Schulti

Cool vid! (personally would enjoy faster pace to get more hands in)

Hand 1: I think supesmall bets on the flop/turn make a lot of sense with range to get some protection. As played on the river, isnt 5900 kind of big? The allin guy will win most of the time vs your bluffs so you have very little incentive to bluf unless you make it very small so versus big bets UTG6 can just fold v often. I guess its hard to debate sizes withouth doing a quantitative analysis but on feel id go 2k or smth . Do you agree that your river sizing should go down the higher % of time UTG7 has the best hand?

More of a conceptual question: Do you have some trick to make sure you keep playing reasonable bluf/value frequencies esp with the mixed hands if you use a lot of different lines and sizings etc? Like in reality everything is mixed but in practice its easy to default to play the most natural line for a particual hand closer to 100% of the time and closer to 0% with the unnaturel hands, in which case if you use lots of different lines/sizings a skilled defender who understands the way you think can resort to trying to soulread/ count combos in every spot and crush. (This goes esp online wehre executing a complex strat is just totally impossible for me)

"This isn't true because we chop with his 6x hands very often when we're value raising"

Youre right. Instead the ratio {He calls with hands <6}/{He jams nuts or blufs}  should be >50% ?

I did some quick calculations on the J7 turn call and im pretty sure its a fold if IP plays well but might be close vs someone who barrels more aggro on turn but vbets correct on river. 

Also looked at near-Gto strats on the river for some several turn barreling strats for IP and the computer agrees(more or less) with the strat that forhayley outlined, 

If i give Ben a range  like this :{74s,75,76,J7-A7,84s-K8s,86o-A8o,44@50,(77,88)@30 (96,t6)@50,(J9cc,JTcc,A5cc-A6cc)@80}

and knigtdabest {84-A8,99-AA,44,77,88,96,T6,(j9,JT)@50,(63CC-K9cc)@80

The computer comes up with: 

 http://gtorangebuilder.com/#share_scenarioHash=f6ac667d5706e0b58233a5d97a6df50

Hey, tnx for your answer. Ok, So lets say villains call ranges on the flop and river arent he same in terms of absolute hand strenght but they are  static in terms of relative hand strenght, so if we rank his range from [0-1] the weakest number he calls with is the same. But the overall succes of the check-check-bluf river line depends on us getting to the river. If OOP leads the turn or river with any non zero frequency the line will have lower ev than the bet flop line. 

-The practical approach makes sense ye, just trying to get a better understanding of the conceptual idea 

@J7 20min You make it 690 into 500, so your bluf should work ~57% of the time, and he should defend around 43% of the time.  Around 30% of his original range should have been blufs, say he folds all those (I think he should balances 69 jams  with blufjams with a 6 instead of pure air?) , he should fold  around 27% of his valuerange.  Assuming he bets the turn with 69 always, in order for him to fold 27% and still call >50% <6 hands (As to make a raise with the 6 for you +ev) his original bet should be very light for value (Include one pair hands for value), or he should bet the turn with very little 6x . I think thats very optimistic. Given theres relatively so much combos of 69 in his valuerange it would be better to go for a minraise on the river against players that valuebet somewhat light, and to not even raise with a bare 6 against a lot of players that would valuebet somewhat tight in this spot. 

@AKQ 12min 

Im having a little trouble understanding the situation. Say its correct to check back with some one-street-of-value hands here on the flop and turn and than bet them on the river. Now you need to play some blufs too for this line too otherwise you would never get any value and it might become more profitable to checkback all air on earlier streets than to bet it. So we checkback some air and we cbet some air. But in a a vaccuum our bluf on the flop will have a higher succes-rate than our bluf on the river because OOP can draw a strong holding with some of the hands that he would fold the flop with. So this implies that in order for villain to make us indifferent between a flop bluf and a check-check-river bluf he should call tighter on the river than on the flop to balance for the EV we lose by giving him free draws? So all our river-blufs should have some positive EV. But this means that given that we get to the river we would always bluf our air,  but because we also need to make villain indifferent on the river to call or fold aswell, this implies we can only checkback flop with just exactly enough blufs to balance for our one street vbets. So far so good, but what would happen if we have a unlimited supply of zero-equity air on the flop, such that we do have to check back a lot of it? 

@J4cc hand 

You mention that its std to cc rang on AQ4cc. My question is, can it be correct for OOP in GTO play to not have a checkraiserange on this flop, but too have a checkraiserange on other flops that are very similair in terms of drawiness but where OOP is uncapped, say on K82cc? In other words, if its GTO to checkraise flops in general but not when one player is capped, doesnt this imply that in the second case IP should simply use multiple cbetsizings and weaken the range for the smaller sizing to the point where OOP should have checkraises against the smaller size again?  

Also given the Meta you describe it seems silly not to just checkjam the J4cc and balance with 4c blufs. Esp because u are sauce :D, people are gonna assume your checkjams are balanced anyway and in that case not jamming cost u a ton.  Also because the most obvious flushdraws to checkback for IP would be 4xcc so when you have the 4c villain its less likely to have a lower flush (The only benefit of raising smaller than allin with nuts is to force villain to raise for value with worse valuehands/corresponding blufs), therefore checkpot makes more sense with KXcc than 4xcc.   

Eitherway imo villain should def check back some flushdraws on the flop because  if we dont OOP is allin on club runouts when he has 2p+ and blufs and that should be valuable enough to click check when we hold like K4cc.  

Checkback with AQ seems bad yea, its worthless as a slowplay on clubs/K/J runouts, and on blanks it only works if OOP is gonna put a ton of money in with one pairs Aces, but if he does we could simply check A7-AT some times instead and be better off.  

So the boardpair gives us near nutted hands sometimes that our opponent doenst have often, and also removes some of the blufcatcher hands out of our range, so it polarizes our range more. Now if we would check our whole range to him IP would check behind most of the time as he now doesnt have much hands that can valuebet all three streets. 

So one idea is to bet big with our trips for value and with some of the weakest hand in our range, and checkcall the rest. This is good for our trips as we get more value this way. Its worse for our blufcatchers as it forces IP to start valuebetting lighter on the turn. So we bet some trips and check some and somewhere in the middle we get some sort of equilibrium. I think this is a good strategy on boards like j82 when the 8 hits on the turn. IP would certainly check behind a lot of 8s so when we have the 8 we have the best hand very often so we need to bet to get value. we can also have some weak hands and draws that we can use to bluf. 

The k73 example from the video is a little bit different. Here IP wil probably cbet the flop more often with some 3x, and we flat less often preflop with 3s. So overbetting a wide range makes less sense in this situation. 

Kevins idea to bet 1/4 with whole range is interesting. It seems like the main purpose of it is to put hands with two overcards to the 7 in a tough spot. A lot of those hands won't  be able to bluf the turn so folding them out seems like a good result for our own 7x.  It also makes sense (I think?) that betting small  with your whole range when you have a small range advantage could be better than checking the whole range simply because we force some money in the pot with a equity advantage.  

Would be nice to see some quantitative arguments why/if betting 1/4 with range is better than mixing it up between checks and bigger bets. 

Checkraising also is a possibility ofcourse. Having a balanced lead, checkcall, and checkraiserange seems like a hard thing to do tho. I think checkraising makes most sense on a board like j74ss with the offsuit 4 hitting on the turn. On boards like that the ranges are a little bit more linear and there is more incentive for IP to bet Jx for protection. His blufs have more equity aswell so theres more incentive for us to raise to punish draws. Its also harder to lead as the IP will have a strategic advantage on a lot of rivers after he calls because our trips wont be as good anymore when draws kick in and IP can have a lot of them. 

^ a goran  a 1/4 donkbet is very close to a check so IP should play almost the same as he would play vs a check. He will fold some of his weakest hand, raise with whatever range he thinks is good enough for it and balance with blufs. Call with medium strenght hands as well as with some slowplays. Facing bigger bets OOP range is gonna be more polarized so we raise less often. We have a lot of blufcatchers and not that many nutted hands so it seems like there is a lot of value in slowplaying. The bigger villain (should) bet with his trips, the less of a raising range we need. We should have some small raising range in theory i think when villain best size is anything under geometric growth of the pot as that would assume that we have the nuts often enough that OOP cant bet superbig with trips. In that case if we would only play flat or fold OOP should bet even smaller on the river giving us a reason to raise the turn. 

I think he says it loses at least 3347 milli bb/hand to the best response strategy, so thats ~300 bb/100. Thats a whole lot but its versus the max exploit, not versus the GTO. Impossible to tell how much it loses versus the GTO strat but it must be way less than this number. It wouldnt suprise me if most humans would lose even more if they get maximally exploited. Max exploit strats do really exploit you hard, especially if your strategy is fixed. They will bluff nothing or everything everytime your defending ranges are slightly off, use differrrent betsizings for valuehands/bluffs etc 

Starting with a scenario where both players are playing a balanced game where they maximize the expectation of each individual hand against the counter strategy 

Your question is, can one of the players improve the expectation of his total strategy by taking a lower EV  line with one of his hands?

The idea is that player 2 would be forced to adjust to the new strategy of player 1, and that the resulting outcome would be more favorable to player 1 than the original game

However, the reason that this can't be true is because player 2 is not forced to make adjustments! If he keeps playing the same strategy as before your total expectation has decreased. (Because you gave in some EV with one hand and the EV of all the other hands are the same). The only reason player 2 would adjust his strategy, is if it increases  his overall  expectation. In that case your overall expectation must be lower than it was before. 

So, no loss leaders in GTO.

However, from an exploitative view your point sometimes makes sense. In practize it might not always be best to aim for the highest EV line with each hand, because giving up some EV with some hands can get your oponnent to make wrong adjustments (Or keeps him from fixing bad strategys)

To give one example, consider an opponent who is folding way too often to 3bets. The max exploit line might be to 3bet any 2. However it quickly becomes obvious that you are exploiting him and he will fix his mistakes. If you 3bet somewhat higher than the theoretical optimum but dont overdo it, thus giving up some direct EV with the hands you do not 3bet, he might not realize that you are exploiting him and therefore he keeps playing bad, thus making your longterm EV of your strategy higher 

Agreed, the ultimate goal is to solve this problem for N=3 as that would give us a model for completely static postflop situations with equal linear ranges which would be very insightfull. But it makes more sense to start with N=1.

So I'm wondering if anyone else has thought about this problem. I don't think its possible to solve it analytically (In MoP (Page 203-215) they solve some easy simplifications and its already hard to follow).

Numerically, would be it be possible to solve this for say increments of 1 and stacks of 100? Seems like an infinite amount of possibilities already but maybe with some smart ideas you can eliminate a lot of possible strategies

Also, intuitively, how do you think the solution would look like?

For the N=1 case, the most interesting question for me is how many different betsizings are used for both players.

For N=2 and N=3 cases, one thing that i'm wondering about is whether you just play aggressively on earlier streets with the strongest portions of your range in a continuous way, or if there will be a lot of mixing where for every possible passive action you have to include some  subrange of very strong numbers to keep ranges on future streets strong enough.

Hi Sauce, why do you think that preflop all hands gets played for each different openingssize at least some % of the time? This can only be the case if we are indifferent between every size with all our hands in the equilibrium. Do you think the same goes for bettingranges on the flop and turn? If so, would that imply that it doesnt really matter what we do against a gto-bot before the river? As long as we know a certain line is taken with a nonzero frequency, we could take that line with all our hands. This seems to make more sense for very deep stacks because the GTO-bot will overbet sometimes on runouts where we should have very few combinations of the nuts, but for shallower play this seems to be not that much of an issue. Do you think there is much practical relevance to playing a fully mixed strategy like this? To me it seems a strategy like this dooes a very poor job of exploiting deviations from optimal play from your opponent. We could also aim for a different kind of balanced strategy that doesnt rely so much on picking off overbets. It will loose some ev against GTO-bot, but exploit devations from  our opponent much harder. 

The allin ev is a better measure of how well you are playing than your actual results. Both are unbiased estimators of your overall ev. (This means that on average, if someone wins 100$,  the amount he was expected to win is 100$ aswell., Same for the allin-ev line , if someone Allinev line is at 100$, the best guess for his true expected winnings is 100$.  On the long run both lines will be the same.   However the allin-ev line has less variance,because it removes some portion of the luck element, namely the allin luck.  Therefore it is a better measure off how wel you play than your actual results . Exactly how much of the variance it captures depends on the game you play and your playstyle. If your edge comes from getting it in on the flop everytime with a small equity advantage  your allin-ev-line will have much lower variance than if you play a strategy that is based on calling down untill the river. In your example, if you can choose between getting it in on the turn for 100bb with 80% equity , or calling the turn and  getting it in on the river with 100% equity 80% of the time  and with 0% equity 20% of the time,both  the Allinev-line and the results line will go up with 60b on average. However, in the first case your Allinev-line will go up with 60bb everytime,while your results line will go up with 100bb if you win  or 100bb down if you lose. In the second case your allin-ev line will just be the same as your results line. 

Its always better to look at your ev-line than at your results if you want to get an idea of how well you are playing. However,because it still has quite some variance,so dont  bother drawing too much conclusions from it on a daily basis. On the long run it does giv some indication of how well you are playing and how lucky you have been in all-in situations. 

 Trying to create  some kind of true EV-line with even lower variance by capturing other parts of the luck element involved in poker would be a bad idea, because it would be prone to biases, and  would give less indication of how well you are playing in a given period of time. For example. i could look up in my database how much money i make on average whenever im dealt 67 on the button. Then i could let the ev-line go up by that amount everytime im dealt that hand on the button, that way i would eliminate the variance of good and bad board runouts for my hand postflop. I could even take it further and  just look up how much money per hand i make on average on the button and let the EV-line go up by that amount. That way i would also  eliminate the variance of getting dealt good/bad hands preflop. But now EV-line would just be linear, every hand it goes up by the amount of BB/hand i make. It would be completeley useless because it does not tell me anything  about if im playing well or not, furthermore the BB/hand i make over time is not static, it might chance over time depending on how I play and how my opponents play, therefore this kind of EV-line would not be unbiased anymore  

Given your assumptions I don't think the river spot is that ugly, I think you can easily fold. However, once Villain realizes that you only have 45 when you checkraise here it becomes easy for him to float with any reasonable  flushdraw, he has equity to improve on rivers that you are not gonna fold that often, and if the board pairs he winst the pot. If he's not doing that but he's just folding a lot to the checkraise the  problem with checkraising is more with not getting value if the board bricks. So if you want to checkraise the turn here I think you want to checkraise with some hands like 95+fd that have blockers or a fh when the board pairs, can have the flush sometimes when its hits and can profitably bluf when the board bricks. I feel like a strategy of checkraising turn should do better than just cc-ing turn because although your turn range might be weak , villain can't bet more than the pot on the river so you don't get that much value by slowplaying the top of your range. Furthermore, I don't think he will fold the turn often with hands worse than your  that can vbet the river if you cc? Im new to plo btw so I hope my thoughts make some sense) 

Hey, im new to this site so sorry for still responding this old video but im watchinig this series and  have a couple of questions:

-5m you defend 58o against a minraise and lead the turn on aq5ss7o after Kanu check backs the flop. Given that you have a lot of draws that you want to bluf with here and only so much hands that are good enough to valuebet, isnt 58o too mediocre to put in your betrange? Eventhough Kanu checked back the flop his overall range is still stronger than yours, (i guess you have more tophands tho so idk maybe it does make sense to ues that as a leverage to bet bottum pair ) Do you think its easier to play your hand on the river facing a bet after you check when you led the turn as opposed to facing a river bet if u would checkcall the turn?  

-12m K7TTQ 1. You are suprised Kanu did not bet the river with the Q8. You said you thought cbetting with complete air without barreling equity was a mistake so i assume you also check back 7x and Ahi a lot here? If so, eventhough he has a lot of straightdraws that missed, there dont seem to be that many hands Kanu can get value from by leading the river with qx?. 2. Kanu has a lot of missed draws in his turn range that he could bluf the river with, and its hard for you to get a lot of value with your Tx on the turn and river (Because if you dont have the ten he has it a lot so its hard for you to tripple barrel bluf on this texture, therefore easier for him to fold if he has a weak hand and you do in fact bet the turn) Would these 2 arguments be enough reason to check the turn back  with some of your Tx and try to incoporate a balanced raising range on the river ?