pasita: Great idea. Tried last night but too tired. Note: The Wikipedia article on Expected Value is an excellent overview of EV concepts.

Screenshots:
EV calculates profit per decision. EV <> EWIn <> Default EV
One way to calculate EV and EV of alternative lines for hands in DB. Math Check.
EV for full set of EV (BNATS) for hands in DB
Effective use of Replayer & Math Check EV (BNATS Call 5Bet All-In).
Optimal Stat Report . Note that stats are assumed to be exploitatively biased by nature. Not necessarily so.

1) The accepted wisdom that EV can’t be calculated is just plain wrong. Although its only one of the dozens of inaccurate, incomplete, implicit or explicit (mis)assumptions or misleading conclusions based on theoretical analysis, its clearly the most damaging: it’s sufficient in and of itself to prevent EV analysis (hands in DB) from ever getting off the ground.

2) The accepted wisdom that EV and EWin are interchangeable is just plain wrong. EV <> EWin. Actually confusing 4 different concepts now that I think about it.

3) There are 2 distinct ways to calculate EV.
4) Theoretical (or brick and mortar hands) analysis calculates EV by looking at all the theoretically possible outcomes of a specific decision for a single hand. EV or optimal equations are generally not solvable or provable as continuous random variables.
5) Decn EV – like all stats – calculates EV by measuring all the actual possible outcomes of a specific decision over a range of hands. Decn EV stats combine statistics, probability theory, the law of large numbers and DB programming techniques.
6) Decn EV stats calculate the EV of a decision and the EV of all of its alternative lines – without exception or theoretical limitation. Preflop, Flop, Turn, River, Showdown, Multi-street – makes absolutely no difference.

7) The definitional issues related to EV aren’t limited to the confusion caused by players using EV and Equity interchangeably – it’s “the unholy trinity of EV/EWin/PotEQ” that does the damage.
8) Unified theoretically consistent EV definitions must be consistent with statistics, probability and decision theory AND both methods of EV calculation.

9) Optimal theory (and stats) can be broken down into 3 types or situations:
A. Both players are playing optimally. They are risk neutral and maximizing EV.
B. Player A is risk-adverse. Player A will not change strategy to exploit opponent – he or she is minimizing risk, not maximizing EV.
C. Player A is altering his strategy to exploit a non-optimal (or unknown) opponent(s) to try and maximize EV. Value of exploitative play is measured by the difference between EV (Exploitative line) and EV (Optimal line).


The Zepper

Current draft of “unified” theoretically consistent definitions and other helpful info:

Expected Value (decision or situation faced) is the weighted average profit (adjusted for Pot Equity) of making a decision to act (or not to act).
Profit <> Win or Ewin
EV <> EWin <> Default EV stats. Apples and oranges. No such thing as an EV graph. EWin graphs.
Player EV <> -Opponent EV. Player EV + Opponent EV = Dead Money in pot. Player EWin = -Opponent EWin.

Win = Net amount won (lost) overall on the hand.
“Expected” in EWin = adjusted for Pot Equity.
Expected Win is the Net Amount Won adjusted for Pot Equity.

3 Types of EWin:
SklanskyBucks (Hand) = Net Amount Won overall on the hand adjusted for All-In Pot Equity in $ or bb.
WR (Total Hands) - Expected Winrate or TotalAdjustedWinnings – SklanskyBucks for Total hands expressed in currency/100 hands or bb/100 hands.
WR (decision or situation faced, position, action, range, anything other than Total hands): Expected Win-rate Contribution.

Profit (decision or situation faced) = Win + Player’s dead money already invested in the Pot.
Value (decision or situation faced) = profit (or loss) adjusted for Pot Equity.
Maximze EV (decision or situation faced) is the point at which marginal EV >= 0.

“Expected” in EV = the product of (all the possible values of a decision * their probability of occurrence). Has absolutely nothing to do with the adjustment for Pot Equity.
The method of calculation of EV depends on whether it’s considered as a continuous or discrete random variable.
EV and Profit can’t be calculated without reference to its decision point, or point of calculation (decision or situation faced).

Pot Equity. Has nothing to do with All-In. Pot Equity cannot be realized until the hand is over. Can be 0%, Split%, or 100% or
Adjusted for Pot Equity means SklanskyBucks (All-in before the River and Saw Showdown).


The Zepper

FINALLY figured out a way to illustrate the distinction between EV and EWin conclusively.





Player in SB Jams (SHV44 = Bet) on the River OOP putting BB All-in. BB Folds.

MATH CHECK:
SB EV: SB Wins Net Pot: $45.50 – Rake ($2.27) = $43.23
BB EV: BB Folds and loses ????? = $0.00

NO!!!!!! Using this terminology with EV causes nothing but confusion - on many different levels. I’m willing to bet that the word “win” can be used misleadingly in at least 6 – 10 different ways.
Ain’t no room for context when defining and calculating a stat.

YES:

Decn EV (SB River SHV FEQ OOP) = $43.23
Opp EV (SB River SHV FEQ OOP) = $0.00
FEQ stands for Folding Equity (Opp Folds)

EWin (Hand) ....... = $20.48
Opp EWin (Hand) = -$20.48

SB’s Expected Win when BB Folds = $20.48.
SB’s profit and EV when BB Folds = $43.23.

Also note that SB’s PotEquity when BB folds = 100% or $43.23
Note that BB’s PotEquity when BB folds = 0% or $0.00


Expected Value (decision or situation faced) is the weighted average profit (adjusted for Pot Equity) of making a decision to act (or not to act).

Profit <> Win or Ewin
EV <> EWin <> Default EV stats
Player EV <> -Opponent EV. Player EV + Opponent EV = Dead Money in pot. Player EWin = -Opponent EWin.
EV and Profit can’t be calculated without reference to its decision point, or point of calculation (decision or situation faced).

Pot Equity. Has nothing to do with All-In. Pot Equity cannot be realized until the hand is over. Can be 0, Split, or 100% or
Adjusted for Pot Equity means SklanskyBucks (All-in before the River and Saw Showdown).


NOTE: All this should be common knowledge, since Sklansky pointed it all out in the December 2012 post I cited earlier – albeit indirectly (the EV’s of $60 and $40 will not change …). See http://www.twoplustwo.com/magazine/i...ploitation.php.
Exception: note that the (decision or situation faced) is always described from the Player’s perspective, whether we are looking at the actions of the Player or Opponent.
We’re trying to maximize Player’s EV – not minimize opponent’s EV. Don’t equate “maximize and minimize” to Player EV = -Opponent EV. Two different concepts. One is average EV, the other is marginal EV.



The Zepper

[QUOTE=Matthew Janda;38797820]I do not think anyone knows that.


#1) There are some bet-sizes which I think get calls or folds a disproportional amount in certain spots, so it's effective to use these sizings to get your opponent to take the line you want.

Bought and loved your book!

I know that Exploitative play is not your cup of tea but can you give me examples of spots mentioned above? Thanks!

[QUOTE=Bietebauw;41780650] It's hard to say since I've played so little since Black Friday and I know a lot of exploitative plays that worked very well for 6-max don't work particularly well for heads up.

Old example:

You open in middle position and the button calls. Flop T 8 3. You bet villain calls. Turn 2. Overbetting here used to get way more than enough folds to make overbetting any two cards profitable. People would just tank/fold there draws/Tx hands the vast majority of the time and they rarely actually had a very strong hand.

But people get better and things change. So while I still like overbetting in this spot in theory (though you of course can't always overbet and multiple bet sizes will make sense), I highly doubt it's a "just overbet and he'll almost always fold" spot like it used to be.

Hi Matt, i really loved your book but can you explain the following in more detail, because in your book you always calculate how often we should defend vs cbets (ie to the 1-alpha threshold) but apparently this thresholds are not that important, as oppose to defending vs raises:

you can also check out this HUNL thread where these thresholds are discussed: http://forumserver.twoplustwo.com/58...-play-1395179/

Think about it like this. If you raise and are folding to re-raises so much that your opponent can re-raise any two cards profitably, why would he ever fold to your raise? He'd always either re-raise (which is +EV) or call.

So, it seems unreasonable for a player to ever be able to profitably re-raise with any two cards. Yet on the flop before any betting takes place, one player's range will often be a lot better than the other player's. In this case, it's reasonable to think the player with the much better range should be able to profitably bet any two cards (especially if he has position as well) and the opponent won't defend very aggressively.

Hello Matthew,

I would very kindly ask if you could clear up something for me. I am pretty much lost on
your chapter about the value to bluff ratio on the river when betting.

If we (Hero) bets the pot on the river we have to construct our range in a 2:1 ratio, where 2 are our value bets and 1 are our bluffs. Because that way we make our opponent indifferent between folding and calling. This is your explentation of the matter. This bluffs to value bets ratio is also called Alpha, right?

In other literature alpha`s formula is S/S+1 where S is a fraction of the pot when bet. You have the same formula on your book but instead of S you have X. Then (1-alpha) or the calling frequency for the player that has been bet into with alpha ratio of bluffs to value bets is 1/1+S.

So for example: I want to make you indifferent between calling and folding on the river. So I will bet with alpha % of bluffs and you will be calling me with (1-alpha).

I decided to bet the pot. Alpha = S/S+1 = 50%
So when I bet the pot on the river I will have half the time bluff and half the time a value-bet.
Now you have to be calling me with (1-alpha) or 1/S+1=50% because if you are not calling half the time I can just bet every time.

The idea seems to be similar then pot odds thing but the construction for both ranges is different. Here by using alpha we have 1:1 ratio of value bets to bluffs on a pot sized bet and not 2:1.

Could you please explain, I am very eager to learn and I am confused now.

Cheers

Hey Matt,

Is the issue of capping ranges more of what happens when your opponent doesn't make a polarized bet? If he makes a polarized overbet, as long as all your bluffcatchers beat his, you just need to defend your top 40% or whatever and it would be 0EV.

In your video you mentioned vs overbets we are forced to put more money in to prevent opponent from betting any two cards.

Your example was 0.5 x 66.7% = 0.335 PSB, 1.0 X 50.0% = 0.5 PSB, etc.

But the thing is its still 0EV no matter what he bets when we both play GTO. Yea we are forced to put more money in, but when we win, we win more money.

Could the problem with capping our ranges be more of what happens when our opponent makes non-polarized valuebets, and also when we get to showdown we usually get crushed?

Thanks

Hey,

What other literature are you referring to? Do you agree with me that if you bet pot on the river you need 2 value bets for every 1 bluff to make your opponent indifferent to calling with his bluff catchers?

Bluffs are still 0 EV in your example whether we overbet or bet small if the opponent calls enough to keep us indifferent. But our value bets become MUCH more profitable when we bet big than when we bet small, so overall our range will be more profitable.

Thanks for your response, but I'm still confused. Valuebets are way more profitable when called, but our bluffs are way less profitable when called.

When opponent bets $20 into $10, he bluffs 40% of the time. We call 33% of the time. The EV is the same as if he bet $5.

Our opponent's EV at the river with these 2 betsizes is:

(0.67) (10) + (0.33) (0.6) (30) - (0.33) (0.4) (20) = 10
(0.33) (10) + (0.67) (0.75) (15) - (0.67) (0.25) (5) = 10

-----------------------------------------

A more real example is this:

On flop Jh8s5c, turn 8c and river 4h. CO opened, you defended button IP.

Say our opponent thinks we are capped, and we are capped:
We have a range that is 1/3 busted straight/flush draws, 2/3 hands capped at AJ, down to 99, some A highs (say we raised all our trips + on the flop or turn). We have enough hands to beat his bluffs. If he knows we are capped at AJ, he's gonna overbet his better hands 2x pot, and overbet bluff 40% of the time. We just call with our top 33%, which actually comes up to mostly AJ (lot of offsuit combos). 0EV vs his overbets. Against his other betsizes, like half pot, he could do this with AJ/KJ, and bluff the proper amount. Maybe our range gets crushed against this range?

This time, we are not capped:
Opponent overbets the same range, however we have slowplayed hands in which our opponent will own himself. Not really sure what to do now based on how wide to call. Calling 33% of the time with hands that will beat his range will make our overall EV in this spot higher than 0EV. Though AJ still loses 60% of the time vs his overbet range. Not sure how to make our opponent indifferent in this case when our range isn't made up of only bluffcatchers.

After all this, seems the key to figuring out why exactly capped ranges aren't good must lie with what happens with our unpolarized range?

Or am I missing something with bluffs are 0EV and its use is to increase the EV of our value hands?

But he calls less often, so when you bluff you lose less often (though of course you lose more when you're actually called). So you aren't on average losing more money on your bluffs when you overbet compared to when you bet smaller, but you are on average winning more with your value bets.

Let me know if that clarifies things as I'm thinking there's just a small understanding and trying to use specific ranges/situations may be confusing you rather than helping you (at least right now).

However, he will call less when we have a value hand as well.

When I plug the numbers into the EV equation, I get the same number. Is this the correct way to go about it?

https://imageshack.com/i/nand1kp

No matter what I make the min defense, the overbetters EV is $10. The only way to get the number above $10 is to bluff less % and have our opponent call. If we bluff too much, our EV will never get over $10.

I bought and read your book twice. But I think somewhere along the way my train of thought got off track.

I THINK I GET IT

I get myself into a spot where I have 10 nuts, 10 air.

I bet 20 into a pot of 10, so i bet 10 times with nuts, 6.66667 with air.

I am making 10 EV 16.66667 times.

I bet pot, so i bet 10 times with nuts, 5 times with air.

I am making 10 EV 15 times.

That works too and I like thinking about it this way as well.

Yes ofcourse I agree with you. However I have come across some Game Theory MIT classes on youtube. https://www.youtube.com/watch?v=VHcr...b6uItXeJVVSeMw . Matt hawrilenko is basically saying that if I bet the bot I need 50% value and 50% bluff bets. I just want to learn the right way. Damn

There are a few mistakes in that video. I've been shown it before as someone else (a very very good player) was also was confused by some things said in it.

Hi guys / Matt,

I'm reading the book at the moment and really like it. Trying to make some range constructions, so maybe you can help me out here and give me some feedback on my thought process. I would like to analyze this hand:

IPoker No-Limit Hold'em, €0.20 BB (6 handed) - IPoker Converter Tool from http://flopturnriver.com/

Hero (BB) (€17.50)
UTG (€44.91)
MP (€18.11)
CO (€14.48)
Button (€10.97)
SB (€63.57)

Preflop: Hero is BB with 8, 9
4 folds, SB raises €0.50, Hero calls €0.40

Flop: (€1.20) 10, Q, 5 (2 players)
SB bets €0.84, Hero....

He cbets 0,7PSB, so I should defend 1-(0,84/(0,84+1,2)) = 59%.
If I use the coldcall range in the SB that the book describes, I have a 45,4% range, which consists of 602 combo's, but on this flop only 534 because of card removal. The hands in my range:

Sets: TT, 55 (6 combo's)
Two pair: QTs, Q5s, T5s, QTo (13)
Top pair: QJs, Q9s, Q8s, Q7s, Q6s, Q4s, Q3s, Q2s (24)
Top pair: QJo, Q9o, Q8o (27)
2nd pair: ATs, KTs, JTs, T9s, T8s, T7s, T6s. (21)
2nd pair: ATo, KTo, JTo, T9o, T8o. (45)
Between 2nd/3rd pair: 99-66 (24)
Bottom pair: 5xs, A5o (27)
Bottom pair + FD: 5xss (9)
OESD+SFD: KJss, J9ss (2)
OESD + overcard: KJs, KJo (15)
OESD: J9s, J9o (15)
FD+gs: J8ss, 98ss, K9ss (3)
NutFD+overcard: Axss, Kxss (16)
Naked FD: 43ss, 64ss, 74ss, 76ss, 86ss, 87ss, 96ss, 97ss, J4ss, J6ss, J7ss (11)

Total of 255 combo's.

If I valueraise 17 combo's (TT, 55, KJss, J9ss, AJss, TQs, QTo), I can raise about twice this amount of bluffs on the flop. Some hands I picked from the remainder of my range, to bluf with: As9x, As8x, Ks9x, J8dd, J8hh, J8cc, (8cc, 97cc, 87cc, 76cc, J7cc, A9cc (20 combo's).

If I defend all hands above, I still only defend 52% instead of the required 59%.

If I defend all these hand on the flop I have to call a lot of weak pairs (5x with no backdoors) as well. Which other hands should I raise on the flop as a bluff? Or should I widen my value raising range?

Any feedback on calc+thought process is much appreciated.

First off, kudos for putting in some work and trying to figure things out. It should help your game a lot in the long run, as you'll remember concepts and ranges much better if you do the work yourself rather than just read what someone else has done.

A few things:

#1) Ask yourself if you have a problem with your opponent being able to profitably bet any two cards here. You have position, but his range is almost certainly (much) stronger than yours here. I'm not sure what % of all possible hands you called with pre-flop, but it should be a very wide range and shouldn't likely include any AQ+/TT+ hands (3-betting AJ and KQ is probably fine too, but meh).

#2) You aren't putting enough emphasis on keeping dominated hands in the opponent's range. Why raise As9x on the flop when you will make him fold worse spades (in case the flush runner runners)? If you raise A9, he'll likely fold worse AX and 9X hands that you'd have outkicked if the turn comes an A or 9. I would for sure call this hand and As8x as well as a few others (stick with getting the easier hands down first though before worrying about exactly where to draw the line).

#3) Since your range is weak, any pair with 5 outs on this flop is going to be a very easy call (or raise). So even the weakest 5x hands are very easy calls.

#4) There are a lot of hands missing from your range that I would either raise or fold. Most hands with two clubs (Jc4c for example) should either be called or raised. What about your AXcc? How about your As7x? Where are these hands?

I think your value raising range looks fine and wouldn't change much. I think you have the right idea here, you're just in general not defending aggressively enough and raising some hands that can be called and folding some hands that can be called or raised.

Hope that helps.

Hello Matt,

Thanks for your reply, greatly appreciated. Coming from a more or less fit or folding strategy on the (even more micro) stakes that I used to play, calling a hand like As9x in this spot is absolutely new to me, since this would be the most easy fold ever for me before reading the book. I think my fold to cbet was always around 50-60% in spots like this and I see a lot of guys on my stakes with defending percentages that are way to low (like mine...). Now it's up to me to construct solid ranges to abuse those mistakes.

#1) I called preflop with the range you describe on page 83 (BB Flat vs SB). So, would flat TT and KQ in this case.

#3) You are saying you would bluffraise with some weakish 5x hands here too? That is to make him fold his equity and protect yourself from having to call multiple streets vs his aggression with a weak hand?

#4) Exactly, it seemed unreasonable to defend with a hand like J4cc, but as you state: this is a mandatory defend vs. his range. I have to go even wider than I already think I am...

Thanks a lot, once again and I will definately post more hands / questions if I get too them. Lots more to learn!

Hey Matt, this may be a ******ed question but when we bet on the flop (or on the turn) a pot size bet and we are all in, what ratio of value to bluffs should we have? We are keeping the 66% value to 33% as if we were on the river? Please consider nuts or air as in pure theory.

Well if you are all-in for a pot size bet and your opponent is getting 2:1 odds, how often does he need to win for calling to be profitable?

You need to bluff enough so that he wins exactly as often as he needs to in order to break even. So if he risks 1 to win 2, he needs need to win 33.3% of the time.

First of all your book is really awesome. That being said i'm a bit confused about the hand charts you give page 84. Let´s take the vs CO ip 3b range which is JJ+/AK, ATo, KJo, QJo, A7s-A2s. I guess you plan on felting JJ+/AK in this spot which makes 40 value combos and 72 bluff combos, giving CO 64% FE therefore letting him 4b ATC and showing an immediat profit.

I first thought that some of the 3b bluff combos need to be 5b bluff (Arag suites hands) but if you make the assumption that CO will felt AK/QQ+ and that in the GTO frame the worst value hand he felts must be 0EV versus our range, this gives no room to put Arag suited hands in our range.

ps: excuse my french, i'm a true one

You would need to 5-bet bluff some AXs.

Why can't he have a BE call with some hands if you 5-bet AK/AA-JJ/AXs? Also if the opponent only 4-bets AK/QQ+ then it's not really a problem if you're not 5-betting AXs (you're exploiting him by not doing it).

FWIW, I would call 4-bets IP and OOP a lot more than is suggested in the book now. So if CO opens and we're in the BTN I'd pretty regularly 3-bet AQ/JJ/TT (though not always) and just flat the 4-bet, which lets you 5-bet an even more polarized range which I like.