Here are my notes on this episode. Hope it helps
Notes on Mathematics of NL Holdâ€™em episode7 by WiltonTilt
This video is about Value Betting, Bluffing, Bluff Sizing, C Betting
Note to self, Watch the UNCONVENTIONAL WISDOM series. (WoT mentions it is awesome).
Changing bet sizes based on different factors is an exploitable strategy but it is also the highest EV strategy provided you are only doing it against people who will not take advantage of our bet sizing.
Most of our profit is going to come from playing exploitabley provided we are not being taken advantage of.
When making a value bet our goal is to maximize our Expectation from the bet.
We are against a player who is a 50/15 with 33% WTSD.
We are on the river with AsAd with a board of 8dJdJs9sAc. We are UTG+2 he is in the cutoff. thereâ€™s $400 in the pot
Our assumptions about this type of player are
1. Might fold a big pair *QQ/KK) to an overbet, but will call pot or less
2. Will call with a medium pair for half pot bet or less, but never calling with an underpair.
3. He is never folding trips or better. If he has a full house/4jacks heâ€™s going to raise all in on the river.
So we then give him a range. We come up with
77+, J8s+, suited semi connected hands with diamonds/pair/OESD
So given the range and thereâ€™s $400 in the pot, take a guess at what you might typically bet here to maximize against this range given our assumptions, then weâ€™ll do some math to find the best bet size.
So I think I would bet all in here based on his range.
Now to determine the correct bet size, we need to think about how many combos of hands there are that call us.
Letâ€™s first organize the hands in his range into 4 buckets that will likely react the same to a given bet size: medium pairs, Big pairs, Monsters (Trips of Better)
Coming up with the ranges here is kinda difficult because in the video we are not told how the betting went for each street.
Medium Pair combos = TT(6), 9Ts(3), 78s(3), 89s(2), Q8s(3), Qd9d(1),Kd9d(1), A8s(1),T8s(3)
TOTAL = 23
Big Pairs: KK(6), QQ(6)
TOTAL = 12
Monsters: 8Js(2),J9s(2),J10s(2),JQ(8), JK(8),AJ(2), 88(3),99(3),JJ(1),10dQd(1).
Because he may react differently with some of the hands in the Monsters category, we can further divide the Monsters category into 4-sub-catagories of
Boats(12),, Straights(1), Quads(1), and Trips(24) for a total of
TOTAL = 38
The entire calling range is 73 combos
Once we have the number of combos, letâ€™s determine the % of the range each category makes. Divide the number of combos of each category by the total number of combos of 73.
Remember this is his calling range
Medium Pairs: 23/73 = 32%
Big Pairs: 12/73 = 16%
Monsters: 38/73 = 52%
Further dividing Monsters into
Boats:12/73 = 16% Straight: 1.73 = 1.5% Quads: 1/73 = 1.5% Trips: 24/73 = 33%
Bet size EVâ€™s
In general the EV calc will go like this:
EV(bet) = bet * (calling%) + stack * (raising%) â€“ stack * (losing%) + pot
Where the calling %, raising%, losing% are the frequency he has those hands on our cheat sheet of (Medium pairs = 32% Big Pairs = 16% Monsters = 52%
Monsters broken down to Trips = 33%Boat 16% Straight = 1.5% Quads = 1.5%).
Ev(halfpotbet) = bet* (Medium pairs + big pairs + trips + straight) + stack*( boats) â€“ stack * (quads) + pot.
Ev(halfpotbet) = 200 *(.32 +.16 +.345) + (800*.16) â€“ (800*.015) + 400
EV(halfpotbet) = 165 + 128 â€“ 12 + 400
EV(halfpotbet) = +$681.00
Now lets do the same equation if we bet full pot. Our equation changes a bit because now we are not getting called by medium pairs. So new equation is
EV(fullpotbet) = bet*(bigpairs + trips + straight) + stack*(boats) â€“ Stack*(quads) + pot
EV(fullpotbet) = 400*(.16+.345+.015) + 800*(.16) +400
EV(fullpotbet) = 208 +128+400
Ev(fullpotbet) = $736
So a full pot bet on the river is more profitable then the $681 we would have made if we just bet pot.
So now lets see what our EV is if we jam. The equation changes because he will not call with big pairs anymore if we jam.
EV(Jam) = bet*(trips + straight + boats) â€“ (stack * quads) + pot
Ev(Jam) = 800(.33+.015+.16) â€“ (800*.015) + 400
EV(Jam) = 404-12+ 400
EV(Jam) = $792.00
So we see that the $792 EV of a jam is higher then the $681 of a half pot bet and the $736 of a pot bet.
Villain on the button raises to 35, sb folds and our hero defends his big blind with Ts6s.
We are the BB and PFR comes from the button.
Flop(80) comes Td8h7c
Hero leads out for 50 and our villain calls
Hero bets 133 and villain this considers this for a little bit and calls
Hero thinks for a second and bets 100, now villain min raises to 200
Do we call or shuv here?
Reads: We have the best hand 90% of the time
We expect to have the best hand 30% of the time when our shuv gets called.
So now we need to figure if we are trying to value bet or bluff? Since villain is probably not folding anything he min 3 bets here we have 0 fold equity so the play is definitely with the intention of value betting, not bluffing.
So now we need to find our EV of a call
EV(Call) = .90(466+100+200) - .10(100)
Ev(Call) = .90*766 â€“ 10
EV(Call) = $679.40
Now we need to see if our EV of a shuv will get us more EV
Our hero estimates the villain calls with a worse hand around 30% of the time. Is this often enough for us to shove?
Since we donâ€™t expect him to call with any worse hand then us, then our EV canâ€™t go up, it can only go down because he will only call with a better hand then us.
Ev(shove) = EV(call) +EV (raise)
Weâ€™re rasing 582 on top so the EV(raise) = 582x-582*(1-x) where x is how often he calls with a worse hand
EV(shove) = EV(call) +582x â€“ 582 *(1-x)
We estimated that villain calls with a worse hand 30% of the time so we then plug this in for x and get
EV(shove) = 679.40 + 582(.30) â€“ 582 * (1-.3)
EV(shove) = $446.60
So Ev(shove) = $446.60 which is less then our EV(call) of $679.40.
So even though we think we have the best hand 90% of the time how can a shuv be worse? Because the thing that matters is not how often he has the best hand but how often he calls us with a worse hand. So this begs the question, how often must he call with a worse hand for a shove to be better then a call?
So now we need to solve for x where EV(shove) = EV(call)
EV(shove) = EV(call) +582x â€“ 582*(1-x)
Then we set it to 0 so we can get x
0 = 582x â€“ 582(1-x)
0 = 582x â€“ 582 + 582x
X = 50%
So he needs to be calling here with a worse hand 50% of the time in order for EV of a shove to be more then EV of a call. So in conclusion, in most situations on the river, where our fold equity is 0, a bet or a raise is only profitably if you expect to win at least 50% of the time.
In a tournament situation on the river with a dry side pot, 3 way pot with one player all in. We want to value bet, how often must we be called by worse?
EV(bet) = 1x-1*(1-x)
Ev(bet) = 1x -1+x
Ev(bet) = 2x â€“ 1
X = .50 So we need to be called with a worse hand more than 50% of the time for us to value bet the river and it be +EV.
So here is another example where we want to know what the best bet on the river is
We are playing against a player who is hyper aggressive and tough and can read hands and is a good strong player. Remember against a donkey who is never folding anything our huge overbet was the most +EV. Now we are not playing against a donkey, we are playing against an aggressive reg. We are on the river after leading all 3 streets into him and we backdoored the nut flush. We have Ah4h and the board is 5h8cJsKh9h. Pot is 500 and we have another 1000 behind and so does he.
We estimate that 70% of his hands are folding on the river because they were draws or weak pairs
30% of hands which are willing to call a reasonably sized bets (say pot or less)
Top 5% of hands will call an overbet (150% of pot)
Top 5% will raise a reasonable bet, but no hands will raise an overbet.
So how do we best attack him?
If we make a standard pot value bet of 500 we think heâ€™ll continue with all 30% of his decent/strong hands and top 5% will raise.
Ev(500 bet) = 500 *.25 + 1000 * .05
= 125 + 50
If we overbet, heâ€™ll call with his top 5% of hands
EV(750 bet) = 750* .05
So overbetting does not work good against this type of good aggressive player.
So now lets see if we bet $200
EV(200 bet) = 200*.25 +1000*.05
= 50 + 50
= $100 so if he raises the top 5% of hands and just calls the next lower 25% of his range then this is not as good as betting pot.
BUT, he may see this last 200 bet as a blocking bet because we have a marginal hand. When he seeâ€™s this weak river bet he could read this as a blocking bet and turn one of his hands into a bluffing hand. He could have 910 that he doesnâ€™t want to turn into a hero call, things like that. So because he has 70% of hands that wont call and we look weak it is not unreasonable that he will try and bluff us.
So we now modify the EV calc to account for this
EV(200) = 100+1000x 100 = the ev of a 200 bet we figured out before
Where x is the % of time he decides to bluff raise us
So now we need to find x so we set up an inequality to see how often he would have to bluff us to make our EV of betting 200 the same as our EV of betting 500.
So he has to bluff shuv on us(x) 7.5% of the time for this play to get us the same EV as betting 500. We would expect a good aggressive regular who can read hands and sense weakness to make this play much more then the 7.5% so the bet of 200 most likely gets us the most EV.
So against the donkey we overbet because they call with much worse hands. Against an aggro regular we know he has many hands that arenâ€™t calling on the river anyway and we bet that gives the villain an opportunity to make a huge mistake after we have made our hand. We are taking advantage that there are many hands we could have that we would fold if he shuved here.
OK so here are some things to think about:
Do you know how often a pot bet bluff needs to work in order to be profitable?
Well, lets set up an EV calc to find out assuming the pot is 100
EV(pot bluff) = 100x â€“ 100(1-x)
X = how often they are folding
Then set it to 0
X = 50% of the time. So our bluff needs to work 50% of the time if we are betting pot in the river for us to be breakeven on this play.
What about a half pot bet?
EV(halfpotbluff) = 100x - 50(1-x)
0 = 100x - 50(1-x)
0 = 100x -50 +50x
50 = 150x
X = 33.33% of the time he needs to fold when we bet half pot in order to break even.
Here is a hard example:
Final board reads: ($100) 9d6d4h2h2c. and we have Td7d we missed and we decide to bluff. How often does he have to fold for this to be EV of 0
We estimate that he can call a pot bet 20% of the time
1/3 pot bet or less he folds an extra 10% of the time and the other 70% of the time he has a busted draw/overcards.
Is this a good spot to bluff?
Since he will mostly have weak hands and busted draws it looks like a pretty good spot to bluff, but how big should we bluff
So final board is ($100) 9s6s4h2h2c
He calls a reasonable bet (pot or less) 20% of the time
He calls extra small bet (1/3 pot or less) = 10%
He folds (busted draw,etc) = 70% of the time
So if we bet pot, that is if we bet 100 to gain 100 our setup is
EV(pot bluff) = 100*.80 â€“ 100 *.20
Ev ( Pot bluff) = $60
So 60$ is good, but what if we bet 40$ instead of $100
EV(bet40bluff) = .8(100) â€“ .2(40)
EV(bet40bluff) = 72$
Since he has so many missed straights and flushes that he has to fold we can risk less and win more by betting 40% of pot.
What if we bet extra small say 25$
EV(25bluff) 100*.70 â€“ 25(.30)
EV(25bluff) = $62.75
So this is still more EV then betting pot, but it is not as high EV because he is going to call with an extra 10% of his hands.
So say we want to figure out the exact optimal bet size could we do this? Yes, even if we have to just start plugging in bets and guess and check.
So lets do it, we know it is around $40 so lets try $45
EV(45bluff) = 100(.80) â€“ 45(.30)
EV(45bluff) = $66.5 so this is not it. Lets go the other way and remember that his threshold of adding 10% to his calling range is at 1/3 pot. Which is $33.33
So lets say we bet $33.33
EV(33.33bluff) = 100(.80) â€“ 33.33(.20)
Ev(33.33bluff) = $73.33
So we found our optimal bet is $33.33 and it give us $73.33 in EV
So now lets switch over to the topic of C Betting
We should be thinking about how the board texture effects whether or not we C bet and how much we C bet vs different players and different board textures
The 3 bluffing axis that are related are
1. bet size
2. hand range related to board texture
3. player type
and we should always be thinking about these 3 bluffing axis when considering how much to bet and whether or not we should bluff at all.
So lets take a classic dry board texture of Ac7h2d and we are thinking about C betting. We should realize right away that this is one of the best board textures to bluff or to c bet because if they donâ€™t have an Ace it is going to be hard for them to continue and there are virtually no draws that they can be considering calling. The thing we need to realize about this board texture and how it relates to hand ranges is that against unaware opponents we can c bet a significantly smaller amount with our air hands and give ourselves a better price on our bluff like we did in the previous example. Giving ourselves a better price on a bluff raises our EV. Then, when we have AK we can bet much closer to pot to get maximum value out of our hand. We of course canâ€™t do this against any of the regulars but if we are up against a 45/10 we can. We will be getting action from another player on this flop mostly when he has an Ace also so we should be c betting max we can because these players arenâ€™t folding anyway.
What about a board of 6d7s8s and we have AK. Many times we should not be C betting this board because even if we bet pot, there are a lot of hands that will be calling here and then it puts us on the turn where we still may be good, but we donâ€™t know where we are at so we are forced to check/fold.