn", because it depends so much on what the pre-flop ranges are.

EDIT: Snowie hates folding to 3-bets, and it often flats KK. Go figure.


Amusingly, it flats the 3-bet with hands that weren't even in your CO opening range.

Are the newer ranges available on their paid platform or anywhere online?

Yeah SCs in particular will probably vary a lot in a mixed strategy

Just on the paid platform. My subscription ran out some time ago, so I can't tell you exactly what they are.

P.S. They also vary according to the stakes (due to rake, obv), but the online pre-flop advisor strats were either for 100NL or 200NL. I can't remember. Can we assume this thread is about a 100NL hand, just to make the maths easy? If it was for micros, then the ranges should be a fair bit tighter.

Sure. NL100 it is. Modifications to the CO flat sound good. We can go with that, now we need to eliminate the combos that would bet turn, since our turn action is checked through.

Whoops, I missed Arty’s addendum wrt to Snowie calls. Arty, please decide on the CO call range so we can proceed.

We can use the flatting range you put up. This is only an academic exercise, after all.

Very true.


It would seem that Snowie preserves nut advantage by keeping lots of combos in range, while not sacrificing too much strength of range by only calling at mixed frequency.

Theoretically, any call in holdem can be made this way, even on the river. Within broad parameters, the caller always reserves the right to mix calls at frequency, while not giving up a shred of EV.

Does Snowie ever make river calls at mixed frequency? Very interesting!

Ok here is an approximation of a CO call range, inspired by Snowie. This is a weaker range than Snowie would actually have, but calculations involving mixed strat combos are too complicated for now. (We can think about doing those after the method for pure combos becomes clear).



Now we are ready for the flop action.

BB has this range below and bets 8bb into a pot of 18.5bb

Flop (18.5bb) is Jc 7h 6d



What combos would oop BB absolutely check, so we can remove those from the oop BB range on the turn.

Do you mean at 100% frequency? All I can think of is the A3s without bdfd, all the other hands would probably be bets.

Also I guess if we're playing polarised we can check JTs to have a top pair in checking range so removing that too?

Worlds of strategies will collide!

I like your strat, but lets keep some polar combos as checks. Maybe those would be checkraised or folded. Any case, *some* strength in check range is needed, imo.

Yes. It happens an awful lot. You don't just call with one bluffcatcher and fold all the others. You need to have several in your range, and that results in mixing.
In most cases, a specific combo on the river will be mixed between a "breakeven call" and a fold, but occasionally a combo will be raised as a bluff at some frequency and folded otherwise. I'm not entirely sure, but I think there might even be situations where one combo mixes all three options: calling, bluff-raising, and folding.

Ha! Fascinating confirmation of original Game Theory as presented by John von-Neumann.

Truly wonderful, the mind of a AI is.

I think it's probably best to c-bet all the sets (including JJ) and 2prs (76s) instead of x-raising. Overpairs also bet. I think I'd almost always check TT, and use JTs or QJs (if not both) as bluff-catchers. I'm ambivalent about A7s (second pair). Could be a bet or check.
AKs with BDFD might be viable check-calls as well, but that board is so good for BB's range that his range can bet it a LOT.