# Poker Video: No Limit Hold'Em by WiltOnTilt (Micro/Small Stakes)

## Mathematics of NL Hold'em: Episode Eight

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### Mathematics of NL Hold'em: Episode Eight by WiltOnTilt

In the final chapter of Mathematics of NL Hold'em, WiltOnTilt focuses on G-Bucks and hand-ranges. We see a comparison between Sklansky Bucks and G-Bucks, and also compare those to actual bucks. In addition, this episode briefly delves into Game Theory and gives an example of a Game Theory Optimal bluff on the river.

#### About Mathematics of NL Hold'em

WiltOnTilt will discuss key concepts related to the mathematics of No-Limit play using Powerpoint. Begin with the basics: probability and pot odds. Then follow Wilt to more advanced arenas: implied odds and reverse implied odds, software tools and mental shortcuts for equity calculations, complex EV calculations, and an exploration of fold equity. And watch this series conclude with a discourse on the ultimate in professional poker math: hand frequencies, valuebetting, and G-bucks.

### Video Details

• Game:
• Stakes: Micro/Small Stakes
• 65 minutes long
• Posted about 5 years ago

## Comments for Mathematics of NL Hold'em: Episode Eight

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#### bottomset

164 posts
Joined 02/2007

there are only 12combos of AKo

so TT has 36.4% equity vs that range

#### WiltOnTilt

2402 posts
Joined 10/2007

bottomset, I apologize for that mistake. ugh... not sure how I missed something so obvious there. Hopefully you guys can still learn from the methodology behind this example even though I mistakenly gave us 4 extra combinations of AKo.

Here's how the new math would look:

QQ = 6 combos, 82% equity
KK = 6 combos, 81% equity
AA = 6 combos, 81% equity
AKo = 12 combos, 43% equity
AKs = 4 combos, 46% equity

For a total of 34 combos.

Now set up the division to get the weighted average:

[QQ: (.82 * 6) + KK: (.81 * 6) + AA: (.81 * 6) + AKo: (.43 * 12) + AKs: (.46 * 4)] / [34 Total Combinations]

4.92 + 4.86 + 4.86 + 5.16 + 1.84 = 21.64

21.64 / 34 = 64% (rounded)

So our range of AK, AA-QQ has 64% equity against TT, so letâ€™s check our EV:

EV = (total pot * our equity) â€“ shove
EV = 415 * .64 â€“ 200
EV = 265.6 â€“ 200
EV = \$65.60

So our range made \$65.50 G-Bucks vs his call with TT. At least the math wasn't so off that it killed the example ;-) but it was a mistake nonetheless.

Good catch bottomset and thanks for watching.

Aaron

#### svidrigailov

10 posts
Joined 02/2007

Hey WoT,
Just wanted to say that I've really gotten a lot from this series and appreciate the work you've put into it. It's helped my fundamentals more than any other resource and the more advanced topics will certainly be useful as I'm getting out of the micros. Great job!

#### Fascist Baby

3 posts
Joined 01/2008

Yea aaron solid series and great last episode. You also come across as a very nice guy. Looking forward to the next series.

#### Millman123

163 posts
Joined 01/2008

Excellent series!!! One of best trainingvideos I have ever seen.

-Millman

#### WiltOnTilt

2402 posts
Joined 10/2007

thanks guys, really really appreciate the kind words.

WoT

#### tjcs

33 posts
Joined 07/2007

Great series, thanks.

Quick question that might be missing the g-bucks point so forgive me in advance.

Re: G-bucks. It seems that as I tighten up my range in a position, the G-bucks would increase. I.E. if I only shove AA/KK in the UTG example, my G-bucks will skyrocket vs his call of TT.

However, it occurs to me that if I do so then the likelyhood of getting a call there by TT decreases.

It seems that including AK in the range decreases my G-bucks, but increases the frequency of call.

Using the example of an uber-tight range UTG just to understand this, especially as it gets into the more likely usage in river bet/bluff scenario. And how it relates to the importance of balancing my range to be sure it does include bluffs signifcantly enough.

#### WiltOnTilt

2402 posts
Joined 10/2007

tjcs, you're exactly right. When your range tightens it will increase the mistake a player makes by playing an inferior hand against you. Seems to make sense right? The stronger the hand I have, on average, the more of a mistake it is for another player to make a call against me without a premium holding. The opposite is true too, if they fold wayyyy too much, your G-bucks will increase when you turn more hands into a bluff.

I talk a little about this towards the end of the video when wrapping up the K4cc example and on that weird looking "mind game" slide. In that section I'm talking about the back and forth that good players will think about in terms of increasing and decreasing their ranges based on recent history and game flow. The same is true if people are using pokertracker stats to play against the tight shortstacker. They may see his stats as 12/8 nit but not know just how their hand stacks up against his UTG shoving range. If they did know how strong his range is, they probably wouldn't call with TT in that spot, etc.

So you are right, you have to think about the overall hand and each street's actions to develop a hand range, then on the river vs predictable players who you think can fold a certain subsection of hands (such as AJ- in the K4cc hand) you should be increasing your bluffing range against them to increase your G-bucks. Vs players who are good or better than you, you'll want to make more game-theoretic optimal bluffs, such as bluffing 30 of the 90 combos as I was talking about in the example so he can't gain a mathematical edge by either calling or folding. Vs other players who arent folding anything, you increase your G-bucks most by simply not bluffing at all and taking advantage of their propensity to call (such as the full house example in episode 7 where we overbet jammed).

Hope that makes sense, let me know if I need to clarify.

Aaron

#### tjcs

33 posts
Joined 07/2007

Perfect, thanks for the clarification.

I remember reading in an old 5-card draw book about similar formulas for balancing ranges, and when I saw the way you laid it out it seemed much clearer, and wanted to understand the effect of tightening ranges have not only on the calculations, but how it influences behavior to require them to call wider so we do get paid off. And how balancing it causes the overall \$\$ to increase even though in isolation a very tight range may seem like a better calculation.

I appreciate the tremendous effort that you put into this series and making the math more digestable.

2402 posts
Joined 10/2007

#### paranoiD

1 posts
Joined 01/2008

really enjoyed ur series!

#### Philbet

2 posts
Joined 09/2007

Top series Aaron,totally differant to any other with some excellant mathmatical points, every no limit player will have benifited from this series of videos in some way, many thanks keep up the good work.
Phil

#### Mendez

810 posts
Joined 02/2008

Great series

A couple of questions:

1. Assuming your opponent is never going to adjust to what you do, and calls river bets slightly too much (slightly more than Game Theory Optimal), am I right in thinking that the best (maximally exploitative) strategy would be to NEVER bluff? And that in general terms if your opponent is veering slightly away from GTO the best way to exploit is not to veer the same amount in the other direction, but to go 100% in the other direction (again assuming that he never adjusts); and, therefore, the only reason for not doing this is if you think your opponent is capable of adjusting and would adjust if you went for a maximally exploitative strategy?

2. So G bucks is the EV of our range against villain's particular hand, and in both the examples in the video villain had a particular hand. But shouldn't we in fact be thinking of the EV of our range against villain's range (seeing as a lot of the time we can't put villain on one particular hand)? What kind of bucks are those?

As an example, I've been trying to analyze a hand I played the other day in a HU match where I raise with A7s on the btn, villain calls and donks into me on K54. I've got the NFD and an overcard so I raise, he calls, and then leads into me smallish me on a 3 turn. I've now got a gutshot as well so I shove. So I've been calculating my EV against his calling range, calculating how often he needs to fold, working out what his turn leading range is and what part of it calls my shove to see if I had the required FE. But after watching your video it seems I need to be thinking about G bucks. How would you go about analyzing a hand like this, just stick range against range into Pokerstove?

#### WiltOnTilt

2402 posts
Joined 10/2007

Hi Mendez, great questions.

1. Yes, assuming your opponent never adjusts and always calls or always folds your maximally exploitative strategy would be to either never bluff or always bluff. I tried to mention something close to that effect in the video but I might not have said it clearly enough. In the K4cc hand, I felt like my opponent was good but not great, therefore I didn't feel like I could go 100% in either direction and therefore left myself a few bluff combos such that even when he did call my bluffs he was losing money.

2. Yes, the "next" step in the ladder of complexity in the real bucks -> sklansky bucks -> g bucks -> XXX bucks would be range vs range calculation. Unfortunately it gets a bit more complex to make this calculation. I thought about attempting to do a video on this for you guys and naming it after myself (lol) but it didn't materialize. Unfortunately calculating the EV isn't quite as simple as plugging in your range vs his range in pokerstove. I'd have to do some more thinking about how exactly we could accomplish that calculation, but just thinking off the cuff it would immediately be harder because of how each number of hand combinations would be different based on which hand it was matched up against in your opponents range. So if your range is {A, B, C} and opponents range is {D, E, F} and you tried to calculate the equities/hand combos, the number of combos of hand A could be different if matched up against opponent's hand D, E or F. See what I mean? I think theres probably at least one other level of complexity that isn't coming to mind right away that also would make the calculation harder.

In any case, I think G-Bucks gets you much closer to being "correct" than the standard sklansky bucks, but it also takes a great deal of self discipline to accurately identify what your true hand range is. For instance if I said I "never" bluffed in that K4cc spot then it makes my opponent's call look super super bad... but then I wouldn't be being honest with myself so the G-bucks calc would just be an exercise in whining...whereas if I'm accurately trying to determine how many bluffing combos I would have (for instance, by making the assumption that I check behind the turn x% of the time) then I can get a much clearer picture about what's going on.

So to kind of wrap it up, although it's important and kind of neat to see mathematically that even if he calls or folds we're still happy, vs a lot of opponents (especially at lower stakes) you're much better off by simply playing a maximally exploitative strategy wherein you either never bluff or always bluff vs a particular opponent because he will never adjust and/or can't read hands.

It's like I often tell my students during our coaching sessions: "It's OK to play your hand face up if your opponent is blind."

In your particular hand in question, if I was analyzing it, I would probably base my analysis off of the fold equity video where you give him a range of hands to donk/call flop then donk turn and figure out how much fold equity you need then figure out of the hand combinations he has in his range, will he fold often enough?

WoT

#### Mendez

810 posts
Joined 02/2008

Aaron, thanks for the response,

Re: my first question, maybe you misunderstood, or maybe I've misunderstood your answer. I wasn't talking about an opponent who always folds or always calls, but rather an opponent that does a mixture of the two but is still exploitable. Using Rock, Paper Scissors as an analogy, against an opponent who does Rock 36% of the time, Scissors 32% and Paper 32%, maximally exploitative strategy is for us to do 100% paper, isn't it? (Assuming he doesn't adjust).

I would definitely be very interested in seeing a video in which you explore range against range calculations. I think I understand the problem you outlined. Would this be an example: our range: QQ, JJ, TT, villain's range: AK, AQ. We calculate the equity of each of the hands in our range against villains range (So, QQ vs AK, AQ; then JJ vs AK, AQ; then TT vs AK, AQ). But then we come to weighting each of those equities to get the equity of our range against his range, we don't know how many combos of QQ to count because there will be less combos when he's got AQ? Couldn't we just say that half time he's got AK so we count 6 QQs, and half the time he's got AQ so we count 3 QQs, for an average of 4.5 combos? Or do we also have to take account of the fact he's not going to have AK 50% and AQ 50% because QQ is in our range, making it less likely for him to have AQ? Or is that already incorporated into the calculation when we calculate the equity of QQ vs AK, AQ?

Now I've totally confused myself.

Anyway, I'll look forward to hearing about W-Bucks in the future. Keep up the good work!

#### Mendez

810 posts
Joined 02/2008

OK- here's my attempt.

I thought it would be better to start with a very simple scenario, and one where we intuitively know the correct answer so we can check to see if the methodology is correct.

So, lets say my range getting it AI preflop is [AA, KK] and villains calling range is [AA, KK]. Intuition tells us that the equity of my range against his range must be 50%.

We know that the equity of AA v KK is 80%, KK v AA is 20%, AA v AA is 50%, and KK v KK is 50%.

First, take each hand in my range and see how it does against his range:

When I have AA, there are 6 combos of KK he can have, and one combo of AA he can have, for a total of 7 combos.

So equity of AA v [AA, KK] = 1/7 (0.5) + 6/7 (0.8) = 0.7571

When I have KK, there are 6 combos of AA that he can have, and one combo of KK, for a total of 7 combos.

So equity of KK v [AA, KK] = 1/7(0.5) + 6/7 (0.2) = 0.2429

Looking at our range only, there are 6 combos of AA and 6 combos of KK, so 50% of the time we have AA and 50% of the time we have KK.

So the equity of our range against his is:

50% x 0.7571 + 50% x 0.2429= 0.5

or 50%, which is just what we expected.

Now a more complex example:

my range: [AK, AA, KK] 28 combos
his range: [AK, AQ, AA, KK, QQ} 50 combos

AK v [his range] = 52% (using Pokerstove)
AA v [his range] =86%
KK v [his range] = 63%

[my range] v [his range] =

[6(0.52) + 6(0.86) + 6 (0.63)] / 28 = 0.6164

If I stick these two ranges into Pokerstove, I get the same answer. Doesn't Pokerstove get its results by running simulations? If so, can we not then see the Pokerstove result as an empirical confirmation of this method?

#### WiltOnTilt

2402 posts
Joined 10/2007

mendez, at first glance this looks solid. let me think about it some more when it's not 4:30am.

i'll get danzasmack (chuck) to try and weigh in as well

#### caracorules

5 posts
Joined 02/2008

You should make a video showing how we calculae our risk of ruin.
Thanks

#### EvilSky

78 posts
Joined 01/2008

Brilliant series m8, I learned a bunch of stuff I thought I already knew lol .
Also I love how it forces you to actually think about poker instead of just miming what somebody says mindlessly.

#### DeMonstrative

76 posts
Joined 07/2008

Just joined DC, and I really enjoyed Math #8. Great Job!

1) Math is foundational in Hold Em'....I'd never argue that. However, I think one argument against the use of Game Theory (at least this type) is that although GT works well in a vaccuum, it doesn't account for metagame stuff (image, reads, game flow). You touch on this when you describe the back and forth constant re-adjusting top players make aginst each other as the game progresses.

So, even if you have, somehow, determined that it's profitable to river bluff an exact percentage of the time in one spot, first of all, you CAN'T, because it's impossible to bluff say 15% on this ONE hand...you either bluff or you don't.

Second, the sea of possible variables is too great and renders every situation completely unique and "this spot" will never come up again. So, maybe a better way to look at it is the expectation of a river bluff over a check.

The innate uniqueness of every hand could be why some high level players excel while knowing little math let alone GT.

2) The combinatorial analysis of the river bluff is very interesting and got me to thinking, that, although most players wouldn't phrase it this way, when staring at an all-in river bluff with AJ on the given board of Ac Tc 5d 9h 8d, what they should be thinking is "there's a lot of combinations that beat me".

So, it's interesting to consider the implications if your hand is 23o and the 8d on the end was the 8c, what would the effect be? Basically, an all in river bluff is saying "Now, there's A LOT of combinations that beat you."

Maybe that's why high-level guys bluff flushes a lot?

3) When calculating G-Bucks do we need an exact hand, or a range (in this case AJ or worse)? I know I'm not good enough to put someone on just one hand...so would it suffice to say top pair or worse?

4) I think you've just scratched the surface on GT and would like to see more!

- Thanks Again

#### WiltOnTilt

2402 posts
Joined 10/2007

Hi DeMonstrative, nice post. I'll try to briefly comment on each of your points:

1) I'm not sure if you've watched all of the math series or just episode 8, but I do agree with you and I'm pretty sure I've made similar comments throughout the series on the usage of math, both strengths and weaknesses. I would never suggest that math is the end all, be all of NLHE and if I wasn't clear on that in the entire series or this episode then let me dispel that rumor now :-)

2) sorry but can you restate your question? i'm not really sure what you're asking here... your hand and the board cards will change the # of combos of bluffs and possible hands based on the range you're giving... it was never the intention of the video to suggest that people would be able to calculate all the combos in their head, but how the different cards can help you estimate the likelihood of the various hands.

3) In general standard ev calcs work for just your hand vs another hand or your hand vs a range (this has been covered a lot in earlier episodes...). you need g-bucks to look at your range vs their hand .

4) i am not at all a game theory expert, there are still some parts of "Mathematics of Poker" by chen and annkenman (fantastic book btw) that i don't fully understand, so i probably won't be doing a whole lot of GT stuff for that reason, and because it's hard to apply it correctly and usefully to a wide range of viewers...whereas I feel like the info i've talked about in this series of videos is both useful, applicable, and at a level of complexity that can be grasped by non-math majors.

Hope that helps...

Aaron

#### DeMonstrative

76 posts
Joined 07/2008

1) We're definitely agreed here.

2) This isn't really a question, more of a concept for discussion to illicit your opinion.

I guess my point was that on the given board Ac 10c 5d 9h 8d, if the 8d is the 8c the number of combos that potentially beat the villian's AJo explode.

Therefore, drawing from your discussion of G-Bucks, if we bluffed with the same frequency (30 times I think it was) our G-Bucks earnings go up tremendously due to the added number of flush combos that beat the villian.

This means, one could achieve equivalent G-Bucks earnings by bluffing MORE OFTEN when the 8c hits the river instead of the 8d.

Conclusion: Bluff flushes on the end for added G-Bucks.

3) I'm asking if I wanted to calculate G-Bucks in-game, would I need to put my opponent on a single hand "he's got AJ" or simply say "he's got AJ or worse...I'm bluffing; because he can't profitably call my range."

4) Thanks for the advice, I'll try grinding through Mathematics of Poker.

Thanks for the discussion,

-TED

#### WiltOnTilt

2402 posts
Joined 10/2007

Hey Ted, Ok I get your question now. Yes if you can credibly represent a flush, we can get our opponents to make much bigger g-bucks mistakes in theory, but we also have to consider that he might "polarize our range" in these type of spots and give us flush or nothing, which in some circumstances could reduce the number of combos we have.

Say for instance the board is 5c 8c Tx Jc Ax, because of the board texture we cannot have 45c,56c,89c,78c,JTc,QJc,AJc,ATc,etc etc Also if we don't fastplay the flop, he might not give us credit for any of the club combo draws like 67c,97c,TXc etc etc. So just consider those things before trying to bluff every flush, especially if you're playing against one of these 2+2 types that don't want to give you credit for value betting "thin" (ie, if you bomb this river, he's only going to say "flush or nothing" and remove all the 2pair and set combos as well).

#3 - G-bucks are primarily used as post game "how did he do vs my range" analysis because it's pretty darn difficult to put someone on an exact holding as well as mentally tabulating all those hand combinations. Ideally we would calculate range vs range (how does his range rate vs my range?) but that's quite a bit more complex and pretty impossible to do mentally.

Hope that helps,
Aaron

#### IceMaiden

3 posts
Joined 06/2008

Just signed DC. And luckly started with this series. It worth every cent I payed. Thanks a lot for tour time and good will. I tried to read nathematics of poker, and gave up to aply maths in poker besides pot odds.
U really enlighted the subject.
Congratulations.
hope n0whereman learned a lot with you bcause I had one session with him and hope to become his regular student... LOL

#### jimike

8 posts
Joined 05/2008

Yay!
Finished the series.
I really enjoyed it,your a brilliant teacher and presented the topics in a way the math illiterate like myself could understand.
I feel this series has changed the way i'm thinking about poker at the table and this is before i've even had the chance to go over my 80+pages of notes.
Well done sir!

Do you have any more series on the horizon?

#### WiltOnTilt

2402 posts
Joined 10/2007

jimike - Thanks for the kind words. I really appreciate it a lot. I don't have any more math based series in the works (it was a ridiculous amount of work to do this series and took a tonnnnnn of time) but I have done a HU series with KRANTZ as well as a series helping small stakes 6max NL players. If you go to the search screen you can select my name in the "author" drop down and see the videos I've done. I'm planning to do some HU Duel vids soon as well.

WoT

5 posts
Joined 08/2008

Wilt, great video

Wilt & Mendez,

I'm interested in your dicussion about moving to the next step, to think about My Range vs. His Range:
1. My hand vs. His range = Sklansky bucks
2. My Range vs. His hand = G-bucks
3. My Range vs. His Range = XXX-bucks

As an aside, it seems G-bucks is really just a standard EV calculation when you switch seats and look at things from his perspective, and so doesn't "get you much closer to being correct" - it's just an evaluation of his play, rather than of yours. XXX-bucks is the real answer

Thinking in terms of XXX-bucks, if you have exactly the same ranges for pushing and calling respectively, your equity is the same. From the caller's perspective, he can actually take slightly the worst of it because of the money in the pot, and so can have a slightly WIDER CALLING range than your PUSHING range. Both ranges can widen the more already in the pot, but because he acts second villian can actually have the wider of the two (the reverse of normal gap-concept thinking)

So the real interesting question is:
- For a certain ratio of stacks to amount already in the pot, in the situation where you push with your range, if villain includes TT in his calling range would his calling range then be too wide wrt your push range to make it +EV in XXX-bucks?

Anyway, I'm not sure if I'm asking a question or making a comment, but really hope danzasmack/wilt/mendez say more on the subject...

#### cubaner

161 posts
Joined 07/2009

firt of all i like to thank you for this great series, itÂ´s really great work!

for reviewing and rereading it would be nice to have the powerponit presentation that is used in your video series, is it posible to download it anywhere?

#### Synchronicity

6 posts
Joined 09/2009

Awesome series dude! Thanks for breaking everything down so well!

#### PygmyHero

4246 posts
Joined 08/2007

You should make a video showing how we calculae our risk of ruin.
Thanks

You almost certainly won't see this post but I do some things with RoR calculations in this video.

#### PygmyHero

4246 posts
Joined 08/2007

Why did you compare TT's equity versus each individual hand rather than just plugging our hand range into Stove? I was working the answer out while watching and got ~\$63.94. Your answer (\$55.23) was sufficiently different that I doubted it was due to rounding. Then I noticed the AKo combos as bottomset mentioned. Your revised answer (\$65.60) is very close to what I got using a range in Stove.

It seems to me that Stoving can help us avoid these types of combo counting errors, so I'm very curious to hear why you break the combos out individually.

#### PygmyHero

4246 posts
Joined 08/2007

I know I'm late to the party but I just finished and wanted to say thanks for the excellent series!

#### WiltOnTilt

2402 posts
Joined 10/2007

Why did you compare TT's equity versus each individual hand rather than just plugging our hand range into Stove? I was working the answer out while watching and got ~\$63.94. Your answer (\$55.23) was sufficiently different that I doubted it was due to rounding. Then I noticed the AKo combos as bottomset mentioned. Your revised answer (\$65.60) is very close to what I got using a range in Stove.

It seems to me that Stoving can help us avoid these types of combo counting errors, so I'm very curious to hear why you break the combos out individually.

the purpose was just to show the pieces of how it's all put together. you're probably right just keeping it more simple and stoving would have been fine and almost as educational

#### st3v3k4hn

3 posts
Joined 01/2010

Wilt
Again on the K4s bluff hand, I'm wondering why you shoved pot? In previous lessons you showed that you can frequently get your opponent off a hand with a smaller bluff, and reiterated that you want to bet as little as possible to get the job done. Did you do some sort of calc in your head that said you needed to shove here in order for the bluff to succeed? Any chance a 1/2 pot bet would be more profitable?

#### FerrisBueller

4 posts
Joined 08/2008

Hi

Im trying to work out how profitable a bluff will be for each extra combo he has that will fold to a shove. Assume he has to fold 35%, we win 70 when he folds and lose 50

Any ideas?

#### gamer078

5 posts
Joined 06/2010

Great video, G-bucks is way above what I should be learning at my level. But I do have a question. Wilt mentions that we should be INCREASING our bluffing range against the player that calls with AJ on the river given the situation (though we thought that he would fold AJ). This seems to be the opposite of what should be done, can someone please explain?

Wilt mentions that our betting/bluffing frequency(range) is exploitable. I can see that with such a disparity in bluffing/value betting frequency, a good player would be folding in this spot simply based on our value/bluffing frequency or range. But I do not see how our bluffing needs to be increased against this player to make our frequency less exploitable when our opponent calls us with a hand that is beaten by our range so often. It seems that since our opponent is making such a huge mistake in the long-run, we shouldn't change anything, or decrease our bluffing to exploit his overcalls. Or is it that we should increase our bluffing range in different situations that I didn't catch?

#### gamer078

5 posts
Joined 06/2010

Nevermind the previous post, futher down the video Wilt says we should be decreasing our bluffing unless our opponent adjusts.

#### kgbmiked

192 posts
Joined 11/2010

Here's my notes on this episode, feel free to use them or ask questions

Our hand vs a range of villainsâ€™ hand and our +EV over that range are Sklansky bucks

G Bucks are +EV when considering our hand range vs villains actual hand.

So now we are no longer concerned about how our actual hand does against a villains range but instead concerned with how our range does against the villains actual range. We are flipping it

So say we are SS with 200 compaired to the standard start of 1000. We are UTG at 5/10 NL and we open shuv with AhKs and the Button calls us with 10d10c and he wins. So we lost 200

So how did we do over the long hail over this matchup. For that we need to calc our EV

We have 43% equity and we lost 200
EV (ie Sklanksy Bucks)

EV = .43(215) â€“ .57(200)
EV = -\$21.55 This is our Sklansky bucks

What about the G Bucks? To determine G bucks we care about OUR hand range. So what is our hand range?

AK,AAKKQQ

So how does our opponents hand do against our range in G bucks? To determine our G Bucks, we need to figure up our hand combos and equity with each hand in our range

QQ = 6 combos with 82% Equity
KK = 6 combos with 81% Equity
AA = 6 combos with 81% Equity
AKo = 12 combos with 43% Equity
AKs = 4 combos with 46% Equity

TOTAL of 34 combos

Now set up the division to get the weighted average

(QQ(.82*6) + KK(.81*6)+AA(.81*6)+AKo(.43*12)+AKs(.46*4)) / (34 combos)

= 21.64/34
= 64% (rounded)

So our range of AK,AA-QQ has 64% equity against TT, so lets check our EV

EV = total pot * our equity) â€“ shove
EV = 415(.64) â€“ 200
EV = \$65.60

So our range made \$65.60 G Bucks vs his call with TT.

According to WoT we should care mostly about our G Bucks.

Example:

We are on the button with 4cKc against the decent regular BB. The board is AcTc5d9h8d. He knows our range of opening the button and C betting this flop is very wide. The pot is 665 and we shove the river for 665.

Our river shoving range is
One Pair: AQ,AK
Two Pair: AT,A5,A9,9c8c
Straight: Jc7c,6c7c,QcJc
Sets: 55,TT,AA
Missed club draws no pair (KcJc+, Kc7c, Kc6c, Kc4c-Kc2c,Qc7c,Qc6c,Qc4c-Qc2c, Jc6c, Jc4c-Jc2c, 7c4c-7c2c, 6c4c-6c2c, 4c3c,4c2c,3c2c)

Count the Combos assuming he has AJ in this situation

One Pair: AQ(8),AK(8) = 16 combos
Two Pair : AT(6), A5(6), A9(6),
A8(6), 9t(9) = 34 combos
Straight: Jc7c,6c7c,QcJc = 3 combos
Sets: 55(3), TT(3), AA(1) = 7 combos

TOTAL Value Bet Combos = 60

Missed club draws without pair (half the time I would check the turn instead of Bet,Bet,Bet) = 25 Combos
25Combos/2 = 13 combos to bluff

TOTAL Combos = 73
So our Bluffing combos are 13
Value Combos 60
Total Combos 73

We make a pot sized shove for 665. Since there are no future streets, the â€œSklansky bucksâ€ and the â€œactual moneyâ€ will be the same. If he makes the call with AJ and wins, he stand to profit 665+665 = 1300

EV (Villainâ€™s call) = 13/73 *1330 â€“ (60/73) * 665
EV (Villainâ€™s call) = 236.85 â€“ 546.58
EV(Villainâ€™s call) = 309.73 G Bucks

So In the long run he made a bad call because he is losing \$309.73 against our range in this spot. So based on what we assumed on our hand range, we made a good bluff.

If we go back and donâ€™t reduce our bluff clubs by 50% because sometimes we check the turn and assume we are opening all suited hands from the button. So essentially we are adding in 12 more combos of bluffs
EV(Vilainâ€™s call) = (25/85) * 1330-(60/85) *665
EV(Villainâ€™s call) = 391.18 â€“ 469.41
EV(Villainâ€™s call) = -\$78.23

So assuming I 100% open all suited hands on the button and 100% 3 barrel all of these non-paired club draws, then his play still loses him about \$78. That means in order for him to be turning a profit, Iâ€™d need to be bluffing many more combos of flush draws that have a pair, like 5cXc or 9cXc. Not impossible, but certainly not highly likely.

So even if I am bluffing with 25 combos of flush draws on the river he is still losing -\$78.23.

Notice what will happen if we find an extra 5 combos of hands to bluff with. Maybe we turn some 5cXc combos into bluffs, notice what happens with the math:

Now at 30 bluff combos and 90total combinations:

EV(Villainâ€™s call) = (30/90)*1330-(60/90)*665 = 0

So when we add in 5 more bluff combos his -\$78.23 EV goes up to 0. If the EV is 0 then we make it so he can not prefer a call or a fold. So this is a game theory optimal bluff. So our game theory optimal bluff is when we shove here with 30 bluffing combos because his EV is 0 and he does not gain anything by calling or folding. It makes it so it doesnâ€™t matter what he does here because he will not make money in the long run even if he knows we are bluffing. However, this is not the highest EV strategy for us. It is just the limit to which we can possibly do it without giving him EV over us. The reason this works is because our bluffing frequency when betting 30 combos here is exactly the same as his pot odds to call. We are bluffing 30/90 combos and value betting 60/90 combines so he is 2:1 against being bluffed. AJ beats no hands that we are value betting therefore he is getting 2:1 on a call and stands to lose twice as often as we wins. Since weâ€™re value betting twice as often as weâ€™re bluffing then it turn out to be the same ratio of 2:1. So even though this is an unexplainable bluffing strategy, this is not the highest EV strategy, especially if we have some sort of read as to what hands he can fold and what hands he will call with. We thought we could get him off AJ, but even if he doesnâ€™t he is making a G bucks mistake by calling assuming our original bluff combos makes us 4.5:1 against bluffing and he is only getting 2:1 pot odds on a call. So he is not getting the pot odds to call. So we really donâ€™t mind if he calls here because in the long run he is making a 309.72 dollar mistake every time he calls because we will be making this same play with a range where we are beating him 66.6% of the time.
So based on our read that he can fold hands like AJ we are fine no matter what he does. So, our shoving range on the river with only 13 bluffing combos and 60 value betting combos is an exploitable strategy. Why? Because he can and probably should just fold every time on the river. In situations like this where we have a missed club draw that is good, but, remember that most times (60/73) times we are value betting him and we want him to call with AJ. So by only bluffing 13 combos on the river we are playing an exploitable strategy, but we are only playing that way because we think we can get him off of AJ. So we are only playing exploitable but it is against a player whom we are assuming will not exploit us.

So since he called we now know that he is making a â€“EV call in G bucks by calling with a bluff catcher so we need to adjust down our bluffing range to maximally exploit his tendency to call. We do still need to have some bluffs in there though so that we are not playing completely straight forward and he canâ€™t exploit the fact that we are never bluffing by making his folding range higher.
Against very good players though, once he makes a call like this with a pure bluff catcher he is going to assume we are reducing our bluffing range because we think he is calling with bluff catchers so he will be less inclined to make a hero call next time and will adjust his calling range up. So against good players we are both making adjustments anytime we both see this situation. It is against people who are not changing their range after we see him make a hero call that we can take advantage by tightening up our bluff range because he is not adjusting his calling range up and now we will be having a value betting hand in this situation more and he will still have the same hero calling range. This adaptation becomes hugely +EV. We want to be putting Villain in situations where even if they snap us off they will be losing money.

Against players who are better than us or just as good, we need to balance our bluffing range to keep them from getting an edge on us. This is defense against players that are better then us and we do it by going back an forth between our 30 bluff combos with an EV in G bucks for our opponent is 0 back to a tighter range. We keep them guessing without going too far into bluffing.

#### KeysorSoze

68 posts
Joined 10/2009

Hi Aaron

You're probably not going to see this message, but just wanted to thank you for taking (Im guessing A LOT) of time to put together this amazing series.

Thank you!

/Christian

#### brainfreeze0

68 posts
Joined 05/2011

Was such a great series WoT. I'm not the greatest at math, but I think I see what you're trying to get us to do. Obviously the majority won't be able to do all of these equations during the hand. However, doing hand reviews and relating situations to future situations will help us make more calculated unconscious +EV decisions when similar hands pop up in the future. I think out of everything I learned on your series my absolute favorite was the TUPAC method (he's still greatest of all time also). I've been concentrating on making up hands in my head at work and where ever else I am to make quick assessments of my EV vs his range and getting pretty good at makin quick rough estimated decisions based on that process. Was an absolute platinum series and I'll be watching it more than once thanks!

BF

#### Makaton

20 posts
Joined 02/2011

Thanks for the great series. It was a lot of stuff to absorb but well worth it. You did a great job and your dedication shows.

T

98 posts
Joined 07/2010

Hi Aaron,

I am through it!

Cu
Christoph

#### KamikazeMan1

11 posts
Joined 04/2011

Hey Wilt,

I went through the whole series without playing a single hand of poker in between. I finished the series, developed a game plan into how I would include math in my critical decision making at the online tables and already it is making me look at everything completely different(the correct way). I'm looking at everything from a hand range perspective which is really making a lot of sense to me already but I have one quick question to ask. Is there any possibility to calculate G-Bucks when your villain has AA? I ran into the dreaded KK vs. AA matchup today and I went through the hand in my head and I found that I made the correct play. In this 6Max cash table villain has shown to be a pretty tight player up to this point (15/15) and he opens 3x UTG. I'm in the cutoff with KK and I decide to 3Bet him 3x his open size. It is now heads up at this point and villain decides to 4 bet about 2.5x my 3Bet size(We both have 100bb stacks btw). In pondering whether shoving against his 4Bet range was correct instead of flatting with the intention to possibly trap I decided that his 4Bet range would most likely call a shove and I was guessing that range included KK+, AKs and AKo for sure, and maybe QQ because I have been somewhat aggro preflop. So I decided a shove was best because all of his AK combos would call as well as QQ. I wanted to make a G-Bucks calculation but I didn't find it to be possible given that villain's holding was AA and so my shoving range to him doesn't matter in terms of deciding whether to call the shove or not. Can you elaborate on whether I can make a G-Bucks calculation here or if I can just make a standard EV calculation of shoving against his 4Bet range.

Thanks Wilt and I love your videos man, you are by far the best

Cheers,
KamikazeMan1

#### WiltOnTilt

2402 posts
Joined 10/2007

Hey Wilt,

I went through the whole series without playing a single hand of poker in between. I finished the series, developed a game plan into how I would include math in my critical decision making at the online tables and already it is making me look at everything completely different(the correct way). I'm looking at everything from a hand range perspective which is really making a lot of sense to me already but I have one quick question to ask. Is there any possibility to calculate G-Bucks when your villain has AA? I ran into the dreaded KK vs. AA matchup today and I went through the hand in my head and I found that I made the correct play. In this 6Max cash table villain has shown to be a pretty tight player up to this point (15/15) and he opens 3x UTG. I'm in the cutoff with KK and I decide to 3Bet him 3x his open size. It is now heads up at this point and villain decides to 4 bet about 2.5x my 3Bet size(We both have 100bb stacks btw). In pondering whether shoving against his 4Bet range was correct instead of flatting with the intention to possibly trap I decided that his 4Bet range would most likely call a shove and I was guessing that range included KK+, AKs and AKo for sure, and maybe QQ because I have been somewhat aggro preflop. So I decided a shove was best because all of his AK combos would call as well as QQ. I wanted to make a G-Bucks calculation but I didn't find it to be possible given that villain's holding was AA and so my shoving range to him doesn't matter in terms of deciding whether to call the shove or not. Can you elaborate on whether I can make a G-Bucks calculation here or if I can just make a standard EV calculation of shoving against his 4Bet range.

Thanks Wilt and I love your videos man, you are by far the best

Cheers,
KamikazeMan1

Really appreciate the nice words!

Yea for that type of situation, you would just want to do some ev calculations about your range vs his range. Given that your situation is a preflop one with fairly defined ranges, you can probably do it without too big of a headache. In some of these postflop situations, it can get really hairy to do range vs range calculations for like 3 barrel bluffing because you have to make a lot of assumptions about hand combos/frequencies for all types of different hands, which hands are folding which are calling, if he turns hands into bluff raises, and all sorts of other stuff I can't even think of right now... so usually instead we would simplify it by doing it as a range vs a hand, but as your example shows when your opponent has the nuts, it doesn't make a lot of sense to see how our range does vs that :-)

Also I should point out there are some tools out there now that help with these things so you dont have to crunch as many numbers by hand (although I still think it's useful to know the process because it helps you think in that way). As you saw in the other thread, we didn't have those luxuries back in the day :-) You might do some searching, I've been meaning to brush up on some of these tools myself. I heard flopzilla is pretty good but I've never used it. If you check it out, let me know what you think of it.

Best of luck!

#### GrimbleGrumble

39 posts
Joined 04/2008

Thankyou for this series. Some of the most important information I have learnt about poker is from here. Now to start really applying it...

#### WiltOnTilt

2402 posts
Joined 10/2007

Thankyou for this series. Some of the most important information I have learnt about poker is from here. Now to start really applying it...