Yeah his mdf argument is comparing the ev of potting to the ev of open folding. I think this was one of otb_redbaron's criticisms of applications of no limit holdem.

Equity does not also directly correlate to how often you'll win/draw the hand if there is further action. 76% equity does not equate to 76% chance of winning.

I've only skimmed it, but this seems like pretty objectively wrong stuff.

There was some disagreements in the youtube comments about this and it devolved into basically the guy challenging pio hu4rolls. There's already training software that allows that if he's actually serious about the bet.

Hey all,
I just finished a PC build and started using pio (I’m brand new to it). I have a 5950x with 64g of 3600mhz RAM and pio BASIC. I ran a sim for SB v BB (85bb eff) in a single raised pot with 3-4 bet sizes, 2-3 donk sizes, 2 raise sizes and add all-in. It took upwards of 4 hours-would upgrading to pro cut down significantly on solve times? Would I also have to disable hyperthreading if I upgraded? Thank you.

lmao, must be some kind of level

guy's saying "that's completely nonoptimal" about calling hands with lower absolute strength than some hands which are folding. not really rocket science as to why GTO sometimes does this, and people have known this since pre-solver time. there are loads of other incorrect BS, simple things which completely ruins his credibility

You can't run Python in Pio but you can use Python to open Pio with a parameter (save file name).

From my experience it's more or less linear. There are trees which are exceptionally difficult for the solver for some reason but in general it's a good guideline.


Hi Mason, thanks for linking it here.
I will address the article in a separate post after I answer support requests in this one.

Thanks for going through this. I am very surprised the author hasn't reached out to us, it would help him understand some concepts you are pointing out. I will go through that article as well so I am likely to repeat some of your points in my incoming post (just to have my answer on the record somewhere).

I pointed out many many times here and in other places that mdf is nonsense. It applies to some simplified toy games with numbers. It doesn't apply to multistreet games with draws (or even multistreet toygames without draws with the exception of some lines).
I was arguing this point about MDF and 1-alpha nonsense years before I programmed a solver. Sadly this catchy simplistic thinking was something many even very respectable and smart people got fooled by at some point. It's probably the most prevalent harmful myth about poker strategies.

It's a very nice computer. Pio pro will be much faster than basic on it (probably around 2.5x times faster). Still you are solving a really big tree. Pio is already very memory efficient so it's possible to fit large trees in RAM (especially that you have 64GB). Sadly those still take a very long time to solve. Our choice was to allow building those huge trees instead of forcing simplifications. Still it's a good idea to consider making it a bit smaller. You can look at how big the tree is RAM and use that as reasonable estimation of time to solve.

Now let me address the article from the 2p2 magazine.
First, the author hasn't contacted us by any point. I will address some of his points. My overall impression is that the author doesn't understand much about poker which makes me write a bit more basic explanations than I would normally do when talking to poker players.

If you could find one tree Pio solves incorrectly you should be able to make an exact description of it (so we can replicate) and point the errors. Instead he chooses to focus on general points below.

I am not sure why he excepts hands with higher chances of winning being played more often than lower equity hands. There are draws in poker and there are blockers in poker. Draws may have low equity but high chance of hitting strong hands. Blockers might be valuable as bluffs. I mean it's poker strategy 101. This point was widely understood way before solvers came out.

The author seems to not understand the definition of optimal play. The optimal play is defined as a set of strategies such that no player can improve their EV by changing their strategy alone (without others changing theirs). In other words it's such a state that you can't exploit your opponent any more than you are currently doing and they can't exploit you any more than they are currently doing.

There is nothing about preventing whole range from betting in there. In fact there are many spots where stronger range bets 100% of the time.
First, the whole spot he is analyzing is very unbalanced. One player has two advantages there: position and huge range advantage. His BB defending range is very loose (K5off vs 2.5bb raise from MP raiser with 14% range). The main reason he struggles in subsequent points is that the way to limit the damage from this ridiculous preflop strategy is to actually fold a lot on the flop.

It seems to me the author is surprised that in a multi street game when you bet the flop having two more betting rounds you should have a lot of bluffs in your range (the more streets left the more bluffs). As this phenomena can be observed in even in toy games with numbers I would expect the author to be familiar with it.

This reasoning is just incorrect. Here is a very simple example showing it:
-river spot
-we have 99% nuts, 1% bluff
-the opponent has 100% bluff-catchers

optimal strategy: we bet 100%, the opponent folds 100%.

While the spot presented by the author isn't that extreme it also features huge range disparity which causes the stronger range to bet all the time (I solved it with my dev version to almost exact equilibrium and Pio in fact never checks the flop).

Pio in fact recommends betting 100% of the time with all hands (the sizings differ). When I solve it with our current public version there is very small % left for checking (less than 1%) but that's just a characteristic of our algorithm (a bit less precision in exchange for saved RAM).
I suspect the author doesn't understand how to use the solver. Maybe he solved it to very bad accuracy. It's really hard to guess because numbers in his table are different that actual numbers Pio produces.

I've solved the author's tree two times. Once with our public versoin of the solver to 0.2% of the pot accuracy and then again with my dev version to 0.05% accuracy.
I tried to replicate the config from the description in the article and came up with this:

https://pastebin.pl/view/7b65e978

Here some screenshots:

1)Betting strategy on the flop from 0.2% solution:
https://gyazo.com/4d1a9d74d39c01f4b5fbb99f7c952914

Pio bets almost 100% of the time. It becomes exact 0 in the dev versoin:
https://gyazo.com/0f06964b96e09721ac5fffc63e0bbb38

2)Flop after IP bets the pot:

https://gyazo.com/596d0bfb99fb750d89c3be59fdb0b821

Pio doesn't fold top pairs (well, it does 0.05% of the time at this accuracy). I am not sure how the author arrived at his numbers.

The author writes:
Well, Pio doesn't.
It probably does at some very bad accuracy till the solution becomes better.

It's the same story at the turn.
In his example the turn strategy after pot size bets on both the flop and the turn (turn 6d) looks like this:

https://gyazo.com/1be95ded41fd41d64119f65094554705

and the dev version:
https://gyazo.com/01bce4c78a2c4c78e8e8fb932d06ea0f

In the less precise solution we are folding ATo some 0.3% of the time while in the more precise one it's 0%. The author is claiming 26% fold.

Here we have a double whammy: incorrect argument about equity and incorrect use of the solver. It's funny how even looking at very imprecise solution he isn't able to form a correct argument saying why those results are not optimal yet.

Now the river is more of the same:
https://gyazo.com/9806a7508ddeeb9d06747391d0485a61

Pio doesn't call "6622" here". Pio doesn't folds some top pair top kicker hands in favor of always calling with top pair with lesser kicker.
It's not a huge secret why: IP's range is very broad-way heavy so the bluffing range is going to have AQ/AJ/KQ/KJ hands. AT/KT block some of those and therefore are a less attractive call than T9. This is maybe a bit too advanced for this particular discussion although it's really not a huge secret anymore.

Btw, pot bet river is more interesting as it seems Pio only wants to choose that sizing at very good accuracy. Let's see:

1)River betting range for IP pot bet:
https://gyazo.com/74e5cb5c53f6b5f967c8dd2c253857eb

River calling strategy vs that pot bet:
https://gyazo.com/e0ae59986011051502afa155f243f5f8

Looking at those two side by side make it obvious why hands like T9 (doesn't block broadway bluffs) and K2 (blocks a set of 2s as well as KK) are better bluff catchers than AT/KT/QT.

Ok this one I am assuming is a not so thinly veiled insult.
Let's look at actual range composition after 3rd river barrel (pot size):
https://gyazo.com/5a4291e86645f196b67cb0f74a768ee4

and again on the flop after the first bet:
https://gyazo.com/25188c3bc02c32846a843d3d2e6c1370

After the flop c-bet top_pair+ is 28.3% of the range and overpair+ 20.8% of the range.
After the river barrel top_pair+ is 63.4% of the range and overpair+ 61% of the range.

It would be funny if it was made after making some kind of coherent argument. Meanwhile we have an author who:

1)Provably doesn't understand basic poker concepts like:
-all-in equity (just because a hand has chances to win vs initial range on the flop if not betting is allowed doesn't mean it's the best hand to continue with when actual betting takes place)

-mdf/betting with the whole rang thing (expected and often correct)
-blockers

2)Not only doesn't understand how to use the software he is criticizing but didn't even bother to contact us so we could help him run the tree he is describing

3)Writes a book about poker without understanding neither the strategy nor the math behind the game. The book brings him to believe nonsense like poker hand rankings leading to calling K5off vs a raise from MP and then being surprised the optimal strategy after that is to fold a lot on the flop.

4)Makes youtube videos about why Pio is incorrect even though he doesn't even understand how to use it.

I mean, I've met a lot of arrogant bold ignorant people during my almost 6 years doing this project but this one takes the price. Usually it's some heated discussion and then careful retreat after being shown mistakes in your thinking, not going around publishing books and youtube videos.
I mean, if they teach you anything in academia it should be some standards of research and intellectual honesty.
When you come to a field you know nothing about, try to use a state of the art tool there and arrive at surprising results there are some obvious steps you could take:

1)Ask around about your results, it's likely you are not the first newcomer having problems or doubts. People may help you.

2)Contact the authors of the tool asking about your results. While we don't have resources to analyze every single tree you have problems with we might point you into proper direction

3)Research just a bit farther. It's unlikely you have just discovered something that goes against what every successful person in the field believes. It takes some humility but in general big claims requires big evidence. It's a good idea to start with your own convictions.

Mr Norman Zadeh has managed to demonstrate that having a math PhD does not garantee that one understands how tools work or how the game of poker should be played. The best math book in that sense, remains The Mathematics of Poker, by Chen and Ankenman.

Mr Zadeh may take a 101 training about how to use PioSolver properly (after all, algorithms and numerical methods are not pure maths), then start solving a few toy-games exercices, and check if PioSolver finds the same results as Chen and Ankenman describe in their book.

I love the disclaimer from 2p2 at the beginning of the article :
"However, he does make some statements which some of our readers may not agree with, and we'll be linking this article to the PIO Solver Thread in our Commercial Software Forum for those who want to discuss the issues that Dr. Zadeh brings up."

Well, let's make it simple : Dr. Zadeh is claiming so many false statements in his youtube videos, that the number of thumbs down is increasing, fast. Most of us are poker & math amateurs, and we don't need a math Ph.D. to provide a valuable peer review to Norman Zadeh's poker videos, poker article or poker book.

lol what was Mason thinking publishing that crap?

Hi memo:

Even though I had much communication with Dr. Zadeh I’m not strong enough in this area to explain to him exactly why PIO is working right. So we at 2+2 decided to publish the article hoping that good answers would come here and everyone would benefit.

Best wishes,
Mason

I definitely see the case for doing that. It's a bit of a pity that the author didn't make effort to use the software correctly. It seems he has some misconceptions about poker theory which would be fun/educational to debate but sadly all this is overshadowed by just not running the simulations to decent accuracy (at least that's my best guess as his numbers are just completely off comparing to what Pio actually produces).

Dear Okra,

Thank you for your interest in my article. The pio output you have posted in your article is the wrong output. If you are interested in where I obtained my numbers, please click on the link in my article to a youtube video which will allow you to replicate my analysis. The pio output you have posted in your article deals with betting issues of the IP. Table 2 of my article, which you are apparently criticizing, deals with calling errors of the OOP. You have not explained in your post how it can ever be optimal to call on the river with 6622 in favor of TT66 (see Table 4 in my article). Nor have you explained how it can ever be optimal to fold a hand like top pair top kicker, which has a 70%+ chance of winning, while calling with a hand that has an 18% chance of winning (see Table 3). By the way, I'm interested in playing in an online no rake game using a program like pokerrr-2. I would run the game. If you or anyone you know would like to play, please let me know.

Thanks,

Norm Zadeh, Ph.D.

Hi punter:

I want to address this one point because in my communications with Dr. Zadeh this was the first thing I told him. I even sent him a direct quote from David Sklansky's The Theory of Poker Applied to No-Limit which addresses this and recommended he get Mat Janda's Applications of No-Limit Hold 'em which goes through this in detail (and Dr. Zadeh said he would). So, I do believe that he at least now understands this point and the importance of it.

Also, thanks very much for your detailed and professionally done post. I think lots us, including myself, will benefit from it.

Best wishes,
Mason

Comments from Norm Zadeh: Here are my responses to some other points that have been made, this time by punter11235. With respect to the calling range of K5 or better for the big blind, I agree that this is loose. But Piosolver is supposed to be able to deal with ranges specified by the user.

With respect to the comment that I did not use Pio Solver to solve to the correct accuracy, I used the accuracy recommended by Pio Solver.

With respect to printout that Punter included, referred to as Gyazo2, that output is non-optimal because it has the big blind folding 41% of the time with TT66A, but calling 31% of the time with 6655

Does 2+2 do any quality checking at all before publishing an article? "Examples of Non-Optimal Strategies Suggested by PioSOLVER" is riddled with claims that are asserted to be "clearly" true despite contradicting the consensus of the modern poker community. For example: "I contend that Pio’s recommendation to fold hands with a 76% chance of winning 26% of the time, while calling (or raising) with lesser hands some of the time, including hands with an 18% chance of winning, is clearly non-optimal." There are well known reasons to sometimes call with "worse" hands over strong ones, such as the nature of the equity distribution for the hand across future runouts, not to mention that the author's measure of hand value here is non-conventional and incredibly misleading. The hands that he claims have a 76% chance of winning, do not in fact have a 76% chance of winning in the context which they are presented. From the article:

"To those readers who may suggest that the ADTS no longer has a 76% chance because the opener’s range has changed by the river, a review of Pio’s strategy shows that, because Pio bets each hand in its range throughout, the fact that the opener has bet on each round does not substantially change the opener’s likely distribution on the river from the opener’s initial distribution."

He is actually measuring a hand's chance of winning facing a river bet by its equity against the preflop raising range!

This claim is utterly ridiculous, ignoring that different actions are taken at different frequencies for each hand, and the 76% chance can easily be demonstrated to be false by calculating the actual equity ATo has against the river betting range.

Another problem is that the author claims that he has provided information in his videos from which readers and viewers can replicate his results, but he failed to do so, not being aware of all of the necessary parameters to replicate a solve with this software.

This is a poorly written critique of a software the author clearly does not know how to use properly. The author shows a minimal understanding of the game theory concepts applicable to poker by making blatantly false, unsupported or unverifiable claims. This article is damaging to the reputation of 2+2, the author himself, and harmful to readers who don't have enough expertise to recognize the many problems with it. If there were only a few errors I'd suggest it be edited to correct them, but it has so many major problems that it's unsalvageable and I believe it should be pulled.

Tx and 55 are both effectively bluffcatchers in this situation. A bluffcatcher is a hand that will beat bluffs and lose to valuebets. The hands exact strength is not relevant. Blockers (holding cards that impact the likelyhood of your opponent holding a bluff or a valuebet) are usually more important when bluffcatching. Of course, you can not always defend the weaker bluff catchers and fold the stronger ones. Then your opponent would be able to exploit you by betting wider for value.

Hi browni3141:

The article was sent to me, I reviewed it, and had much communication with the author. Because of this, some fixes were made.

However, I'm not a PIO expert and assumed that the numbers were that were told to me were correct. For instance, I could not explain to the author why AT would be folded 8 percent of the time. Now I see that it's not folded at all.

As another example, which I have already mentioned, I tried to explain to the author that because of the nature of GTO betting, hands that can have over a 50 percent probability of winning on the flop should sometimes be folded on the flop and even sent him stuff from one of our books that addresses this issue.

But I made the final decision to run the article and to link it here for discussion. One of the purposes of 2+2, and perhaps the most important purpose, is vigorous debate. I knew that by having discussion here any errors in the article, assuming there were some, would be explained, and I thought that lots of people, including myself, would benefit, and I think that's exactly what happened.

As for pulling the article, I'll have discussion with David Sklansky tomorrow and we'll make a decision. However, it would certainly be wrong to also pull the posts that now appear here as well. And without the article, there would be no frame of reference for posts like yours.

Best wishes,
Mason

Response by Norm Zadeh. Where's the proof for anything you are saying? I realize that AT might not have exactly a 76% chance of winning, on the turn with a board of Tc 6h 2s 6d, but its got a lot better chance of winning than 9 high, which Pio is recommending raising with. Would you rather raise with TT66A or nothing? If you claim that AT does not have a 76% chance on the turn, what are its chances vs. the chances for nine high? If I'm such an idiot, lets play some poker.

Response from Norm Zadeh. Are you saying that you'd prefer to call or raise with a hand that has an 18% chance rather than a 76% chance? Let's play some poker. In the future, I would very much appreciate you're identifying the "loads of other incorrect BS." Just saying this means nothing. You tell me, why does Pio fold really good hands and call or raise with crap?

I thought the articles were intended to be instructive, authored by people with some expertise in the topic being presented. I don't mean that as a dig, I'm just saying that with respect to the topics of game theory (at least applied to poker) and solvers, Norm could learn a lot more from the forum than the forum can learn from him. A few of the people in the Poker Theory sub forum here are fairly knowledgeable.

I'm happy to debate in situations where the correct answer isn't clear, or when someone has an alternative perspective which is interesting. All of the issues presented by Norm have straightforward answers.

I will reference the graphic from punter's sim here:


It seems to me that your issue is that 98, 97 and 87 are raised 100% of the time, while AT is raised 0% of the time, only called.

There are many different incentives in play for different types of hands to take various actions. AT has a higher incentive to call than 98 because it can win at showdown against many of the hands in the bettor's range. 98 just doesn't have enough of a chance of winning to call profitably. The reason that 98 can raise is that it retains some equity against the range of hands that will call a raise, allowing it to bluff profitably where a hand like K5 can't. Note that while 98 can raise, the expected value (EV) of the action is still close to zero. We can't verify this from the graphic, but this is something you can check in PioSOLVER, and in general we already know that most bluff raises will be close to neutral EV.

The reason AT doesn't want to raise is that it does very poorly against the range of hands that choose to continue. You will either get 3-bet and face a roughly 0 EV call, or you will get called by a hand that beats you. Hands like JT aren't really strong enough to value bet two streets for pot sizing and call a check/raise. It's more profitable to call and try to win a showdown than it is to turn the hand into a bluff.

In short, it is often better to call with hands that have a good, but not great chance of winning. This leads to things happening like here where AT just calls, and 98 raises as a bluff.

PioSOLVER has features to tell you the exact equity and expected value AT has facing a pot sized bet on the turn here. I can't be so precise, so this is only a visual estimate based on the range we can see IP betting on the turn for pot in the video that was linked from your article (T626 board)

[hand] ([combos]*[frequency]): [AT's equity]

Vs. value hands:
AA (3*.5): .0455
KK-JJ (18*.45): .1136
TT, 66 (2*.2): 0.0000
22 (3*.3): .0909
AT (6*.05): .5000

Vs. bluffs:
A8, A9 (15*.1): 1.0000
AJ-AK (36*.1): .9318
KQ, KJ (32*.1): .8636

Sum of weights: 19.5

Other combos aren't bet at an easily discernable frequency.

Weighted average:
(3*.5*.0455+18*.45*.1136+3*.3*.0909+6*.06*.5+15*.1 +36*.1*.9318+32*.1*.8636)/19.5 = 45.5% equity. Nowhere near 76%

To answer questions regarding why you'd call with a lower ranked hand over a higher ranked hand, if we look here on the turn, the worst hand IP bets pot for value in your video is AT, and even that is at a low frequency. The strongest hand that gets bluffed at a meaningful frequency is AK. This means that any hand that loses to AT and beats AK is relatively equal in showdown strength. Since there's little to no difference in showdown strength between a hand as strong as KT and a hand as weak as 33 we differentiate hands based on their blocking effects and equity. Hands like KT have better equity because they improve on more cards and dominate some of the bluffs.

Since apparently many of these hands take more pure actions when solved to a high accuracy in this scenario, I am talking generally for this last point. The reason that you will see many hands mixing instead of only the "best" hands always calling and the "worst" hands always folding is that if your calling range is dense with particular cards, that can be counteracted by the bettor using bluffs which block the bluffcatchers that call, resulting in more frequent folds when bluffing. The bettor can't do this if the distribution of cards in the callers range is uniform. This doesn't always happen, as sometimes there are bluffcatchers which are better enough than others they always get called, or worse enough that they always get folded.

Response by Norm Zadeh: There are numerous errors in this analysis, using the same outputs you provide. While I agree that your outputs are not exactly the same as what I obtained, I used your default recommendation for accuracy, which was .5%. If your users need more accuracy, you shouldn't recommend a default value of .5%. I did truncate the run at 100 seconds but I have a brand new, very fast machine. On the flop, you claim that Pio bets almost 100% of the time, which is what I claimed it should be doing, because the big blind is folding so frequently. However the gyazo.com file you refer to ending in 38 does not show that. In fact, it shows that Pio is checking with A9 suited 25% of the time, AJo 25% of the time, etc.

Later on, you say, the river is more of the same, referring to a file ending in 61. You say, "Pio doesn't call "6622" here. Pio doesn't fold some top pair top kicker hands in favor of always calling with top pair lesser kicker." But according to your own file, ending in 61, the big blind is folding AdTd 50% of the time, while its calling with pocket fours 25% of the time. This is roughly the same thing, calling with 6644 while folding with TT66.

Finally, on the turn, your own file ending in 05 shows that Pio is recommending calling some of the time with A2o, 44, and 55 while folding some of the time with QTo, and raising with 98s (nine high). How can it be right to raise with 9 high a hand that can't make a straight or a flush, but not raise with TT66 and instead throw the QT away?

Hi browni:

To answer your question, we sometimes run articles because we think they’ll create excellent discussion. I think the rest of your post is a good example.

Best wishes,
Mason

0.5% is fine to get an idea what optimal play looks like. Strategies aren't likely to drastically change beyond that. If you set a time out of 100 seconds and solved to 0.5%, then you did not solve to 0.5%. You can check what exploitability it got to before reaching the timeout.

The 25% that you are referencing are the expected values of each action (25 chips). The image is showing that every hand is bet 100% of the time (mixing between a small and large bet size). There are small differences in expected value among actions that are mixed because the solution is not an exact equilibrium. If exploitability was 0%, then the expected values for any mixed actions would be exactly the same. The numbers over the matrix in other files shown may represent strategy. The feature to toggle which you want to display is an option in piosolver. If the values sum to 1.0 then it is showing strategy.

See my last post for why this is strategically optimal

98s can make a straight, the board is T626.

I believe the theory behind why it is folding some Tx and defending with weaker pairs is a bit beyond the depth of this discussion. In general it is common to see stuff similar to this as it allows you to have strong hands in your range on different runouts.

I think you should spend more time using the software to fix misunderstandings you may have about game theory. If you disagree with some of the output that you receive, it would be beneficial to try to learn from it. Poker theory has advanced dramatically over the last decade, and anyone that hasn't been using software has been left behind.

I think there must be some big misunderstanding when reading the results.

I am not sure why you assume it's our default recommendation. I answered a question about recommended accuracy many times. It's 0.35% for most purposes, 0.25% if you want something more precise and 0.5% for a good outline. I don't think you solved it to 0.5% either though as at this accuracy Pio is not folding top pair on the flop with any significant frequency (0.1/0.2% for some combos)

This is "strategy + EV" view. This means strategy (frequencies of actions) are shown with colors but the numbers are EV. Please notice that those numbers don't add up to 100 and they are very close to each other (as in equilibrium mixed actions have the same EV).
It looks like you are using "Strategy + EV" view and reading EV numbers as strategies which is the root of all the confusion. If you want to see Strategy as both the colors and numbers click "Strategy" button. You can also look at all-in equities of both ranges at any given time which will allow you to verify your statements about higher equity hands folding the river (it's not true).

Please consider that this is also avery big tree if you in fact provided options to use 2 bet and raise sizes at all points (with the exception of flop bet of OOP). Those trees take a long time to solve to good accuracy. For example on my 32core machine using faster dev version it took me 150 seconds to solve the tree to 0.5%. Even though Pio uses an algorithm which uses around 2.5x less memory than popular CFR+ (used for Science publication about solving Limit Holdem) the tree still takes about 8GB of RAM. Usual way to cut down on the size is to use only one raise size (for example "3x")

You seem to very convinced that a stronger (in hand ranking categories) hand should always call more often than a weaker hand on the river. While this is true for toy games with numbers where in fact the number 6 calls at least as often as the number 5 it's not true in a game where card removal matters. Most of the calling and bluffing decisions in optimal play are influenced by blockers. You call with hands which are unlikely to block bluffs of the opponent but more likely to block their good hands. You bluff with hands which block their calling range.

Please consider this quote from my long response above:
I explained why seemingly weaker hands are better calls on this river. If you run it yourself and actually look at the equity after the river bet is made you will see that T9s has higher equity than ATo. The reason for this is so called card removal effect. When you hold T9 and your opponent bets he is bluffing more often than when you hold ATo. If that doesn't make intuitive sense to you please consider the toy example:

We shuffle a deck of cards and draw one card each, now two scenarios:
1)You hold 5 of hearts and your opponent says: "I have the Ace of spades"
2)You hold Ace of spades and your opponent says: "opponent says "I have the Ace of spades"

You don't know if they are telling the truth in the first scenario but you know they are lying in the second one.
Coming back to poker when you hold a hand that makes some of the opponents bluff hands not possible you know they are bluffing less frequently. This allows you to chose the best bluff-catching hands. It's not a guess game of "sometimes call sometimes fold". There are actually objectively better and worse bluff catchers even though they all lose to value bets and beat bluffs.

Of course this should be reflected in equity (chance of win as you call it) after the river bet is made (not before!) and in fact in this exact spot:

-3pot size barrels from IP
-board: Tc 6h 2s 6d 3h

When we look at average equity of 3 interesting hand groups: ATo, T9o and K2s we get this:

https://gyazo.com/7dde58aa3afb8e20f96cb910a14f4860

Even though K2s only forms K6622 it has average chance of win at 38.197%
ATs that forms ATT66 has average equity of 37.664%
T9o that forms 9TT66 has average equity of 38.409%

I say average because under ATs umbrella there are 3 possible hands, under T9o there are 12 and under K2s there are again 3.

The point about blockers is something a lot of strong players realized before solvers became popular but I think only recently (last few years) we appreciated fully. It's in fact not a guessing game. There are better and worse hands to bluff with/bluffcatch with and the reasoning when choosing those is far from obvious. It's surely not as simple as just looking at "ATT66 vs K6622".

EV of those hands is 0 for both folding and calling. It's very frequently the case in equilibrium to have various hand types being marginal calls/folds. If you accept that possibility then it's easy to imagine how it's possible that the best strategy is to have a mix of hands instead of just one (so it's possible to have 50% of QTo and 50% of A2o instead of 100% QTo, 0% A2o). The reason is that it makes the opponent's life more difficult. It might be possible to form a more exploitable strategy again more predictable range. Those things are sometimes difficult to understand (especially from the turn or the flop).

Pio makes tools available to make this kind of analysis easier. We can create a tree starting from the turn with the ranges from the original solution and then force the solver to adapt a different strategy to see what adjustments vs that would be and how much EV it would cost.
This is usually a lot of work to understand just one spot. I can go through that process if there is interest although for now I think it's just too detailed for this discussion. It seems we are still stuck at "higher equity hands should always call before lower equity hands" argument. It seems to me that your view is that "it must be so". Please consider though that there is betting left and it's entirely plausible that some hands which do worse if no betting was allowed might do better with actual betting. We even have a measure for that exact case. It's called equity realization (defined as EV/equity).
Looking yet again at your tree this time after IP pot bets on the flop the equity realization table for OOP hands looks like this:

https://gyazo.com/0d91a5a26b109530ac847e31c76c02aa

Let's look at some specific hands:
1)JJ, equity realization of 106%, it gains just a little bit from subsequent betting
2)A2o, equity realization of 1.98% - it already has to make a very marginal call on the flop, it would definitely prefer if the betting wasn't allowed

3)TT, equity realization of 619% - almost the nuts, gains tremendously from subsequent betting

One example of hands where equity/EV cross (one of the hands has higher equity but lower EV) is KTo, for example KsTh vs 9s8s (or any 98s different than 9d8d).

After IP bets the pot on the flop we get this:

KsTh - equity: 65.570%. EV: 21.53chips
9s8s - equity: 33.303%. EV: 27.92chips

Although I would love to take the credit for discovering this concept it was actually discovered before I was born. It's called "implied odds" by old-timers.

While the exact case isn't obvious (98s never gets there in the first place as it bets the turn and the EVs for a raise are close) the principle you are using is just incorrect. It's very easy to see why. Consider this toy example:

As 7s 5h 2c

Your opponent bets the turn and you raise-allin, your opponent only calls with top pair top kicker or better. Do you prefer to have Ts8s there ("nothing") or AhJh (top pair!).
What actually matters is your equity when you are called (and the frequency they call you with). If you are only called by stronger hands it doesn't matter what your equity was before your opponent called you. Hands with potential to make stronger hands have higher equity against strong calling range than weak made hands. This really isn't rocket science or any kind of solver vodoo.

And one more point I've just noticed:

It handles it. It's just that the optimal strategy when one player has too loose a range consists of a lot of betting by the stronger range and a lot of folding by the weaker one.

"With respect to the calling range of K5 or better for the big blind, I agree that this is loose. But Piosolver is supposed to be able to deal with ranges specified by the user".

Jesus... It is almost disrespectful to this forum and the software creators... If it was from a person that is learning poker strategy... But it is from a person that wrote poker books and claims he is right (but has not a clue about several points of GTO strategy and the software. And it seems, so far, he is not open to a real and open discussion that would be great. An awesome opportunity lost).

Enviado de meu SM-G610M usando o Tapatalk

I think we need a new solar system to support the weight of this guy Norm's ego