Is range balancing mathematical or is it just a non mathematical concept?

I always here about people saying balanced ranges are like, if the villain gets 33% pot odds then we want to have a value bluff ratio of 2:1, but when I put some ranges into equilab with 20 value bet combo's and 10 bluff combo's it turns out that the equity each player has is random numbers, not 67% and 33%.

So how does balance work? How do I bet the river with a balanced range? Is it mathematical or just something else?

Thanks for any help.

The example you describe is only true on the river where the opponents range is clear value bets and air hands. If were holding a medium strength hand(bluff catcher) then our opponent would want to balance his range to have the ratio of value bets to bluff equal to the pot odds were getting just like you described.

Things change very significantly when were talking about other streets than the river and where different hands will have different equities

Why talk about the situation from the opposite angle? lol that is confusing.

I think I get what you mean. You're saying that if we are on the river and we know that villain has a bunch of bluff catchers and we are only betting with value hands better than his bluff catchers, then we would just add in appropriate amount of bluffs and achieve a balanced range?

And that if some of our value bets aren't hands that when called have 100% equity, then things change.

How does that work then? Is balance a matter of our ranges equity otr vs what we assume villain's calling range is and then adding in bluffs appropriately to achieve balance?

So for eg. if we have a river value betting range that has 80% equity vs villain's bluff catchers when he calls us, then if we bet pot, we could add some bluffs dependent on the amount of value combo's we have, to the point were his equity become 33%?

And this is all affected by the fact that he could raise river in some situations, and as previously said, earlier streets are different because it's more difficult to quantify what will happen on later streets? (like we might get bluffed off our equity or something?)

Theoretically, it could be a mathematical problem, if playing against a theoretically optimal opponent. However, since there will be worse players at the table, it still wouldn't matter since neither player will bother optimizing a few BBs against the other and better spend their time focusing on the weaker players around them.

Instead you just need to roughly balance vs the opponents weaknesses. i.e. vs a calling station, it drops to hardly any bluffs but against a very tight range that balance may be on the high side on later streets since there will be many more cards that hit your range while missing their's.

Well, that's kind of the objective. The theoretical optimum is for bluffing to balance out around neutral EV so that your folds are providing 100% equity.

However there are better ways to balance than for optimal around a specific street or hand. Instead you should be balancing more for image to optimize a session or longer if including regulars.

In the later case, you could actually even include making some amount of -EV plays to better optimize for the larger picture and result in a higher net EV overall. Not saying you should do this when learning as there will be plenty of room to improve without including this tactic and your actual mistakes will more than create enough of this same kind of effect anyway. It's just an extreme example to illustrate the power of hyper exploitative play.

Bluffing the river will be tougher, in general, than bluffing on earlier streets depending on effective stacks. On earlier streets, facing aggression includes the possibility of facing tougher calls for larger amounts on later streets.

In general, you needn't bother with optimal numbers. Instead just consider, if your consistently not getting paid on your value bets, you should be bluffing more. If you're consistently getting paid often, you should be bluffing less.

Just keep applying this concept towards smaller and smaller samples, session, players.

It will very often be the case that players will be looking at your aggression level overall rather than any level specifically against them due to too small a sample size. When you are the one bluffing vs player tendencies, sample size will be far less of a concern.

Therefore balancing optimally is even more unnecessary since your mix of play vs various opponents will be self balancing to some degree.

Another words, in your strategy, you may be bluffing at double the frequency vs player A than vs player B. In the short term this will not be very apparent to either player since the short term variance may actually result in just the opposite result. Also any value bets that don't get showed adds to this confusion. Instead they need to rely on your frequency vs the entire table.

I really don't understand this, could you please explain?

I sort of get what you mean by this. You're saying that it's ok to make a minus EV bluff sometimes just to develop an image so that you get paid later?

Good point.

A problem I have here is that I don't know where my starting point with my range is. I don't know when to bluff in river spots and not be doing it super often and end up really exploitable. Like I know when to value bet and could adjust based on information I get from that, but I am always paranoid I might end up bluffing too much if I do decide to try and bluff and could be losing a lot of money because my range isn't "balanced" and the player calls me.

I get this last bit as well, another good point.

Thanks for sharing your views so far, I would appreciate it if you could help clarify these others points for me.

In just a little bit of a hurry right now so i will do my best to answer these questions better later. Not that I'm an expert at answering them, but just to give my two cents.

Anyways, yeah what you are saying is completely right about the situation where the bettor has nuts and air, and the caller has bluff catchers.

So on the river if both players have a pot sized bet left and Player A holds either a hand that has 100% equity or 0% equity and player B holds bluff catchers. Then Player A would bet all in for a pot sized bet since a polarized range on the river always benefits from betting as much as possible and since player B will be getting 2-1 on his call, then the value bet to bluff ratio for Player A should be 2 value bets to every 1 bluff.

If we are talking about a situation where there are still streets to come but the ranges of both players are the same, then things will become different because Player A will be able to add more and more bluffs into his range because he has multiple streets to bet. This is a tough concept to understand and it would benefit a lot for you to work through the math on it. There should be articles online that discuss balancing nuts and air when there are multiple streets to bet.

These are important concepts to understand and definitely help us when deciding how to construct our ranges, but they are incredibly simple in comparison to actual poker situations. Once we start talking about ranges that aren't perfectly polarized against ranges that aren't just bluff catchers and we have multiple streets to play, it all gets really complex and I'm not the best person to answer all those questions cause they are really difficult.

This is based on hearsay knowledge since I never felt it was necessary to actually work out the theoretical maximum for bluffing percentage for reasons I posted previously. It works out intuitively though. If someone spots an error in my explanation please point it out.

So while it does work, it's not something I myself would subscribe to since there are better ways to balance that's more profitable and safer which is a rare combination.

So with disclaimers given, I'll explain the concept:

The percentage you should be bluffing, semi-bluffing, value betting, etc. should all combine to a range that is neutral EV when compared to your opponents calling range.

It kind of makes sense if you think about it because if you start adding bluffs to a neutral betting range vs calling range, you start losing EV, because the player is not folding any wider.

If you start removing bluffs from a neutral range vs calling range you are passing on folds that would have generated net 0 or better equity from folds.

But if you are choosing a range that only ties your opponent when he calls, it seems like we are targeting a range that makes the villain more correct so how do we profit?

It's actually true that by targeting villains range, we are actually making his calling range more correct at the same time. However, we profit because if we are neutral when he calls, we are essentially free-rolling the times he folds.

However if he is re-bluffing a significant amount than we are suddenly way over balanced on the bluff side and must significantly adapt to this rebluff strategy.

This is all a part of the idea that as we optimize exploitative strategies, we also open ourselves up wider for counter exploitative strategies. Therefore taking a direct approach to optimizing each bet hand by hand is risky, and somewhat easier for an opponent to recognize and adapt to.

I believe it's far better if we look at balancing for an entire session or longer. It's far more difficult to identify and adapt to and I believe more profitable over-all.

With session balancing over hand balancing, then later adding street balancing concepts, and finally another new concept of stacking opponent error when knowing something no one knows.

Most of this stuff is among the most advanced thinking today but exploiting new knowledge is by definition, something new. I call the entire package a hyper-exploitative style.

When you discover something new, players will all be making the same uniform mistakes. The more uniform the more you can adapt to stack this error from all players and can lead to unreasonable profits. When using other advanced concepts to conceal your play, it virtually makes you invulnerable to counter strategies it so effectively disguises your play to make it hard to identify what your focusing on when searching for play that's exploiting a weakness you don't even know exists based on current understanding.

For this reason I like to call it part of a "hyper-exploitative" strategy. In fact the better the table the more effectively you'll be beating it. This is due to the fact that the error is so well aligned to all players you can safely maximize exploitative play and when combined with concepts like session balancing and such, the chances of a player finding the leak or figuring out your play and coming up with a correct counter strategy are remote without making the same discovery independently.

it's like every player has an identical tell that only you can see. It's extremely rare to come by but the profits can be crazy. The down side comes from the added threat of over fishing the waters and killing the game. Or being mistaken for a cheat which could have even more dire consequences.

Correct. It's more essential when playing within a more regular group of players who will have time to learn your game very well. Or at higher levels where "correct thinking" may end up unifying into a more firmly defined optimal.

I found a comment repeated over and over by Galfond when making a video quite funny but at the same time very true when applied with this concept in mind. Basically, every time he made a mistake, either by exploring the edges too wide or even thinking on the wrong level. He always seemed to dismiss it by saying something to the effect of "That's OK, it's important to do that sometimes" So -EV play is there to keep them guessing as far as what your thinking. By removing the certainty, logic no longer holds up so well.

"He must realize that X therefore there's no chance he could have Y here."

I think this is fine. As I said river bluffs are the riskiest due to the size involved, the lower FE since it just one call without the prospect of facing more, and the tendency of most opponents to error on the call side. That is, if you are including thin value bets in the mix that makes your bet frequency "look bluffy" anyway. Just don't completely eliminate river bluffs because that suddenly becomes hugely exploitable.

Ok thanks Polsk33 and TakenItEasy for the help so far.

Tbh TakenItEasy, most of that stuff is just a million miles over my head lol, I barely understand basic balance so that stuff is very difficult to get my head around.

Where did you learn about that stuff?

You said, "The percentage you should be bluffing, semi-bluffing, value betting, etc. should all combine to a range that is neutral EV when compared to your opponents calling range. "

This means that I have to guess my opponent's river calling range and also try to guess his tendencies so that I can construct a balanced range?

I.e. I have to guess my equity vs his calling range and guess whether or not I think he can raise the river to put me range in a bad spot?

Thanks for the help

Im not totally understanding this and in my head it doesn't seem like this would be true.

Our opponent should never have a hand in his calling range that isn't either 0EV or +EV. Because he would benefit from just folding. Now its obvious how we can see situations where some of the hands in our opponents calling range will be +EV without incentive to use them in his raising range.

Id love to hear you explain some of these concepts more in depth though and maybe provide some examples?

Yea any more input, links or examples would be greatly appreciated.

I can't actually find any decent articles online about how to balance ranges.

When people talk about balance in poker they mean two things:
1) GTO balance
2) estimated balance

In GTO balance, the way to balance is to make yourself unexploitable. Often that will mean making your opponent indifferent to his decisions.

Here is an example of a GTO balanced river play:

River shows 3 3 2 2 2
There is $100 in the pot and both you and your opponent have $100 left.
You are first to act with JJ
You and your opponent have a mutual understanding that his range is 55+

A gush of wind comes out of nowhere and exposes your hand as JJ.

You check and your opponent bets all-in. How often should you call theoretically?


You should call 50% of the time. 50% is the correct frequency in order to make your opponent indifferent to bluffing. When your opponent bluffs $100 he is risking 100 to win 100 which means a bluff must succeed 50% of the time in order to break even.

From his perspective he must bluff half the time he value bets. Why? Because in order to make you indifferent, he needs to consider the odds you are given. If you call 100 to win 200 you need to win 33% of the time in order to break even on a call.

His value range includes QQ+ which is 18 combos. To make you indifferent he therefore needs 9 combos as bluffs sicne 18/2 = 9. Doesn't matter what cards he chooses just as long as it is 9 combos. Lets make his bluff range 88 when there is a diamond in the mix and 66 all the time.

Villains range: QQ+, 88(1/2 the time), and 66
Hero's range: JJ(1/2 the time)

These ranges have reached an equilibrium.


The reason the equity is different in equilab is because you are looking at hands specifically instead of ranges as a whole. Also if you are talking about turn and flop play it gets way more complicated and you got to make some estimations.



In real life though, people have to make decisions much quicker and you have to do some rough balancing. Perfect balance is too hard to do.

In general, when people talk about balance, they mean playing different hands the same way to make them less predictable. This is estimated balance. Say you raise preflop and the BB calls.

The flop comes Q J 2
Your opponent checks.
Your strategy is to never bet your str8 and flush draws and always bet Qx, Jx and better.

When you bet, your opponent will be able to make it difficult for you when another heart comes. When you check your opponent will be able to bet aggressively knocking you off all your nothing hands if you chase and hit a flush he'll know to stop since those are the only hands you are calling with.

Your strategy is face up and easy to read and beat. To make yourself tougher you have to balance.

To balance you have to bet your flush and str8 draws some of the time and check back Qx or Jx some of the time. This wont be exactly 50/50, in fact, often it won't be, but to balance your play you have to do it some of the time.


In real life though you can probably bet the flop 100% of the time as an exploit while starting out and from then you can adjust from there as the game continues. Just because you can balance doesn't mean that you necessarily should. Just be aware of what it looks like roughly.

Think about your collection of hands as a whole and how you can re-arrange your ranges to make it balanced as a whole. It doesn't have to be perfect just good enough so that you aren't ripped apart by an observant opponent.

I feel this thread has veered off into dangerous territory. The reason I went into as much detail as I did theoretical optimums was because I thought you were asking more for theoretical concept but not necessarily as a guide to actual application. Besides, as I said, I'm no expert on that particular topic as I've spent almost no time studying it.

I'd like to stress that you should never strive to achieve any theoretically optimum in poker because they are basically how you would play if you had a perfect read on all of the input data required which will never be the case. It may even assume a perfect opponent in this case.

What this means is if you try to bluff to the theoretical optimum every mistake you make in your opponents calling range or your calculations or even your own ranges overtime in this spot will have a large negative impact on the results and I'm guessing could easily actually make you overbalanced in bluffing and -EV against most players.

So, all those concerns of your's are quite valid but that's not a problem because that's not really the way to achieve balance in your game.

The fact that your opponents will be making mistakes means that you would do much better by adjusting to the players themselves which in general means, if they call too much, bluff less and if they fold too much bluff more.

The safest way to read that would not be by gauging how effective your bluffs are. Indeed bluffing actually has more value in getting you paid off more often when you have the best hand than for the value of the bluff itself.

Therefore the guage you should be paying more attention to is how often your value bets are getting paid off. Another words if they are calling value bets and paying you off very often there's no need to bluff much because they would fail too often and your already getting paid off for value anyway.

On the other hand if they were not paying your value bets off very much at all, then you could bluff more to both win with those bluffs and more importantly, induce the player to call more of your value bets.

This starts to hint at how balance is actually used by most advanced players and is an area which I really should have focused much more on for the best way to balance your game in general so I feel that by not focusing on how to best apply balance in a game has allowed the topic to drift into less practical applications so for the balance of this post I'd like to focus on that aspect.

Lets start at one extreme: If you only made value bets with good hands and never bluffed, players would easily pick up on this and you wouldn't be getting paid enough over time. If you started bluffing in similar spots, you would at least find bluffing those spots to be pretty successful and improve your results.

IF you continued to edge your bluffing rate up eventually they would also start paying you off more on your value bets while starting to pick off your bluffs slightly more but still with a net improvement between the two.

If you continue to bluff more then eventually your net value would start to drop because you are getting paid off but you are getting picked off more. Therefore a good balance is when you are getting the best EV from the net of both value bets and bluffs in similar spots.

The most frequent example is you raise pre-flop then bet the flop on an Axx board both when you have a big ace and when you only have variance that missed or have some kind of draw. Note that for this to work your bet sizing should be similar as well so that your opponent can't tell when you have it or when you don't

What this ratio should be is mostly based on the player's stats online or your history with the player live.

But it goes much further than that. For any kind of action there should be an opposite kind of action that looks the same in the same situation with a similar bet size but with opposite goals. So another example might be, a player may lead the flop against you and maybe you would raise with 2 pair or better. To balance the monsters you flop, you could also be raising with draws or thin information raises with over pairs.

Now it's difficult for your opponent to re-raise because of your balanced monster range vs semi-bluff range and it will also grant you a free turn card which you may elect to draw for free.

But even this check for a free draw might be balanced with your information raise to now pot control your over-pair against a check/raise while still giving you 2 streets of value.

Or if you hit your draw, you can balance your value bet with a blocker bet sometimes when you made a decent hand on the flop but now want to bet the danger card to block a bluff lead out on the river.

So now you can see another benefit of balance because by balancing strength with defensive betting you can make it more dangerous to bluff against you and you can more easily identify when your actually behind and make better lay downs.

Other action you can balance would be smaller inducing bets with defensive bets. Or River thin value bets, value bets, and bluffs with larger bets. Eventually you will be able to balance every action with an opposing action to a good degree.

Just remember that the bets must be made under similar situations, with similar bet sizing but with opposing goals and the frequencies of each should be balanced for optimum effect. So it's difficult to balance only monsters and semi-bluffs for example because monsters occur with much smaller frequencies and the proper bet sizing would be a little higher as well.

Sometimes to achieve this you may sacrifice the maximum value you get from a single monster hand due to bet sizing but with a net benefit of supporting your other bets for maximum over all value from all hands your balancing with.

Also remember this should only be done at levels where players are paying attention to your stats or the lines you take for the types of hands you play, or in live games where there are many regulars who are getting history on your play.

On the other hand, if you are playing games with a very large player base where your own stats on players are thin than balance may not be worth it and simply optimizing each hand for hand is going to produce better results.

wow ok guys thanks a ton for the information, really appreciate your efforts to contribute to this thread, it has helped me learn a lot.

Anybody else with any other input feel free to chime in.

Cheers

I need to add that the idea of balance goes ever further into other aspects of the game. I've already shown how merging your ranges will support your balancing game and vice-versa.

Also it's clear that styles play a huge role in balanced play but not only with the different approaches with regards to action in the game but also range selection plays a huge role in balanced play but I'll leave these topics to be covered at another time.

It works because your opponent is not balanced. You offer him even equity and he decline it by not being balanced.
When you play GTO you win from the fact that your opponent is not playing GTO game. You exploit the non perfect game of your opponent.

Enjoying this thread some great stuff in here

Hey!

You have been talking a lot about balancing the river range but how about balancing a preflop-shove-range when the equities are not 100 versus 0 like they are in the river situations?

Here's an example that I've been thinking and I think I haven't been thinking it right:

We 3bet our opponent to 10bb and he 4bets to 33bb. We know that he 4bets the top 5% of hands which is {88+,AJs+,KQs,AKo} and we shove a balanced range X. Theres 133 in the pot and our villain has 67 to call so he gets about 2 to 1 odds. Should we now find a range X that is 2 to 1 -favorite against this 5% -range? According to PPT that would be approximately the top 2% of hands: http://propokertools.com/simulations...5%25&s=generic

Now if we shove, the villain has about the same EV when he folds his whole range and calls his whole range. But I think the problem is that he gets more than zero EV when he calls the top of his range and folds the bottom of his range. How should the balanced range be constructed in this kind situation?

[The sizings are not the most usual ones, and they are made to get easy-looking odds and we also consider push and fold for the only choices to defend against this 4bet.]

In the preflop all-in scenarios things are a bit different than at a river situation where hand equities are static. In the preflop situation making your player indifferent will not necessarily be the best play. In this spot you make the best EV line possible given that your opponent knows your range exactly. Not your exact hand, but your exact range. You could probably do this by brute force testing out good possible shoving ranges.

So if you push with QQ+ assume your opponent knows this. You can calculate what his best calling range is by anything that can call according to pot odds. This would be KK+ and AKs. Now we calculate are EV by pushing with QQ and getting called by KK+ and AKs.

Compare that range with pushing JJ+, AKs, AKo. His perfect calling range, knowing your pushing range, would be to defend with 88+, AKs, and AKo. Now we calculate the EV of this situation and compare it with the other one in the paragraph above.


I haven't calculated it, but I believe the EV of the second one is better than the first. Notice that we really haven't folded out much of his range in the second. The GTO play is to take the best EV line and not make him indifferent to calling or to make him fold. I believe the reason for this is because this is not a zero sum spot. I just made that assumption right now so I may be talking out of my ass. Either way that's what I'm posting so deal with it.

Now try out all combinations of a pushing range and develop the perfect response and the one with the best EV is GTO... sort of.


A quicker way of solving this is through fictitious play using computers to calculate it for you which would be much faster.


You assume your opponent knows your range and not just your hand, because that is how it would be if your opponent knows you are going to play the best strategy. He can't know the exact hand, because there is no way for him to know. The range should be known though do to logic process of elimination. He now only needs to find what is perfect to defend it.

To simplify, consider the strategy Matrix below:


_____D_____E_____F

A****1*****9*****2

B****6*****4*****3

C****8*****2*****7

In this game Player 1 can only choose one letter 100% of the time. Player 1 wants a higher number. Player 2 wants a lower number.

If player 1 picks A then player 2 picks D resulting in 1 point.
If player 1 picks B then player 2 picks F resulting in 3 points.
If player 1 picks C then player 2 picks E resulting in 2 points.

Player 1 should pick B all the time.

Now if we solve this for a situation for where player 1 can choose one letter as a mixed strategy we get. A and C versus D and E where each one makes the other indifferent.

The mixed strategy scores better with 4.5 being the EV. However for a static range the first one would be optimal. So I guess I didn't really figure out GTO in my previous post after all, but only for a static range where ours can't change, but our opponent's can.

Also, what I said about not being zero sum. That was actually not true. What I should have said is making your opponent indifferent doesn't matter when there is a dominant strategy.

Oh yeah, I think I read something somewhere about making the bottom of our opponents range indifferent to calling so maybe that's where we are actually suppose to start.

Thanks a lot! Maximizing the EV when both players know each other's ranges can't probably be wrong when looking for the GTO.

I'm also wondering how to insert bluffs to this kind of GTO-pf-shove-range, but your posts maybe gave some good ideas where to start from.

Balancing can be both mathematical and non-mathematical , the mathematical balance is called GTO and none can use this yet or has ever perfect it and it can be -ev in a multiplayer game , other forms of balance exist and usually are about mixing your ranges or frequencies or both to become unpredictable and get paid when you have something good or force some hands to fold with junk or prevent bluff profit , this is not gto and it can lead to a terrible result if you dont use it well.


If for example someone calls your 3 bets 100% then its pointless and very -ev to 3 bet wide vs him , you can valuetown him when you have it , on the other hand if you are playing vs a tough opponent who can read hands well then 3betting a 2% range is not ideal , he will always fold , therefore you add more hands and you increase the chance to get paid when you have something good this is the non-gto balance , also you get money because you force folds with junk hands. Same applies with bluffs too , if none bluffs then bluffcathing is very -ev but tough opponents will try bluffs on you if you fold a lot therefore you have to call more with worse hands to prevent this from happening , Generally i believe the balancing of ranges or frequencies and so on is more of a strategy to play against tough villains. Think of it as an adjustment strategy rather than a get go strategy.

Not sure if anyone will find this cool, but I made this a long time ago based off of some theories that I read online:

http://forumserver.twoplustwo.com/15...right-1385754/

It is a little ridiculous and unrealistic in an actual game of poker, but the math is at least interesting.