What is theoretical best sizing on river?
Posted by Zachary Freeman
Posted by
Zachary Freeman
posted in
High Stakes
What is theoretical best sizing on river?
Live Deep $5/$5 w/ $10 Rock. Rock is a forced straddle by winner of last pot. Hero has the rock this hand in the CO. $3k effective.
Btn is Limon well known veteran live pro. Btn limps, a couple more limps nad Hero checks option on CO with
9c8c4s5d
Flop Qc 3d 6h. Checks all the way through.
Turn. Jc. Hero picks up a FD and a gutter to go with the flopped OESD.
Hero bets pot $70, Btn raises to 200 total.
I could 3bet turn but not interested in discussing that. Hero calls. I assume LImon's range is some J's up, set of JJ, occasional QJ (although I think he bets flop nearly always with QX), and some draws.
I planned to x/decide flushes and lead a T and bluff an 8 or 9 probably. I'd cr a 2 or 7.
River 2d.
I check and Btn pots it for $470. What is my theoretical best sizing?
Part of me felt that I should be c/r pot because I have a polarized range of only the nuts for value and by going large I can maximize the number of bluffs I have here.
Other part of me felt that I should go smaller because, his range is not all bluff catchers. He can have the nuts here so my bluffs should be on the cheaper side. Also, if I choose to bluff (I'm not sure how often I should be here) some of the hands I'm targeting to fold are his own missed draws given all of his J's up are checking back river and his only worse value range on river is JJJ. Vs a range of missed draws and JJJ Id probably be best off going small because he is never calling with missed draws. But perhaps a pot c/r will fold out his entire range enough?
Perhaps somebody better than I in the sizing theory and provide some input.
Thanks
Zach
Loading 33 Comments...
I don't think putting a big raise in here makes a ton of sense at least from an exploitative pov. Villain is likely to be overbluffing that spot, so putting a raise in that takes the chance of him spazzing away is not good imo. Also - as you said _ villain can have the nuts, which makes bluff potting unattractive and villain should know that.
Ofc it'll be much more interesting to discuss a GTO like solution. Not really qualified to comment on that, maybe somebody else can.
I chose a min raise becasue I felt that given he was uncapped and that I probably am not bluffing that spot often (not sure if that is a mistake) I felt the optimal size would be small.
I also thought that it worked best against his air but I'm more interested in the best GTO based sizing.
Thanks for your input.
If you are only check-raising the nuts and some bluffs, I think your sizing should theoretically be pot. When you take that line, his JJ hands become bluffcatchers. So if his river potting range is very JJ heavy, he will have to be folding some of those in order to not be calling too much (and thus allowing you to exploit him by only check-raising nuts).
The fact that he can have the nuts here doesn't matter too much for our sizing, since we are only check-raising nuts or air. If we were to check-raise non-nut hands for value, the fact that he can have the nuts would have been very relevant, and we would potentially go for a smaller check-raise.
Another reason to go for a smaller checkraise would be that you don't have enough bluffs in your range. When you check-raise for pot you need 33% of your hands to be bluffs, and if you don't have enough bluffs then you would use a smaller sizing to make him indifferent with his bluffcatchers.
Since in this particular spots your are:
1. Raising only nuts for value
2. Have enough bluffs in your range to fill up 33% of your checkraising range
I think that the optimal GTO sizing would be pot.
Would you still raise pot if his range is:
1/6th JJJ
1/2nd 23456
1/3rd air?
GT, not sure about those exact compositions but that was along the lines I was thinking, which made me want to cr small. That said this subject isn't my strongest point thus I asked for help from you guys which you have provided very well.
Thank you GT, Midori, Ph33roX, Midori, all.
Hi Zach,
Interesting hand. I think this spot is complex enough that we need to know his range breakdown, as well as ours, in order to figure out the best bet sizing. Also, this deep we might not want to get it in on turn with Q6 or even QJ, which are good bluff catchers on the river. This means we have a check/calling range as well as check/raising range, making this spot even more complicated.
If you can give me an approximate range breakdown for both players on the river, I will try to solve that spot and compare two bet sizings, namely pot and min raise. You don't have to give me his betting range on the river; rather, give me the range(s) that he and you get to the river with.
midori
Sorry Im late to the party and you have already run the sims. Our exact range breakedown isnt too important imo becasue I dont think it will change the result too much.
My range would be
draws:
any FD+OESD, NFD+any GS, NFD+pr w/ overkickers to Q, any wrap with a club blocker for bluffing, double oesd
made hands, any 2pr+ prob (maybe not bottom 2)
His Range: Made hands:
any turned 2pr with J. Id heavily reduce QJ given he bets flop mostly, JJ.
Draws: any combo draw like J+turned OESD, pair+FD, wrap, double oesd
As for his strategy I just want to know the best GTO strategy because I dont know his exact ranges nor his river strategy. I assume he would b/c JJ for a min raise but not sure how often he folds it to a pot raise.
Zach,
Thanks for your reply. I think we can count combos for each part of his range to get the percentage and look at our GTO strategy. I am rather busy now, but will get back to this by tonight.
Before modeling this and crunching any calculation, two things I would like to point out:
a) This is similar to AKQ poker without card-removal effect. A = 54 (straight), K = JJ for him and some 2p for you, and Q = missed draws. It's not quite the same thing because of the range asymmetry involved (his K beats your K), and the A/K/Q proportion might be different for each player as well. However, it's a similar spot nonetheless, and that's how I am (roughly) gonna model this.
b) This goes without saying, but staying the obvious: we don't want to take the GTO line unless we knew he is playing with GTO strategy himself. If he is not, we are burning money by taking the "GTO line" instead of exploiting his frequency weakness.
Thing is, he has to bet with JJ (again, corresponding to K in the AKQ game) at least with some frequency. This is due to the fact that we have bluffcatchers in our range that lose to JJ. If our range were only 54 and air, it doesn't make a ton of sense for him to be betting JJ here. That's not the case here though, and his JJ can get value from our bluff catchers.
The point I'm getting at is, his strategy should be fairly complex as he has two decisions to make: a) how often to bet/call and bet/fold with JJ, and b) how often to bluff with air. And because of that, I think this is a spot where he is likely to take a suboptimal line unless he has "done the homework." If so, we are not really interested in the GTO play, and it is possible that we don't even need a mixed strategy with our bluffcatchers and/or air.
However, I am assuming that by "theoretical best sizing" you were implying either of the following: a) he is a good player so his range is GTO, and we are seeking for our GTO strategy too, or b) his play is not GTO, so we want the maximally exploitative line, instead of GTO. Please correct me if I am wrong.
At any rate, I can solve the spot with the frequencies you give me and come up with the ~GTO play for both players, and tweak it a bit to see how we can deviate from it to maximally exploit him (if he deviates from it first, that is).
That was kind of rambly of me, as usual, but hope it made sense.
midori
Game Theory, Good question.
If his range composition consists of 50% nuts then obviously we cannot pot with our bluffs as this would be burning money. I think In this case we'll be forced to raise very small, so we allow ourselves a break-even bluff if villain folds JJ. In the case of him potting, we will be forced to minraise our 53 and add bluffs that will consist up to 20% of our range. I think this should create an indifference equation where he is indifferent to calling JJ and we are indifferent to minraise bluffing.
My first comment assumed that after raising the turn and betting the river, villain's value range consists of a lot more JJ than 53, but you brought up a good point nonetheless.
Unless I am missing something, I don't think we can bluff against a range that has 50% nuts in it, can we? Our mr bluff has to work 50% of the time, which only works if he is betting his whole range and folds all JJ to our raise.. He simply doesn't have enough air.
This is very similar to AKQ game with PSB instead of 1/2 PSB, where the bettor should never bluff and caller should never bluffcatch.
Well, I thought Game Theory meant his river betting range, not his river starting distribution. In any case, if two thirds of his range is JJ+, he can bet everything and make us indifferent with our bluffcatchers, so it's likely he's gonna do just that (bet everything).
If we only checkraise the nuts he can exploit us by always folding his JJ. The way we will adjust will be by adding bluffs to our range up to the point where he is indifferent with JJ. Our success rate with our bluffs will be 50% when he is folding JJ, and slightly higher if we use blockers when choosing our bluffing hands.
I believe 50% nuts in villain's range is the borderline where we can still be bluffing. At 51% we can no longer bluff with any sizing, except when there's a particular blocker effect.
I am not 100% confident about this, so please challenge my assumptions.
Its a little away from the question but will get to it at the end.
I think his river pot bet is just bad and therefor very hard to figure out a GTO raisesize.
Here is my reasoning. Without putting any numbers to it but heros and btn's range should be very made hand heavy given that the pot is multiway and they are both capped or at least have fewer strong made hands and the oop players are still uncapped at the turn.
Since hero is only calling the raise on the turn his range is slightly more draw heavy then villains range who is risking more money by bluff raising their and still could face a raise from the oop players.
So if villain plays the turn right he should not have many bluffs at the river which also means he cannot value bet to thin given because the board has changed a bit.
This seems a little bit weird that he cannot valuebet light because his range is strong but what is heros range on the river?
- missed draws, no SD value
- weak bluff catchers, missed draws with little SD value
- rivered straight
If villain wants to bet very thin for value, hero could not call any bluff catchers at all because villains range contains less bluffs and villain gets only called by better. This could still be +EV, because villain succeds with all his bluffs, but its not the best play.
Like I said i dont have numbers for that given I dont know their ranges but lets assume
Hero has 70% air, 15% bluffcatcher and 15% nut straight
Villain has 25% bluffs, 60% made hands, 15% nut straight.
If villain wants to value bet all his made hands hero can only call with the nutstr8 and like Midori says its like the AKQ game where you never call bluff catchers and villain should never bluff the K. This leads to the conclusion that by betting pot villain can only bet straights and bluffs cause he will only get called by better with his non straight made hands.
Just another example to prove my point.
Lets say the board in a 6 handed game is AAK45 and after the 3rd barrel Hero gets raised. Whats the optimal 3bet bluffing frequ. It is very obv that its 0% given the strong ranges and maybe even the bluff raise frequ of villain should be 0% if u do not want to turn the best blocker hand KK into a bluff.
That all being said its hard to get a gto raise size on the river which is bigger than a minraise. And since we are not allowed to raise smaller, even a minraise is an exploit.
To summarize this.
- villain should be betting smaller and can value bet wider which increases his ev with his made hands and with his bluffs because they are cheaper and he has enough made hands to back this up
- hero can bluff raise against a smaller bet at least some of the time because villains value range do not contains nuts only and he gets to bluff raise cheaper (risk-reward)
An excellent post.
Yeah, I don't think villain should be potting there either, that's along the line of what I was implying by saying
However, let's not kill ourselves by trying to figure out what's villain's optimal bet sizing here. What we know is he potted, and we have a decision to make, against which he sometimes has another decision to make (if we raised). In other words, villain's pot bet on the river is likely suboptimal, but we can still seek for the optimal strategies for both players after he potted.
With your range assignment of
I have obtained the following strategy and value of the games:
a) If Zach choose pot as his bet sizing
Limon should always pot/call with his nuts, always check back his JJ, and always pot/fold his air. Against this, Zach doesn't have a raising range; he always calls with nuts and SDV, and always folds his air. It is interesting to see that Zach should never be bluffing, whereas Limon should always be bluffing yet always checking back JJ.
The value of this game is 4.23P for Zach and 4.77P for Limon. The pot is P on the river, so this means Zach captures about 23% of the pot at GTO.
b) If Zach choose min raise as his bet sizing
Limon's betting range is all nuts and 3/5 of air, and he checks back JJ. Against this, Zach calls with 2/3 nuts and 100% SDV and folds air. Zach also has a x/r range of {1/3 nuts and 1/28 of air}.
The value of this game is 4.23P for Zach, and 4.77P for Limon. Zach captures about 23% of the pot at GTO, which is about the same to x/pot.
c) Why is this the case?
Limon captures ~77% of the pot in both cases, and the reasons are twofold: a) he does the betting, Zach the bluff catching; b) his range is stronger because he has wayyyy less bluff and his SDV beats our SDV.
It might seem counter-intuitive that Zach can bluff 1/7 with a smaller sizing, and never with a larger sizing. That's because of how often it has to work. When Zach is potting with his bluff, he risks 4P to win 2P, meaning Limon should fold at least 66% of the time. Since Limon's betting range consists of {15% nuts and 25% air}, he can only fold 25/40 = 62.5% of the time facing a pot, and Zach's bluff just doesn't work often enough. When Zach is min raising though, he risks 2P to win 2P and Limon should fold 50% of the time. Now, if Limon bets with all his air, we know he has to fold 62.5% of the time, which is overfolding. Thus, Limon should bet with his air just often enough, and give up with the rest. To be exact, he should be betting with {15% nuts and 15% air} and fold all his air when raised (=50% of the time), and 15/25 = 60%, hence Limon with bluff with this frequency.
I hope this helped, and if you would like to try a different range composition for both players, just let me know and I'll crunch a quick calculation.
midori
"c) Why is this the case?
Limon captures ~77% of the pot in both cases, and the reasons are twofold: a) he does the betting, Zach the bluff catching; b) his range is stronger because he has wayyyy less bluff and his SDV beats our SDV.
It might seem counter-intuitive that Zach can bluff 1/7 with a smaller sizing, and never with a larger sizing. That's because of how often it has to work. When Zach is potting with his bluff, he risks 4P to win 2P, meaning Limon should fold at least 66% of the time. Since Limon's betting range consists of {15% nuts and 25% air}, he can only fold 25/40 = 62.5% of the time facing a pot, and Zach's bluff just doesn't work often enough. When Zach is min raising though, he risks 2P to win 2P and Limon should fold 50% of the time. Now, if Limon bets with all his air, we know he has to fold 62.5% of the time, which is overfolding. Thus, Limon should bet with his air just often enough, and give up with the rest. To be exact, he should be betting with {15% nuts and 15% air} and fold all his air when raised (=50% of the time), and 15/25 = 60%, hence Limon with bluff with this frequency.
I hope this helped, and if you would like to try a different range composition for both players, just let me know and I'll crunch a quick calculation.
midori"
Are you saying Limon is best betting with 15% nuts 15% air for a 50:50 value:air ratio? That seems off to me because now his range is way over weighted towards bluffs and with Hero getting 2-1 on a call he should call with all of his bluff catchers(though it seems like they are a significantly small portion of Hero's range). Doesn't it seem like Limon instead should bet 15% nuts and 7.5% air to make villain both indifferent with his bluff catchers and make him unable to make a profitable x/r bluff? Another question, is our goal to make villain indifferent to bluffing or indifferent with his bluff catchers?
It's an interesting situation for Limon because with one ratio we make his bluff catchers indifferent but make him have a clear play with the rest of his range, with another ratio, we make him have a clear play with his bluff catchers but indifferent between x/r bluffing or x/f'ing.
Thanks everyone for contributing. I will respond tomorrow. Crashing for the night.
Also, one thing I forgot to mention in the above post. If the ranges are as following
and there was no betting on the river, what will happen? Well, we can do some quick math to calculate the EV of both players.
a) When Zach has nuts (15%), he wins the pot against Limon's made hands and bluffs, and ties with his nuts. Zach's EV = 0.85 * P + 0.15 * (0.5P) = 0.925P.
b) When Zach has bluff catcher (15%), he wins the pot against Limon's bluffs, and loses to everything else. Zach's EV = 0.25 * P * 0.75 * 0 = 0.25P.
c) When Zach has air (70%), he ties with Limon's bluffs and loses to everything else. Zach's EV = 0.25 * (0.5P) + 0.75 * 0 = 0.125P
Overall EV of Zach's = 0.15 * 0.925P + 0.15 * 0.25P + 0.70 * 0.125P = 0.264P
Thus, Zach captures 26.4% of the pot at showdown, which is quite large compared to the best he could achieve when there was betting (0.21P). This comes as no surprise, because Limon's polar betting range should allow him to capture more of the pot than dictated by his SDV, even if Zach has an option to x/r on the river.
Thanks for calculating midori. Did you use CR or just a tree with programm like Gambit?
Looking at the numbers I think it is a little bit unlucky that I gave Zach exactly the same amount of bluff catchers than the nuts.
This means if Zach is continuing against Limons pot bet, he will have 50% nuts and thats the reason for Limon to not value bet his JJ. But since Zach has no bluffing range (with a Pot raise) Limon will always realize his EQ with JJ and is going to be indifferent with betting JJ.
So the programm probably could not tell that propper and just used the limit value of this static function.
But if Limon choose to value bet all his JJ combos, Zach should not be able to bluffcatch at all because Limon has a value range of 75% (Pot odds 33% against Potbet).
Maybe I got something wrong but this should leed to a higher EV for Limon since he gets away with all his bluffs which are now backed up.
Thanks MorePower. I used Gambit for this analysis.
As for Limon's checking back with JJ - I think a similar case can be found in AKQ game with card removal and pot sizing (instead of 1/2 pot sizing). GTO strategy for the bettor is to never bluff with Q. I'm running a bit short on time but please ask if this is unclear; I will come back later tonight and elaborate.
midori
I quickly ran the math manualy on this and got 26,625% of the Pot for Zach, so I probably got something wrong if your Results are accurate :-(
This is for the a) where Zach only Pot raises and therefor never bluffs.
EVZach = Pnutz * (Pbluff * 2 + Pmade * 2 + P_tie * 0,5)
= 0,15 * (0,25 * 2 + 0,6 * 2 + 0,15 * 0,5)=0,26625
But feels weird to me, because this EV is worse than checking for Limon and this could not be the cause because every single part of his range has a positive EV.
Zach's EV in this case would be 0.252 + 0.62 + 0.15*0.5 = 1.775. Granted, this happens 15% of the time, but I don't think we have to multiply this by 0.15 when calculating the EV of this independent scenario.
I think its different from AKQ cause if Limon bets JJ, Zach is calling with worse half the time and he only is beat half the time. So he exactly wins what he looses and this should be 0.
In AKQ you cannot bet the K cause you only get called by worse. I'm not 100% sure but I think with card removal and only 3 cards in the deck thats the case. Maybe you refer to a 12 card model but I am not familiar with this. Please correct me if I am wrong.
Last example, I multiplied with 0,15 because I calculatet the EV of Zachs whole range and like I said, he cannot bluff catch anymore so is he only getting a part from the Pot when he has the nutz so the other portions of his range are multiplied with zero.
Again this is for changing your model a) to Limon bets Pot his whole Range.
To compare the results I calculated your a) and got a different one, too :-(
EV_Zach = 0,15 * (0,25 * 2 + 0,6 * 1 + 0,15 * 0,5) + 0,15 * ( 0,25 * 2 + 0,6 * 0 - 0,15 * 1 ) + 0,6 * 0
= 0,22875= 0,17625 + 0,0525
Where is my fault?
MP,
I'm sorry that I made a typo and was unclear about my reference to AKQ game. I meant to say the bettor shouldn't bluff with Q (not K, that was my typo) in that toy game. You are absolutely right that betting with K serves no purpose.
About the EV question, your calculation seems correct, so I'll quickly check my tree and see if I can notice some error. :O
Oh, it's weird, I ran the same calculation again and Zach's EV was 3383/800 = 4.22875P, which is equal to his initial stack of 4P plus your 0.22875. I think it was my typo, and thanks for pointing this out. :)
I did the same test for check-minraising scenario and found some error. I will modify my posting above shortly.
Just modified my above posts.
Cliff: Zach can capture ~23% of the pot, and it doesn't really matter if x/pot or x/mr.
This is when assuming the following range: Zach = {15% nuts, 15% SDV, 70% air}, Limon = {15% nuts, 60% SDV, 25% air.}
I just tried to use gambit myself but found no tutorial :-(
Maybe this is an easy solution for these Riverspots:
http://gtorangebuilder.com/#home
But I am frightened how much good software is out there. Poker is gonna be solved...
If my memory serves me right, there was some tutorial and sample input files on the Gambit webpage. Also, Gambit has been around since forever ago, so I doubt poker will be solved anytime soon.. although people have been heading towards that direction for sure.
exactly what the fuck is the point in trying to figure out a theoretical best sizing when in practice a player will just have to use their judgement to pick i sizing in real time? you could argue that analysing these theoretical best sizings over and over again could help you more accurately make that judgement, but i think human instinct is more efficient than any amount of mathamatical anaylsis ever could be.
furthermore, why would they not corrupt the game?
Understanding the theoretical best size allows you to adjust off of that for game and player specific characteristics. Without a baseline you are just stabbing in the dark.
You suggest just using instincts. Humans do not have bet sizing instincts. We have instincts like fear of snakes, fear of heights, etc.
poker instincts are just a way of describing somebody making quick reactionary decisions. However those decisions are based on his/her knowledge. The better you are informed on the theory the better you can make game time decisions.
"human instincts is more efficient than any amount of mathamatical analysis ever could be"
Sorry Phil, site can get closed now :-(
Always I find the best sizing here to a be a minraise vs almost all players. It's also best for your range seeing as tho a lot of the range he could be perceived to have are str8 draw flush draw combos and GT wise we want told fold all those out by paying the least amount to do so. I would do something really annoying to the villain here and tank for a solid amount of time before minraising. Even tho it's transparent what you are doing by tanking it can look like you came to the conclusion he had a missed draw and u can easily rep the nuts here so min raising is the best option you came up with. In spots where my hand is pretty face up I think you have to get a little creative to get paid off in live poker. I've played thousands of hours of live PLO and I can only remember minraise bluffing in a spot like this once and it working which makes me feel like I've missed some opportunities or forgotten some times that I've done it. In standard spots when hands are more or less face up I think going outside the box and trying to think of something semi creative is your best option "not that tanking is really that creative" I just think the value that comes from this spot is going to have to come from any way you can come up with to try to level the Villain.
All that being said... Can we pretend you had KT9c4c and u went for a minraise bluff on the river and poll the audience if we thought it was a profitable play? I vote yes. If not immediately profitable... Possibly it could hold some future equity in showing the table that your willing to go a little crazy in some super thin spots...
I've done min-raise bluffed successfully in situations like this about 5 times in 75 hrs. of live plo play.
All have gotten through, never gotten looked up. I don't think anybody was even suspicious about it, (I play 5-5 with a rock as well).
I'm inclined to agree that doing it more often would be more profitable, for all the reasons you described, so long as you have the relevant blockers.
Be the first to add a comment