CE - Certainty Equivalent, what exactly is it's use?
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sagefr0g
Well-Known Member
thank you for the answer regarding the interpretation of the term Kelly wager with regards to full Kelly betting and ROR.DSchles said:
another question, if I may.
would it be appropriate to treat the parameter B, in the following manner:
for a proposition where the bet size is limited to some given amount, would it be allowable to state that B = ev/bet amount for use in the f* equation?
sagefr0g
Well-Known Member
if I restated what you already wrote, then I feel confident that my restatement was indeed correct. problem was I truly didn't realize that I did indeed restate what you wrote (until you just informed me). I definitely have a problem with nomenclature, hence my use of the term ev in place of win b units and asking if it was proper. in other words, I truly wasn't sure one could equate ev with the term win b units. my questions are based in genuine lack of understanding (especially when it comes to nomenclature), for which i'm trying to overcome.DSchles said:
edit: not to worry far as trying to reinvent the wheel, it's just that I dabble in gambles other than blackjack, hence questions arise because the games are not so well documented as blackjack. end edit.
I do have a further question i'd like to ask that's related to this earlier post of mine: https://www.blackjackinfo.com/community/threads/ce-certainty-equivalent-what-exactly-is-its-use.16357/#post-496340
if I may.
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London Colin
Well-Known Member
b is the odds, a.k.a. the payoff. It comes from the rules of the game you are playing, nowhere else.
E.g.,
even money (1:1), b = 1.
two to one (2:1), b=2
three to one (3:1), b=3
three to two (3:2), b=(3/2)=1.5
But blackjack is an example of a game with multiple payoffs, and in which you can add to your initial bet (via splits and doubles) . So the formula f = (bp-q)/b can't be applied to blackjack (nor to any game that is not a simple 'bet 1 unit, either lose 1 unit or win b units' proposition).
An approximation (f = ev / variance) is used instead.
E.g.,
even money (1:1), b = 1.
two to one (2:1), b=2
three to one (3:1), b=3
three to two (3:2), b=(3/2)=1.5
But blackjack is an example of a game with multiple payoffs, and in which you can add to your initial bet (via splits and doubles) . So the formula f = (bp-q)/b can't be applied to blackjack (nor to any game that is not a simple 'bet 1 unit, either lose 1 unit or win b units' proposition).
An approximation (f = ev / variance) is used instead.
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"An approximation (f = ev / variance) is used instead."
And, an approximation to the approximation, which further simplifies calculations, is to substitute the average squared result of a hand for the variance. Since the latter subtracts the square of the expectation from the average squared result, and the e.v. is so small for most games, the square can be ignored without doing any damage to the calculation. Griffin was fond of expressing the optimal bet in that fashion.
Don
And, an approximation to the approximation, which further simplifies calculations, is to substitute the average squared result of a hand for the variance. Since the latter subtracts the square of the expectation from the average squared result, and the e.v. is so small for most games, the square can be ignored without doing any damage to the calculation. Griffin was fond of expressing the optimal bet in that fashion.
Don
London Colin
Well-Known Member
It's whatever the payoff is, so yes, of course it does not have to be a whole number of units, and can be less than one. (bearing in mind that this is in addition to getting your stake back.)sagefr0g said:
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sagefr0g
Well-Known Member
edit: the hypothesis below is erroneous, negative f* doesn't happen when ev is positive.
so don't bother reading further if you like end edit
maybe, the final question, maybe, lol:
your statements (Don & London) further de-foggify my mind, with respect to f*, lol.
permit me to digress a bit if i may, or tell me to just shut up if you wish. be assured i have no interest in argumentative discussion, as i pretty much despise yaking for yaking s sake. even though i admittedly have a very shallow understanding or knowledge of Kelly ROR, CE, f* and the like, i none the less realize that you gentlemen do, hence my hopes that i can be enlightened a bit. also, i have no qualms what so ever, with regards to any egotistical sense when it comes to whether or not i’ve made any incorrect assumptions. in other words, not a troll here, my questions are sincere. i could see where one might wonder about that, from my behavior. end rant.
as i stated, my minds been de-foggified a bit, far as f* goes. but, that only brings about more questions with respect to the essential problem that i’m trying to solve. i’ve yet to state the essential problem that i’m trying to solve, my error, apologies. so, here goes, i’m not in the least concerned about ROR, but only am concerned with the degree to which a particular bet is worthwhile in the most minimal sense, that minimal sense being up to a point, such as like unto when a blackjack player decides to use a risk averse index play instead of a basic strategy action (that normally is plus ev, sorta thing). but the gamble is not the game of blackjack and it does have a multitude of payoffs, but requires only one bet.
so anyway, far as the real problem that i’m trying to solve (stated above). since i now know that the formula f = (bp-q)/b can't be applied to blackjack (nor to any game that is not a simple 'bet 1 unit, either lose 1 unit or win b units' (uhmm & i believe also one gets their original bet back on top) proposition, then imagine the following. imagine, that one doesn’t care about any other payoffs existent other than the 'bet 1 unit, either lose 1 unit or win b units'. that being because, those other payoffs are nothing but greater profit (icing on the cake, sorta thing), and there are no other bets required such as doubles, splits or insurance. and (imagine this) the 'bet 1 unit, either lose 1 unit or win b units' portion of the payoff scheme is always positive ev, when you make the bet. so, the idea would be, ignore the other payoffs and just compute f* from the above described payoff. (note: far as the value of f* arrived at, i'm not so much interested in the fraction of Kelly bank to bet, but more so, is the value negative or positive and it's relative size). but here’s the rub, there are instances (dependent upon the state of the game) where you can know that f* is either negative (even while ev is positive) or f* is positive. meaning, (too me at least) that in the case where f* is negative, one might not want to make the bet (even if ev is positive), sorta thing.
is that too much of a stretch, in other words would doing what i’m suggesting, ignoring the other payoffs destroy the integrity of f* for the purpose i’m alluding to ?
so don't bother reading further if you like end edit
maybe, the final question, maybe, lol:
your statements (Don & London) further de-foggify my mind, with respect to f*, lol.
permit me to digress a bit if i may, or tell me to just shut up if you wish. be assured i have no interest in argumentative discussion, as i pretty much despise yaking for yaking s sake. even though i admittedly have a very shallow understanding or knowledge of Kelly ROR, CE, f* and the like, i none the less realize that you gentlemen do, hence my hopes that i can be enlightened a bit. also, i have no qualms what so ever, with regards to any egotistical sense when it comes to whether or not i’ve made any incorrect assumptions. in other words, not a troll here, my questions are sincere. i could see where one might wonder about that, from my behavior. end rant.
as i stated, my minds been de-foggified a bit, far as f* goes. but, that only brings about more questions with respect to the essential problem that i’m trying to solve. i’ve yet to state the essential problem that i’m trying to solve, my error, apologies. so, here goes, i’m not in the least concerned about ROR, but only am concerned with the degree to which a particular bet is worthwhile in the most minimal sense, that minimal sense being up to a point, such as like unto when a blackjack player decides to use a risk averse index play instead of a basic strategy action (that normally is plus ev, sorta thing). but the gamble is not the game of blackjack and it does have a multitude of payoffs, but requires only one bet.
so anyway, far as the real problem that i’m trying to solve (stated above). since i now know that the formula f = (bp-q)/b can't be applied to blackjack (nor to any game that is not a simple 'bet 1 unit, either lose 1 unit or win b units' (uhmm & i believe also one gets their original bet back on top) proposition, then imagine the following. imagine, that one doesn’t care about any other payoffs existent other than the 'bet 1 unit, either lose 1 unit or win b units'. that being because, those other payoffs are nothing but greater profit (icing on the cake, sorta thing), and there are no other bets required such as doubles, splits or insurance. and (imagine this) the 'bet 1 unit, either lose 1 unit or win b units' portion of the payoff scheme is always positive ev, when you make the bet. so, the idea would be, ignore the other payoffs and just compute f* from the above described payoff. (note: far as the value of f* arrived at, i'm not so much interested in the fraction of Kelly bank to bet, but more so, is the value negative or positive and it's relative size). but here’s the rub, there are instances (dependent upon the state of the game) where you can know that f* is either negative (even while ev is positive) or f* is positive. meaning, (too me at least) that in the case where f* is negative, one might not want to make the bet (even if ev is positive), sorta thing.
is that too much of a stretch, in other words would doing what i’m suggesting, ignoring the other payoffs destroy the integrity of f* for the purpose i’m alluding to ?
as an aside (sorry folks):
far as another question that pops into my gourd, if it’s ok (in the case of blackjack) to use f = ev / variance as an approximation of f* = (bp-q)/b then does that mean (at least in some instances) that (bp-q)/b ~= ev / variance ? i ask that, because one of my problems with the plays i make is that i don’t know the variance, standard deviation and have no simulation (that i know of) that can help with that. such that even an approximation of variance would be of interest.
far as another question that pops into my gourd, if it’s ok (in the case of blackjack) to use f = ev / variance as an approximation of f* = (bp-q)/b then does that mean (at least in some instances) that (bp-q)/b ~= ev / variance ? i ask that, because one of my problems with the plays i make is that i don’t know the variance, standard deviation and have no simulation (that i know of) that can help with that. such that even an approximation of variance would be of interest.
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It would help if you simply described the payoffs. You say to ignore all the others except one, where you win b units. But I don't know if we can trust your judgment to eliminate all the others, because, even though they might not occur often, if the payout is very large, that not only contributes to the positive e.v., but also, potentially, to increased variance. So, ignoring those payoffs might produce erroneous answers for f*.sagefr0g said:
Anyway, it's not practical to answer your questions without specifics.
Don
sagefr0g
Well-Known Member
with all due respect sir (and truly there is a hell of a lot of respect due! imho), i'd prefer not to describe the payoffs on an open forum.DSchles said:
that said, I pretty much understand your reticence with respect to my judgment to eliminate the others (payoffs) and the possible ramifications as you have alluded. very astute considerations, imho.
so it's great, fore warned is fore armed and I can now study up on those aspects and hopefully determine if what i'm hoping to do is viable or not. I had reflected upon such matters a bit previously but me thinks now perhaps not enough.
edit: probably, i'll get stuck studying up on these matter, which probably means more questions
end edit
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London Colin
Well-Known Member
If the bold part is literally true, then you can ignore the other payoffs if you wish. ( I think Don's reply is assuming that these other payoffs contribute to the ev you have calculated.)sagefr0g said:
What you seem to be saying is that you have -
p = prob. of wining b units.
q = (1-p), taken to be the prob. of losing 1 unit, but in reality includes some probability of winning other payoffs.
such that the e.v. (given by bp-q) is positive, even without factoring in the other payoffs.
Assuming that to be tue, it seems to me, you can only ever be underestimating the value of f, since what you are effectively doing is modelling the situation in which, whenever you win one of these other payoffs, you use it to tip the dealer or something, rather than add it to your own bankroll.
If the ev is positive then so, by definition, is the Kelly fraction, f. Think about what it signifies. You have an advantageous bet; the only question is what fraction of your bankroll should you devote to that bet. A negative value for f would equate to a negative ev.sagefr0g said:
As well as being common sense, this is demonstrated by the formla -
f = (bp-q)/b [where bp-q is the ev.]
hence,
f=ev/b
b is positive, so f has the same sign as the ev.
To be frank, I'm worried that you can claim to have an analysed a game to the point of identifying a +ev situation, while at the same time the type of questions you ask (in this and many other threads) imply a very limited grasp of the fundamentals of probability and statistics (even more limited than my own limited graspsagefr0g said:
).I hope this doesn't come across as too condescending, but, are you 100% sure that you have identified an actual advantage?
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Nice post, Colin. My concern is that I'd prefer to have the facts, rather than simply be told, "there are other payoffs, but you can ignore them." Again, the concern is that these payoffs do NOT just contribute, one way or another, to e.v., but especially to variance. And so, it could actually be that they DECREASE optimal f, because, while perhaps not contributing to increased e.v., they DO contribute to increased variance. And that is NOT a good thing!
Don
Don
sagefr0g
Well-Known Member
shall reply to the ........ out portion after some study.London Colin said:...............
To be frank, I'm worried that you can claim to have an analysed a game to the point of identifying a +ev situation, while at the same time the type of questions you ask (in this and many other threads) imply a very limited grasp of the fundamentals of probability and statistics (even more limited than my own limited grasp
I hope this doesn't come across as too condescending, but, are you 100% sure that you have identified an actual advantage?
thank you for being frank, sir, that's the ticket, far as i'm concerned. and worrying, nothing wrong with that (up to a point) since it denotes concern and genuine caring (similar to how Gronbog expressed that he wouldn't jokingly advise someone to do something stupid). it's been my experience that when someone (especially someone who excels at competence in a given field of knowledge) asks you if your sure, then just maybe you are missing something critical. so, 100% sure? pretty much I don't think i'd ever say that about anything, (it's a weird world) lol. how about this though, on a scale of 1 to 10 is she a pretty girl? she's a 10 (even better) sir, no doubt about it, none, lol. nuff said, her identity deserves to remain a mystery.
limited grasp of fundamentals of probability and statistics? you have me pegged, sir. never had an academic course in either, darn it, that's why I have so much trouble with nomenclature. but, i'll give myself this much, I've been schooled (non-academic) by undoubtedly some of the best of the best in those fields (people, who could in one simple sentence, maybe even just one word bring about for one an understanding of some really complex mysterious stuff), such that there's been more than a few ahh hah moments. there's one thing that I do excel at, the ability to recognize people who know what they are talking about. your grasp sir is not so limited, imho.
sagefr0g
Well-Known Member
shall reply to these matters, I believe you have awakened me to some aspects of the assumption far as ignoring the other payoffs that I didn't take into consideration. working on it as and when I can. unfortunately, dealing with a few other issues currently.London Colin said:
but i'll reply again, good, bad or indifferent. or maybe land up asking more questions, lol.
sorry, gonna take a bit of time...........
sagefr0g
Well-Known Member
as simply as I can put it.London Colin said:If the bold part is literally true, then you can ignore the other payoffs if you wish. ( I think Don's reply is assuming that these other payoffs contribute to the ev you have calculated.)
What you seem to be saying is that you have -
p = prob. of wining b units.
q = (1-p), taken to be the prob. of losing 1 unit, but in reality includes some probability of winning other payoffs.
such that the e.v. (given by bp-q) is positive, even without factoring in the other payoffs.
Assuming that to be tue, it seems to me, you can only ever be underestimating the value of f, since what you are effectively doing is modelling the situation in which, whenever you win one of these other payoffs, you use it to tip the dealer or something, rather than add it to your own bankroll.
If the ev is positive then so, by definition, is the Kelly fraction, f. Think about what it signifies. You have an advantageous bet; the only question is what fraction of your bankroll should you devote to that bet. A negative value for f would equate to a negative ev.
As well as being common sense, this is demonstrated by the formla -
f = (bp-q)/b [where bp-q is the ev.]
hence,
f=ev/b
b is positive, so f has the same sign as the ev.
To be frank, I'm worried that you can claim to have an analysed a game to the point of identifying a +ev situation, while at the same time the type of questions you ask (in this and many other threads) imply a very limited grasp of the fundamentals of probability and statistics (even more limited than my own limited grasp
I hope this doesn't come across as too condescending, but, are you 100% sure that you have identified an actual advantage?
you've wised me up.
back to the drawing board.
thank you London
sagefr0g
Well-Known Member
I added an edit to the text in which I posited the erroneous belief that f* could be negative even when ev is positive in order to warn readers in the future. https://www.blackjackinfo.com/community/threads/ce-certainty-equivalent-what-exactly-is-its-use.16357/page-2#post-496380
as a further note, I believe I was able to determine f* for the lowest payoff only while ignoring all other payoffs for the game in question. turns out the ev would only be positive at higher than normally observed states of advantage, and indeed f* would then be positive not negative. so my hypothesis to use f* to evaluate the 'worthiness' of a bet was incorrect. perhaps, there is a way to apply the idea of Critical f as written about in BJA page 372 but that requires information about variance, for which is lacking for this game.
as a further note, I believe I was able to determine f* for the lowest payoff only while ignoring all other payoffs for the game in question. turns out the ev would only be positive at higher than normally observed states of advantage, and indeed f* would then be positive not negative. so my hypothesis to use f* to evaluate the 'worthiness' of a bet was incorrect. perhaps, there is a way to apply the idea of Critical f as written about in BJA page 372 but that requires information about variance, for which is lacking for this game.
London Colin
Well-Known Member
As a general principle, I agree. There's too much scope for confusion and mutual misunderstanding if all the facts aren't known.DSchles said:
But, as to this specific point, he's not just saying we can can ignore these other payoffs; he's saying he has ignored them. They do not contribute to the ev he has calculated.
Which means...
The worst case scenario for any period of play (with regard to variance and the frequency with which these other payoffs occur) is that they never occur. But that is the very scenario that is being assumed and used as the basis of sagefr0g's calculations.DSchles said:
Any actual occurrences of these payoffs mean the bankroll is growing faster than has been allowed for (assuming the one payoff that has been analysed occurs with its expected frequency).
Hence my assertion that he can only ever underestimate the optimal fraction.
London Colin
Well-Known Member
Well, yes, I imagine that's what we were all assuming. The alternative would be a game with a player edge 'off the top'. (Not unheard of, but rare!)sagefr0g said:
This seems to be piling confusion upon confusion!sagefr0g said:
Certainty Equivalent, etc. is not something I know a great deal about. Critical f seems to be a total red herring. It's a measure of how much you would have to have bet in order for two alternative actions (such as hitting or doubling) to have the same C.E. Whereas, the C.E. itself is how much cash I would have to offer in order to induce you to sell me the hand. (I think.)
I'm not sure it's worth worrying about. If you know how to calculate your f* value, then you should probably just get on with betting whatever your customary fraction of that fraction is.
If you are still ignoring all the other payoffs, then -
- I don't know if there is a alternative formula for C.E. that refers to the payoff, b, rather than the variance. (or if it is close enough to simply substitute b for the variance)
- You can simply calculate the variance for any given hand (or whatever it is). If you can calculate the ev, then you must have all the pieces you need.