I’ve been reading Applications of NLH by Matthew Janda recently and I can’t pin down 2 concepts. I know they’re fundamentally related but I don’t know how.

#1. MINIMUM DEFENSE FREQUENCY

This is the frequency we must defend with to prevent our opponent to have a profitable bluff with any 2 cards. Calculated by the following formula:
(Pot size) / (pot size + bet size)

Ex. If we face a $75 bet on the river with $100 in the pot. We must defend:
(100)/(100+75) = 57%
At the minimum to prevent our opponent from having a profitable bluff with any 2.

#2. VALUE TO BLUFF RATIO

This is the ratio of value bets to bluffs we ought to have when we make a bet such that our opponent is indifferent to calling in a given spot.

Ex. We bet $75 on the river into a $100 pot. Our opponent now can now risk $75 to win $175.
So we better be value betting:
(1.75) (1-X) - (.75) (X) = 0
X=70%

Question:
What is the relationship between these 2 number here? And what is the relationship between the 2 formulas?

There isn't really a relationship (or at least not a linear one) between the numbers or formulas. It's just that if villain bets a perfectly balanced/polarized range and you call with your bluff-catchers at the optimal frequency, villain's bluffs will break even, as will your bluff-catchers. As such, both players are 'indifferent' at equilibrium and cannot improve their strategy.
If one player alters their strat, it becomes exploitable. e.g. If villainunder-bluffs, you should stop calling with bluff-catchers, and if he increases his bluffing, you should call more because your catchers will actually be profitable (not just breakeven).
For any bet-size there will be an optimal ratio for the bettor, and an optimal calling frequency for the caller, but there's not a linear relationship. It all depends upon the pot size and the pot odds. If a large bet is chosen, then the amount of calling goes down, which means more bluffs can get through.

You should be able to learn/remember the most common ratios and frequencies.

2x pot => MDF = 33.33% and v:b ratio is 3:2 (60% value)
1x pot => MDF = 50% and v:b ratio is 2:1 (66.7% value)
3/4 pot => MDF = 57.14% and v:b ratio is 7:3 (70% value)
1/2 pot => MDF = 66.67% and v:b ratio is 3:1 (75% value)

Since MDf, V:B and V frequency (V%) all depend only on the pot and bet sizes, they are obviously related:

For example,

V:B = Pot/(Bet*MDF)

MDF = (Pot/Bet)*(1/V% -1)

I don’t see much value in knowing this, however, except perhaps to quickly see that the value to bet ratio decreases as MDF increases, and MDF increases as the value frequency decreases.

By relationship, do you mean how they can be used together?

MDF and V:B can be used to determine a balanced range you should be betting with on certain rivers. First, determine what your opponent's MDF is and choose the best hands for calling. Second, pick the value hands that have greater EV than checking. Third, use the V:B to determine how many bluffs you can add to make your opponent indifferent with his weakest bluff catchers.

For example, your opponent checks back on river. You go all-in for 2x the pot. Your opponent's MDF is 33%. Versus that range lets say only 18 hands have better EV jamming than checking. Our value to bluff ratio is 3:2. (18 / 3) * 2 gives us 12 bluff combos to choose from.

I always liked 1 - (bet / (bet + pot)) for determining MDF. The equation without the "1 -" at the beginning tells you how often your opponent needs to fold in order to bluff with any two cards. Adding the "1 -" flips the situation around for how often they have to defend.

MDF and V:B have to be used together to construct ranges that make sense. Without MDF you won't know how many value combos to pick. Without V:B your opponent can exploit you by either calling more often or folding more often than MDF.

I'd like to add that MDF isn't always set in stone. In some situations it may be more profitable to check back with hands that could be value bets. In these situations your opponent would be optimal to defend less than MDF. For simplicity, it should be alright for balance.

Ok this is what i was asking but i realize now that its kind of a pointless question. Thanks for the response tho. i went a ahead and graphed both equations on the same graph here to show the relationship.

MDF= (PS)/(PS+BS)

V% = (PS+BS)/(2BS+PS)

I set the Pot size as a constant of 1 so that we can see bet sizes more clearly as fractions of the pot.



So right away we can derive the 4 examples ArtyMcFly gave.

hey OP i wanted to add something to this discussion because there used to be one aspect of MDF that used to trip me up: whatever frequency we derive from the equation only applies to the portion of our range that can beat a bluff. Artymcfly and godson already said this basically

I recommend you check out the AKQ toy game in mathematics of poker because it illustrates the point well

For example if we arrive on the river, and villain makes a pot sized bet MDF=50%, but we do not call with 50% of our total range, we call 50% of the region of our total range that can beat a bluff. For example if 60% of our total range can beat a bluff, we call half of that 60% (30% of our total range)

maybe you already knew this but i thought i would mention this just in case

I used GTO+ trial version to simulate play for the AKQ game. Using a board of T 9 6 2 2 and auto checking to the river. I used a range of TT, 99 and 66 for both players to simulate the game accounting for blockers and used a range of AKs-AJs of different suits for both players to simulate the game excluding blockers.

With blockers:
OOP player always checks
IP player always bets A, always checks K, and bets Q 23.7%
OOP player only calls with A and folds everything else

Without blockers:
OOP player always checks
IP player always bets A, always checks K, and bets Q 50%
OOP player always calls with A, calls with K 33%, and folds Q

I did not already know this.

I’m still not really convinced. Why is this?
Isn’t the whole point of MDF in your example that villain risks 1 unit to win 1 unit, so we better defend 50% of the time or he’s printing.
If we only defend 50% of 60% then we’re effectively only defending 30%... so he printing!

I was under the impression that the only time it’s ok to over fold these spots are spots where we have a range disadvantage. (We don’t simply over fold on account of the fact that all the hands in our range aren’t bluff catchers).

For example facing a pot sized bet if only 40% of our range can beat a bluff then we just call with the whole 40% and acknowledge that villain has a profitable bluff there and this is the best we can do.

But if 40% of our range can beat a bluff and we acknowledge a significant enough chance that villain is bluffing to make calling with 1/2 the bluff catchers profitable, surely calling with the other 1/2 of the bluff catchers is also profitable!

I would appreciate if if you explained a bit better why we only call with 50% of our bluff catchers

If the spot is not polar, and you decide not to bluff or bet, then by definition you do not holding a combo that is “bad enough” to bluff. This leaves you with a weak value range, and most of that can beat a bluff if you have made it to the river, for instance.

So, the point +EV is making is valid, but crunching it down into discrete combos is difficult. For instance, there are probably unaccounted combos that should be thin calls on the turn, but are folded on the turn in practical application of real world poker.

If you count every possible combo you could ever have at the river, then MDF check-call combo vs pot sized bet is exactly 50 percent.

This is incomplete wording. Should read:

I really don't think MDF is a useful concept. I think it does more harm than good in a lot of spots. Even in river spots it's often more important to think about what are your best calls due to blockers rather than MDF, and for non river spots I think MDF shouldn't even be considered.


Artie's post is the best explanation of the simple math behind it though.

yes, and mdf only gives us half the picture, the other half is determining how our distribution interacts with our opponents distribution (on the river, where this is relevant)

hyperknit im afraid that i will not be able to do these ideas justice if i try to explain them myself, the AKQ and 0-1 games in mathematics of poker are the best way to visualize them, IMO they should be mandatory reading for every player. here is the online PDF link

http://www.pokerbooks.lt/books/en/Th...s_of_Poker.pdf

there are several versions of both toy games, but the simplest are the half street limit versions which begin around 137. they also explain the nuance of OBF (opt. bluff freq) as well, OBFand MDF need to be understood together

if you are willing to break off a piece of your roll i would recommend in the strongest possible terms that you invest in a RIO subscription and watch some of the theory videos, specifically those by Ben Sulsky about toy gaming

2 reasons why this is true

When computing MDF you are only focusing on making villain's bluffs indifferent. Other parts of villain's range may have positive expectation no matter what you do.

If you defend with hands that can't beat enough hands in villain's range (bluffs or thin value bets) then the hand you are calling with will always loose money and therefore would be better off folding.


Not exactly. Again the villain only risks 1 unit with his bluffs. His other hands aren't risking they are valuebetting. So the balance is defending enough to prevent villain's bluffs from profiting while not paying off villain's valuebets too often. It's why in the AKQ game the K only defends 50% of the time. It just happens to be the results based on the relative frequency the A/Q bettor shows up with value and air.

Calling with an individual bluff catcher is always profitable if villain is bluffing at the correct frequency.

However if your calling frequency is too great then villain can STOP bluffing with any frequency and make more money.

Similarly if villain only bets the nuts you can stop calling.

Then if villain notices this he can bluff all of his hands.

If you notice that then you can start calling all of your bluff catchers.

It's the balance in the betting players range construction and the callers calling frequency that determine the equillibrium. If there is deviation on either side the other player can make more money by changing their strategy.

ok i just read the AKQ game section. but the upshot is that the caller defends with 1/3 of his Kings to a 1/2 pot bet.

"Similar reasoning works for X. He wants to call often enough that the opponent is indifferent to
bluffing with his queens. When his opponent bluffs successfully, he wins two bets, while
bluffing unsuccessfully costs just one bet. So X must call twice as often as he folds in order to
make the bluffer indifferent. This fraction is 2/3 (leaving 1/3
of hands to fold). But half of the
time that the opponent is bluffing, X will hold an ace. So he needs to call with an additional
fraction of his hands equal to 2/3 - 1/2, or 1/6 . Since kings make up 1/2
of his hands when the opponent has a queen, he needs to call with 2/6 or 1/3
of kings. " (pg 142, P2)

according to my math, player X should never call with a King to a pot size bet. (he can just call with his Aces and never get exploited).

similarly, if faced with a 10x pot sized bet (or a 1000x pot sized bet), player X should only call with Aces.

SO u dont just call with 1/2 your bluff catchers! you use MDF to find out how much u have to call, then U take: MDF - (% of range is aces) = Z. Then if Z>0 you add Z bluff catchers so that u reach the MDF!

Yes, and even in river spots where you "want" to think about MDF there are often spots where we don't defend MDF on the river because of our overall range disadvantage or because we've been betting a super polar range on previous streets and now check the river oop and when IP bets we do not defend anywhere close to MDF.

I really feel like MDF is a huge hindrance to an understanding of poker theory; focusing on MDF does more harm than good in most spots imo.

What about your thoughts on value to bluff ratio Jarretman?

Have u read application of NLHE? Because he specifically addresses both points u made (which collapse to the same point in most cases).
The cliff notes is that spots will exist where bluffing any 2 is profitable, and so MDF is not applicable in those spots. However we must make sure that villain paid the price on an earlier action so that the profitable bluff spot doesn’t come up often enough that his previous float is printing.
For example if we pot the flop and he floats. And a rare turn card rolls off (say with 8% probability) then it’s not so bad to c/ over fold in these spots so long as it’s a rare occurrence.
However apart from the corner cases like this MDF is a reliable tool and should be a default

Word. I've been saying this for about 3 years.
Outside of toy games, and spots where ranges are very clearly defined (nuts/air vs bluffcatchers), MDF is pretty useless.

Arty, what method do you use, on the river, to decide what combos to call vs what combos to fold?

Thanks

I just use my experience (which forms a "gut feeling") to decide whether calling makes money or not. Knowing roughly where I am in my range helps with that. Sometimes it's just a case of "Top pair is a good hand in this spot and I don't like this guy's avatar. I call". If my feeling is "I hate this guy's avatar" then I raise.

-1

Agreed. Janda devotes a section in his latest book to a discussion of MDF and how it's often misapplied/misundertood. Check out "Clearing Up Some Confusion" (pp 159-164).