I have been experimenting with an AKQ like game in GTO+ and CREV, in order to compare results with knoe theoretical concepts, such as Alpha bluff frequency and Minimum Defense Frequency. Solvers seem to confirm the Alpha value but not much the MDF value.

My AKQ like game setup goes like this:

It begins at the river. Board is 22234 rainbow. Effective stacks are 100bb. Both players' starting ranges are AhAs, KhKs and QhQs.
Player OOP always checks, then player IP can bet an amount of his choosing or check and see a showdown. Player OOP can only call or fold, not raise.

Now, for a pot sized bet in this setup, the CREV and GTO+ solvers say that IP player must bet 100%, check KK 100%, and bet QQ 50%. This is consistent with "alpha" value for bluff frequency (bet / pot + bet) as stated in Mathematics of Poker.

However for IP, solvers say he must always fold QQ and KK, for an overall calling frequency of 17% and 83% fold. Now, according to MoP and other theoriticians there is something called "Minimum defense frequency", and that means that when we are facing a pot sized bet from a players who plays perfect GTO, in order to make his opponent indifferent between bluffing and checking, IP must call [1 - alpha] which in this case equals 50%. However I can't find this percentage anywhere in the solution given by the solvers.

Same thing for a scenario where IP player bets half pot (remember, player oop can't raise). In this case solution says IP player must bet QQ 33% of the time, which is consistent with the Alpha value. However, again, here the "minimum defense frequency" should be [1 - alpha], that is 67%. However the solution says OOP player must call 33% of the time with KK. Again, I can't find the "minimum defense frequency" value anywhere.

What am I missing here?

Edit: I think I brain-sharted a moment ago, but in words: what you're missing is that they're also calling with AA.

Oh what I initially said was right, so I'll rewrite it in case you didn't see it before (but with some more clarity, lest I confuse myself again).

In the pot-sized bet example, only calling w/AA is a 50% frequency when Villain is bluffing, because V blocking QQ narrows your range to {AA,KK}, of which AA is half.

In the half-pot example, all your AA + 1/3 of your KK is a frequency of 1/2 + (1/2)(1/3) = 2/3

Minimum defense frequency applies to your continuance range.

Using a simple example:
Pot is 1, stacks are 1
Player 1 (IP) has AA, QQ
Player 2 (OOP) has KK, 99, 88, 77, 66, 55

P1 will bet AA 100% and QQ 50%, in order to make the bluff-catching region (KK) of p2 indifferent. P2 will fold 55-99 since these hands can't even beat a bluff, and it will call with the bluff-catching region half the time to make P1's bluffs indifferent. If p2 only had KK in range, the strategy would be exactly the same.

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Your example is more complicated because there are blocker effects, but the same principle applies.

QQ is NOT part of the "continuance range" since it always loses anyway, so MDF only applies to KK-AA. The continuance range is only two combos.

Facing a half pot bet, we always call with AA and call with KK 1/3 of the time. 1.333/2 = 66.6% = MDF.

Things get weird facing a full-pot bet or larger, since AA makes up half of your continuance range. 1/2 = 50% = MDF, so you can simply fold everything less than the nuts, and you can't be exploited by bluffs.

You'll get different results from different solvers in the full-pot bet spot, because there are multiple equilibria that all have the same EV. For example, betting only AA returns the same EV as betting AA and 50% of QQ. Using anything larger than a pot-sized bet, the correct strategy is to only bet the nuts and only call the nuts.

Bravo, sir!.

I just checked this by widening OOP's range to AA-55, and indeed the frequency call for KK doesn't change and number of the continuance range add up to MDF.

Much appreciated.

What’s a “continuance range?” To me it means the range that continues, but that’s not how you’ve used it here. It’s clear what range you’re talking about but not how it generalizes.

I would define it as the part of your range that can actually consider not folding.
EV calling/raising >= EV of folding

So for example, facing a shove your continuance range would just be hands that have at least pot odds to call.

Well hands that are clearly in the folding range can count towards MDF also. QQ "doesn't count" for OOP in this example because it loses to to IP's bluffs, giving the bottom of his bluffing range a profitable check-back, and requiring a bluff to be equally profitable to remain indifferent between betting and checking.

I think Lezaleas put it well in the other thread:
Edit for clarity:

You would never knowingly call a river shove without sufficient pot odds, just to try and meet mdf. MDF generally applies to the break-even + profitable part of your range

I think we can summarize like this:

Example 1:
OOP (the pot bettor) has:
1 combo of AA
1 combo of QQ

IP (the caller) has:
1 combo of KK

OOP is betting 100% of AA and 50% of QQ. In this case IP calls 50% of KK vs the pot bet, according to MDF.

Example 2:
OOP (the bettor) has:
1 combo of AA
1 combo of QQ

IP (the caller) has:
1 combo of KK
10 combos of JJ-22

OOP is betting 100% of AA and 50% of QQ. In this case IP calls 100% of KK and folds everything else.

So in Example 2 IP will call as much as he can vs the pot bet to get as close as possible to defend 50% of range. If IP doesn't have enough hands that can win, he just has to fold those that have 0% equity.

Obviously this doesn't work out in equilibrium. If OOP knows IP has to fold everything else than KK OTR, then OOP will bet 100% of AA and 100% of QQ. Given that IP has enough 0% equity hands that OOP can bet entire range.

I think I know what you're getting at. But before elaborating, lets clear up something about position. It's the player IN POSITION who can either bet or check in this half street game, because when he checks he sees a showdown immediately. So for clarity's sake we will name playert IP Ivan, and player OOP Oswald (the association of initial letters may help).

Ok, now that we got that out of the way, let's go to the point at hand.

I think I see what you are trying to get at, but not sure whether you are taking bet sizes into account, but anyway, for a pot sized bet, alpha is 50%, and for a half pot bet, it is 33%. Now if I run this scenario with a half pot bet in either gto+ or crev, the equilibrium solution por Oswald is to call with KK 33% of the time, regardless whether his range is AA,KK,QQ, or AA-55. From this we can deduce that MDF (67%) applies only to the part of the range that beat a bluff. So in this case, Oswald should call 67% of hands that beat a bluff, regardles of the amount of combos in his range that don't.

From this I can infer that what you said is correct if Ivan was bluffing with something worse that AA-55, but not if he is bluffing with QQ.

JJ-22 are not part of the continuance range. Those hands are not entitled to any part of the pot. MDF only applies to KK in both examples. Just to belabor the point a bit:

IP makes a pot sized bet with AA and 50% of QQ.
OOP vs a pot sized bet with 2 different ranges:





Either way, KK is the only real bluff-catcher here, so it will call according to MDF.

Actually right! I had a small error in my solve and got confused. So the "making the WORST bluffcatcher indifferent" is always valid. In this case the WORST (the worst hand that has enough equity to bluffcatch) bluffcatcher KK is completely indifferent in both cases.

Thanks tombos21 :-)

I still don't think the term continuance range is appropriate. Hands which are pure folds can still count towards MDF. The reason the hands in this example don't count is because IP can check back QQ and beat them. The hands which lose to bluffs are the ones that don't count.

"Continuance range" is stricter than "hands which beat all bluffs." Not all bluffcatchers go into the continuance range.

Yeah, maybe "all hands that beat a bluff" or "nuts + bluffcatchers range" is more accurate.