I remember in one of Bart's podcasts (DPP or SO) that he talks about the appropriate % of the time that a good, thin value bettor should be value-owning (VO) themselves. I seem to recall the stat at something like 60-70% for maximum profitability. Does anyone know if/how this has been calculated? I've been trying to work on it with limited success. Here's what I have so far:

Net won per bet (EV) = Amt won*%won + Amt lost * % lost

=(1 unit)*(1-VO%) + (-1 unit) * VO%

=-2VO%+1

This makes sense because your ROI will be 100% if you never VO, 0% if you're 50/50, and -100% if you always VO. This works for a single bet. But overall value bet ROI (the optimal 60-70% that was quoted in the podcast) needs to take into account frequency and sizing. You can plug in numbers for VO to look at spot ROIs, but VO% itself is a function of both frequency and sizing. So VO is related to frequency and sizing with something like:

VO ~ frequency*sizing

I think this makes sense, since the greater frequency you value bet, the more likely it is you will VO, and the bigger the bet sizing, the more likely that you will VO. We could plug in some different sizings; maybe something like a different curve for average thin value bet sizings as a % of the pot. But how to determine frequency? Maybe frequency would be a function of the relative strength of your "thin" hands to possible better hands? Something like if you're betting with a hand that beats say, 10% of possible hands on the river, vs 20%, 30%, etc? Obviously this is quite situationally dependent IRL, but is it possible to generalize for math's sake?

The ideal outcome to this exercise would be to get some sort of peak overall ROI values based on VO% to show the optimal point that Bart discussed (or find that, based on sizing, etc, that it's at a different spot!)

Thoughts?

## Comments

I've always struggled with putting this exact concept into a mathematical formula and would welcome some answers here. I think that there a definitely variables that come into play such as frequency of being check raise bluffed, sizing, hand strength etc.

Bart

752Membernot sure if I am missing something, so please don't take offence by anything that I am saying. Your equations are correct IMO. I am just not sure if your attempt to put this concept into a mathematical equation makes things rather complicated instead of more understandable? Or to put it differently, I guess I don't understand what kind of new information you expect to gather by this exercise. My feeling is that you will get only results that provide the same information as your underlying assumptions already contain.

The reason I am saying this is the following: In order to valuebet "perfectly", i.e. with the optimal frequency, we have to tailor our betsize in a way that makes villain call with worse hands 50.000001% and call or raise with better hands 49.9999999% of the time. That's all we can say for certain. But everything else really depends on the specific situation, villain's tendencies, your image, etc.....

One example: you said that "the greater frequency you value bet, the more likely it is you will VO, and the bigger the bet sizing, the more likely that you will VO". To me, this assumption is going to be outright wrong in many situations (the reason is that your assumption implies villain's calling range to be static). E.g. if you generally valuebet thinly and with large sizings, villain will perceive you as an aggressive and polarized (as Bart would say, since he doesn't see anybody else in the player pool do that), so he will adjust by calling down lighter. Therefore, the percentage that you will get called with worse hands might actually go UP against some villains, despite (or even because) of a larger sizing!

(I am aware that opponents on lower stakes don't really adjust properly, but calling an aggressive player down lighter is something that I see frequently, it's a kind of intuitive adjustment made many players make.)

This is why I believe that all other parameters (i.e. input variables you would need for your formulas) are so situational dependent that you won't be able to calculate anything you don't know yet beforehand.

Hope that makes sense to you. Again, please be aware that I don't want to bash your idea. Everything I said is meant in a critical, but factual way (I don't know if that's the correct way to phrase this in English).

702Subscriber'The Mathematics of Poker' must cover some of this material and I know that this is the approach that they take for at least some cases. I own it but haven't gotten around to actually studying it yet. Maybe someone who has looked at it in more detail can comment.

29SubscriberProfessionalNo worries, I definitely appreciate all feedback and critique, and your well-reasoned response. And I understand your main point, that since we "know" that one's ROI/hourly will go up if you make thin value bets, that the complex variables involved in really "solving" the different cases might not be worth the time. Still, I just thought it was an interesting idea to try and come up with some mathematically-based optimal range for most cases. For example: if you use the basic EV equation, your return on value bets with a VO% of 49.999999 is .000002 units. But if you were at 60/40, your EV is 0.2 units for the same bet size. So for the VO of 49.999999 to have a better long-run ROI than the VO of 40, its frequency needs to be .2/.000002 = 100,000 times greater to show more profit. Clearly this is an extreme case, but you see the point. And a VO% of 0 seems great because your EV is 1 unit, but compared to the 60/40 case you would mathematically be (intuitively, are) losing money if your bet frequency is less than 1/.2 = 5 times as frequent. And this doesn't even take into account sizing differences. Maybe the VO of 0 always bombs river thinly because he is, say, overplaying an overpair and gets looked up by top pair all the time. Then the VO of 0 is doing even better than the 60/40 who uses smaller sizings, unless the 60/40 makes a lot more bets.

So given those above examples, I was trying to find out, say, is it LIKELY that your frequency of value betting at a 70/30 VO clip will show more profit than a 60/40 VO (maybe keeping bet sizing fixed, or changing bet sizing, etc). True, I was assuming a static calling range by villians (and very good point by you with a variable calling frequency based on experience!). But the changing perception of your average villian could also be a function to be examined! For example, if you could say that vs some villians increased frequency is actually inversely proportional to VO, then maybe you adjust the overall equation with a "villian factor", or a "perceived aggressiveness factor" that actually shows your winrate going up vs this type of villian. Again, complicated and perhaps not worth all the work it would take to account for the variables (for example, as Bart pointed out, maybe if your frequency increases too high against other villians they start c/r bluffing you more often!), but still an interesting thought experiment if nothing else IMO.

OminousCow: I too own this book and haven't really read it cover to cover. I don't remember seeing any equations for value own percentage, though there is lots of stuff with EV, ROI, etc. I'll have to skim it to see if there's anything additional there.

702Subscriber752MemberSo, I might be contradicting myself here - I think such thought experiments are generally helpful. I am just not sure which information we expect to get as a result in this example. But maybe that's the wrong way to think about it. A better way to approach this might be "the journey is the reward"!?

752Member752MemberYour equation -2VO%+1 basically can be read as a straight line with a slope of -2. It is zero for VO%=0.5. This means that every bet with which we value own ourselves less than 50% is +EV (positive part of straight line, left hand side of graph), all other ones are -EV (negative part of straight line, right side of graph). However, not all bets have the same profit (compare with other thread started by Heisenberg lately). Let's assume we are able to valuebet perfectly. The very first be we make is with the stone cold nuts, so it will get called a 100% with worse only. Therefore, it's EV is (bet amount) x (% being called), and we can't valueown ourselves with it (VO%=0, ROI is 100%). Contrary, the very last bet will be break-even, since we will get called or raised with better 50% of the time (VO%=50%, ROI=0). That's why a thin value bet can be called "marginal".

But be aware these bets are only specific data points on your straight line. Now, the arithmetic mean (and also median) VO% of all bets should be 25% (ROI of 50%) I think. But this information doesn't really help us make better decisions. (The total value of all bets should be equal to the integral of the straight line, which is -VO%^2+VO% I believe. The maximum of this function is at VO%=0.5)

IMO all this stuff only proves that in order to maximize your total EV over all situations, you should make every bet which you expect to get called with weaker hands at least 50% of the time (this is the "last" or "marginal" bet).