Ok, this is one for you solver gurus.

Hero is in EP in a 6-max game with 99 and AQs. All players have 100bb's.

Scenario 1 and 2: Hero opens to 2.5bb, MP folds and CO 3bets to 8bb with a GTO solver approved range. BTN, SB and BB fold. Hero calls the extra 5.5bb with both 99 and AQs.

Scenario 3 and 4: Hero opens to 2.5bb, MP, CO and BTN fold, SB 3bets to 11bb with a GTO solver approved range and BB folds. Hero calls the extra 8.5bb with both 99 and AQs.

What is hero's EV in each of these 4 scenarios if he plays like a solver?

What is hero's EV in each of these 4 scenarios if the 3bet sizings are switched around and CO 3bets to 11bb and SB 3bets to 8bb?

Scenario 1 99: 0 EV
Scenario 2 AQs: Pretty good EV
Scenario 3 99: 0 EV
Scenario 4 AQs: Pretty good EV

Flipped sizings: 99 stays around 0 EV, but this is because villain will play drastically different strategies potsflop depending on what 3bet sizing he uses, playing more passively the bigger he 3bets. He plays more passively because he knows our range is way stronger. Is this relevant in real life? Up to you to decide.

AQs gains significant EV the smaller the 3bet

Wow, why is 99 0EV? That seems rubbish. I thought 99 was a slam dunk call of most 3bets, hence why I chose that hand specifically. I guess this is because vs non-GTO humans 99 does really well?

Against a simple range of TT+,AKo/s, AQo/s then AQs has 37.82% equity and 99 has 37.26% equity, so why such a disparity in EV?

I would have thought that 99 would be easier to play in the sense that it is dominated by TT+ but shouldn't lose too much post-flop to those hands if there is no 9 on the board, whereas AQs is dominated by QQ+ and AKo/s and it is much easier for it to lose reasonable money post-flop to those hands when an A or Q flop.

I have some solved monker ranges and they list the EVs for the situations as follows

1. with AQs = -1.37 bb

2. with 99 = -2.465 bb

3. with AQs = -0.525 bb

4. with 99 = -2.46 bb

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These would have to be taken as +2.5bb to each kinda, so the actual ev's are (compared to folding)

1. 1.13bb (aqs)

2. ~0 (99)

3. 1.975bb (aqs)

4. ~0 (99)

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For scenarios 2 and 4 I listed the ev's that are given, but the frequencies are mixed with folding which shows that there is atleast some inaccuracies here as if it was perfectly solved to equilibrium and there was still mixing then it would be listed as -2.500 and effectively be exactly 0ev.

With 99, some of the only good flops are the ones with a 9, in almost every other flop you will face a bet and lose your entire share of the pot.
With AQs, there's a decent amount of flops where your EV is going to be good.
I don't think in real life 99 performs particularly better than in equilibrium, you will face a range bet on flop and have to fold or make a marginal flop peel that's worth 0 on average.
As for the relationship between equity and EV, you must take into account how that equity is realised.
With AQs, every time you flop a gutshot, flush draw, pair, you have high EV lines that get you to the river and allow you to hit your nutted 2p+ (Nutted hands are the biggest source of EV in poker).
With 99, you can rarely flop any back up equity that will allow you to continue and realise that potential turn and river equity, and even when you are able to continue, it's in a very low or 0 EV line which basically forces you to lose along the way everything you win when you hit.

Well put. Also, even though 99 is 0ev, calling it too often can potentially hurt the ev of hands like AQs (as well as 99 itself). Our range becomes saturated with weak bluffcatchers on lots of flops, which causes villain to increase his cbet frequency. So even when AQs flops something like 2 overs with bdfd, it has much less potshare because we don’t get to see a free turn as often.

You can check out the EV's of these spots in GTO Wizard for free.

In your first example, assuming HJ opens 2.5, CO 3bets 7.5, Here's HJ's EV:



You can see that AQs is 1.64bb whereas 99's is 0EV, relative to folding.

Here's an important point though. You've already invested 7.5bb into the pot. So when you call with AQs, your EV, as measured by your stack size at the end of the hand, will be 1.64bb better than folding. In other words, you expect your average stack to be (100 - 7.5 + 1.62) = 94.14bb. This is still better than 92.5bb that'd you'd get folding AQs.

I think it's important to put this into context. "+EV" just means better than folding, but realistically you're still picking between several losing options and trying to pick the option that loses the least.

Really interesting how 22-44 are higher ev calls than the mid pocket pairs. My interpretation is that the flop is more disconnected on average with a 2 on it, which tends to increase CO's betting frequency. And this gives us better implied odds the times we hit a set with 22.

On the other hand, any board with a 9 on it is going to be somewhat connected. On top of that, HJ's range is still quite weighted towards 88 99, despite these hands folding to the 3b sometimes. (Unlike 22-44, 99 is openraising all the time.) Altogether CO will probably cbet less often when there is an 8 or 9 on the flop, which decreases our implied odds the times we hit a set with 99.

Not sure if there is a way to check this with Wizard?

That is interesting! I suspect it has to do with blocking less of the bottom of CO's range. It could also come down to the bunching effect. The deck should contain more low cards given the preflop action. Therefore, these hands should be very slightly more likely to hit a set.





Here's CO's betting frequency across all flops, grouped by high card:



So it's also possible that 99's sees less action. Though this just looks at the high card not the middle or low card. Interesting observation anyway!

Yeah if only there was a way to see frequencies for middle and low card too.

Right, I saw your article on bunching and thought it was very well written!

Thanks!

Minor error. 7.5 should be 2.5. Your stack size after folding is 97.5BB.

I always get nervous when browni3414 comments on one of my posts because he's one of the few ppl to check my math lol. Thanks man, you're right, I must have thought I was looking at a 4bet spot for some reason.

Let me rewrite that point:

Honestly- I'm like 4 days into learning what "solvers" were- and trying to use the programs and trying to read this thread and sort of get the reasoning is really tough. maybe one day haha. hopefully.