CE/WR ratio

MJ1

Well-Known Member
#1
Why do people say if the ratio of CE/WR is < 0.5 that means you are overbetting and if the ratio is > 0.5 than you are underbetting?

As a matter of fact, if CE = WR, wouldn't that mean the counter is extremely prone to risk and will not settle for anything less than his WR? For example, if his WR is $50/Hr with BJ, and he could earn $49/Hr at a regular job, he would forgo the $49/Hr and take the risky proposition of BJ in order to have a chance of realizing his $50/Hr expectation.

Now, could somebody please explain to me what about the aforementioned example implies the counter is 'underbetting' his BR? As he is prone to risk, wouldn't this mean that he is overbetting his BR?

MJ
 

blackjack avenger

Well-Known Member
#2
In Short

Overbetting is very bad, you will go broke or your bankroll will shrink.:joker::whip:

A $25hr job is roughly equal to $50hr bj. The reason is with BJ you risk money.

$25hr job vs $70hr ev take bj
$30hr job vs $50hr ev take job
roughly:joker::whip:

This is why CE/WR ratio is .5
 

NightStalker

Well-Known Member
#3
BJ is not a job

It is a business and it can be compared with other business models..

You have certain investment(bankroll) -liquid(trip br) and long-term(safe)..
You play on certain risk say 1% to earn 1/500 times your investment per hour or play..

Separating investor Vs players:
Now player's are doing a job which can be paid hourly or based on the results. In this case,player's share of EV(26$) at bj > Ev(25$) at other job..

And mostly the ratio of splitting between investor and player is 1:1..
Hence, probably the number 0.5
 

MJ1

Well-Known Member
#4
No offense guys, because I appreciate your effort, but neither of you answered my question. Why should the ratio of CE/WR be 0.5?
 

KimLee

Well-Known Member
#6
MJ1 said:
Why should the ratio of CE/WR be 0.5?
It's an approximation. CE=WR for small bets, because risk is negligible. But as you raise your bet, risk increases and the incremental CE declines. At the optimum, you get no additional CE for raising your bet.

So, the marginal CE ranges from WR to zero, and the average effect is around half the WR.
 

Nynefingers

Well-Known Member
#7
My understanding (possibly flawed?) is that the CE represents a fixed income rate (no variance) that results in the same utility as a risky wager with a given winrate and variance. As KimLee pointed out, when the wager is small compared to bankroll, the variance is insignificant and the CE is almost equal to the winrate. As the wager grows relative to the bankroll, the variance becomes more of a factor and causes the utility of the wager to grow slower than the winrate. At some point, as the winrate continues to grow, the utility will begin to shrink, and this corresponds to overbetting. We are betting more, getting a higher winrate, but it is actually worth less to us because of the increased variance. CE is maximized when CE is 0.5*winrate. As you bet more, the winrate will increase and the CE will decrease. You won't see the CE decreasing if you are solely focused on CE/WR (the ratio will be decreasing though), but CE/WR isn't the important stat. The CE itself is what matters.
 

KimLee

Well-Known Member
#8
Nynefingers said:
CE is maximized when CE is 0.5*winrate.
No, CE is maximized when marginal CE is zero (the first-derivative condition of calculus). I explained why this will make total CE approximately equal to WR/2.
 

sagefr0g

Well-Known Member
#10
muppet said:
what is CE again? :eek:
erhh i think it's a qualitative utility sort of thing that's been made quantitative by applying math...

"CE is a useful notion, which provides the intuitive quantitative understanding for accepting risk for greater return (so-called price of risk). The definition of CE is relatively simple: it is the minimum profit (percentage or absolute) one would require to forfeit an advantageous and risky investment opportunity. Equivalently, it is the maximum profit one would be willing to pay for entering such an investment opportunity."
from:
http://www.bj21.com/bj_reference/pages/certaintyequivalentanalysis.shtml
 

sagefr0g

Well-Known Member
#12
muppet said:
the qualitative aspect of CE is probably something that we should evaluate more often than we probably do.
it's the question of, 'is this gamble worth it to me?' sort of thing.
the math part can help but only we can answer the question regarding the worthiness of the gamble.
 

aslan

Well-Known Member
#13
sagefr0g said:
the qualitative aspect of CE is probably something that we should evaluate more often than we probably do.
it's the question of, 'is this gamble worth it to me?' sort of thing.
the math part can help but only we can answer the question regarding the worthiness of the gamble.
So is it to say, a guaranteed $100, say a wage, is better than say, a $150 dollar wager that I would on average win, because of the risk factor? I hate to keep beating this dead horse, but at least he doesn't feel it.
 

KimLee

Well-Known Member
#14
aslan said:
a guaranteed $100... is better than ... a $150 dollar wager?
You you prefer a free $100, or a 50% chance at $300? It's a personal decision you make whenever you size your bet. This example requires a Kelly bankroll of only $100. Unless you are a sub-Saharan slave, you would be nuts to refuse the gamble. Counters make big bets with a 2% advantage, not a 50% advantage.

Who died on June 15th?
 

aslan

Well-Known Member
#15
KimLee said:
You you prefer a free $100, or a 50% chance at $300? It's a personal decision you make whenever you size your bet. This example requires a Kelly bankroll of only $100. Unless you are a sub-Saharan slave, you would be nuts to refuse the gamble. Counters make big bets with a 2% advantage, not a 50% advantage.

Who died on June 15th?
My mixed breed terrier, Aslan, 19 years, 8 months.
 

MJ1

Well-Known Member
#16
KimLee said:
It's an approximation. CE=WR for small bets, because risk is negligible. But as you raise your bet, risk increases and the incremental CE declines. At the optimum, you get no additional CE for raising your bet.

So, the marginal CE ranges from WR to zero, and the average effect is around half the WR.
Hi Kim, thanks for weighing in on the matter. Okay I did some studying and I think I understand it, sorta.

CE = EV - [Variance / (2 * BR * k)]

-or-

CE = EV * Bet - [Var * Bet^2/ (2 * k * BR)]

I prefer the second equation because it includes the bet size. If all the other parameters are fixed and you vary the Bet, then CE is maximized when it equals 50% of EV.

MJ
 

sagefr0g

Well-Known Member
#17
aslan said:
So is it to say, a guaranteed $100, say a wage, is better than say, a $150 dollar wager that I would on average win, because of the risk factor? I hate to keep beating this dead horse, but at least he doesn't feel it.
lol, i dunno about your example, i just know i dread and hate losing $5, nay five cents, even if i know it's only an interim sort of thing towards earning some nice loot, while others lose thousands and hardly blink an eye.
so it's (this CE stuff) got a dose of 'different strokes for different folks' about it, far as i can see.:rolleyes:

http://en.wikipedia.org/wiki/Risk_premium
http://www.gametheory.net/mike/applets/Risk/
 
#18
Certainty Equivalent of Kelly

sagefr0g said:
lol, i dunno about your example, i just know i dread and hate losing $5, nay five cents, even if i know it's only an interim sort of thing towards earning some nice loot, while others lose thousands and hardly blink an eye.
so it's (this CE stuff) got a dose of 'different strokes for different folks' about it, far as i can see.:rolleyes:

http://en.wikipedia.org/wiki/Risk_premium
http://www.gametheory.net/mike/applets/Risk/
With a continuous resizing bank:
If you bet over 2Kelly your bank will shrink.
If you bet 2Kelly your bank will churn up and down.
If you bet 1.5Kelly your bank will grow at the same rate as betting .5Kelly.
If you bet Kelly your bank will grow at max rate.
If you bet less then Kelly your bank will grow at a submaximal rate.

Of course it is a matter of choice what one does:joker::whip:
 

sagefr0g

Well-Known Member
#19
blackjack avenger said:
With a continuous resizing bank:
If you bet over 2Kelly your bank will shrink.
If you bet 2Kelly your bank will churn up and down.
If you bet 1.5Kelly your bank will grow at the same rate as betting .5Kelly.
If you bet Kelly your bank will grow at max rate.
If you bet less then Kelly your bank will grow at a submaximal rate.

Of course it is a matter of choice what one does:joker::whip:
ok thanks bja, and with the above the errhh 'rate' of risk or i dunno maybe level of risk varies some way as well i guess?
i've never done a study of how that goes, but i believe the risk can go down as you whack away at full Kelly which i believe has that 'magic' number of circa 13.5% ror, sorta thing.
easy enough to see how it goes with some sims, i guess.
 

Nynefingers

Well-Known Member
#20
KimLee said:
No, CE is maximized when marginal CE is zero (the first-derivative condition of calculus). I explained why this will make total CE approximately equal to WR/2.
You are correct. I probably missed an "approximately" in that statement. I don't know the formula that relates CE and WR (or CE, edge, and bet size) so I didn't realize the math didn't maximize CE at exactly 0.5*WR.
 
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