Adreno

Well, one thing a larger sizing achieves is that it allows us to have more bluffs in our range. When we put some of our nuts into the smaller sizing range, we have to check/fold more of our bluff hands.

Sorry, let me elaborate:

Intuitively it makes sense to have multiple bet sizes on a single river, as opposed to a single betsize. We can make thinner value bets by using a small sizing and we can polarize part of our range with overbets. While we can balance all sizings by bluffing an appropriate amount, we typicly end up with having all our nuts in 1 range, thus capping our other ranges. 2 problems arise from this:

1) Villain can exploitatively bluff raise a lot when we are capped

2) If we bluffcatch villain's raises the appropriate % of the time, villain will be able to extract a lot of thin value with hands that would "just call" against an uncapped range.

Galfond has a video where he advocates using multiple bet sizings on the turn from a game theory perspective (and by extension, other streets as well). Sauce appeared to disagree in the comments section, but it didn't really evolve into a debate. I certainly see the merit of using exploitative sizings on the river, but it's a more complex question from a game theory standpoint where we face the perfect opponent: Does the added benefit of using the "optimal bet size" for each hand outweigh the cost of allowing villain to put pressure on our capped ranges?

Using multiple sizings allows us to valuebet more hands, and therefore also bluff more hands. It also improves the EV of some individual hands considerably (for example, the EV of our overbet range). However, the EV of some of our other hands must decrease, since villain is now able to valueraise more thinly.

Another way of solving the problem would be to mix some of our nutty hands into our capped ranges, and I think it would be a good practical solution to induce mistakes from our opponent. However, against a theoretical nemesis opponent, that's going to greatly reduce the EV of our nutted hands. And the more mixing we do, the more we should ask ourselves why we want to split our river betting range in the first place.

PokerSnowie is really, really bad. You can beat it HU by playing 100 VPIP, betting any equity for 3 streets and by never folding to its raises. I would expect any HU pro to get over 20BB/100 on it.

Typicly the PF raiser hits the high and low cards better while the PF caller hits the medium cards (6-T) better. Same applies for both SRP and 3bet pots. I ran some numbers with Pokerstove back when it used to work and in SRP pots the raiser was typicly 52%-55% ahead the caller, but could be as low as 48% on some flops. In 3bet pots the numbers vary much more if 3bettors range is polarized and doesn't have good board coverage.

Hey teunuss. I stopped reading after this paragraph: "(1) We bet all our AA and check all our 55. The optimum strategy for the OP player is to simply fold to all our bets, which has an EV of -1. But wins vs all our 55, which has an EV of 1. Thus, this strategy would result in 0EV overall"

The EV of a fold is always 0. Thus, the EV of OOP strategy in that example would be 0.5 (half the time we win nothing, half the time we win 1 pot = we win half the pot on average)

When a midstakes reg uses multiple bet sizes on a specific river, I think they are typicly trying to maximize the EV of their current hand. That means they bet larger with better hands, smaller with thinner value, and usually pick their bluff sizing exploitatively (they decide bluff sizing by what they think gives most EV.. usually a larger bet representing big hands + minimizing cry calls and re-bluffs). This means for any given river they have multiple betting ranges, but only one of their ranges contains bluffs! If their opponent has "would he pick that sizing for a bluff?" in their thought process, they can make some sick laydowns and hero calls on the river.

I can't personally imagine a human player balancing multiple bet sizes for a given river. A computer might be able to do it, but if we want to develop good, balanced strategy for a human player, I think 1 river sizing is a must.

"I think our sims differ a bit on this point.  OOP has to call around 40% versus a 1.5x pot bet, and on my sims, he'll hold trips around 21%.  Because of card removal he only has to call around 36%, so he'll add another 15% or so bluffcatchers, with the best candidates being 9x hands that don't block any of OOP's straight draws or flush draws."

I think we have calculated card removal effects differently. I'm probably in the wrong here since I haven't done many calculations like this, but I can't really figure out how. I tried to answer the question "what range does villain need to call with in order to stop us bluffing profitably?" - so I calculated villain's combos against our bluff range (where we are not blocking any JX). Did you calculate villain's combos vs our betting range (where we partially block some JX)? In any case, the combos I used for IP when we are bluffing were 82 combos of trips+, 80 combos of bluffcatchers and 80 combos of missed draws. If IP calls trips+ and KK,AA without diamonds then he is calling 88/242 = 36%. If I understood correctly, you got 21% for trips+, whereas I got 34% (82/242). That's a huge difference, where does it come from? The combos I used for trips were J2s,J3s,J6s,J7s,J8s,JTs,QJs,KJs,AJs (18 suited trips combos) J6o,J7o,J8o,JTo,QJo,KJo,AJo (42 offsuit trips combos) and the combos I used for boats were J9o,J5s,J4s,J5o,J4o,99,JJ (22 boat combos).

"Remember, that in optimal play our opponent knows our strategy and maximizes his value against it, so if we bet 1/4th pot with a range of [J9-J5] for value, and [Kxdd-87o] as a bluff, then how will our opponent play against us?"

I'm not sure how you split your range on the flop, so I can't really run numbers on this (IP's range is more easily defined in this spot imo). If I may venture a guess I would say that we have way too many bluffs, so raise bluffing us becomes +ev. If villain doesn't care about being exploitable himself, he will proceed to raise with all hands which gain more ev by raising than by calling. I would guess the threshold would be around Q high / K high. Villain would also call ace high and better and raise high jacks for value.

"usually it's a nice way to model the situation by adopting some criterion for "value bettableness," for example that we have to be ahead of 45% of IP's calling range, and then sizing a bet such that our range's betting+bluffing volume is maximized.  This works because in balanced play, the EV of most of villain's range is near zero when facing a bet, therefore if we can maximize the frequency at which villain faces a bet we get close to minimizing the value of his range."

Thanks! I ran some numbers from this angle:
If we bet JT+ for value @ 1.5x pot, we can bet 18 value combos + 11 bluff combos = 29 (J9o,JTo,JTs,J9s,J5s,J4s)If we bet JX+ for value @ 1/4pot, we can bet 30 value combos + 6 bluff combos = 36 combos (J8,J7,J6s,JT+)If we flat KJo and QJo instead of 3betting them, 1.5x becomes more attractive again in terms of maximizing betting frequency with a balanced range. What's appearing to me intuitively now is that because we don't have any thin value hands in our range, 1/4 is too small of a betsize for a balanced range. I now think the optimal sizing is somewhere above half pot. If river was, say, King of hearts, a smaller sizing might make more sense, since we would have many KX in our range and we would like to get thin value from 9x,TT-QQ. On a 5h river we are too polarized to bet as small as 1/4, but not polarized enough to bet as big as 1.5x.
"What sort of analysis is required?"
I think we should be able to actually calculate the optimal bet size for this spot, given that villain knows our ranges and exploits us maximally. I have some ideas how to approximate this and eventually draw a graph that shows our bet size on the X axis and our EV (for our whole range with that betsize) on the Y axis. It would also be interesting to see how much IP's frequencies and ranges change when IP is also forced to play a balanced range. I guess that would be a measure of error in our model? (eg. if IP gains more EV by playing exploitatively against OOP than by playing a balanced strategy, that means OOP isn't playing a balanced strategy?) Is there any software out there that could do this?

"we will always make more money betting bigger with a polarized range that is balanced correctly than betting smaller assuming we choose the right value hands to include."

Except this is not one of those textbook scenarios where we hold [nuts,bluffs] and our opponent holds [bluffcatchers]. Our nutty hands aren't nutty enough and our opponent is also somewhat polarized (he has plenty of bluffcatchers, making him less polarized than us, but he still has plenty of nuts and air). If we only have 1 betsize on the river and we choose to make it 1.5x, then we can valuebet very few combos and therefore bluff very few combos, leaving the majority of our hands to check. This is not ideal in a scenario where our range is polarized. When we bet smaller, we can valuebet more hands and therefore bluff more hands(I think, might be wrong).. while also giving villain the opportunity to re-bluff us.

I challenge you to find any reasonable scenario on this board where a balanced bet with 1.5x gives more EV for our range than 1/4pot! ;) Regardless of whether we assign IP actions by GTO, best-response or "population tendencies", I don't think you can find a scenario where 1.5x gives more EV.

Ben, I appreciate the toy game example of how a valuebet is sometimes our best move even if we are getting called by better more than 50% of the time. Gave me a lot to think about and it's definitely something I have overlooked in my game. Let's say a generic situation where we have a value hand on the river that we are thinking about betting thin: if we are IP, we have the option to go directly to showdown by checking - so when we decide to instead put more money into the pot, we have to win more than 50% of the time that villain doesn't fold. Let's say that villain has a balanced X/R range, so that our EV of calling is 0: then more than 60% (depending on sizing) of his hands win a showdown. Because we are putting money into the pot at less than 50% when we are raised, we have to win SIGNIFICANTLY more than 50% when our bet is called. Therefore, we can in general bet more thinly OOP than IP, and we can bet more thinly in spots where we can't get raised compared to spots where we can get raised. Let me know if you spot any flaws in my thought process.

Now back to the hand that we were discussing, I'm not implying I have any special insight into population tendencies. I just think *any human player*, especially regs, will make fewer mistakes against a 1.5x shove. Consider it from this perspective:

Against 1.5x shove the approximately correct play is to call trips+, fold everything else. If you sometimes hero call other hands, it's not going to be a big mistake, and may even be the correct play if we tweak our assumptions slightly.

Against a 1/4pot bet the correct play is to shove a portion of missed draws, all of the boats and some of the jacks, but call the lowest jacks, along with any pair and some of the Ace high flush draw combos.

It should be obvious from that comparison that any reg is going to make fewer mistakes against a 1.5x shove than against a 1/4pot bet. That said, looking at this 1/4pot sizing from a GTO perspective makes for an interesting exercise - one that I probably should have done a long time ago.

Daniel, having multiple betsizes on the river does solve the problem I presented (if the sizing for all our bets was 1.5x pot then we were either valuebetting too thin or our bluffs were -ev). It's a way to solve the problem in theory, but in practice I find that players who use multiple sizings tend to give away a lot of information about their hand. It's just really tough to balance multiple river sizings in-game. If we do choose to take the simpler strategy, one sizing for all bets, then any sizing below pot would be closer to optimal than 1.5x shove.

Ben, wrt "We'll fail to get the right amount of value with our Jx range" [with 1/4pot compared to 1.5x] -I showed earlier that if IP is calling with optimal frequency, his calls will beat J6 more than 80% of the time. That means we actually get more value for our weak jacks by betting smaller than 1.5xpot.

 "Because of our smaller betsize, we can't bluff as often"

I ran into a very interesting situation when I tried to formulate a balanced 1/4p range. Basicly, I thought that villain had missed draws so often that he wouldn't be able to exploit us by calling wide, only by raise bluffing. Assuming the most common raise size would be shove, this would allow us to jump our bluffs from 17% (unexploitable by calling) to 56% (unexploitable by raising). When I calculated the combos I actually concluded that we DON'T HAVE ENOUGH BLUFF COMBOS on the river :D Then I read your post about the 5x/4x and realized that some flush draws in both players' ranges hit a pair on the turn and river. This gives villain enough bluffcatchers to exploit us by calling wide and means our bluffs in the 1/4p range come down from 56% to 17%, which obviously sucks.

However, while we are in theory land, we might as well bet-call FDs that hit a pair on the turn/river, possibly even Ace high FDs. I might calculate later how much EV our entire range would get with that strategy against a nemesis who plays best response against our strat.

"If a small bet is best in optimal play, then we'll want most of our value bet-able hands to be clustered in a region that beats roughly 60-85% of IP's range, and we'll also want to have a lot of bluffs and few bluffcatchers."

I didn't understand this part of your post. 60-85% against entire range or calling range? I think we need to look separately at the range we are getting to fold and the range that we are getting called by.

Daniel, with regards to 93 being a losing call.. *edit* oh you used 28% instead of 40% as the defend range as Ben pointed out. That explains it. I had a paragraph here about how 93 is a threshold hand that becomes a call or fold depending on assumptions we make about IP's flop checkback range.

"I would be inclined to think that the check and bluff catch line on the river would be most profitable with weaker Jx hands because people probably wont balance bluffs correctly with their bet sizing and leave us with an easier exploitable decision."

You seem to just disregard the "small bet" option without any analysis. I'd be willing to bet that a balanced 1/4pot bet is better than checking EVEN when we give our ranges to villain & let him play best response against our ranges. At least 1/4p weak JX lets us use more bluffs than checking.

Wall of text, so I'll start with the punchline: First we shove 1.5x pot with a bluff, then we do it with J6 for value. In order for the bluff to be +ev, villain has to fold most overpairs. If villain folds most overpairs when we shove J6, we are going to be beat more than 50% of the time that he calls. Therefore, 1.5x pot can not simultaneously be the best move for both a bluff and for J6. Therefore, it can not be the optimal sizing for a balanced strategy.

"Why do you think that J6 isn't strong enough to shove for 1.5x pot?  Do you think it's beaten by more than half of IP's calling range?  Or do you think another sizing maximizes its value?"

You were surprised in game to see Syous call with a 9, even though he was not blocking diamonds or straights, making it one of the best nines to call with. You said he's losing to your range with that call. Let's try to extrapolate a calling range from that: If he shouldn't call with 9 no blockers, he shouldn't call with TT/QQ (blocking straight draws) or KKd/AAd either. That leaves 6 combos of PP's, ~7 combos of boats and ~38 combos of trips (when we are blocking one jack). J6 wins only 20% against that range. In reality people are going to find hero calls (or call with a wider range to begin with), so we're probably going to be good more than half the time we are called. However, if that is true, then our bluffs are explicitly -ev:

We have a missed draw and we shove for 1.5x pot. Villain has to call >40% of the time to stop us from bluffing profitably. When we are not blocking the jack, he has ~74 combos of trips, ~10 combos of full house, ~80 combos of bluffcatchers and ~80 combos of missed draws. If he calls with the range I talked about in the previous paragraph (trips+ and 6 combos of PP's), he's calling 37% of the time. Now add in 9's that don't block flushes or straights and we've surpassed the 40%. Anyway, if our bluffs are profitable, then J6 gets called by better hands 80% of the time.

Compare this scenario to the next:

When we have a missed draw, villain also has a missed draw about 30% of the time. If we size our bet as 1/4pot, villain has to fold more than 20% of the time for our bluff to be profitable. Assuming he never folds a 9 or better, this sizing forces him to play back with at least a third of his busted draws if he wants to keep us from bluffing profitably. He's going to be unsure how wide to raise for value, unsure how wide to call (A high? K high?), and unsure if bluff raising is profitable.

The second scenario is more likely to induce mistakes from our opponent. The first scenario involves mainly trivial decisions for villain (call with nuts, fold bad hands). The 1/4pot bet bluff is also more likely to be profitable in a vacuum, since it only requires villain to fold his MISSED DRAWS more than 67% of the time. In order for the 1.5x pot bet to be profitable in a vacuum, villain actually has to FOLD OVERPAIRS - which is unlikely from a pop tendency perspective.

"how did you arrive at 1/4th pot as a near optimal betsize to balance in the first place?"

I think the optimal sizing would be somewhere between 1/5pot - 1/2pot. I prefer 1/4pot, because I think it's big enough to be significantly different from checking, yet small enough to induce a lot of raises. In other words, I think it's good against population tendencies. I have no idea what the GTO sizing would be.

Really interesting spot! I have a comment with regards to our "15 very clean outs":

Out of the 9 cards that make a flush, 9d and 3d potentially give him a full house. If river is a T or 5, villain should actually be check/folding his sets, 2pairs and overpairs (especially on Td or 5d). When we make a straight and he bets, we should be expecting to split the pot most of the time.

Here is how I would view all potential rivers:

8 cards that make the nuts & kill our action

5 cards that make the nuts without killing our action

2 cards that make semi nuts & don't kill our action

3 cards that make a decent bluff catcher & don't kill our action

43 cards where we have to fold to a bet & can't bet if checked to & very rarely win a checkdown

That's a total of 7 river cards that are really good for us. The fact that our opponent is more likely to check/fold rivers that complete our draws should weigh down on calling and lift jam as an option. I don't have an opinion which is better, just thought this was missing from the analysis.

I think it's fair to say that OOP's range is more polarized than IP's because high PP's and 9x shouldn't be in OOP range. How much of OOP's range is missed draws on the river depends on how we split our range on the flop and how many JX,99 combos we 3bet preflop. I would be guessing too much if I tried to break down your OOP range on that river (in comparison, reading IP's range is much easier as we can expect any reg to cbet most of his draws on that flop & expect them to not really have a significant flop 3bet range IP)

"Remember, that the more polarized OOP's range is relative to IP's, the more aggressive he should be, and the more bluffcatchers in his range the more he should check and call."

OOP is more polarized than IP with very few hands that may want to x/c the river (perhaps KQo and A high fd). Because we are so polarized, we should be agressive with our range. If we use a big sizing, like 1.5x pot shove, we can bet more of our bluffs while being balanced (compared to a small sizing where we have to x/f more of our missed draws if we want to be balanced). Is this what you were getting at? If it is, I have 2 counter points:

1. When we shove for 1.5x pot we are valuecutting ourselves way more often compared to when we bet small. (As a sidenote, does this reduce the amount of bluff combos we can add to our 1.5x pot range? I thought it did but the more I think of it maybe it doesn't. Why am I confused?)

I don't think a hand like J5 is strong enough to shove for 1.5x pot (I'm sure shoving is +ev but just about any other line is going to be more +ev). Even our best hands for shoving (KJ,AJ) our probably getting more EV by a smaller sizing. Do you think shoving a weak Jack is a good play in a vacuum or do you agree with me that we would be sacrificing hand EV to balance our range if we shoved a weak Jack for 1.5x pot? Shoving J9,99 is also very bad since we are blocking the bluffcatcher part of IP's range, causing IP to be more polarized, giving us more reason to bet small (or check) to induce action.

2. Let's make some reasonable assumptions about how IP plays his range vs. a 1/4pot bet and how IP plays his range vs. a 1.5x pot shove. We can then calculate how much EV the OOP player can get for his range with each line. I'm pretty sure the small sizing is more +ev for OOP's entire range regardless of whether OOP bets all of his missed draws on the river or only a balanced amount of them.

edit: one more thing. You suggested "Let's also simplify the analysis by assuming that trips+ always calls any sized bet." If we did that, we would probably reach your conclusion that betting big is better -- but it's not realistic. If we bet 1/4pot on the river, trips+ is never actually going to call. The whole point of betting 1/4pot with good hands is to induce raises.

I assumed the majority of villain's range would consist of missed draws but it seems I was wrong. If we have AJ on the river, villain should have 70-100 combos of missed draws, ~38 combos of trips, ~7 combos full house, ~60 combos of 9x and 21 combos of TT+.

Even if missed draws only make up around 40% of his range, that's still a lot. Here's some arguments I would use to support a smaller river sizing:

1. A smaller bet makes sense for both our value hands and our bluffs. Shoving only really makes sense as a bluff. If we shove JX it's almost guaranteed to extract less value than betting small (since all busted draws just insta-fold to a shove) and feels more like we're shoving JX just to protect our bluffing range, rather than taking the line that yields max EV for our hand in a vacuum.

2. Inducing mistakes. That 40% of the time villain has a busted draw, a small river sizing like 1/4 pot puts him in a tough spot and can induce mistakes. Is he supposed to call down KQ high? How often is he supposed to bluff raise when he has no SD value? In comparison, shoving 1.5x pot yields an insta-fold from all busted draws and induces no mistakes from our opponent.

3. Strategy simplicity. We can bet all JX for a small amount, but we can't really shove Jack+weak kicker. We could use 2 bet sizes depending on our hand strength (and place appropriate amount of bluffs to each range). However, having a single betsize for this spot is a simpler strategy and means we're more likely to be balanced and less likely to make mistakes.

I've put a lot of thought into this and really look forward to your reply!

Hi Ben, thanks for the video. Throughout the video I would have used different sizings in multiple spots. I was wondering if I could get your thoughts on this for example:

36:00 you CR 9JJdd with T8, bet turn and shove river for 1.5x pot. You justify the shove by saying "he has enough air". When I'm OOP on a brick river and I believe my opponent's range is draw heavy I tend to bet small (1/5pot - 1/2pot) with the idea that it's a cheap bluff to make missed draws fold & it can also work as an inducer when I have a good hand like trips. I imagine shoving JX for 1.5x pot would be a huge mistake since we're not giving our villain's draw heavy range the opportunity to put money in the pot & we're also blocking many of the hands that he could call with.

While 1.5x pot might make sense as an exploitative bluff, I can't see how it makes sense as part of a balanced range.

Hey Mushmellow, great points on both minimizing the rake in small stakes and maximizing hourly by folding marginal hands. If you want to max your hourly against fish those are very important things, althought when playing regs you probably need to take every edge you can and play those marginal hands.

I was thinking the same thing. Would love to get a comment on this.

At 36:00 you weigh various options for extracting value with Q2 on 2s 5h Qc (3bet pot). You end up cbetting the flop, calling the raise, and instantly check the 9s on the turn. Would it make sense to lead the turn to charge his drawing hands that he might check back with? As you said yourself calling a raise on the flop doesn't really look weak, so how often would he continue betting on the turn?

Not splitting your range makes your HAND less faceup, but your RANGE more faceup, if that makes sense.. 

An example: suppose you have a flush draw. If you don't split your flop range, your opponent will know precisely what range you are calling the flop with, but your range has a variety of different hands so your fd is more protected (less likely he will be able to put you on a fd). Compare that to splitting your flop range: Your opponent has no idea what your ranges are (since there are so many different ways to split them) - but he will have the opportunity to figure it out during the course of the match. Perhaps after 500 hands he will notice that you play most of your flush draws with line x and then he may be able to read your hand better & exploit you.

However the first few hundred hands AT LEAST a strategy that splits flop ranges in a smart way is likely to induce way more mistakes from villain than a strategy that puts our range faceup (always x/c flop). It also allows us to take max EV lines with many hands rather than x/calling everything "for balance". Of course it's much easier to play a strategy where we only have 1 flop range that continues into turn & river decisions, so I can certainly see the merits of not splitting our flop range.

On a sidenote, I'm not advocating check/raising all strong hands, but I do think a x/call range is supposed to be weaker than a donk or x/r range for the same reason a preflop flatting range is supposed to be weaker than a preflop 3betting range.

While never x/raising flops keeps your strategy more balanced, it also puts your ranges more faceup to your opponent. Example: Suppose you call 60% preflop and check-fold 30% on the flop. If the 70% of hands you play go in your x/c range, it's obvious what your range is and a good villain can make perfect decisions against you on the turn.

Now compare that situation to a flop strat where you x/r 20%, donk 10% and x/c 40%. There are many ways to build those ranges, so your opponent will be in the dark and always guessing what you might have. It takes a huge sample for him to figure out even part of your strategy. We can induce a lot of mistakes from our opponent by splitting our range on the flop, but the downside of such a strategy is that it is very difficult to balance, especially for a variety of board runouts.

Is PLO2 really beatable?

It's easier to code a 6max bot than a HU bot, because tighter preflop ranges lead to fewer postflop decisions (we're talking about creating a winning bot, not about solving an entire game)

I would guess the 6max bots are at least partly based on static rules (eg. preflop charts for most common stack sizes, how to play top pair, how to play draws, how to play from position vs oop, etc.) It would be extremely hard to do the same for HU.

Villain's range shouldn't have 67,55,44,TT,T5,T4 after he checks back flop. We could have some of those combos. We could also make a weak pair fold (on the turn or on the river) and we might get called by draws that have 30% equity against us on the turn.

Great video! I haven't played ring games in years and when I saw that AK hand I thought "has nl100 come to this already?" - but judging from the comments it hasn't. Phew. The thought process behind the hand was solid, but like others pointed out it requires a good bet size read to make the math work.

Which screen names on Microgaming do you suspect are bots?

May I ask which site has HU bots?

Oops I read the 8 as a club.. my bad

After you check back the flop your range is capped and it's easy for him to rep a better hand. That said, I think he would expect you to fold a bare ace to the big turn raise and once you call it might seem like you are making a stand (as you said you had always folded before, now you call and he STILL overbets the river). I'm guessing you called the river and he showed a flush?