Advanced Theory Principles
Posted by Sean Lefort
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As an example we know if you min raise the btn HU, the "minimal defence" threshold as you call it in the BB is 50%. However this is not taking into account the BTN's natural positional advantage postflop, the BTN's equity in the hand, etc...
So in reality he gets a discounted price on the pot odds because the hand doesn't just end PF and hence we need to defend more then 50% to not get exploited. My question is how would you accurately go about estimating/solving this discount that exists due to the multi street nature of the game?
The short answer is, we can't calculate it precisely. But the long answer is that we can come to some solid conclusions with a few approximations. Let's try. :)
First of all, keep in mind that if our defense range happens to be all {3bets} and no {flat-calls}, then villain receives 0% equity on his raise/folds. So we're only looking at the times we flat and he sees some R% of his equity post-flop.
If we're defending say, 70% to a BTN MR with {20% 3bet, 50% flat}, then here's villain's EV Calc for opening 72o:
EV = 0.30*(+1.5) + 0.20*(-1.5) +0.50*[(R*E*4.0)-1.5]
ie. 30% of the time we fold and he wins 1.5bb, 20% of the time we 3bet and he folds and loses 1.5bb, and 50% of the time we see a flop with a pot-size of 4.0bb where he has Equity (E)% that he Realizes (R)% after risking 1.5bb.
E =~ 30% for 72o vs. a flatting range.
R > 100% given that he has position and initiative assuming both players are equally skilled post-flop.
Let's estimate R = 100% for curiosity.
EV = 0.45 - 0.30 - 0.15 = 0bb EV = folding
Thus, we've made him indifferent to opening or folding 72o with these variables.
However, R is likely as great as 120%+ given post-flop position and initiative.
If R = 125%:
EV = 0.45 - 0.30 + 0 = +0.15bb
ie. In this case, villain would be making a mistake by folding 72o pre-flop.
Hope this helps! It should be clear how to manipulate the numbers to find different thresholds that you're looking for. In the case of the 50%+ (1-a)% for defending the BB HU vs. a MR, it's clear that this is just a threshold and we need to be much wider if we want to ensure that 72o isn't played for a profit.
it would be good to have some hands played example to extrapolate the maths to scenarios in real play.
I think it could make ur video way more effective.
Since I'm coming from a place where I don't really get behind game theory, I was surprised that you talked about things in a way that will still allow me to add some new ideas to my game. Whether or not I fully get behind it, I feel like I've learned something new, and will continue to learn by watching your videos.
Very interesting videos you are making!
I play HU cash-game myself. Mainly at NL400-NL600.
I was wondering if you could be interested in making a video where you review my game?
I got some HU cash footage which I think would make for a very interesting video
ie. say you arrive at the river with a range that's: {100 NUTS combos, 100 AIR combos}
If you decide to jam all-in with your whole range on the river for 300 into a pot of 100, you're bluffing frequency is 50%. However, that's not the optimal bluffing frequency and thus it's not the optimal strategy choice. To solve for the optimal bluffing frequency, we look at our opponent's EV with bluff-catchers:
X*(+400) + (1-X)*(-300) = 0
When you're bluffing X% of the time, he wins +400 chips. When you're not bluffing, he loses -300 chips. If we make the value he receives from both calling and folding = 0, we solve for the optimal bluffing frequency. In this case:
400x -300 +300x = 0
700x = 300
x = 0.43
X = 43%
So in this case, you want your jamming range to be {57% NUTS, 43% BLUFFS}. And given your original arrival range of {100 NUTS combos, 100 AIR combos}...
(BLUFF COMBOS) / (NUT + BLUFF COMBOS) = 0.43
B / (100 + B) = 0.43
43 + 0.43b = b
b = 75.4 =~ 75 COMBOS
So in this case with this bet-size, we can jam 75 of our bluff combos but have to chk and give up with 25 of our bluff combos.
This example specifically is a good one for showing the reason why overbetting rivers with a polarized range {NUTS / BLUFFS} against a capped range is extremely effective, in that it means that it minimizes the number of times you have to give up the pot (while playing optimally).
"X = 43%
So in this case, you want your jamming range to be {57% NUTS, 43% BLUFFS}. And given your original arrival range of {100 NUTS combos, 100 AIR combos}...
(BLUFF COMBOS) / (NUT + BLUFF COMBOS) = 0.43
B / (100 + B) = 0.43
43 + 0.43b = b
b = 75.4 =~ 75 COMBOS"
Okay so we've solved for X = 43% = the % of the time we want to be bluffing when we bet the river. Now if we know our range, we can calculate B = the # of bluff combinations that we're allowed to bluff with as it will be directly dependent upon N = the # of nut combinations we have that will also be betting. Thus, our TOTAL number of betting combinations = B + N.
In this case, X = 43% = 0.43
Therefore:
X = 0.43 = B / TOTAL
Where TOTAL = B + N
Thus:
0.43 = B / (B + N)
In our case, N = 100.
0.43 = B / (B+100)
To solve for B, we do some simple algebra. First, multiply the left side by the denominator on the right side.
0.43* (B+100) = B
Now multiply the 0.43 through the bracket on the left.
0.43B + 43 = 1B
Now group the like variable (B).
43 = (1-0.43)B
43 = 0.57B
B = 75.4
Hope that helps.
Also, you can't really "calculate" R. These are all approximations, and the best we can do without some more data/information.
Especially early in a match, you could argue that solid players are more apt to call in bluff-catching situations so that they get to showdown and gain more information on you more often. Ie. anything they deem "close", they may be inclined to call.
So if you're default strategy is to bluff too much in these spots, you're immediately making large mistakes in the present time against a lot of these players. I don't think it's appropriate to assume most players won't be calling enough right off the bat. It could be true, but I'd need some empirical data before I'd be willing to accept it as truth. :)
Looking forward to your future videos!
have a nice day
I have problems with figuring out what should be my ratio for 4betting bluffs and value hands. Can you tell me please what's the right GTO ratio for 4b?
the only point i want to add is a tweak to your math, or the math you presented. it's not so much the combinations which matter bur the ratio of the combinations in our range. this helps greatly with the math as you no longer have to calculate how many bluff combinations you need to be shoving, but instead multiply your "nut combinations" factor.
you can see the ease of this term and use by taking your bluff percentage equation and tweaking it by dividing through by "bluff combos" which then yields X=1/((nut combos/bluff combos)+1). this is where the ratio shows its significance. we can then define "bluff combos" as n*Nut combos, n [0,oo], substitute this back in and the bluff frequency equation cleans up to: X=1/(1+1/n) which makes the math cleaner. From there analysis becomes easier as players can look at their range and realize they have "20" nut combinations and easily derive how many bluff combinations are needed.
Thus after a bit of math one can obtain the relationship of frequencies of GTO combinations of bluffs and Nut hands PER BET SIZE (defined as b1= Bet size/Pot size): as
Bluff combos= (b1/(1+b1))*Nut combinations
thus for each bet size we choose well have a different range composition of "bluffs" and "nuts".
whats more important in than determining our combinations of bluffs is determining our betsize given a range consisting of "bluffs" and "nuts". Using the same n from above (n=bluff/nut) and doing some algebra we can flip the "bluff combos" equation to . read and thus tell us our GTO optimal bet sizing:
b1=n/(1-n) here again by the constrains of our equation we see we need to be value betting at least as often as we have a bluff. doing some more mathemagic and using a side equation of Wins+losses=Range and putting back in, n=bluff/nuts, then dividing thru by range (i can map out the math if requested) we can arrive at a final equation of
b1=(1-A)/(2*A-1), A is defined simply as the percentage of nut hands of our range. the resultant graph looks can be seen here: http://fooplot.com/#W3sidHlwZSI6MCwiZXEiOiIoMS14KS8oMip4LTEpIiwiY29sb3IiOiIjMDAwMDAwIn0seyJ0eXBlIjoxMDAwLCJ3aW5kb3ciOlsiMC41IiwiMSIsIjAiLCI1Il19XQ--
The X axis is the percentage of our range which is the nuts, thus the nuttier our range, the smaller our GTO bets and the air-eyer range the larger our bets need to be
I just watched your Advanced Thoery Principles video and was wondering if you might clear somethig up for me. I don't understand all this fuss about 'minimum defense frequency' and why we should aim to make villain's worst opens neutral EV PREFLOP. If villain's open of 23o on the button is neutral EV preflop, he should still be making a profit on the open given he will have +EV spots to cbet or barrel and can occasionally value bet post flop. Thus villain's worst opens are still +EV and villain can continue to open his entire range profitably. Aiming to defend at the MDF seems rather arbitrary.
Also, it seems possible that defending at or above the MDF may be less profitable than a strategy that plays tighter oop. Couldn't the instant profitability of raising the button with ATC simply come from the structure of the game?
I'm not a HU player but I do have a math background.
However, for PF situations it does still tell us something. It's stating that if the defense a raise faces is a frequency < than the MDF, that raise will auto-profit preflop. So it still holds as a reasonably useful tool to use as a threshold when applicable. It's not used as a guideline we're looking to match (as like you mentioned, villain will still see equity {and profit} when we see flops), but moreso a threshold we want to make sure to surpass when applicable. When applicable would be a situation we *know* that villain shouldn't be profiting with the bottom of his range, ie. somebody opening 75% from the BTN 6max. HU of course is trickier, but I think at the very least we can find some way of proving that there is a better BB defense strategy than letting 72o auto-profit preflop.
I am VERY interested in your proof of - or idea of proving that - letting 72o auto-profit preflop (including postflop-equity) is suboptimal in HU on the button. I also assume this statement is true and am basing my "GTO"-BB-strategy on that assumption, i.e. having a merged / semi-merged 3-betting range between 15-20% and folding about 10-15%, which forces the button to play only 80 to 90% with a minraise-strategy.
im late with this, but really great video
nice vid
Was a discussion of R originally in this video?
Can anyone help me how Sean comes to the follwing numbers (the math behind that)?
½ pot, Z = 25%
2x POT, Z = 40%
Ex. Hero bets river with {NUTS, BLUFFS} and Villain has {BLUFFCATCHERS}
Assume Hero’s river range (before betting) is {50 combos of NUTS, 50 combos of POTENTIAL BLUFFS}
At ½ POT, Hero reaches Z by bluffing 17 combos of his POTENTIAL BLUFFS
At 2x POT, HERO reaches Z by bluffing 33 combos of his POTENTIAL BLUFFS
-Thus, with the 2x POT sizing Hero can now bluff with 66,6% of the hands with which he needs to bluff in order to win the pot, vs. the ½ POT sizing that allows Hero to bluff with only 33.3% if these hands.
Am I understanding this correctly? 17min mark of video
Optimal Calling Frequency(Also MDF)
If Villain bets pot and is bluffing 33% of the time we must defend 50% of the time? I don't understand why we have to defend that amount? Is that because we have some hands that beat Villains value hands? Or are we assuming that Villian has nuts or air?
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