Clarifying bet and raise sizing in relation to the pot once and for all

This question is regularly asked in the forum or irc, so this is an attempt to clarify the issue. If you are more confused after I answered the question than you were before, I will consider becoming a politician.

One of the reasons for betting or raising to a certain amount is to lay pot odds of our choice for our opponent

This is easy to do in unopened pots (when nobody has opened the betting yet on the current street). We simply decide the odds we want to lay and bet the corresponding fraction of the current pot. For example say the pot is $5 and we want to lay 30% pot odds for our opponent. According to the table below, we simply bet 3/4 of the $5 pot ($3.75). Easy game.

Here are a few classic bet sizes, and the corresponding pot odds:

2x pot: 40% (3:2)
1.5x pot: 37.5% (5:3)
1x pot: 33.3% (2:1)
3/4 pot: 30% (7:3)
2/3 pot: 28.6% (5:2)
1/2 pot: 25% (3:1)
1/3 pot: 20% (4:1)
1/4 pot: 16.7% (5:1)

When thinking about raise sizes in relation to the pot size ("pot sized raise", "2/3rd pot sized raise"), we have to twist our mind a bit. The idea is that we want a 2/3rd pot sized raise to lay the same odds as would a 2/3rd pot sized bet in an unopened pot. I will first give you the shortcut to calculate the raise size, and for those who are interested I demonstrate the shortcut at the end of this post.

Let's call pot the actual pot size just before we make our play, including any bet(s) from our opponent(s).
Let's call B the bet size from our opponent. In multiway pots, this is the bet size of the opponent who last bet or raised.
Let's call C the amount we would have to call when it is our turn to play.
Let's define P=pot+C. This amount P is the "pot" in "2/3rd pot sized raise".

To make a 2/3rd pot sized raise, we raise 2/3rd of P extra on top of B. So we raise to B + (2/3) * P.

Here are some examples to make this clear:

Example 1:
pot is $1 (from previous streets)
villain bets $0.8 ($1.8 now in the pot)
-> pot=$1.8, C=$0.8, P=$2.6
-> to make a 2/3rd pot sized raise, we raise to $0.8 + (2/3) * $2.6 = $2.53

Example 2:
pot is $1 (from previous streets)
villain1 bets $0.8 ($1.8 now in the pot)
villain2 raises to $3 ($4.8 now in the pot)
-> pot=$4.8, C=$3, P=$7.8
-> to make a 2/3rd pot sized raise, we raise to $3 + (2/3) * $7.8 = $8.20 (by doing so we lay 28.6% pot odds for villain2 and, incidentally, more than 28.6% for villain1).

Example 3:
pot is $1 (from previous streets)
we bet $0.8 ($1.8 now in the pot)
villain raises to $3 ($4.8 now in the pot)
-> pot=$4.8, C=$3-$0.8=$2.2, P=$7
-> to make a 2/3rd pot sized reraise, we raise to $3 + (2/3) * $7 = $7.67

Note that some poker rooms' interfaces have buttons to help you bet or raise predetermined fractions of the pot. These conveniently perform the above calculations automatically. For poker rooms without these buttons, you can use software such as TableNinja to do the same job.

Now for the mathematically inclined, I will demonstrate all of the above in equations.

Let p be the size of an unopened pot.
Let f be a fraction

When we bet fp into p, fp is the amount our opponent has to call to continue, and p+2fp would be the total size of the pot after he calls. So we are laying fp / (p + 2fp) = f / (1 + 2f) pot odds (or (1+f):f for those who like this notation).

Conversely, to lay X% pot odds we have to bet a fraction f of the pot p such as:

f / (1 + 2f) = X/100
100f = X + 2fX
100f - 2Xf = X
f = X / (100 - 2X)

Now looking at already opened pots:

Let pot be the current amount in the pot just before it is our turn to play including any bet(s) from our opponent(s).
Let B be the bet of our opponent. In multiway pots, this is the bet size of the opponent who last bet or raised.
Let C be the amount we would have to call to continue when it is our turn.
Let R be the total amount that we need to raise to in order to lay the same f / (1 + 2f) pot odds as in the case of the unopened pot. Note that in multiway cases we assume that we want to lay these pot odds for the player who last bet/raised in the hand.

Just after we raise to R into pot, the total money in the pot will be pot+R-(B-C) (B-C being the money that was already in front of us from this round of betting before we raised). Now our opponent would have to call (R-B), and the total pot after he calls would be pot+R-(B-C)+(R-B) = pot+2R-2B+C. So we get this equation which we have to solve to find R:

(R - B) / (pot + 2R - 2B + C) = f / (1 + 2f)
(1 + 2f) * (R - B) = f * (pot + 2R - 2B + C)
R - B + 2fR - 2fB = f*pot + 2fR - 2fB + fC
R = f*pot + fC + B
R = B + f * (pot + C)

If we now define P = pot + C as the "pot" in "f pot sized raise", we find again the equation:
R = B + f * P

er... clear?

Right. Nice formula too.

It's a little counter-intuitive to me that p includes villain's last bet, but it does yield a really nice formula. Of course you can do it either way.

So, to illustrate the case of a pot-sized open raise, small blind's already bet .5, big blind raised to 1,
so p = 1.5, B = 1, and we want to make a full pot-sized raise, so f = 1.
Plugging in to R = B + f(p+B), we get R = 1+1*(1.5 + 1) = 3.5, correct.


If you don't want to think of the pot as containing the last uncalled bet or raise, then daviddem's formula becomes
R = B + f(pot + 2B)
ie, match the bet, then add another f times all that, and that's an f-times pot-sized raise, laying odds of 1+f:f to the original bettor.

edit:
We should note that this formula works whether
*hero is the first to bet that street, (in which case, B = 0 in the formula) and then the first caller behind is getting pot odds of 1+f:f, or
*villain is the first to bet and it folds to us, or
*there has been previous action that street and villain is either calling or raising (previous action before villain is just part of the pot), and this will lay odds of 1+f:f to villain, assuming everyone else in between folds.

had to re-edit the edit lol

Nice post

Bump for recognition people should really take the time to read and write this down

Great post. Needs to be stickied in Beginners Digest.

Originally Posted by couriermike

Right. Nice formula too.

It's a little counter-intuitive to me that p includes villain's last bet, but it does yield a really nice formula. Of course you can do it either way.

...

If you don't want to think of the pot as containing the last uncalled bet or raise, then daviddem's formula becomes
R = B + f(pot + 2B)
ie, match the bet, then add another f times all that, and that's an f-times pot-sized raise, laying odds of 1+f:f to the original bettor.

mmmh, but then how does your formula work in case of a reraise (example 3 in OP) or a reraise with calls after the raise? looks like you need some unnecessary mind twisting here, esp. in complex situations.

TBH I like mine better because:
1) the total pot size before it is out turn to play (and including villain's uncalled bet or raise) is always displayed in the middle of the table or wherever it says "total pot"
2) B is always the amount that villain last bet or raised to, which is also readily available (displayed in front of him)

Easy.

Make sure you do this:

R = B +(f * P)

and not this:

R=(B+f) *p
Big difference

most would shit their pants anyway as they have to add with a fraction

bump

printing this out today. Thanks alot for the post. Two thumbs up!

vnh daviddem, will favourite and reread later.

I am interested in using dadivdem's formula to calculate proper BE FE for Raises as well.
Is it possible to take your formula and adapt it to make the BE FE% of a (R) calculation as well ?
If so how would we setup the calculation to get BE FE of a Raise ?

This is a different story, but the easiest and most elegant formula to calculate BE FE for a semi-bluff has been shown by Spoon a long time ago. Works for bets and raises alike:
1) calculate your risk. This is the total amount you are about to put in the pot (whether it is a bet or raise), decreased by your equity in what the total pot would be if your opponent called. Don't forget to use effective stacks in case of an all-in.
2) the break even FE is risk/(risk+pot) where "pot" is the size of the pot just before you play

See an example here, post #1 for situation, post #9 for solution

thank you so much for the reference. This math post is awesome too basically everything i need to double check the situations i am involved in to make sure i am making the right decisions.

davidem,
Start a blog/operation. It should be interesting.

That is all...

I'm missing something here.
Let's say I'm in the SB w/ JJ @ 10NL. It folds to the HJ who opens to $0.35, CO & BTN fold...

pot = $0.50
B = $0.35
C = $0.30
f = 0.375

I have 1.67:1 direct pot odds. Using your formula, to find what I need to raise to in order to lay Villain 1.67:1 would be:

R = B + f*(pot + C)
R = $0.35 + 0.375*($0.50 + $0.30) = $0.65

which is incorrect, because the appropriate amount wound be to raise to $1.55, which makes the pot $2.00, and Villain needs to call $1.20, giving him an f of 0.375 to call.

So where did I go wrong?

Originally Posted by MadMojoMonkey

I'm missing something here.
Let's say I'm in the SB w/ JJ @ 10NL. It folds to the HJ who opens to $0.35, CO & BTN fold...

pot = $0.50
B = $0.35
C = $0.30
f = 0.375

I have 1.67:1 direct pot odds. Using your formula, to find what I need to raise to in order to lay Villain 1.67:1 would be:

R = B + f*(pot + C)
R = $0.35 + 0.375*($0.50 + $0.30) = $0.65

which is incorrect, because the appropriate amount wound be to raise to $1.55, which makes the pot $2.00, and Villain needs to call $1.20, giving him an f of 0.375 to call.

So where did I go wrong?

In the formula, f is not the pot odds you lay, f is the fraction of the pot you want to bet. If you want a formula with the pot odds in it, you have to replace f with X/(100-2X) where X is the pot odds you want to lay (in %).

However this makes the formula too complicated to work out during play. In practice, it is much easier to memorize the pot odds corresponding to common fractions of the pot (see table at the start of OP).

So if you want to lay 1.67:1 (37.5%) pot odds for villain, you should remember from the above table that this corresponds to a 1.5 pot-sized bet. So f=1.5 and as per the formula you should raise to:

R = 0.35 + 1.5 * (0.5 + 0.3) = $1.55

So as you said, you raise to $1.55 and the pot becomes 0.50 + 1.50 = 2
You raised to $1.55 so Villain has to call $1.2. So his pot odds are 2:1.2 or 1.67:1 or 1.2/(1.2+2)=0.375

edit: I made the formula as it is because all the numbers it uses are readily available at the tables (pot is displayed as total pot, B is displayed in front of your opponent, C is displayed on your call button). All you have to remember is the table in the OP to know which f yields the pot odds you want to lay.

Just for info, I wrote to Party Poker support about the behavior of their bet buttons because the raise sizes they come up with seem incorrect.

Hello,

Can you please explain how your bet/raise sizes are calculated when using your bet buttons post flop? For example, on the turn:

pot=$0.30
Hero checks, Villain bets $0.12, Hero hits the 1/2 pot button and raises to $0.32 because that is what the half pot button does

However a correct half pot raise in this spot is a raise to $0.39. So why does your half pot button make a raise to $0.32? How do you even get $0.32 in your calculations?

Can you also explain how the x2, x3 etc buttons calculate preflop when 3betting, because I don't get it either.

All these buttons work as expected on Full Tilt and Pokerstars.

Regards.

PP support response:

Dear David,

I am writing to let you know that the case you had previously reported regarding the what occurred with possibly incorrect bets has been escalated further and is now being looked into. Please allow us some more time to investigate this. Your understanding is much appreciated.

In case you have further questions, we will be more than happy to assist you. We are available 24/7.

Thank you for choosing us as your online gaming site.

Best regards,

Al

Customer Service

PartyPoker buttons have always been a mystery to everyone, they do the same weird stuff in MTTs

Am I wrong for not using the x/pot buttons on any site ever?

I find them a convenient way to quickly get standard bet/raise sizes without having to calculate. Sometimes I adjust manually from these base values. I like having 1/2, 2/3, 3/4 and pot buttons.

Originally Posted by Lucothefish

Am I wrong for not using the x/pot buttons on any site ever?

The customisable ones on Stars/FTP are very useful

I shall investigate said buttons forthwith, thanks

I have

Pre flop 1/2 2/3 3/4
Post flop 2/3 3/4 POT