You still need each individual hand in your calling range to be plus EV. You can't just pick a frequency to model calling and folding frequencies for all of your hands in your river range if some of them will lose money when they call.

Optimal play requires a lot of mixed strategies (actions and frequencies) with various parts of your range, such that your overall strategy maximises EV in the long run. Have a look at any of Snowie's range recommendations, or a screengrab from Pio. There's tons of mixing going on. Unfortunately, this mixing is impossible for a human to remember, but it's very easy for a bot with an RNG and a look-up table of previous results.

e.g. If there was a spot where JJ and TT were both in your potential bluffcatching range and you wanted to call with half of the combos, you can't "optimally" call with JJ 100% and fold TT 100%, because that would be (slightly) exploitable. An opponent could improve his strategy by bluffing more often with hands that block JJ rather than TT, for instance. If you "threaten" to call with TT sometimes too, then villain needs to be prepared for that.
If the optimal strategy requires calling with both JJ and TT at some (non-100%) frequency, you can call with one hand at X% frequency and the other at slightly less than X% until your overall range and frequency makes them both break even against the opponent's strategy.

FWIW, Snowie's river frequencies are seldom close to the theoretical MDF numbers, and the EV figures for the river usually don't make much sense either. (It often tells me I have a precisely breakeven bluffcatcher, when its entire betting range has my actual hand beat).

I never said we should call with -EV hands. We should always fold (or sometimes raise) when the call is -EV.

In the JJ example, calling with JJ seems +EV on the river.
But Snowie sometimes folds in this spot.
And the only reason I think is not to be exploitable in the long run by calling 100% of the time in this spot.

So I guess we can come to the conclusion that even when the call is +EV on the river, one should sometimes fold in order not to be exploitable later. It's a kind of "Fold for protection".

And now, my question is : how to calculate this "Fold for protection" frequency with a hand that seems profitable to call?

Of course I'd say that the least profitable the call, the more frequent the "Fold for protection". But I wonder if there's some kind of math formula behind that.

Agree.
Note that the decisions are often very different between Pio and Snowie (for instance, in the JJ example, Pio recommends to bet the Flop 81% of the time, when Snowie always checks behind).
I'd say I am more confident in Pio's advices in vaccuum because I understand how Pio is built and where the results come from. But Pio's advices are correct only if you play against a GTO machine which is never the case in real life. So in practice, Pio's advices seem very far away from the best plays in today's online games. It's a great tool to study the GTO mechanics in theory. On the other hand, Snowie's conclusions seem closer to the real games, but not great though imo.

In the JJ example, with my exact JJ combo, Pio says to call 99% of the time and fold 1% of the time (which is far away from Snowie - calling only 52% of the time). With other JJ combos, Pio says to fold between 14% (with no J of hearts) and 99%.


yes, it's not calculated like MDF, but seems to depend on how profitable the call is (highly profitable -> call very often or always / slightly profitable -> call less). I am looking for a decent formula to understand how this works exactly.


So you don't trust Snowie's recommendations on the River?

Also, as we are talking about how reliable Snowie's recommendations are... The ones who are interested in this might have a look at this post (http://forumserver.twoplustwo.com/45.../#post51152575) and feel free to post their thoughts about it.

This is what prompted my comment. You seemed to be asking whether or not EV was relevant and it is unclear what range you are referring to.

OK, I meant that EV matters (to exclude the -EV calls from our calling range), but there seems to also be another factor to take into account which is some sort of Defense Frequency for each combo (probably depending on the profitability of the call, as I suggested in the previous posts?).

Mixed strategies become necessary precisely because your opponent is using a properly mixed strategy. The properly mixed strategy will cause you to be indifferent with many hands in your range, particularly your bluffcatchers and thin value hands on the river.

I agree. Do you have any idea how to calculate the Call frequency in such an example?

No, but luckily I play vs human opponents. I estimate ev and act accordingly.

You won't be able to derive the frequencies analytically. You have to compute them using a bot and simulations.

Yup. There is no secret magical formula for poker. The future of asymmetrical range game theory is very unclear. Put your opponent on a range and act accordingly.

http://forumserver.twoplustwo.com/17...27/?highlight=

I feel honoured that you posted your post #7K answering one of my questions in this thread, ty :-)

... but if you called 100% of the time with JJ in this spot, this would be a mistake... Snowie says ;-D

Right it's a mistake playing against an opponent that has a perfect counter strategy to it but in reality humans will never be able to play that way. They can approximate that play but only to a very limited degree and more often it's better not to play perfectly but to play exploitively.

Does Pio use precisely the same pre-flop ranges, 3-betting frequencies and bet-size options as Snowie?
If you start with different pre-flop ranges, then the river "solution" can be radically different.
If the call is profitable, Snowie calls 100% of the time. If it loses, Snowie folds 100% of the time. If it's breakeven (0.00bb EV) then Snowie usually mixes between folding and calling (and occasionally bluff-raising). Whilst Snowie is not completely GTO, it follows tenets of the theory. It only makes plays that are at least breakeven. (It never makes what it believes is a -EV call). The bottom of its range for putting money in the pot will be breakeven at least.
Especially with regard to the EV numbers, I take them with a pinch of salt. On early versions of Snowie, there were many hands that supposedly won or lost 0.01-0.10 bb as river calls, and this slight difference from zero made them "clear" calls or folds. After an AI upgrade some time ago, many of the "roughly breakeven bluffcatchers" magically became precisely breakeven. It seemed a little suspicious to me, as my gut says they haven't run enough sims to find true equilibriums for so many spots.
FWIW, I'm not even sure if GTO would actually lead to so many combos being precisely breakeven, but I've not really thought about it, or looked deeply at Pio/GTORB to see the calculated EV of every hand in a range. It just seems a bit weird to me that Snowie often has a very wide range of supposedly 0.00 EV hands. It's hard to make what Snowie calls an "error" if almost every hand can call at some frequency.

I didn't know this feature.

So I ran it into Pio again with the same PF ranges (I have added JJ to the 3bettor's range because for Snowie, JJ is not in the 3bettor's range in such a spot).

Pio's conclusion with this exact same combo of JJ :
1st of all, Pio cbets JJ on the flop (Pio cbets 99.36% of the 3bettor's range for a 60% pot sizing).
Anyway, the River spot is the one we are interested in :
Pio calls 66% of the time with our exact combo of JJ (and 17% with the combos of JJ with no heart ; Snowie never calls with JJ with no heart).

So again, the results are different.
(maybe the postflop raise sizings are different in Snowie, so maybe it changes a little bit the postflop ranges...)
(I believe Snowie's method to build the results is drastically different from Pio and GTORB : Snowie run billions of random experiments for each possible spot and collected the winning decisions)

+1
in the JJ example, Snowie says the call is 0EV. So Snowie calls 52% of the time (not 50%, 52%...). So on one hand, it's very accurate (frequency), and on the other hand, many many spots are 0EV which doesn't seem accurate at all.

"many many spots are 0EV which doesn't seem accurate at all."

Seems right to me as long as the bettor is correctly polarized on the river.

Any difference in sizings (both pre-flop and post) changes the entire strategy; sometimes only by a small bit, but sometimes hugely. I built a quick scenario like the hand you posted, but for 100NL where Snowie would 3-bet JJ pre 10% of the time. (Which means jacks make a very small part of its range). Snowie seems to prefer 1/4 pot for the range it c-bets on that board, and JJ would sometimes be in that betting range. When it bets small, Snowie expects to get more calls, which means the turn and river ranges would be different in comparison to the big bet that Pio suggests.
In the scenario I'm looking at JJh calls 52% of the time, but it's impossible to say that is "accurate" without a complete solution. There isn't a "minimum defence frequency" for JJ specifically, if that's what you're implying.
Snowie folds 38% of its range vs villain's 70% of pot bet (so it's calling/raising a little bit more than the MDF number of 59%) but with that size villain should be able to beat QQ/JJ about 99% of the time, because Snowie would only bluff with 77 (and at a low frequency). This is why I think the 0.00 EV figure makes no sense. The pot odds on a call require you to win 29% of time (or more, due to rake) to break even, but vs Snowie you'd only win 1% of the time, because it hardly ever bluffs in that spot.
Perhaps Snowie's EV numbers are wrong because it simply hasn't trained enough with the sizes chosen by your opponent on this board texture to get a decent sample size. In villain's shoes, Snowie would jam 1.86x pot, since its range is so strong (boats and nut flushes) and give you a very clear fold with anything worse than a flush.

yes, in fact it makes sense if Snowie considers we face a Snowie Villain, then this Snowie Villain should bet a range that makes our bluffcatchers indifferent.
I guess this is what happens.

Your post made me realize that Snowie's advice changes from one stake to another one !!! wtf?

Rake has a small but significant affect on which hands are profitable.
Consider that in the lowest micros, the rake "steals" about 5% of the final pot. That means there are some hands that won't make money at 50NL that would at least breakeven at 500NL. It's partly why nosebleed players are LAGs, and 2NL winners are mostly nits. (Or why AK is happy to go all in pre at 500NL due to low rake, but not so much at 2NL when 'flipping' actually means you lose 5bb). You have less incentive to play a hand if the house takes a significant cut of your winnings. Snowie kind of "worked this out" and built different ranges for different stakes by factoring the rake into its simulations.

ah ok, ty