Not sure
"Correct (in all cases): EV of calling >= EV of folding"
is true in all cases. Depends on your equity and the size of the bet you have to call, doesn't it?

Hi tombos - yes, your post makes perfect sense to me - and I thought it through carefully and did some numbers. I was a little perplexed until I realised that:

"EV of calling >= EV of folding"

and

"EV of bluffing >= EV of checking"

may mean that all values are zero.

However, this does not clear up the confusion I have about minimum defence frequencies.

I've always considered EV as the potential/expectation of increasing your stack IF you make an 'investment' (bet/call). We use the amount of chips in the pot, the amount of chips we need to put into the pot and the (estimated) chance/percentage of winning the pot.

Yes, IMO OP is moving around the reference points and mixing the terms EV and 'profit'.

"Don't chase bad money with more bad money"
"Don't get married to your hand"
"Well, there was so much 'out there' ... so I called"
"Well, I came this far I might as well go the distance"

EV is 'expected value' if I TAKE a certain course of action and should not be influenced because I took a previous action to get to this decision point.

This may not be a direct correlation but let's take a look at a set mine, which is something we need to get 8 to 1 on our chips to 'break even'.
1) We see a Flop for 5 with 4 other players .. great, 4x (20) our bet is already covered in the pot.
2) We miss the Flop and FTA donks out for 20 and it folds to us, LTA. Do we still set mine?

Are we calling 20 to win 45 or are we calling 20 getting 8 to 1 on our initial 5 call PF?

Out there yes, but I don't look at this from a 'profit' standpoint based on our stack at the start of the hand and what I've already invested. I base 'this' decision on what needs to come out of my stack 'right now' and how much might come back when I make that action.

Your reference points change during a hand all the time. We raise with JhJd (for value) PF and see a Flop 4 ways. Flop comes AsKc4s .. Are we still evaluating the spot as 'value'? No, this just turned into runner/runner, set mine or bluff catching spot (if that). I'm certainly not going to fret over my PF raising chips when making this decision.

So I can't pick and choose when 'my' chips in the middle matter in my decisions, I can only make decisions based on what I do with my 'remaining' chips going forward.

Yes, there are spots in poker where Players are 'pot committed', but there's no suggestion of that in this thread as of yet. To me EV is not 'profit', but I do calculate EV to see if a decision is 'profitable' in the long run. GL

From UpSwing Poker

How to use minimum defense frequency
Minimum defense frequency (MDF) describes the portion of your range that you must continue with when facing a bet in order to remain unexploitable by bluffs.

In other words, MDF is the minimum percentage of time you must call (or raise) to prevent your opponent from profiting by always bluffing you. If you fold more often than the MDF indicates, your opponent can exploit you by over-bluffing when they bet.

The formula to calculate MDF is simple:

pot size / (pot size + bet size)

Then, multiply the answer by 100 to express it as a percentage.


Not sure .. but perhaps you were thinking MDF was a 'times in the spot' calculation?

As described above .. it's the percentage of YOUR RANGE that you should call with. So when facing 1/2 pot bet you should be defending the upper 2/3 of your range for the spot, not 67% of the time you're in the spot.

This again comes down to how you define the terms. Granted over the long term 'time' will ultimately match up to 'range'. Even within this short blurb, they switch to 'time' from range in the explanation. GL

Telemakus, do you understand conditional probability?

Example, I want to set mine with pocket pair. If I choose to hit set or fold on the flop, then I have 12% chance of hitting my set. If I choose to always call a flop bet, I have 16% chance to hit my set. However, if instead of looking at the strategy as a whole, I consider my decisions in the moment, the flop comes and I missed my set, now I have 4% chance of hitting my set on the turn. All those possible sets I could have hit on the flop are gone now, they didn't happen, so I need to consider my situation now.

I think it's a similar thing with your previous bets. If you consider the strategy as a whole, yes previous bets matter. If you consider the decision in the moment when something has already happened, you have to disregard what you already put in the pot. It's not yours anymore

@Telemakus

MDF is acheived when you make your opponent "indifferent" to bluffing. Note that "indifferent" does not mean greater than 0EV (unless you consider previous bets a sunk cost). Indifferent means that the EV of bluffing = EV of checking back, or facing a bet indifferent means that EV of calling = EV of folding.

I've made a tool that I think will help clear up your confusion. It's based on Browni's example.

OOP has KK
IP has AA or QQ
pot is 10bb, each player has contributed 5bb
IP shoves pot (10bb into 10bb pot), action on OOP...

You input IP's bluffing frequency, and OOP's calling frequency. It outputs the expected value.

If OOP calls 50% of the time (MDF) no matter what, then IP's EV of bluffing is the same as their EV of checking. In other words, you have made IP's bluffs indifferent.












If IP bluffs exactly 50% of the time, then your EV of calling KK is the same as your EV of folding KK. They have made your bluff-catcher indifferent. This is true regardless of how often you call.







------

It does not matter whether or not you model this with or without sunk costs. The optimal decision is the same either way.



It is true in all cases; your EV on a call is defined by the pot odds laid, and your equity in the pot. You would never knowingly make a call that was lower EV than folding.

Your EV calculations are correct here. Now if OOP calls 33% of the time like you claim he should, what is the EV of IP's strategy if he chooses to bet 100% of the time instead? Give two answers: What you think it should be before you calculate, and the result of your calculation.

Now my follow up will be, what if OOP calls 50% of the time (MDF)?

Okay – I think I realised what I was doing wrong.

Let’s say it’s the river and there is 100 in the pot with 100 effective stacks.
OOP goes all in. How often does IP need to defend?

Using the MDF formula, we can see that the answer is:
pot size / (pot size + bet size) or 100/200 in this case – so 50%.
I.e. IP must call 50% of the time to prevent OOP from profitably bluffing with any two cards.

The area where I had some confusion was this:

Before the river bet goes in, both players have put 50 into the pot. Therefore, when OOP’s bet succeeds, his total profit for the hand is 50. When he is called and his bet fails, his loss is 150. So what happens when IP calls at 33% frequency – far below the MDF? OOP wins 50 two times out of three, and loses 150 one time out of three. IP, on the other hand, loses 50 two times out of three, and wins 150 one time out of three. Shock horror – OOP is losing money, despite the fact that IP is defending at below the minimum defence frequency.

The EV of OOP’s bet is (0.66 x 100) + (0.33 x -100) = 33.
The EV of IP’s call is (0.33 x 200) = 66.

How can OOP be losing money when the EV of his bet is positive? Because the EV equation operates from the point of view of the current pot, and does not consider the money that either player has put in there.

When we instead use the profit that the players make for the EV equation, it looks like this:

The net average profit for the hand for OOP is:
(0.66 x 50) + (0.33 x -150) = -16.67.

The net average profit for the hand for IP is:
(0.66 x -50) + (0.33 x 150) = 16.67.

So what is going on here? How can IP defend at below the MDF and still be winning?
The answer – and the reason for the confusion – is that the above assumes that OOP is always bluffing when he bets. Of course, this is not the case – and if it was, then it would explain how IP can make money calling at well below the MDF. In fact, as we all know, when OOP bets the pot – if he is playing correctly – then he is value betting 66% of the time and bluffing 33% of the time. How does this impact the EV equations?

The EV of OOP’s bet now becomes (0.66 x 100) + (0.33 x 200) = 133.334 when he is value betting – two thirds of the time.

And the EV of OOP’s bet now becomes (0.66 x 100) + (0.33 x -100) = 33 when he is bluffing – one third of the time.

So the actual EV of his bet is (133.334 x 0.66) + (33 x 0.33) = 100.

The net average profit for the hand for OOP is:

50+50+150 (when value betting) 50+50-150 (when bluffing)
Two value bets to every bluff = 50+50+150+50+50+150+50+50-150 = 450/9 = 50.

The EV for IP’s call is (0.66 x 0) + (0.33 x -100) = - 33 when OOP is value betting – two thirds of the time.
The EV for IP’s call is (0.66 x 0) + (0.33 x 200) = 66 when OOP is bluffing – one third of the time.

So the actual EV of his call is (0.66 x -33) + (0.33 x 66) = 0.

The net average profit for the hand for IP is:
-50-50-150 (when OOP is value betting) -50-50+150 (when OOP is bluffing)

Two value bets to every bluff = -50-50-150-50-50-150-50-50+150 = -450/9 = -50.

If IP defends at the actual MDF of 50%, then:

The EV of OOP’s bet becomes (0.50 x 100) + (0.50 x 200) = 150 when he is value betting – two thirds of the time.

And the EV of OOP’s bet now becomes (0.50 x 100) + (0.50 x -100) = 0 when he is bluffing – one third of the time.

So – the actual EV of his bet is (0.66 x 150) + (0.33 x 0) = 100.

The net average profit for the hand for OOP is:

50+150 (when value betting) 50-150 (when bluffing)

Two value bets to every bluff = 50+150+50+150+50-150 = 300/6 = 50.

The EV for IP’s call is (0.50 x 0) + (0.50 x -100) = - 50 when OOP is value betting – two thirds of the time.
The EV for IP’s call is (0.50 x 0) + (0.50 x 200) = 100 when OOP is bluffing – one third of the time.

So the actual EV of his call is (0.66 x -50) + (0.33 x 100) = 0.

The net average profit for the hand for IP is:

-50-150 (when OOP is value betting) -50+150 (when OOP is bluffing)
Two value bets to every bluff = -50-150-50-150-50+150 = -300/6 = -50.

So to summarize – in the first (very imbalanced) instance where OOP is bluffing 100% of the time:

OOP EV: 33, Net Average Profit: -16.67.
IP EV: 66, Net Average Profit: +16.67.

In the second instance where OOP is bluffing 33% of the time and IP is calling 33% of the time:

OOP EV: 133.334 (value bet), 33 (bluff), 100 combined. Net Average Profit: 50.
IP EV: -33 (OOP value bet), 66 (OOP bluff), 0 combined. Net Average Profit: -50.

In the third instance where OOP is bluffing 33% of the time and IP is calling 50% of the time:

OOP EV: 150 (value bet), 0 (bluff), 100 combined. Net Average Profit: 50.
IP EV: -50 (OOP value bet), 100 (OOP bluff), 0 combined. Net Average Profit: -50.

So what’s the deal here? It seems that OOP has more EV in the second instance compared to the third instance (a total of 166.334 compared to a total of 150) – but when the value and bluffing EVs are combined, they equal 100 in both cases? Also – why is the net average profit the same in both cases?

I assume the difference in EV is because OOP is no longer profiting from his bluffs (the whole idea of MDF when facing a bet) but why is the combined EV of value betting and bluffing the same in both cases (100) – and why does this difference in the separated EVs not result in a larger net average profit?

@answer20

Thanks for the response, and yes, what you're saying most total sense - I've got that part of it nailed, I just don't understand some of the mechanics of MDF.

@SmallBallCall

Thanks, what you're saying also makes a lot of sense and I agree with it, but it doesn't address MDF.

@tombos 21

Thanks a million for going to that effort, I had a look at the spreadsheet and it's great. Is there any chance you could hook it up to show the numbers for the situation that I detailed in the post above, where OOP has already made the bet and IP is deciding whether or not to call?

@browni3141 -

Thanks, I will take a look at that tomorrow. Have been trying to figure out the above most of the evening!

Cheers guys, I really appreciate the help and that you would share your knowledge.

T

I swear Tombos is the man at answering questions. Helped me out of my biggest maths block I ever had once.

But yeah long story short, if the call makes sweet ev gains but ultimately still returns no “profit” you still have to make that call. Why? Cos it makes more than a fold.

Sure. Here's an updated one that lets you enter the pot, effective stack, as well as how much IP and OOP have contributed prior.

OOP shoves AA and some % of QQ, action on IP with KK.

https://docs.google.com/spreadsheets...it?usp=sharing





@Telemakus, you should definitely give this a go as it gets at the crux of your issue. Try to solve it by hand.

Much appreciated @Cfoye!

@tombos that's so great - thanks a million.

The first thing that is immediately obvious when I start putting numbers into the spreadsheet is that when OOP is bluffing 33% of the time his betting frequency is 66%, and when he is bluffing 50% of the time, his betting frequency is 75% - I had not accounted for this yesterday when I was trying to work everything out.

Instead, I was saying that OOP bets 100% of the time and that: In the first instance, he is always bluffing, then in the second and third instances he is value betting 66% of the time and bluffing 33% of the time. I get the feeling that this must be impossible though, I.e. in the first instance he has arrived at the river with only bluffs, and in the second and third cases he has arrived at the river with too many value bets. Is that correct?

Or is it that he must balance the bluffs with checks some of the time - this is another thing I did not consider yesterday.

Thanks again!

T

Before I calculate, I believe at first glance of course that IP’s EV will go down and OOP’s EV will increase, as IP is bluffing at a greater frequency and IP is calling at the same frequency. Although, on the other hand, it’s important to note that the only time that OOP is making money is when IP checks with the queens. Essentially what will change is that the EV of OOP’s calls will increase by 50, but he will no longer win when IP checks (as that is now removed from the game tree) – so he will lose his EV of 100 in those cases, and overall his EV will actually decrease as IP increases his betting frequency to 100% which – I think – makes OOP’s EV 50 when he calls, but this is only a third of the time so his total EV is 50/3 = 16.667. IP’s EV should increase as he is longer checking the queens, so I think that would make it 83.334.

Okay so now calculating:

IP goes all in with his aces and all of his queens.
OOP calls 33% of the time.

Assuming a current pot size of 100, IP's EV is 83.334 because:

When he is value betting, I.e. 50% of the time -
his EV is (0.66 x 100) + (0.33 x 200) = 133.333

When he is bluffing, I.e. 50% of the time -
his EV is (0.66 x 100) + (0.33 x -100) = 33.333

Total EV = (133.333 + 133.333 + 33.333 + 33.333) divided by 4 = 83.334.

OOP calls 33% of the time when IP bets and his EV is 50 because:

(0.50 x -100) + (0.50 +200) = 50.

OOP folds 66% of the time when IP bets and his EV is 0.

Total EV = (50 + 0 + 0) divided by 3 = 16.667.

So it’s as expected – IP gains EV because OOP no longer wins when the hand checks through, and the bluffs are making profit.

If OOP instead calls at 50%:

IP goes all in with his aces and all of his queens.
OOP calls 50% of the time.

Assuming a current pot size of 100, IP's EV is 75 because:

When he is value betting, I.e. 50% of the time -
his EV is (0.50 x 100) + (0.50 x 200) = 150

When he is bluffing, I.e. 50% of the time -
his EV is (0.50 x 100) + (0.50 x -100) = 0

Total EV = (150 + 150 + 0 + 0) divided by 4 = 75.

OOP calls 50% of the time when IP bets and his EV is 25 because:

(0.50 x -100) + (0.50 +200) = 50.

OOP folds 50% of the time when IP bets and his EV is 0.

Total EV = (50 + 50 + 0 + 0) divided by 4 = 25.

Hey guys! Do you have any feedback on the above please?

Cheers

T

Your answers are correct! But use a weighted sum instead of an average. The only reason an average works here is because IP's value:bet ratio is exactly 50:50. For example:

EV = 50% valueEV + 50% bluffEV
EV = 50%(133.33) + 50%(33.33) = 83.33

So now you see that changing the point of reference doesn't matter, yes?

Great, thanks tombos!

How does a weighted sum work? Using the sample example as above, where:

IP's value:bet ratio is exactly 50:50. For example:

EV = 50% valueEV + 50% bluffEV
EV = 50%(133.33) + 50%(33.33) = 83.33

If IP's value:bluff ratio was 2:1, I would have done it like this:

EV = 66%(133.33) + 33%(33.33) = 88.888 + 11.111 = 99.9

Or, in the other examples from above, again assuming 2:1 value:bluff and assuming a current pot size of 100, IP's EV is 100 because:

When he is value betting, I.e. 66% of the time -
his EV is (0.66 x 100) + (0.33 x 200) = 133.333

When he is bluffing, I.e. 33% of the time -
his EV is (0.66 x 100) + (0.33 x -100) = 33.333

Total EV = (133.333 + 133.333 + 33.333) divided by 3 = 100.

OOP calls 33% of the time when IP bets and his EV is 0 because:

(0.66 x -100) + (0.33 +200) = 0

OOP folds 66% of the time when IP bets and his EV is 0.

Total EV = (0 + 0 + 0) divided by 3 = 0.

If OOP instead calls at 50%:

Assuming a current pot size of 100, IP's EV is 100 because:

When he is value betting, I.e. 66% of the time -
his EV is (0.50 x 100) + (0.50 x 200) = 150

When he is bluffing, I.e. 33% of the time -
his EV is (0.50 x 100) + (0.50 x -100) = 0

Total EV = (150 + 150 + 0) divided by 3 = 100.

OOP calls 50% of the time when IP bets and his EV is 0 because:

(0.66 x -100) + (0.33 +200) = 0.

OOP folds 50% of the time when IP bets and his EV is 0.

Total EV = (0 + 0 + 0) divided by 3 = 0.

Yes, I do see what you mean about the point of reference, and I am still trying to figure out what was wrong with my maths when I tried the first time around – I think, with your help, I may have figured it out. Did you see my post from before:

“The first thing that is immediately obvious when I start putting numbers into the spreadsheet is that when OOP is bluffing 33% of the time his betting frequency is 66%, and when he is bluffing 50% of the time, his betting frequency is 75% - I had not accounted for this yesterday when I was trying to work everything out.

Instead, I was saying that OOP bets 100% of the time and that: In the first instance, he is always bluffing, then in the second and third instances he is value betting 66% of the time and bluffing 33% of the time. I get the feeling that this must be impossible though, I.e. in the first instance he has arrived at the river with only bluffs, and in the second and third cases he has arrived at the river with too many value bets. Is that correct?

Or is it that he must balance the bluffs with checks some of the time - this is another thing I did not consider yesterday.” ?

I think once I have an answer to this, I will be able to redo the maths from the first time I tried, and – hopefully – it will work out correct.

Thanks again so much for your help!

T

If we're using a 2:1 value:bluff ratio, then we're checking back sometimes and losing, you need to account for that.

Here's a more intuitive way to calculate EV:
1) Write out all the possibilities of the game tree.
2) Assign a probability to each one, such that the sum of the probabilities = 1.
3) Assign an EV to each one.
4) Use a weighted sum to add them all together.

Example:
IP bets all AA and 50% QQ (2:1 value:bluff ratio) and OOP calls 50% KK

Here are the possibilities:

1) IP bets, OOP calls, IP wins(p = 0.75 * 0.5 * 0.666 = 0.25) (+$200)
2) IP bets, OOP calls, IP loses (p = 0.75 * 0.5 * 0.333 = 0.125) (-$100)
2) IP bets, OOP folds (p = 0.75 * 0.5 = 0.375) (+$100)
3) IP checks back and loses (p = 0.25) ($0)

Now we put them together with a weighted sum:
IP EV = (0.25 * $200) + (0.125 * -$100) + (0.375 * $100) + (0.25 * $0) = $75

You may have mixed up a few EV calcs, but the original issue was this:

Your mistake was only using +$0 bluffs, instead of bluffs that have greater EV than giving up. You decided sunk costs matter, which means giving up is now -EV, yet you still only bluffed with hands that have >$0 EV. That's the crux of the issue.

"If we're using a 2:1 value:bluff ratio, then we're checking back sometimes and losing, you need to account for that."

Sure thing - I had not accounted for that in most of my previous calculations - I will redo them when I know how I am meant to approach them.

So - how do we calculate the frequency with which we should bet/check? In the spreadsheet that you sent, it says that if IP is bluffing 50% of the time then his betting frequency is 75% (and the betting frequency seems to increase as the bluffing frequency increases). Is there a formula of some kind to show how often a player should bet/check in relation to how often he is value betting/bluffing?

I understand your example totally and it makes perfect sense - thanks a million. I think it's clear now that the problem I was having before was that I was not looking at the entire game tree - I was only looking at the section where betting occurs, and omitting the section where there is checks is of course going to affect the EVs of the players significantly. Also, it means that the MDF calculations will appear incorrectly.

"Your mistake was only using +$0 bluffs, instead of bluffs that have greater EV than giving up. You decided sunk costs matter, which means giving up is now -EV, yet you still only bluffed with hands that have >$0 EV. That's the crux of the issue."

Well, kind of - it was not that "sunk costs matter" exactly, it was that I couldn't see how the increased EV that IP has when OOP is only calling at 33% frequency instead of 50% frequency translates into actual profit (because IP's value + bluff EV was 100 in both cases, despite the bluffs being positive EV when OOP is calling at 33%). This is what I could not see - how that extra EV converts into extra profit, and that is still something I don't understand. It makes sense mathematically (with the EV calculations) but when you add up the profit/loss, it doesn't appear to be there.

However, it does seem that in these setups, IP is incentivized to bet at a high frequency. For example:

IP is bluffing 50% of the time and OOP is calling 50% of the time. Betting frequency is 75%.

Value of the game for IP:

Bets 75% of the time and the EV is: (0.50 X 100) + (0.50 x 50) = 75.
Checks 25% of the time and his EV is 0.

Total IP EV: (0.75 x 75) = 56.25

Value of the game for OOP:
Facing a bet (75% of the time) he:

Calls 50% of the time and his EV is (0.50 x -100) + (0.50 x 200) = 50
Folds 50% of the time and his EV is 0.

It checks through 25% of the time and his EV is 100

Total OOP EV: (0.75 x 25) + 25 = 43.75

If instead IP bets at 100% frequency:

IP is bluffing 50% of the time and OOP is calling 50% of the time.

Value of the game for IP:

IP bets 100% of the time and the EV is: (0.50 X 100) + (0.50 x 50) = 75.

Total IP EV: 75.

Value of the game for OOP:

Facing a bet (100% of the time) he:

Calls 50% of the time and his EV is (0.50 x -100) + (0.50 x 200) = 50
Folds 50% of the time and his EV is 0.

Total OOP EV: 25.

I never really delved into the mechanics of mdf because it always seemed so much less important than learning how to calculate ev when combined with things like fold equity. To me it just doesn’t seem very applicable very often and if I tried to apply it I’d probably fall into fancy play syndrome where I make big losing calls because “I’m unexploitable man”. Besides, very rarely are people straight up bluffing places like the flop. There’s usually some equity in their holdings that skews the calculations.

Yes. IP uses a bluff:value ratio designed to match the pot odds, so that the QQ bluff-catcher is indifferent. In this example the pot odds are 33.3%, so the correct strategy is to use 2value:1bluff. Obvious AA is always a bet, so you just have to balance with the appropriate amount of QQ, and check behind whatever is left. This is always true for "perfectly polarized" situations, but it gets way more complicated in practice.

The "bluffing%" just represents how often you bluff with QQ. It assumes you always bet AA. So the total betting frequency is just (1 + bluffing%)/2

Well half their range is the stone cold nuts, so it's always gonna be at least 50%. They'll add bluffs to balance that according to the pot odds laid.


MDF has its place in abstract game theory. Though the more I study it the less applicable it seems in practice.

I agree that learning how to calculate EV is so much more fundamentally important. MDF/Pot Odds/betting frequencies/etc can be derived from EV equations, after all.

Ah, of course! This does make perfect sense. It's interesting to see how the EVs change when IP increases the betting frequency:

IP is bluffing 50% of the time (I.e. betting his whole range of AA and QQ) and OOP is calling 50% of the time. Betting frequency is 100%.

Value of the game for IP:

Bets 100% of the time and the EV is: (0.50 X 100) + (0.50 x 50) = 50+25 = 75.

Total IP EV: = 75

Value of the game for OOP:

Facing a bet (100% of the time) he:

Calls 50% of the time and his EV is (0.50 x -100) + (0.50 x 200) = -50 + 100 = 50.

Folds 50% of the time and his EV is 0.

Total OOP EV: (0.5 x 50) + (0.5 x 0) = 25.

IP is printing money. Why? Because OOP is not calling enough. If IP is betting the pot with 50% value and 50% bluffs, then OOP must adjust and call more of the time as his calls are plus EV because IP is bluffing too much. If OOP instead calls at 75% frequency, then:

Value of the game for IP:

Bets 100% of the time and his EV is: (0.25 X 100) + (0.75 x 50) = 25+37.5 = 62.5.

Total IP EV: = 62.5

Value of the game for OOP:

Facing a bet (100% of the time) he:

Calls 75% of the time and his EV is (0.50 x -100) + (0.50 x 200) = -50 + 100 = 50.

Folds 25% of the time and his EV is 0.

Total OOP EV: (0.75 x 50) + (0.25 x 0) = 37.5

IP is still winning because OOP is not punishing him hard enough for over-bluffing. As soon as he knows his calls are +EV he should call every time, then it looks like this:

If OOP calls at 100% frequency then:

Value of the game for IP:

Bets 100% of the time and the EV is: (0.50 x -100) + (0.50 x 200) = 50.

Total IP EV: = 50

Value of the game for OOP:

Facing a bet (100% of the time) he:

Calls 100% of the time and his EV is (0.50 x -100) + (0.50 x 200) = 50.

Both players break even. IP loses value because he is bluffing too much, and OOP exploits him by increasing his calling percentage.

Of course, IP should instead bluff at the optimal frequency of 33% (as he is betting the pot), which means he must check half of the time instead of bluffing every time with his queens:

IP is bluffing 33% of the time and OOP is calling 50% of the time. Betting frequency is 75%

IP’s EV when he bets is (0.33 x -100) + (0.66 x 200) = 100.
IP’s EV when he checks is zero.
Total EV = (0.75 x 100) + 0 = 0.75

When there is a bet (75% of the time)

OOP calls 50% of the time and his EV is:

(0.66 x -100) + (0.33 x 200) = 0.

OOP folds 50% of the time and his EV is 0.

Where there is a check (25% of the time)

OOP checks and his EV = 100.

Total EV = (0.25 x 100) = 25

So - as expected, IP makes the most money when he bluffs at the optimal frequency - assuming that OOP is also playing optimally.

Another thing - I figured out how EV translates into profit! It may seem simple to you, but I had not fully appreciated that EV is equal to the amount of the pot to which the given player is entitled. So for example from the above example where:

IP is bluffing 50% of the time (I.e. betting his whole range of AA and QQ) and OOP is calling 50% of the time. Betting frequency is 100%.

Value of the game for IP:

Bets 100% of the time and the EV is: (0.50 X 100) + (0.50 x 50) = 50+25 = 75.

Total IP EV: = 75

Value of the game for OOP:

Facing a bet (100% of the time) he:

Calls 50% of the time and his EV is (0.50 x -100) + (0.50 x 200) = -50 + 100 = 50.

Folds 50% of the time and his EV is 0.

Total OOP EV: (0.5 x 50) + (0.5 x 0) = 25.

IP is making an average of 50 per hand (after 8 hands he is 400 up)

Or, in this example:

If OOP instead calls at 75% frequency, then:

Value of the game for IP:

Bets 100% of the time and his EV is: (0.25 X 100) + (0.75 x 50) = 25+37.5 = 62.5.

Total IP EV: = 62.5

Value of the game for OOP:

Facing a bet (100% of the time) he:

Calls 75% of the time and his EV is (0.50 x -100) + (0.50 x 200) = -50 + 100 = 50.

Folds 25% of the time and his EV is 0.

Total OOP EV: (0.75 x 50) + (0.25 x 0) = 37.5

IP is making an average of 12.5 per hand (after 8 hands he is 100 up).

Very cool! Thanks again

So, to return to the original post:

"Button vs. big blind and the pot is 6BBs. The big blind checks and the button bets 2BB.

I read in the books that as this is a third pot bet (the button is risking 2BB to win 6BB) that the minimum defence frequency is 75% in order to prevent the button from profitably betting any two cards."

Assuming the button is playing optimally, in theory this is a value bet 75% of the time and a bluff 25% of the time (although in practice this is unlikely). If it was true that this is a 3:1 value:bluff frequency, it would mean that the button would have to also check some portion of his range in order to remain balanced - in this case 33.34% (I believe).

So - the button is betting 66.67% of the time, with three quarters being (presumably very thin in some cases) value and a quarter being bluffs. However - anyone who knows any theory at all is aware that the flop is bluffed at the highest frequency of all streets (in order to be balanced when getting to the river). So - doesn't this turn things on their heads a little?

"However, what is not mentioned is the fact that half of the money in the pot has been put in there by the button, and therefore his actual profit from the hand is 3BB when the bet gets a fold. Doesn't this change the MDF, as it means his bet has to succeed more than 40% of the time to profitably bet any two cards - which would make the MDF 60% instead of 75%?"

The difference here (and presumably why it's not possible to use MDF accurately on the flop) is because when the big blind calls (and there is still money behind) it does not end the hand. There are two more cards to come, and they can change the EV of the hands in play considerably. Right?