Interpretations of "R"
Posted by AF3
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AF3
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Interpretations of "R"
If that's the case, then for R > 1, this would mean that we win the pot more than our pure equity would have us winning, so for example, if our pure equity was .35 and our R was 1.2, then we would entitled to the following amount of the pre-flop pot: 1.2 * .35 = .42 or 42% of the pre-flop pot that we create.
When constructing a range based on R, we measure R for each specific hand, is that right? We do this because R fluctuates across hand-type.
What's kind of interesting is if you think about what it means for R to be 1, it means that for every time we fold the best hand, we need to have another instance where we make more than equity.
I'm having trouble seeing how we make more than equity when we do anything other than make our opponent fold out their equity share. I guess the idea is like, if we gain 5 BBs postflop by value betting and being called down, then our R for that instance would be like:
5BB/(Raw Equity * Pre-Flop Pot Size)
This would give R >> 1 in many cases, which makes sense since it kind of explains why unpaired high-card hands do well. Namely, we have a very large gap of which to value bet into upon making our pair, since Villain must be calling down with worse in order to neutralize our bluffs.
If anybody can confirm what I'm asking/stating or point out where my ideas are not good, I would really like to hear.
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Another question: This would mean that every time we end up calling down with the worst hand, we will have obtained less than our equity for that instance, correct? This is a corollary of R being defined by our equity in the pre-flop pot that we create by calling.
Is anybody out thurrrrrrrrrrrrrrrrrr (?)
I think there's at least one, possibly 2 videos discussing "R" : first is Lefort's (don't remember which one exactly), and I've seen a recent video discussing it too (again, can't remember which one)
Edit : found other Lefort videos where he discusses "R" :
6-Max Concepts Series Finale (Part 7: Extended Analysis) @20:21
Exploitative-Based Hand Analysis @14:35
HTH
Thanks Robert.
Those videos don't really address these questions, though. They just introduce "R", a concept that every player intuitively understands. Nothing Lefort does actually defines it in terms of an actual pot size though.
I´m afraid don´t really get your questions.
"Which "pot" is R actually measuring?"
R does not relate to any pots, it´s just a factor that increases or decreases our equity. Something like "true equity". So, if your equity is 50% and your R is 80%, then your "true equity" is 40%. Then you can take this number and do whatever you want, calculate equity share, compare odds ...
"I'm having trouble seeing how we make more than equity when we do anything other than make our opponent fold out their equity share."
That´s actually the point. If nobody would ever fold, your "R" would be 100%, regardless of your holding. R is similar to the "error rate" of each player, and errors happen when we fold our equity share (or the best hand) though getting the correct price.
we fold our equity share (or the best hand) though getting the correct
price.
If this was the case, then you could make a very convincing argument that the R of AA would not be over 100%. If R is actually a measure of the error rate, then it will also take into account the inverse type of error (calling with the worst hand).
"If this was the case, then you could make a very convincing argument that the R of AA would not be over 100%. "
I´m pretty convinced that it´s < 100% in most situations. Why do you think it has to be > 100%?
I´m pretty convinced that it´s < 100% in most situations. Why do you think it has to be > 100%?
I'm not claiming it is, but if you watch his video, Lefort seems to think so.
I'm near 100% sure that the conventional definition of R is as follows:
If no betting occurs, then the EV of a hand is just EV=Pot*Equity
With betting, each hand has some EV, and we define R to be such that EV=Pot*Equity*R
R is the "% of our equity we realize" and can range from 0% (bluffcatcher vs a correctly balanced river betting range of nuts/air) to inf (the nut low with a profitable bluff due to overall range).
Generally speaking on the river R<1 for bluffcatchers, R>=1 for value bets and bluffs. Every hand has a different R and it depends on position as well (R(ip)>=R(oop) for ~all regular situations I think.)
Extending to preflop, hands that make more value bets/bluffs will have R>1, hands that make few strong hands and lots of bluffcatchers will have R<1. This is just a (not very) mathy way of saying the standard idea that suited connectors and pairs play well postflop (=R>1) and hands like offsuit weak aces and other random junk are hard to play postflop (R<1).
I´m surprised that we´re discussing this topic in such depth when I thought we were all talking about the same. :) Even moreso as OP claimed there was nothing groundbreaking new in Leforts Video.
"With betting, each hand has some EV, and we define R to be such that EV=Pot*Equity*R"
No, that´s wrong. As mentioned, R is the factor we use to "adjust" our equity against Villain´s range. That said, we don´t use your formula, but we use the regular EV-formula (which includes betsizings!) and then adjust our equity:
EV = (EQ * (Pot + 2*bet)) - Invest
So, when the pot is $100, Villain bets $100 and we have 33% equity, then our EV is Zero:
EV = (33% * $300) - $100
Now, we bring "R" into play and it´s simply a factor we multiply our equity with:
trueEV = (33% * R * $300) - $100
So, if our calculated EV was zero (bluffcatcher vs. balanced range), we always realize 100% of our equity - because we can´t make any mistake!!
That even shows that it´s pretty useless to think about R "once" the betting occurs, R is important for further streets, like we are on the turn, face a potsized bet and think we have 39% equity, so we could easily call. Now we have to decide how much of this equity (that might include marginal overcard hits or even A-high being best) we realize against a potentially uncapped range - and when we get to the decision that our R is 80%, then we had to fold:
calcEV = (40% * $300) - $100 = +200
trueEV = (40% * 80% * $300) - $100 = -4
"R is the "% of our equity we realize" and can range from 0% (bluffcatcher vs a correctly balanced river betting range of nuts/air)"
As shown, this is not correct - and easy to "prove". When R would be 0% with a bluffcatcher against a correctly balanced range - we could never call.
I think we're mostly saying the same thing, I just was slightly unclear. When I say "with betting" I don't mean after someone has bet, I agree that R doesn't really make sense once betting occurs. What I mean is in a game where betting is allowed. If no betting is allowed then EV of any hand is just pot*equity. If there is betting allowed in the game then we use R to adjust the EV to take into account how much of our equity we realize EV=Pot*Equity*R
So if the pot is $100, and villain has 50 bluffs and 50 value hands, then our bluffcatchers have 50% equity. If no betting was allowed then our EV would be $50. But if villain has 1PSB and chooses to bet 50 value hands and 25 bluffs, then our EV is $25, so our R is 50% (our equity in the pot is effectively reduced to 25%, or 50% of 50%.)
If instead we had a hand that beat 50% of his value hands, then our equity is 75% and our EV=0.75*(100+100)+0.25*(-100)=125 and our R is 125/75=1.67. We realize much more than our equity in the pot because we make money on the river action.
It is possible that Lefort uses a different definition of R (it's been a long time since I watched that video) but this is the version I have seen used and that seems most useful to me, though it's obviously still not super useful since it's near impossible to figure out R for any hand in a non-trivial situation.
edit:
Steve-"R is the "% of our equity we realize" and can range from 0%
(bluffcatcher vs a correctly balanced river betting range of nuts/air)"
Bigfiszh-"As
shown, this is not correct - and easy to "prove". When R would be 0%
with a bluffcatcher against a correctly balanced range - we could never
call."
Exactly! If we can never call then we realize none of our equity in the pot! If he gets to the river with 2 value bets for every bluff and bets 1psb then even though we have 33% equity, we effectively have 0% equity and thus R=0. I think I confused the issue by inadvertently implying I was calculating R after he bets, which I agree is silly.
OK, I see where you come from and I fully agree with you! Like, say
we face a turnbarrel and estimate we need 30% equity to make the call -
knowing (clairvoyant) that Villain´s range consists of 50/50 nuts, air,
we still couldn´t call because we will only realize 50% of our equity,
so with 25% equity eff. - while needing 30% - we had to fold (or bluff).
Fair
enough - but your last statement is still wrong, our equity doesn´t go
down to zero, it´s still 33% - we don´t win money, so our EV is zero,
but our equity isn´t:
EV = (33% * R * 300) - 100 = 0
=> R = 100%
Otherwise we weren´t indifferent to call (while being break-even) but indeed would lose by calling:
EV = (33% * 0% * 300) - 100 = -100
Agree?
I think stevejpa is kind of conflating equity with expected value. As far as I know, there is no way to actually convert them, and it's the fact that R exists (and does not always equal 1) which actually makes this the case.
I'm not saying our equity isn't 33%, of course it is. But it is equivalent to having 0% equity because our 33% equity is wasted. The existence of betting rounds has caused our EV to go from 1/3*P to 0, having 33% equity has no benefits over having 0% equity. R is the term that we use to convert between equity and EV (or we would if it were possible to calculate for a given hand - in fact we can do this for very simple cases like the nuts/air one I showed above.)
So if R=0 then EV=Pot*equity*R=0 and it doesn't matter how much equity we have, we can't realize any of it and it's effectively 0.
If R is 1 then the betting rounds have no impact on our EV and the simple equation EV=Pot*Equity holds; this is equivalent to us being all in, we realize all of our equity.
If R>1 then we profit from the existence of the betting rounds, EV=Pot*Equity*R which is > Pot*Equity
If R<1 then we lose from the existence of the betting rounds, EV=Pot*Equity*R which is < Pot*Equity
"R is the term that we use to convert between equity and EV"
Strongly disagree with that, but let´s agree on disagreeing ... that doesn´t lead to anything constructive.
R is a way of representing our share of the pot after some point in the hand, usually in CREV where we don't feel like closing out a big game tree. When we don't simulate the whole game tree, sometimes it's convenient to take a guess at our pot share (and measure this in a consistent way) by multiplying our equity by some constant, where the amount above or below 1 of that constant represents our implied odds or reverse implied odds.
"R is the term that we use to convert between equity and EV"
stevejpa and BigFizh: I was hoping you guys would participate in the discussion, since both of your ideas are always stimulating and/or correct.
With regards to converting equity to EV, it's impossible, and not in the "humans just don't know how to do it" type of way.
You can construct numerous counter-examples to show EV and equity are almost completely de-coupled. Chapter 7 in M.O.P. has some examples showing how and why this is the case.
expected value if, and this is a very big if, we know the value of R for a given hand in a given situation.
This actually illustrates why it's impossible to convert equity and EV. Equity depends on the cards. Expected Value depends heavily on bet-sizing. I agree with you that R is some kind of in-between term since it is influenced by both, but stating it is the conversion factor between Equity and Expected Value is a little presumptuous.
If you don't see why this is true, then consider the following:
Suppose we could convert between equity and expected value. Then for the purposes of converting, there is absolutely no difference between knowing R and knowing our equity.
Unless we are all-in, however, we are not able to convert equity to expected value. Therefore, R does not give us our expected value.
Lefort defines conversely. He defines it as:
Equity/R = Adjusted Pot Odds
I don't think it's possible to have an incorrect or correct definition. Definitions are invented by useful (or not). I'm glad that you frequently point this out.
I also like the way that you frequently characterize ideas as helpful or not helpful as well. It helps keep things on topic.
What I'm saying is that R is by definition the conversion factor between equity and EV because as Sauce says: "R is a way of representing our share of the pot after some point in the hand"
In all in situations our equity represents our share of the pot, ie EV=Pot*equity
In non-all in situations equity does a pretty terrible job of representing our share of the pot. So we use R to represent our share of the pot, and we define it such that it allows us to convert equity to EV, where EV=Pot*equity*R. Because R is different for every hand/situation you are of course not able to say things like "40% equity is worth x" but in a solved spot you can say that a particular hand has 40% equity and an EV of x and R (by definition) allows you to convert between those two.
edit: if you knew R for a given hand/spot you could convert between equity and EV but since to know that you need to have solved the entire game that's not really particularly useful
"EV=Pot*equity*R"
This formula just is not true ... If betting occurs then our EV is
EV = ((pot + bet + call) * EQ) - call
Or, if you adjust EQ by R it's:
EV = ((pot + bet + call) * EQ * R) - call
There is no such formula like "EV = Pot * EQ" - unless it gets checked through, which you were explicitly not referring to.
I
don't think it's possible to have an incorrect or correct definition.
Definitions are invented to be useful (or not). I'm glad that you
frequently point this out.
I also like the way that you frequently characterize ideas as helpful or not helpful as well. It helps keep things on topic.
I think that stevejpa is on to something as well. He's not explaining it with the most useful terms, because new terms and new questions have to be invented. The "equity" he's using is all-in equity.
Rather than asking how often we get our all-in equity, the more interesting question in my view is:
How much does it cost us to realize our all-in equity?
This introduces a way characterize various equity types, since we could see that "nut" equity and "bluff-catch" equity will have very different values.
I'm not sure what new terms need to be invented and of course I'm referring to all in equity...??? I'm also not asking how often we get our all-in equity and don't think that's useful in any way. How much does it cost to realize our all-in equity is a needlessly complicated question imo.
The question I'm asking, and the question Lefort asks in his videos, is what % of our (all in) equity do we realize?
Take an example from Lefort's 6max concepts part 3 video at 9:30. He's looking at defending vs a minraise and what does R need to be to defend 72o vs top 50% of hands? 72o has 29.5% equity vs that range. We're getting 3.5:1 pre and so need to win 1/4.5=22.2% of the flop pot to breakeven. So he calculates the breakeven R as 22.2%/29.5%=0.75. Ie we need to realize 75% of our preflop equity in order to breakeven.
Looking slightly closer at this, we have for Equity*R=22.2% where 22.2% is the point at which EV=0. In other words:
EV=Pot*Equity*R
EV=4.5*0.295*0.75
EV=4.5*0.222
EV=1
Our EV on the flop is 1, we called 1 preflop so the preflop call is breakeven. If you prefer to put a -1 at the end so that
EV=Pot*Equity*R - 1
EV=4.5*0.295*0.75 - 1
EV=1 - 1
EV=0
that's fine, both tell you the same thing and are equivalent.
Lastly, and this post is already a bit rant-y so I apologize for that, the main take-home from the whole R discussion is that bluff catcher hands return less than their equity (R<1) and nut hands (and bluffs) return more than their equity (R>1). This is not groundbreaking stuff, this is exactly what everyone has always said, whether it's by saying some hands are easy/hard to play, some hands have implied odds/reverse implied odds, or some hands realize more/less than their equity. It's an attempt to assign a value to those vaguer, hand-wavy ideas.
If you're really interested in this stuff, I'd highly recommend Will Tipton's 2nd book. I haven't finished it but he uses "capture factors" (calls them R, but they're defined slightly differently, basically equity is the % of the pot you'd win if you were all in, capture factor is the % of the pot you actually win) and solves some spots and is able to give R values for various hands in those spots.
I'm going to use a specific example because somewhere along the line what I'm saying is getting confused and hopefully this will help. All numbers chosen semi-randomly.
Say we are looking at a completely solved game. The button minraises, sb folds and we defend 74o which has 33% equity preflop. If no betting were allowed, then our EV on the flop would be
EV=.33*4.5=1.5bb (note this is a 0.5bb gain since we called 1bb pre, but our "share of the pot" on the flop is 1.5bb.)
Now we look at our GTO solution and we see that we actually only win 0.7bb back on the flop with 74o in this spot (ie lose 0.3bb by calling pre). We now define R to be such that
EV=Pot*Equity*R or 0.7=4.5*.33*R which leads to R=0.47
In other words we only got back 47% of our "share of the pot" because our hand flops a lot of weak pairs that have some reverse implied odds. The rounds of betting have cost us a bunch of money.
R isn't anything inherent to the game. It's just a number we defined to compare the EV of a hand when betting isn't allowed to the same hand where betting is allowed. We make estimates of R to let us make some guesses about what hands can be profitably played in some spot.
Totally agree. Now take your example and imagine, our R would be 67%, ok? That means, we would make 33% * 67% * 4.5bb = 1bb. That is exactly our invest (excluding dead money), so we lose 1bb net (the big blind we already posted), which is the same result as if we would simply fold.
OK? Now that is what I wanted to show ... an EV of zero doesn't mean that our R is zero as well, it's 67% in this case and our EV is still zero. That should show that you can't simply convert Equity into EV by multiplying it by R.
Do we agree now?
In this way R does convert equity to EV!
If R=0.67 our EV on the flop is $1 and we called $1 to get there so the EV of our preflop call is 1-1=0
If R=0 our EV on the flop is $0 and we called $1 to get there so the EV of our preflop call is 0-1=-1
I think part of the confusion is I've been a bit loose with not specifying what EV I'm talking about in my previous posts. Hopefully this clarifies!
Agree to disagree I guess
"we defend 74o which has 33% equity preflop"
Why does it have 33% equity? if it is v random hand, then does it not have 38.5% euquity?
I think 3 years ago me was just picking a number for the sake of the example
"I think 3 years ago me was just picking a number for the sake of the example"
Oh ok, thanks Steve. I was confused enough already up to that point lol.
Hey Steve,
I don't know if this is proper forum etiquette, but would you be so kind as to take a look at a post of mine please?
http://www.runitonce.com/nlhe/comparing-openfolding-v-flatting-v-raisi/
EV calcs depend on where u start looking. For instance, when looking at a call in the bb we suppose for simplicity that the blind we posted isn't a loss. Then we play any hand that's EV is bigger than 0. We can also suppose the bb is a loss and choose to lessen that by playing any hand that has an EV of less than -1bb. These are equivalent. And similar equivalent calcs happen at any other point in a hand. How to calc EV is independent of any points about "R".
How much does it cost to realize our all-in equity is a needlessly complicated question imo.
Seems like a pretty important question.
of course I'm referring to all in equity...???
This is done by convention, but there's no reason equity needs to refer to your all-in equity.
I'm all for nitpicking but this all in equity one is too far even for me. When someone says equity with no qualifier then they mean all in equity.
Any comments on the rest of my post?
I'm all for nitpicking but this all in equity one is too far even for
me. When someone says equity with no qualifier then they mean all in
equity.
Right. I'm contending that it's very incomplete in situations when we have an SPR of 30.
I wasn't nit-picking. I was making the point that people rarely think about turn equity going forward from the flop. By "think about", I mean actually reference it. This isn't a trivial thing. Chapter 7 of M.O.P. makes the point that turn equity can have a major influence on flop bet-sizing.
Does R also relate to 'how well you play'? Is that obvious? For example, say I call flop with 44% equity, but am then prone to donk shove turn for 3x pot with unreasonable hands, is my R on the flop then decreased?
Yes, R's are strategy dependent. You can only really calculate R once you have a full game tree and know both the equity and EV of a hand in a given spot. If one or both players make a strategic adjustment then their EVs change but the hand equity doesn't so R changes.
To take a ludicrous example that shows the point: say you get to the turn with AA vs an opponent whose range is {22} on AA22. There is 1psb behind. If your strategy is to jam and his is to call then your equity is 100% and your EV is 2psb. EV=Pot*Equity*R, and R=2 (this makes sense, your all in equity entitled you to the whole pot, but you won 2x pot, or twice the amount of your all in equity.) If instead your strategy is to open fold, an expert adjustment, then your equity was 100% but your EV=0, so R=0.
Cliffs: yes, how well you and your opponent play affects R values.
Nice nitpicking in thread :) I agree with stevejpa's definition, only addition is that r < 0 is also
possible when we're not at equilibrium. Obviously we should strive to keep it zero or better.
I have implemented an equity realization tab in my software Flop Inspector and I'm looking into best ways of estimating R for different hand types. Software calculates the overall (preflop) R for a hand (vs range) when user inputs R for each flop type. The flop types are user configurable. Examples are quads, pair + flush draw, or more elaborate stuff like middle pair with backdoor straight draw on Q
high boards with one club and no straight draw possible.
What do you guys think would be a good way to do estimate these partial R's? I've been thinking about deriving them from villain stats(like cbet), or calculating R from expected villain contribution to the pot. What do you guys think? I appreciate any suggestions you guys might have!
Also, I haven't read any of Will Tipton's books. Is it OK to start with the second one?
A minor adjustment I would make to the definition I gave here is instead of defining R as the % of our equity we realize (ie EV=Pot*equity*R), it is more useful to just define it as the % of the pot we win (ie EV=Pot*R, where R=equity * some factor). The reason for this is that the first definition gives some screwy numbers for R when your equity is very low but EV very high (say for instance an air combo in a very strong range). This was not my idea, see Will Tipton's capture factor.
Other than that I agree with past me :)
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