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 let’s check our EV:

EV = (total pot * our equity) – shove
EV = 415 * .64 – 200
EV = 265.6 – 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

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

thanks guys, really really appreciate the kind words.

WoT

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

Thanks tjcs, glad you're finding it helpful :-)

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

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

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?

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

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

A few comments/questions:

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

Aaron, thanks for your thoughtful reply.

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

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 - 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.

Glad you enjoyed the series!

WoT