GTO simplified (OTR)
Posted by Robert Johnson
Posted by
Robert Johnson
posted in
Gen. Poker
GTO simplified (OTR)
In river spots where the bettor has a polarized range (only nuts and bluffs) and the caller only has a range of bluff catchers (can only beat a bluff), you can use GTO frequencies to maximize your EV, if you can't exploit or don't know Villain's tendencies.
Using these frequencies,
- the polarized range always wins the Pot, EV = 1P
- the bluffcatching range always has an EV = 0
I have this simple table memorized, so I don't need to solve equations in game :
(P = Pot size; A & B : pot odds expressed in rounded % values)
A : Minimum Fold Equity to Bluff (MOP p113 : "alpha") :
- ratio of bluffs to value bets
- minimum fold equity required to make a break-even bluff bet
B : Minimum Equity to Call : pot odds offered; minimum equity required to make a break-even call
That's all you really need to know to instantly derive most of the other important GTO
frequencies :
C = B : Optimal Bluffing Frequency : the required portion of bluffs in the betting range for a given bet size; always equal to the pot odds offered; used to make an opponent indifferent to calling or folding on the river with his bluff catchers; EV(river call/fold) = 0
D = 1 - C : Optimal Value Betting Frequency : the portion of value bets in the betting range for a given bet size; used to make an opponent indifferent to calling or folding on the river with his bluff catchers; EV(river call/fold) = 0
E = 1 - A : (MOP p113 : "1- alpha" )
- Optimal Calling Frequency : used on the river to prevent the bettor to make a profit on his bluffs and thus, to make him indifferent to betting or checking his bluffs; EV(river bet) = 1 P
- Minimum Defense Frequency : used on the flop or turn, to prevent the bettor to make an immediate profit on his bluffs; same as above, except that it is used as a minimum threshold and is not static (more streets to play, other factors relevant)
F = vc / D - vc : Number of Bluff Combos to Bet given a known number of value combos and the optimal value betting frequency
G = bc / C - bc : Number of Value Combos to Bet
given a known number of bluff combos and the optimal bluffing frequency
These shortcuts are much easier for me than solving the equivalent equations.
The 2 sets of frequencies (A and B) are the most important to remember, because they are the GTO target frequencies to make the opponent indifferent :
The polarized range is always betting and sizes his bets to target column B for his optimal bluffing frequency to make the bluff-catching range indifferent to calling or folding.
The bluff-catching range targets column A to determine his optimal calling frequency to make the polarized range indifferent to betting or checking.
Example (from the "Value Betting" video by Felipe Boianovsky, but also elsewhere like in Lefort's series) :
You have 7 bluff combos in your river range and decide to make a 2/3 pot sized bet. So you need to calculate the GTO number of corresponding value combos to bet.
1. deriving the Optimal Bluffing Frequency : C = B : 29%
2. deriving the required number of value combos : G = bc / C - bc : 7 / 29% - 7 = 18
Now you have your GTO river betting range of { 18 value combos : 7 bluffing combos}.
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Everyone should thank me for messing up repeatedly, so we have this info available now.
I do thank you because I learned a lot in the process of trying to give you correct answers and I must have made about 100 corrections to this post before posting it.
I had these things written down for me, but everytime I was ready to post it, I would find a new inacuracy or mistake.
It looks pretty correct now and so much more useful :)
So thanks again and GL to you !
Seems like an imperfect strategy - being used as of now (which doesnt make much sense). Hasnt this strategy been discredited for doing anything useful?
dynosoar,
this is not a suggested strategy but simply a set of shortcuts to use THE strategy, which is known as Game Theory Optimal (GTO) in poker.
It's just math and thus cannot be "discredited".
This is not the place and I'm not qualified to discuss why or when to use it or not, but I suggest you check some theory videos where it is explained in details (Ben Suslky, Sean Lefort, and others).
Thanks appreciate it!
Very nice work!!
Only (minor) thing I´d disagree with though is the following:
"You have 7 bluff combos in your river range and decide to make a 2/3
pot sized bet. So you need to calculate the GTO number of corresponding
value combos to bet. [...]"
Obviously, your math is totally correct and you´re most likely meaning the right thing, still (as I´m a nitpicker :D), I´d suggest NOT to work out your ranges off the bluff combos, but the other way round.
It´s not your bluffing-range which is fixed, but the value-range is fixed (in theory). Your technical valuerange consists of all hands that have > 50% against Villain´s callingrange (there are some factors that influence that but let´s take it for the sake of simplicity). And derived from that you can derive your bluffing-range (by the formula F you´ve given).
If you go the other way round you´ll very likely end up valuetowning yourself. I guess it´s obvious and you probably just made an example (or copied it?) but I think it´s important.
Thanks for your remark BigFiszh.
This was made after notes I made trying to understand the calculations in the excellent videos of
Lefort, Boianovsky (especially his vids) and others.
Sometimes the starting point to find indifference frequencies was the bluffing hands, sometimes the value hands.
Your point is valid and even more interesting, because that was something that I realized was generally not taken into account (IIRC, let me insist) in the reasoning and equations.
So, it looks like a "yeah, ldo", but I need to review the different examples to make a more appropriate comment on your remark (which tells me I don't have a good enough grasp on the whole GTO framework).
For now, I only have a "feeling" that it's not that "relevant" (for lack of a better word) to the examples used, and that it would be more of an essential condition to keep in mind.
I'll be back when I can make a more articulate comment, and hopefully get a better understanding in the process :)
> Your technical valuerange consists of all hands that have > 50% against Villain´s callingrange
that's what bothers me :
this statement implies that we know villain's calling range or at least should estimate it and adjust accordingly; but in the GTO framework, with no exploitable reads / against the nemesis / with our strategy exposed, we shouldn't care about his calling range and have an optimal strategy against his GTO play.
I get from this that we should not care about a particular Villain's calling range (unknown anyway, lol) , but only about an optimal calling range; however, I don't remember having seen this taken into account as a starting parameter for deriving a GTO betting strategy.
I will go over the examples again to see if/where I'm confused.
Then I just need to sort out the remaining 999 thoughts your post induced :-}
Haha, it seems as if I accidentally opened a can of worms. :D But maybe I just expressed myself poorly at the end ... ;)
"I'll be back when I can make a more articulate comment, and hopefully get a better understanding in the process :)"
Please do, sounds like an interesting discussion!
"this statement implies that we know villain's calling range or at least
should estimate it and adjust accordingly; but in the GTO framework,
with no exploitable reads / against the nemesis / with our strategy
exposed, we shouldn't care about his calling range and have an optimal
strategy against his GTO play."
That´s completely right! But think about it this way: GTO-ranges don´t drop from the sky ... they´re the result of two strategies exploiting each other until none of both can increase his EV anymore by unilaterally deviating from the chosen strategy.
Now, consider the following scenario:
We´re at the river, pot is X, we have one psb left and our opponent checked to us.
We have a range of {2, 2, 4, 4, 6, 6, 8, 8, 10, 10, 12, 12} (higher numbers being better), our opponent has a range of (7, 7, 9, 9).
Round 1
We know (by GTO), that when we bet pot (for value) that we can add 50% bluffs (in relation to our valuebet). As we don´t know how wide we can bet, we´re starting with the absolute top of our range and add the very worst combo for a perfect 2:1 ratio, so our range looks like {12, 12, 2}.
Our opponent knows (by GTO) that he has to defend 50% of his range to deny us auto-profit, so he calls with a range of {9, 9} and folds the rest {7, 7}.
Our EV when we bet is:
EV(bet) = (1/2 * p) + (1/2 * (2/3 * 2p + 1/3 * -p)) = p
Our EV when we check is:
EV(check) = (2/9 * p) + (2/9 * 0.5p) = 0.33p
Our total EV with the given strategy is:
EV(total) = 1/4*p + 3/4*0.33p = 0.5p
The rest (0.5p) is the EV of our opponent.
Round 2:
We try to increase our EV and start to bet lighter for value, so we add two more valuecombos and another bluff-combo:Bet = {12, 12, 10, 10, 2, 2}
Our opponent can´t do anything, so he´ll keep his optimal strategy.
EV(total) = (1/2*p) + (1/3*0.5*p) = 0.67p
Round 3:
We increased our EV in round 2, but we strive to MAXIMIZE it, so we take another step and bet {12, 12, 10, 10, 8, 8} for value and {0, 0, 2, 2} as a bluff, so our total range looks like:Bet = {12, 12, 10, 10, 8, 8, 4, 4, 2, 2}
EV(total) = (4/12 * 1/2 * p) + (4/12 * 1/2 * 2p) + (6/12 * 1/2 * p) + (6/12 * 1/2 * -p) = 0.5
Ooops ... we overdid. :) Obviously we reached the treshold in Round 2, so we get back to that valuerange (and keep it for ever).
Now you can see two things:
1) It´s absolutely essential (!) to know or estimate Villain´s calling-range. And by that I don´t mean that we have to know what % of his range he´s calling / folding, we´re indifferent to that - but we have to know the strength of his bluffcatchers!! Replace the range of Villain in my example by {5, 5, 3, 3} and you´ll see that you can comfortably bet any2!
I hope it got clear now, what I was trying to explain with my first "sloppy" post. :)
Round 2:
EV Bet: (1/2)p + (1/2 * (4/6 2p + 2/6 * -p)) = p
EV Check: (2/6)*0.5 = 0.165
(It makes sense that our weaker checking range would make less than in Round 1)
EV Total: (1/2)p + (1/2)(0.65) = 0.5825
Round 3:
EV Bet: (1/2)p + 1/2(4/102p+6/10-p) = 0.6
EV Check: 0
EV Total: (10/12)0.6 + (2/12)0 = 0.5
1. deriving the Optimal Bluffing Frequency : C = B : 29%
2. deriving the required number of value combos : G = bc / C - bc : 7 / 29% - 7 = 18
Now you have your GTO river betting range of { 18 value combos : 7 bluffing combos}.
I am new to this site. Whole new world for me. Just working through this equation myself with a calculator and I get 17 value combos. 7 divided by 0.29 = 24.1, subtract 7 = 17.1. Did I do that right? or do you round 17.1 up to 18?
the % values have been rounded for ease of memorization :-)
for a 2/3 pot size bet, the exact value for the Minimum Equity to Call (or Optimal Bluffing Frequency) is 28.571429% ; so it's rounded up to 29 % and when you divide 7 by the exact value, you get 24.5, which gets rounded up to 25; the rounded values give you an acceptable result in game, while you should use the exact values when analyzing a spot afterwards.
I suggest you watch Felipe Boianovsky's "Value Betting" video for some nice practical examples.
Hi! I'm going to revive this post xD
If I'm on the river on a river completing flush:
As bettor I can bet pot 67% of my Ass and 33% of my As, and make him indiferent between calling and folding, and
As a caller if my range is Kss and Qss I have to call 0.5% so call all Kss and fold all Qss to make him indiferent between betting and checking.
Say he bets 100 on a 100 pot.( he is beting 100 to win 200) 0.5% (and we ned to call100 for a 300 pot) 0.33%
But what happens if I bet the river with my Kss and Qss and now he raises me the pot! What now?
Say I bet 100 and he raise me 400 (now he is betting 400 for a 600 pot) 0.66% its like a 2P (and we need to call 300 for a 900 pot) still 33%
As bettor what now? We stil bet pot 67% of my Ass and 33% of my As to make him indiferent?
As caller what now? We only call 33% of our range now?
Im totaly lost with this! Please somebody help me! thanks xD
2x YES
Could someone explain this to me.
If we set up our ranges so that villain is indifferent to calling or folding does this not mean that if we have value then villain can just fold because they are indifferent and if we have a bluff villain also folds because they are indifferent...so we never get paid off on our value hands but our bluffs automatically pick up the money in the pot? My head is very confused about this concept. Usually if I have a strong value hand I just bet. If I think the villain is capable of folding using their stats and the board runs out in a way that I think favors my line more than theirs then I bet/shove or w/e as a bluff.
This "indifference to calling or folding" is a bit interesting concept but I don't fully understand it.
His bluff catchers are indifferent to calling becasue when he heros overall he breaks even on his calls. He breaks even on his calls becuase we are bluffing at the optimal frequency. If our bluffing strategy was fixed, then yes he could fold every bluff catcher and our value bets would never get called by his bluff catchers. our ev would be the same as if he called every time he had a bluff catcher.
In reality, though our bluffing strategy isnt fixed. If we observe he isnt hero calling much we would deviate from optimal frequency to bluff more often and if he heros a lot, vice versa.
I hope that provides clarity.
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