I am starting to get the hang of the idea that in GTO poker, the number of value bets vs. bluffs needs to track the pot odds that one offers.

So, the classic situation is the pot-sized river bet. This offers two-to-one odds, and hero needs to have a minimum defensive frequency of 50%.

So what I was wondering is whether hero should be maintaining this MDF across ALL pot-sized river bets, or whether hero should only be counting those pot-sized river bets where the proper ratio of value bets to bluffs exists in villain's range. I.e., the bets that appear to be at equilibrium.

If hero catches villain making a mistake constructing his river betting range, and exploits the mistake, should that hand be counted in the denominator of the MDF the next time a properly-constructed river range shows up?

If villain underbluffs, you fold every bluff catcher. If villain overbluffs, you call every bluff catcher.
If villain seems balanced, the strategy with least downside is MDF. Having said that, any strategy would perform the same unless villain himself adjusts

I'm really not thinking of this in practical terms. This was purely a theoretical question. But yes. There is no practical reason for me to make this so complicated.

It's theoretical. Just calculate the EV of each play amd see what numbers come up

MDF assumes that every hand in the bettor’s range holds either 0% or 100% equity and the caller holds equity greater than 0% and less than 100%. Obviously in real life, the equities will not be so clean. So, rarely will the calling player be able to call at MDF overall as many hands in their range simply won’t qualify as a bluffcatcher. For example, you arrive to the river with a 4 high missed draw. You wouldn’t begin calling such hands to meet the MDF. You would fold all of these hands and call each bluffcatching hand at the MDF.

Example:

OOP has 88, and 43s. IP has AA and 73o. Board is AQ962r. Pot is 1, Stacks are 1.

IP bets 1, and now OOP will call only 32% of the time, not 50%. OOP will call 88 50% of the time and fold 43s 100% of the time. OOP will not call 88 more often to compensate for the folding of 43s.

This is innaccurate. MDF does not only apply to bluff catchers, it applies to the entire range. If villains range contains enough complete air (0 SDV), and we have a range composed of 0 SDV hands and less bluff catchers than MDF, we will call every bluff catcher always.

The reason in that hand you fold 88 at some frequency in your example, is because villain does not have enough 0 SDV air, and he is forced to bluff with hands that have checkdown value.
Therefor, to make him indifferent between betting and checking with those SDV hands, we need to overfold in order for his bet to be +EV.

MDF dose applay only to hands that beat all the bluffs.
When you face bet on the river you are trying to make highest equity bluff indifferent between checking and betting and you do that by defending MDF of your range that beats that hand .

Example
pot=1
EQ=10%
EV(check)=0.1
90% of range that is facing a bet beats this hand, so in order to make it indifferent we defend 50% of that range. Then we have
EV(bet)=(1-0.5*0.9)*1-0.5*0.9*1=0.1

In this case all bluffs with equity lower then 10% will have +ev bluff.

Only in special case where best bluff has 0% equity we defend mdf of entire range.

Yeah I actually blundered that explanation
In the case we had a portion of our range that doesn't beat bluffs, that would inherently make villains bluffs +EV checks and therefor we would under defend MDF to make them just as +EV of a bet.

I guess what I'm getting to is that the reason we under defend MDF in those scenarios is because villains bluffs make money on a check

We do fold more then MDF when you look whole range, but we defend exactly MDF of range that beats the bluffs. So it's not like MDF is not applicable in that case.

I look it like this
If he bluffs hand with equity, ofc we fold all hands that are weaker then his bluff and we make sure he dosen't win any additional money vs hands that beat that bluff by defending MDF of the portion of our range. That way he wins the pot when he has the best hand (like he should) and has 0 EV bet or check if he is beaten.

Of course this is true within the toy game models of MDF and PvBC. However, the assumption of those models is that the caller's range consists of 100% bluffcatchers. Obviously, this doesn't translate to real life at all as the equities of the ranges will never be this clean. You'll see something similar if you change my example slightly to account for a blocking effect:

OOP has 88, and A8s. IP has AA and 73o. Board is AQ962r. Pot is 1, Stacks are 1.

IP bets 1, and now OOP will call only 41% of the time, not 50%. OOP will call A8s 100% of the time, call 88 25% of the time. OOP will not call 88 more often to meet MDF due to block effects.

It seems these models fall apart quickly once you begin to changes that ranges such that different equities exist. As we saw in my first example, allowing the BC/caller to hold a hand other than a bluffcatcher causes the P/bettor to no longer hold a truly polarized range.

This logic is a bit messy because OOP still defends half (MDF) of their range.

A8s: calls 100% of 3 combos
88: calls 25% of 6 combos

1.5 + 3 = 4.5

OOP calls 4.5/9 = 50% of their range


You could argue that you'll have A8s less often in this line, because villain's betting range is more likely to block your A8s, but that doesn't change the fact that you're defending half your range.

Nice catch here Tombos.

I see that the Card Removal option in GTO+ along with my own laziness caused my pain here. With Card Removal ticked on, the overall call is 41%, and 50% with it turned off, despite no change in the combos (4.5 calls of 9 for each)

I didn't know that Card Removal button works that way. Still oop is calling 50% when IP bluffs and calls less when he value bets. So MDF completely explains these results as well.

Yeah, I don't understand what this card removal button is doing. I wonder how it's calculating the 41%.

IP has AA 3/15=33% of the time and 12/15=67% has 72o.
I think solver gives you average calling frequency. If IP bets with AA it gets called by one come of A8s and 1.5 combos of 88, so calling frequency is 2.5/7=36%(there is only 7 combos in OOP range because AA block A8s).
Then when you average it you get how often OOP will call vs bet
0.67*0.5+0.33*0.36=0.45=45%
Probably in your sim AA block all of the A8s combos. In that case you get calling frequency of 41%.

How often oop calls it's not the same number as what percentage of his range he defends vs bet. I wasn't aware of this, but it's better to turn removal effects off, because you are interested how much of your range you need to defend.

The idea is to adjust your overall frequencies according to the actual probability of the line occurring.

IP's betting range consists of:

1/3 73o
2/3 AA

so 2/3rds of the time, OOP's range consists of (1 combo A8s + 6 combos of 88) - OOP defends 2.5 / 7
and 1/3rd of the time, OOP's range consists of (3 combos A8s + 6 combos 88) - OOP defends 4.5/9

P(OOP's call | IP has bet) = (2/3 * 2.5/7) + (1/3 * 4.5/9) = 17/42 = 40.4%

----

This doesn't actually give me the same results though. Haizemberg93 also gets different results. I don't get it. I think it must use matchups somehow. Either that or something lazy.

Side note - if you change the view to "Percentages" then it will adjust the percentage of each hand class you have in range when toggling the card removal button.

I didn't use correct formula, that's why we don't match with results xD

Difference probably has something do do with sim setup or accuracy.