Hi all,

I have a question about MDF on the flop. Let's say it's 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.

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%?

Often in books etc whatever is in the pot is considered 'dead money' and has no bearing on anything else - but doesn't that overlook the above?

Anything you put into the pot on prior streets is a sunk cost and irrelevant to the current decision. MDF seeks to deny profit only only for the current street.

Regardless, BB would normally be able to defend less than MDF since he is often at a range disadvantage due to heading to the flop with a wider range after getting a discount preflop, and being capped due to declining to 3-bet. BB would have to defend some very weak hands to meet MDF, which isn't profitable.

BU still won't always range bet because although he can profitably bet even his lowest EV bluff candidates will be +EV check backs. He can always delay c-bet or realize his equity by checking back.

MDF applies very poorly facing bets on streets other than the river for these reasons and others I haven't talked about. Most of the time it's ok to allow the opponent to realize a small profit by bluffing.

Thanks for the quick response.

So - why is money put into the pot on prior streets considered a sunk cost? Surely it's essential to include this money in any profit/loss calculations that are made?

For example, let's say there is 100 in the pot on the river (Hero and Villain have both put 50 in). Villain bets 50 - what is the minimum defence frequency that Hero needs to call at in order to prevent Villain from bluffing with any two cards?

When Hero folds, Villain wins the pot and profits 50. When Hero calls, Villain loses the pot and loses 100 overall - the money he has already invested, plus the bet on the river. So - the bluff needs to work 67% of the time in order to make a profit (a little over two wins for every loss). Wouldn't this make the MDF 33%?

The reason I ask is because, when facing a half-pot bet, the MDF is given as 66%, for example here:

https://upswingpoker.com/minimum-def...y-vs-pot-odds/

But this really doesn't seem to add up?

Because it's not your money anymore. That money is in the pot and either you or the other player is going to win it. (Unless there's a split pot)

You can't just tell the dealer that you changed your mind and want your preflop bet back. So at that point, it doesn't matter who put the money in the pot, you or the other player.

Imagine you put a $5 chip in the pot preflop and the dealer breaks it into five $1 chips from his rack. At this point, your chip is technically in the dealer's rack and you can't win it back. But you can win the five $1 chips that he put from his rack into the pot. Does that change anything for you?

Sure, I get what you're saying, but surely it is something that has to be kept in mind when calculating things such as defending frequencies etc?

To return to my first example:

Let's say it's button vs. big blind and the pot is 6BBs. The big blind checks and the button bets 2BB.

As this is a third pot bet (the button is risking 2BB to win 6BB) that the minimum defence frequency should be 75% in order to prevent the button from profitably betting any two cards, and this is certainly true if you ignore the fact that both players have invested money into the pot. The problem is that they have, and that fact results in the net profit for the hand being 3BB (and not 6BB) when the bet results in a fold - which has significant implications for the MDF - making it 60% if the button stands to win 3BB and 75% if he stands to 'win' 6BB.

I think you're getting too caught up in an 'overall' stat. Each poker decision is based on the spot you're currently in.

1) Pot Odds
2) Perceived EV based on your and opponent's ranges with 'this' Board texture
3) Stack sizes moving forward, pending different courses of action taken

As indicated above, it's OK to let your opponent take down a few pots. Folding IS NOT losing. Every session is a 'war' and you wont be able to win every battle, but you can set up play to maximize the overall battle, so to speak.

'Overall' stats are good to look at to see trends in your style of play. But to focus on gearing your game 'to' those stats would be cumbersome to me anyway. Opponents and position on those opponents change every session.

IMO every opponent has their own MDF that I would use on them. There are opponents that I would never fold to in certain spots but with others I would never see a Flop with the same holding. While I agree that 'all of that' can be reflected in a quantitative set of data I'm not going to go around worrying about it.

Yes, understand the stat ... but don't become overwhelmed by it. GL

Related to OP’s question about prior investment is the EV of folding. Most often a fold is assumed to have zero EV. Why? Well, lets first define EV as follows:

In a long series of identical poker betting situations, EV is the expected or average change in stack size as a result of making the same decision each time.

If the baseline stack size is just before the betting decision, then clearly a fold has zero EV, for there is no change in stack size. The OP might reasonably ask, “What about money I put in on earlier bets?”

Fair enough. Here is the answer.

We do an EV analysis to earn the most profit (or least loss) from our decision. Therefore, we have to consider all decision alternatives. Each alternative must use the same baseline stack as the starting point. If that baseline is the stack prior to preflop for example, then clearly a fold costs the bettor the amount he put in prior to the fold. But, for all alternatives, that prior bet amount is also a cost so we have decision consistency, i.e., the EV difference between two alternatives will not change because of a baseline stack change.

Thanks statmanhal, that does make sense.

But to reiterate - when we ignore the money that players have put into the pot, the MDF to a third pot bet is 75%. When we account for the money that players have put into the pot, the MDF is 60% - surely this has some implications for the optimal way to play the hand?

You just said that EV=0 for a fold decision makes sense when the baseline stack is that at the current decision pont. That results from not considering the money you put in earlier as a loss (actually that amount becomes part of the winnings.)

Why would another poker decision such as one based on MDF require you to consider earlier money you invested?

It's more about the money that our opponent has invested in the pot, and what he gains when his bluff is successful. In the examples above, his 2BB bet wins a 6BB pot - and this is what creates the MDF. What I am having an issue understanding is why this calculation does not account for the fact that half of the money he wins is money that he put into the pot himself. The MDF is pot divided by pot plus bet - in this case 6 divided by 8, or 75%. But his profit is only 3BB for the whole hand. This means that when he bets the 2BB it is only to gain a profit of 3BB, meaning it must work 40% of the time to break even, resulting in an MDF of 60%. This is what I don't get - the clear difference between 75% and 60%.

Thanks for taking the time to explain these things, I appreciate it a lot.

Why do you think that’s case?

Are you playing a 6BB pot differently on the flop if 1BB came from a dead blind instead of villain and you? Assuming you and villain played the same way preflop.

No I'm not playing the pot differently and that's a good point - I would play the pot the same way. My question is simply why is the money that the player put into the pot accounted for when making the MDF calculation, as I detailed in the last post.

It can be illustrated quite easily - using the same example again, if Hero bets 2BB to win a 6BB pot, then MDF for Villain is 75% to prevent Hero from profitably betting any two cards. If Villain folds:

60% of the time, then Hero makes 3BB profit 60% of the time, and loses 2BB 40% of the time (although he may of course win it back later in the hand - but for now I want to focus on the expectation of this specific bet). In this case hero is making 1BB on average from his bet.

50% of the time, then Hero makes 3BB profit half the time, and loses 2BB half the time. In this case Hero is making 0.5BB on average from his bet.

40% of the time, then Hero makes 3BB profit 40% of the time, and loses 2BB 60% of the time. In this case hero is breaking even from his bet, and so we can see that Villain must call at least 60% of the time in order to prevent Hero from profitably betting any two cards.

30% of the time, then Hero makes 3BB profit 30% of the time, and loses 2BB 70% of the time. In this case Hero is losing -0.5 BB on average from his bet.

25% of the time (the MDF as given in many online articles and books etc) then Hero makes 3BB profit 25% of the time, and loses 2BB 75% of the time. In this case Hero is losing -0.75 BB on average from his bet.

So - against a third pot bet Villain is meant to have an MDF of 75%, but in fact he can reduce this to 60% and Hero is not making any instant profit from his bet.

If, however, the money that Hero put into the pot is counted as 'profit', then indeed the MDF is 75% - even though he is not making profit if Villain defends at 60%. This is why it's confusing to me.

Another thing to consider is, for example, in a 12BB pot - there is of course a difference in terms of the profit of the winner if it is a heads-up or three-way pot. If it's a heads up pot the profit is 6BB to the winner, but if it's a three-way pot then the profit is 8BB to the winner.

You shouldn't be using MDF before the river, so these questions are really moot.

You have repeatedly made similar statements. When you refer to “whole hand,” you are effectively using the stack size prior to the first deal as the baseline. You can do that, but then you have to use that same baseline for all alternative decisions and if so, you will come to the same conclusion.

Saying “5 is greater than 2” is the same as saying “5-1 is greater than 2-1.”

Look up Sunk Costs on Google.

From a theoretical point of view - of course you can use MDF before the river, although it is well-known that in practice it will result in making negative EV plays with the bottom of your range, especially OOP on the flop - but it's still interesting with regard to working things out.

In any case, I figured out what I was doing wrong in the calculations above.

So I'll leave it as a challenge for anyone who is interested in showing off their skills:

The pot is 6BB on the flop and each player has invested 3BB. If Villain bets 2BB, assuming he always loses the pot when he is called, what is the minimum defence frequency at which Hero must call to prevent villain from profitably betting any two cards?

Yep. It’s like hero made a decision to win the blinds preflop. That decision was not successful and so now you are both at a flop with a completely new decision to make and the past is completely irrelevant.

In some cases, it is relevant.

As detailed above - there is a significant difference in multi-way pots in comparison to heads-up pots, for example when you are making EV calculations.

For example, when a player wins a 20BB heads-up pot, his profit is 10BB. But when a player wins a 20BB four-way pot, his profit is 15BB - and this difference obviously has an impact on his decision-making process.

Let's say there is 20BB in the pot on the flop in both cases, and Hero is the preflop aggressor. What is the EV of him betting 5BB if it checks to him, assuming all villains are folding 40% of the time to bets, and that he always loses when he is called?

If you were using standard EV calculations, then it would be as follows:

In the heads-up pot, it is (20 x 0.4) + (-5 x 0.6) = 5BB.
In the four-way pot, it is (20 x 0.064) + (-5 x 0.936) = -3.4BB.

A difference of 8.4.

If you instead do the calculations from the point of view of how much profit/loss is made from the hand when the bet wins/loses then it would be as follows:

In the heads-up pot, it is (10 x 0.4) + (-10 x 0.6) = -2BB on average.
In the four-way pot, it is (15 x 0.064) + (-10 x 0.936) = -8.4BB on average.

A difference of 6.4 – as it accounts for the fact that there is more profit to be made in the four-way pot.

I do have the feeling that I am making a mistake here - but can someone explain how?

You're just moving the point of reference.

If you reference your stack at the current decision point the EV of folding is zero. If you reference the beginning of the hand the EV of folding is your cost up to that point, negative.

Referencing the current decision point is cleaner because it removes an unnecessary variable, the difference between our starting and current stack, and the EV of folding is normalized to 0, which is easy to work with.

The difference between these scenarios is irrelevant, but if it will help you understand...


Correction

The difference between the first result of 8.4 and the second of 3.4 is 5BB, which is exactly equal to the difference in cost between the 10BB contribution to the heads up pot and the 5BB contribution to the 4 way pot.

Also note that the EV of bluffing in HU here is 5BB better than the cost of arriving here, and 4-ways it is 3.4BB worse than the cost of arriving here. The same numbers that we calculated before.

You are too hung up on past decisions. How the money got into the pot doesn't matter with respect to the profitability of the current decision. It could be 100% yours, or it could be a gift from Gozag. In the context of MDF, it is only aiming to make the current decision neutral EV for the opponent, not past decisions.

You’re obviously an intelligent guy asking the right questions and with a bit more really deep thought you’ll wrap your mind around this one. I personally can’t articulate it properly but it seems to me as though you’re using your original chip stack size as a base anchor of relativity and a result that finishes below that is a fail and a result that finishes above that is a success.

The reality is you can have a result where you end up with less than your baseline starting chip stack and that result is still a success.

Say you have kqs and you call 50 preflop and see a flop. Now there’s 100 in the pot. Now say you flop a nut flush draw and he gives you a great price to call. If you fold then you’ll lose that 50 from your base starting point.

But let’s say you call and the ev of the call is 40.

You still lost 10 from your base but it was a success not a fail. Think of it like you’ve crashed your car into a tree. Villain has offered to help you repair the damage (when he priced you in to the flush draw). He has offered you a chance to repair the majority of the damage. If you had of folded to his bet that priced you in then that’s like declining his offer to repair most of your car and you just abandoning it and leaving it be as it is damaged and mangled up.

Hi browni, thanks for the correction, that does make a lot more sense now.

Can you clarify what you mean by:

"Also note that the EV of bluffing in HU here is 5BB better than the cost of arriving here, and 4-ways it is 3.4BB worse than the cost of arriving here."

Specifically - what does "arriving here" mean?

Also, with regards to MDF, if we consider this case:

In the heads-up pot, it is (10 x 0.4) + (-15 x 0.6) = -5BB on average.

As Hero is betting quarter pot, the MDF is 80% right? But doesn't the above show that Hero is breaking even when Villain calls at 40% frequency?

I.e. Hero makes 10BB when his bet succeeds, and loses 15BB when it fails. So it needs to work 60% of the time to break even (three wins for every two losses).

Why then is the MDF 80% and not 40%?

Thanks again for your help, I really appreciate someone taking the time to listen and explain, as I really want to understand these things better.

I don't feel like I have anything to reply with that's not repeating myself, so let me disengage and try a different approach. Let's analyze a scenario:

HU poker
Board: 2h2d2s3h3d
OOP's range: {KK}
IP's range: {AA, QQ}
Effective stack is 1 pot sized bet, action on IP. IP can check or go all-in.

What is the optimal strategy for both players and what is each player's EV?

IP should go all in with his aces and a half of his queens.
OOP should call 33% of the time.

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

Average EV = two thirds of 133.333 plus one third of 33.333 = 100.

OOP calls 33% of the time and his EV is:
(0.66 x -100) + (0.33 +200) = 0.

Wait a second - I made a mistake, as I forgot to account for the times when IP checks the queens. So:

IP should go all in with his aces and a half of his queens.
OOP should call 33% 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 (0.66 x 100) + (0.33 x 200) = 133.333

When he is bluffing, I.e. 25% of the time -
his EV is (0.66 x 100) + (0.33 x -100) = 33.333

When he checks the queens, I.e. 25% of the time, his EV is 0.

Total EV = (133.333 + 133.333 + 33.333 + 0) divided by 4 = 75.

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

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

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

OOP wins 100% of the time when IP checks and his EV is 100.

Total EV = (0 + 0 + 0 +100) divided by 4 = 25.

Note that OP must call some of the time when IP bets, in order to prevent him from profitably bluffing every time with queens.

Also note that as the EV of betting aces is higher than the EV of checking aces, IP must bet aces for value in order to profit when he is ahead.

This also means that he must bluff some of time (otherwise OOP would simply fold every time there is a bet) and as it is a pot sized bet, he should be bluffing 33% of the time.

@Telemakus, Imo This is the thing you're not quite getting:

If you don't consider previous bets a sunk cost, then you need to compare the EV of actions to other actions, rather than compare it to $0.

For example facing a bet you need to compare the EV of a call to the EV of a fold. (Folding is no longer $0, but will actually cost whatever you've put into the pot so far). So sometimes, that means making losing calls, because it's less losing than folding. You're trying to make the least -EV decision.

Correct (in all cases): EV of calling >= EV of folding

Incorrect (unless previous bets are considered sunk cost): EV of calling >= $0

Similarly, if you're bluffing IP, sometimes the value of a bluff is worth more than the value of a checkback, despite losing money either way.

Correct (in all cases): EV of bluffing >= EV of checking

Incorrect (unless previous bets are considered sunk cost): EV of bluffing>= $0

It all comes down to your point of reference.