Why you can't bet to give yourself better pot odds!

[loonychune]

Let's say it is the final round of betting. The pot is $10, your equity vs your opponents range is 20% (a probability of 0.2 that we win) and you don't know what to do. You are first to act vs one opponent.

1) You check and your opponent checks:

Your Expectation = 0.2*($10) = $2.00

2) You bet $2 and your opponent calls:

Your Expectation = 0.2*($10 + $2 + $2) - $2 = $0.80

3) You check, your opponent bets $2 and you call:

Your Expectation = same as in 2)

4) You bet $2, your opponent raises it to $4 (he calls your $2 and adds $2 more), you call:

In this case, the raised pot totals $16 ($10 + $2 you add + $4 opponent puts in) and it costs you $2 to call -- that's 12.5% pot equity and you have 20% equity to win! But...

Your Expectation = 0.2*($10 + $2 + $2 + $2 + $2) - ($2 + $2) = -$0.40

If you want some sort of equation for Expectation, it's basically:

Expctation = (probability we win)*(size of new pot) - (what it costs you)

Things get more complicated when we're looking at catching draws and things on later streets, but i'll post on that eventually.

N.B. This is for those times are equity is less than 50%, since we want to bet/call (or bet/raise) when we have better than 50% equity.

[kb coolman]

Build a straw man and tear it down.

You can bet to give yourself pot odds. It's called semi-bluffing for fold equity.

Let's say you have flopped the nut flush draw in a $3 pot OOP with one opponent. If you check, he bets $2.50, and you cannot profitably call.

Now, let's say you flop that same nut flush draw, but this time the Ace is on the board and you hold the King. You are still beat here, but you have excellent fold equity against your opponents likely high pocket pair.

The pot is still $3, but you semi-bluff $2. If he folds, you win the pot. If he plays back at you, raisng to $5, you are now faced with calling $3 into a $10 pot.

You have just given yourself odds to call profitably.

[LuckySlevin]

the mistake was at 2 =)

[loonychune]

slevin, ffs, explain what you mean!

[LuckySlevin]

no explanation needed, it's obv.

But to clarify I'm contesting betting at 2) =)

[Robb]

If you raise smallish with a draw oop, villain will often just call rather than raising you out of the pot. If you can get away with a $2 bet rather than checking and having him bet $3.25 and needing to call that, you can "price yourself in." Now, I'm not the world's biggest fan of "blocking bets" 'cuz I think they're overused at the micros by non-thinking regs. But that's the beauty of it. We can profit by getting reads for villains who try small blocking bets on us. Let's say our read is that they often bet 1/4 PSB oop with an 8- or 9-out draw. It's time to raise them and make sure they pay a premium to see another card. A lot of these regs at 25nl have such an unbalanced "small" flop bet line that you can raise it with almost any two cards.

[bjsaust]

2) You bet $2 and your opponent calls:

Your Expectation = 0.2*($10 + $2 + $2) - $2 = $0.80

3) You check, your opponent bets $2 and you call:

Your Expectation = same as in 2)

You're not betting to win the money that you're betting.

[bjsaust]

You also completely ignore the main point, which is that we bet small to avoid being faced with a large raise. I.e,. if he bets $7 into that $10 pot.

[loonychune]

I have looked to get my head round this myself. You do this problem your way and i'll do it my way:

The pot is $800, we check, our opponent bets $100 and we call expecting to win 10% of the time (that's 9:1 against).

We expect to break even:

i.e. EV = 0.1($800 + $100 + $100) - $100 = $0.

I think the confusion here comes from working in terms of ratios a lot.

When our opponent bets $800, the pot is $900. We have 10% equity so that makes us 9:1 against. We're getting 9:1 pot odds so we call and break even.

[loonychune]

OK. I shouldn't have posted in the blocker bets thread. Let's look at this as just a nice theoretical exercise on why you don't make things better for yourself by betting!!

[bjsaust]

Hmm, I think you're right. I thought you were trying to say it was 10:1 because you were including our $100 in the winning portion, just misread the formula.

[Stacks]

I didn't have much time to look this over, and I really don't have tremendous experience in ev calcs (at least not recently so have forgotten quite a bit), but you should be factoring in the % of the time villain calls your bet (and your equity/expectation when called) and also the time when he folds (and your ev then).

If the pot is $10 and a $2 bet from you will have villain folding 30% of his range. When called you have 20% equity, then this changes your equation a bit from earlier.

Now, the formula is:

Ev = [Ev(fold)] * (x) + [Ev(call)] * (1-x), while x = % of fold

In this example:

Ev(fold) = $10 That is if we bet and villain folds we win the pot, or $10.

Ev(call) = (equity in pot)(total pot) - (bet made)
Ev(call) = (0.20)($14) - $2 = $0.80

Ev = [Ev(fold)] * (x) + [Ev(call)] * (1-x)
Ev = [$10] * (0.30) + [0.80] * (0.70)
Ev = $3 + $0.56
Ev = $3.56

Note - I could very well be incorrect here, or not addressing what you were intending to address. Like I said, I haven't studied this in many months.

[loonychune]

Well you aren't the only one who said i'm wrong! I'm glad you said something more constructive than just "the problem is with 2" and a big smily face.

I'm actually glad you brought it up though, since it's the first time i've ever articulated why i think the equations just don't 'seem' right.

And I hope I've not come across as a wanker after posting in the blocker bets thread -- I got sidetracked with this exercise. It wasn't meant to be a refutation, not once i got started on the math anyway (it's just fun!).

[Stacks]

In this case, the raised pot totals $16 ($10 + $2 you add + $4 opponent puts in) and it costs you $2 to call -- that's 12.5% pot equity and you have 20% equity to win! But...

One other thing I felt was important enough to address. You state in the earlier examples that we have 20% equity when our bet is called, or when we check, he bets, and we call. However, this is a different situation. We are now not facing his betting range or calling range. We are now facing his raising range, which will likely differ from the above. Therefore, our equity is likely to also change. So it is very plausible that we no longer have 20% equity in this spot when we are facing a raise from villain, and this should be considered and factored into the calcs.

[loonychune]

1) The expectation values adjusted for when our opponent folds is a nice addition. I actually posted some maths on this the other night and someone posted to say "but it's not fold equity if he's folding a part of his range that doesn't beat you"... whatever the case, I think the maths works where we say: "he folds here x% of the time, he calls (100-x)% of the time".

2) Yes, it's all very much a simplified exercise. The idea simply was: If our hand has x % equity and we bet to make the pot bigger and so our equity is then much bigger than the pot equity, we don't actually profit more. Just a simple mathematical meandering that uses equities and our chances of winning in probability form.

This principle works the same when we bet to give us better outs to draw to a flush on a later street (it's just more complicated).

[Stacks]

In this case, the raised pot totals $16 ($10 + $2 you add + $4 opponent puts in) and it costs you $2 to call -- that's 12.5% pot equity and you have 20% equity to win! But...

Your Expectation = 0.2*($10 + $2 + $2 + $2 + $2) - ($2 + $2) = -$0.40

And yet here's something else. In the above, why are you subtracting $4? You have already made a bet, and you are now determining the ev of your call. Your bet has a different expected value than your call here.

So, the pot is $10. You bet $2, and villain raised to $4 (calls your $2 and puts in another $2). The pot is now $16, and you must call $2. Let's say you do have the 20% equity you stated when you call (just for example purposes. I've already stated that is might be incorrect to assume). Therefore, the ev of your call can be factored by the following formula:

Ev(call) = (equity)*(total pot) - (amount it cost)
Ev(call) = (0.20)*($18) - ($2)
Ev(call) = $1.60

OR

Ev(call) = (win %)*(amount we win) - (lose %)*(amount we lose)
Ev(call) = (.20)*(16) - (.80)*(2)
Ev(call) = (3.2) - (1.6)
Ev(call) = $1.60

The only difference is the extra $2 you added in. I don't see why adding that in would be correct.

[drmcboy]

stacks is right, I'm also not sure assuming we have the same equity vs his betting and calling range is useful - this is of course player dependent but usually one will be higher than the other - but I may not understand the math point you're trying to get to.

Let's say you have flopped the nut flush draw in a $3 pot OOP with one opponent. If you check, he bets $2.50, and you cannot profitably call.

Now, let's say you flop that same nut flush draw, but this time the Ace is on the board and you hold the King. You are still beat here, but you have excellent fold equity against your opponents likely high pocket pair.

The pot is still $3, but you semi-bluff $2. If he folds, you win the pot. If he plays back at you, raisng to $5, you are now faced with calling $3 into a $10 pot.

You have just given yourself odds to call profitably.

no, your opp did when he bet 1/3 of the pot instead of 5/6 like he did in your first example. if he bets whatever 5/6 of 10 is you have the same price but less implied odds so your situation is a bit worse. this doesn't mean it's a bad play, just that we haven't given ourselves anything.

[kb coolman]

no, your opp did when he bet 1/3 of the pot instead of 5/6 like he did in your first example. if he bets whatever 5/6 of 10 is you have the same price but less implied odds so your situation is a bit worse. this doesn't mean it's a bad play, just that we haven't given ourselves anything.

yes, opp gives you odds to chase. But he bets pot in my example. Where are you getting 1/3?

And what I was building was a case where we can semi-bluff with fold equity to profitably call a pot sized bet.

[drmcboy]

Where are you getting 1/3?

you are not adding your bet or his call into the pot

The pot is still $3, but you semi-bluff $2. If he folds, you win the pot. If he plays back at you, raisng to $5

after you bet 2, the pot is 5. He calls your 2, pot is 7. The he raises 3 more, or 1/3 of the now 10 dollar pot.

[Stacks]

no, your opp did when he bet 1/3 of the pot instead of 5/6 like he did in your first example. if he bets whatever 5/6 of 10 is you have the same price but less implied odds so your situation is a bit worse. this doesn't mean it's a bad play, just that we haven't given ourselves anything.

yes, opp gives you odds to chase. But he bets pot in my example. Where are you getting 1/3?

And what I was building was a case where we can semi-bluff with fold equity to profitably call a pot sized bet.

In your second example where villain raises, if the pot is $3, and you bet $2, the pot is now $5. If villain raises the $5, the pot is now $10, and you must call $3. Therefore, you are getting >3:1 on a call. Which is basically the same as betting 1/3 of the pot. I believe this is where drmcboy is getting that.

[kb coolman]

ah...gotcha.

[loonychune]

In the above, why are you subtracting $4? You have already made a bet, and you are now determining the ev of your call. Your bet has a different expected value than your call here.

Your EV on the call is positive, yes. But if you're betting just to make the pot bigger and give yourself +EV on a call then it's costing you $2 to do this and you're only $1.60 +EV. I'll look to post something interesting later tonight

[Stacks]

Your not betting just so that a call is +EV, in this situation your betting because betting is +ev. And then when you are faced with the raise, the call is also +ev. Therefore, you make both decisions. Like I was saying, the ev of your bet and the ev of your call is separate of one another.

In this case, the raised pot totals $16 ($10 + $2 you add + $4 opponent puts in) and it costs you $2 to call -- that's 12.5% pot equity and you have 20% equity to win! But...

Not sure how many more times I will quote this paragraph, but anyways. Can you explain where "12.5%" comes from? If the pot is $16 and it costs you $2 to make a call, then you need:

Ev(2call) = (amount to call) / (pot + amount to call)
Ev(2call) = (2) / (16 + 2)
Ev(2call) = 2/18 = 0.1111111 or 11%

Therefore, you need 11% equity in the pot to make a breakeven call. Given you have more than the needed equity in this situation, then a call is +ev.

I would just like to note, that it wasn't you that gave yourself correct odds to call, it was villain. His raise size just laid you too good of odds to pass up on. Had villain raised to a reasonable amount, then a call from you would have been -ev; however, the bet you made would have still be +ev. For example,

You have 20% equity. The pot is $10. You bet $2, and villain raises to $9 (calling your $2 and putting in $7 more). The pot is now $21 and you must call $7.

Ev(call) = (equity)(total pot) - (what it cost)
Ev(call) = (0.20)(28) - (7)
Ev(call) = -$1.40

[loonychune]

Not sure how many more times I will quote this paragraph, but anyways. Can you explain where "12.5%" comes from?

This is an oversight on my part. I just threw this paragraph in to sorta tie in with what i meant.

I would just like to note, that it wasn't you that gave yourself correct odds to call, it was villain.

I feel like the purpose of my post has still been missed, but at least it's incurred a fair deal of discussion.

Just to kinda re-illustrate my intentions (i can't work out whether everyone is being constructive or just trying to say "you're wrong wrong wrong"):

Let's pretend we're playing limit. The pot is 5 Big Bets (BB) large. Your hand is 8:1 against to win. Our opponent always bets or raises on the river.

If you check, the pot will be 6BB when opponent bets so we're getting 6:1 pot equity which just isn't enough.

You figure, you'll bet, opponent will raise and the new pot will be 8BB - now we're getting 8:1 pot equity and our hand is 8:1 against vs his range, so we break even.

But betting just to make a call profitable on the end has negative expectation. If you're betting solely to make the pot large enough to call after you get raised, you must take into account that money you bet since you're, in this instance, breaking even on the call-to-raise but losing the money you put in to give yourself the correct odds.

[drmcboy]

you are correct, I'm not sure why you think everyone is saying you are wrong. The confusion ITT I think is coming in because the example is too complicated and the math is too close. Here's an easier way:

You're playing NLwith 1 million dollar stacks and $1/2 blinds. You are the SB and it folds to you. You raise $999,999 with 72o. Your opponent moves all in. You call.

It's clear that even though you have given yourself pot odds to call the raise you've made a bad play. This concept doesn't change no matter what the bet sizes are, it's just easier to see this way.

If you like you can pretend we're on the river with a good but non nut hand and you way over bet the pot leaving yourself just a few chips behind.

It's OK to make some -EV plays if you think your opponents will make even more -EV decision later in this hand or future hands, but you have to evaluate the plays together and compare them to the EV of your other options. To refer back to my 1M stack example, there is no mistake villain can make that will allow us to recoup our lost EV, unless maybe we both have 25M more in our pockets.

Flip it around - It's a lot like calling a raise with A8o from the BB. You will usually have the correct price to do so based on your the hot and cold equity vs your opponent. But, OOP with a hand like A8o you will often make a bunch of mistakes later in the hand which will overwhelm the small amount of equity in your pre flop call. If you or your opp is all in when you call these considerations don't matter and we would play.

Making one +EV play during a hand does not mean every other play you've made in the hand was also +EV, nor does it mean you played the hand well.

[Stacks]

You could simply rephrase that as your bet is -ev, but your call is +ev (0ev in this case).

So yes, it is a bad idea to take a -ev decision to set up a future +ev (0ev) decision.

If you bet has an ev of -$2, but after you bet and are faced with a raise and the call has an ev of +$1, then you are still -$1 EV on the river.

[loonychune]

You're playing NLwith 1 million dollar stacks and $1/2 blinds. You are the SB and it folds to you. You raise $999,999 with 72o. Your opponent moves all in. You call.

It's clear that even though you have given yourself pot odds to call the raise you've made a bad play.

I think, yeah, it's better to illustrate things in this manner. I don't think you NEED a load of difficult maths to point something out like this. But it's nice to introduce it I suppose.

As for table image and inducing -EV on later streets, well, that's why I stick to the bare assuming maths I just ain't that good yet! Nice point though!

You could simply rephrase that as your bet is -ev, but your call is +ev (0ev in this case).

So yes, it is a bad idea to take a -ev decision to set up a future +ev (0ev) decision.

If you bet has an ev of -$2, but after you bet and are faced with a raise and the call has an ev of +$1, then you are still -$1 EV on the river.

Yep, nice way of putting it

I think a fair bit of maths has been introduced in this thread so i'd hope it can be useful (though I wonder, since most posters are FTR regs, but maybe...)

I'm going to try to explain the maths I use more when I post in the future.