# Score

#### London Colin

##### Well-Known Member
I understand how the definition of SCORE, EV^2/Var, gives a numeric value by which one game can be meaningfully compared to another, but I don't understand how it also equates to the \$ win-rate per 100 hands, under the SCORE assumptions of \$10K bankroll and fixed, full-Kelly betting.

I'm not doubting the truth of it; it's just I've not been able to find an explanation in BJA3 (though it may well be in there, evading my gaze ). Could anyone clarify this for me?

#### blackjack avenger

##### Well-Known Member
The Essential Elements

The games you are comparing have to have some similar variables or you are comparing apples to oranges.
So:
10g bank
100 hands an hour
13.53% fixed ror, not kelly

The higher the SCORE the better the game when considering risk vs reward.

#### London Colin

##### Well-Known Member
blackjack avenger said:
The games you are comparing have to have some similar variables or you are comparing apples to oranges.
So:
10g bank
100 hands an hour
13.53% fixed ror, not kelly

The higher the SCORE the better the game when considering risk vs reward.
I understand all that. It's one thing to say that a SCORE of 100 means you win twice as much as a SCORE of 50; it's another to say that a SCORE of 100 means you win \$100. It's the reason for the latter that I don't understand.

#### blackjack avenger

##### Well-Known Member
SCORE & EV Do Not Flock Together!

SCORE and EV can be the same, but mostly not.

In your example it's easy to pick the \$100 SCORE over the \$50 SCORE. Let's see if we can complicate things.

Game 1
EV \$100
SCORE \$50

Game 2
EV \$80
SCORE \$60

Which is the better game?
Game 2 when considering risk vs reward.:joker::whip:

SCORE looks at more then just raw EV. It is quite useful because the highest EV would be to bet all on each hand, this however would have a very very low SCORE.

#### psyduck

##### Well-Known Member
I find SCORE is a tricky parameter. For the same game, SCORE will change depending on the betting strategy. Can one say changing betting strategy makes the same game a better game? I guess not.

#### London Colin

##### Well-Known Member
blackjack avenger said:
SCORE looks at more then just raw EV. It is quite useful because the highest EV would be to bet all on each hand, this however would have a very very low SCORE.
I don't wish to seem ungrateful, but you keep telling me what I've explained I already know.

I'm running out of ways to rephrase the question: Why does the measure EV^2/Var equate precisely to the win-rate, rather than just being a useful metric, an indicator of win-rate on a linear scale?

Obviously, by squaring EV we get squared units as both numerator and denominator, which I guess is what gives us the linear scale, but beyond that small insight my lack of mathematical knowledge leaves me shrugging my shoulders.

#### London Colin

##### Well-Known Member
psyduck said:
I find SCORE is a tricky parameter. For the same game, SCORE will change depending on the betting strategy. Can one say changing betting strategy makes the same game a better game? I guess not.
Well there is 'SCORE' and there is 'score', if you adhere to Don Schlesinger's wishes. By definition, I don't think there is any flexibility to change the betting strategy for SCORE.

D.S. does talk about the need to establish the 'maximum tolerable bet spread' for the game in question, which I suppose is something you might change your mind about, thus changing the SCORE. But that could be said to be an admission that the game is indeed a different one to the one you first thought it was, and hence your original SCORE was inaccurate.

On the other hand, you can create a personal score regime, based on your own bankroll, rather than \$10K, and perhaps also the limits of bet spreading that you yourself won't go beyond, and use that to compare games. Two games compared by SCORE might lead you to prefer one, whereas when compared by your personal 'score' you prefer the other.

At least, that is my understanding, based on having read the the relevant chapter a couple of times. :grin:

#### blackjack avenger

##### Well-Known Member
Yes, To SCORE Can Be Very Hard to Do!

If one were using the exact definition of SCORE it does need to be:
10g
optimal bet ramp
13.53% ror
100 hands an hr

In practice SCORE or score is often used without the strict guidelines. It has been used to compare same or different games using different bet ramps. So score can be used to compare same game bet ramps.

Some other ways to compare games are the DI and NO:
DI is the square root of SCORE or score.
NO is a long run measure.

I think both these methods are not as rigid as SCORE so both can be used when considering same or different games with different bet ramps.

I personally use score :joker::whip: to compare same or different games with different bet ramps which for me is very close to SCORE and has a high DI and low NO.:joker::whip:

Some have a harder time with SCORE or score then others!

Probably score to compare same or different games with different bet ramps would work for you because it can be personalized.

#### London Colin

##### Well-Known Member
London Colin said:
It's one thing to say that a SCORE of 100 means you win twice as much as a SCORE of 50; it's another to say that a SCORE of 100 means you win \$100. It's the reason for the latter that I don't understand.
Is there really no one willing to explain this to me?

#### blackjack avenger

##### Well-Known Member
Another Chance to SCORE

SCORE, score or DI are all measures of risk vs reward. No, a SCORE of \$100 does not mean you will win \$100. We still have to contend with SD.

comparing 2 games:

EV \$45
SCORE \$20

EV \$40
SCORE \$25

The first game has a higher return but with greater risk. If one is concerned with the most return vs risk then the 2nd game is preferred.:joker::whip:

Which game has the better score? Which would you prefer to play?

To win \$2 you risk \$100,000
To win \$1 you risk \$100

I would hope no one wants to play the first game even though it offers twice the reward over the second. So it appears risk vs reward should be considered.

The second game has the better score.:joker::whip:

#### London Colin

##### Well-Known Member
blackjack avenger said:
SCORE, score or DI are all measures of risk vs reward. No, a SCORE of \$100 does not mean you will win \$100. We still have to contend with SD.
You're continuing to miss the point, I'm afraid.

Here's a quote from BJA3 -
What makes this measure so appealing is twofold: First it puts an absolute value on an hour's worth of play. We see, immediately, from the charts exactly what our hourly "wages" will be. Second, and this is crucial, it relates more realistically the comparisons we are trying to establish between games and conditions. If two DIs are given as, say, 7.07 and 5.00, the point has been made that we really don't grasp from these numbers, whose ratio is 7.07/5.00 = 1.41, that the first game is actually worth twice as much as the second, on a risk-adjusted basis. This is because the SCORE squares the DI values, permitting us to understand that (7.07/5.00)^2 = 2.00, and that we win precisely double choosing the first game over the second.
It's the first part, which I've highlighted in bold, that I have been asking about. But for some reason, you keep focusing on the second part, despite the fact that I've explained more than once that I already comprehend that part.

Obviously it is far from true that each 100 hands you play will magically return the exact amount decreed by the SCORE. Nevertheless, SCORE is not unitless, as one might think; it is measured in "dollars won per 100 hands". All I'm asking is why? What's the mathematical basis for interpreting EV^2/Var in that way?

#### blackjack avenger

##### Well-Known Member
Don Knows How to Score

In the quote Don gives the DI's of:

7.07 = SCORE of \$49.98
5.00 = SCORE of \$25

Now Don is writing for the unwashed masses. The information provided in SCORE is more easy to understand then the same info given in DI.:joker::whip:

As far as will you make that much, yes if you follow the guidelines and play long enough, most will earn that much give or take.:joker::whip:

He answers the question of why SCORE,
it simplifies and makes the DI clearer.:joker::whip:

If you are contmplating 2 games the one with the better personalized score is probably superior, given same or similar real world conditions.:joker::whip:

#### London Colin

##### Well-Known Member
For Crying Out Loud!

blackjack avenger, I give up. It seems you are being deliberately obtuse.

I still live in hope that someone might respond to the question I actually asked, but I've given up any hope that it will be you.

#### SleightOfHand

##### Well-Known Member
London Colin said:
blackjack avenger, I give up. It seems you are being deliberately obtuse.

I still live in hope that someone might respond to the question I actually asked, but I've given up any hope that it will be you.
I'm not sure what you want, it looks like BA answered most questions that you may or may not have asked about SCORE. Try this, open up CVCX and turn on the option to automatically calculate optimum bets. Drop the minimum chip size to 1, use minimum chip size for bet sizing, set your BR to 10k, hands/hr to 100, and kelly size to 1. The WR of the game should be ~= to the SCORE.

Optimum bets for TC are calculated by BR*Advantage/variance, which is then multiplied by its associated TC's frequency and advantage, which is finally summed and mutliplied by hands/hr to get WR. So like BA said, while this does give you a theoretical WR, this is only under precise conditions , which is why SCORE is used for comparing games rather than obtaining WR. Personally, I like CE becuase you define your own BR and risk utility to more accurately compare games

#### sagefr0g

##### Well-Known Member
London Colin said:
You're continuing to miss the point, I'm afraid.

Here's a quote from BJA3 -
First it puts an absolute value on an hour's worth of play. We see, immediately, from the charts exactly what our hourly "wages" will be.

It's the first part, which I've highlighted in bold, that I have been asking about...
first off i don't understand the concept of SCORE at all, except that it can be used as a way of measuring the relative value of blackjack games.

but anyway looking at Kasi's spreadsheet, SCORE is DI squared and DI is 1000[ (winrate/hr divided by 100)/(sd/hr divided by 10)], i believe this is where it's by convention assumed that there are a hundred hands of play for each hour of play and the DI has to do with being one thousand time the ratio of a game's per-hand-seen win rate to the per-hand-seen standard deviation.

finally did you read page 203 3rd edition Blackjack Attack? errhh the stuff on DI. also the stuff about the Sharp ration and the ("M-squared") model?

trust me i realize i'm not making any sense here, just throwing out some stuff, wondering if you considered that stuff.
i sure don't understand it.

what ever ... |a| = sqrt a^2

#### psyduck

##### Well-Known Member
Let me give it a try:

For a \$10,000 bank, win of 100 hand = 100*(betsize)*adv. =

(DI)^2 = SCORE

Note: EV and SD refer to per hand.

Any error?

#### Kasi

##### Well-Known Member
London Colin said:
I understand how the definition of SCORE, EV^2/Var, gives a numeric value by which one game can be meaningfully compared to another, but I don't understand how it also equates to the \$ win-rate per 100 hands, under the SCORE assumptions of \$10K bankroll and fixed, full-Kelly betting.

I'm not doubting the truth of it; it's just I've not been able to find an explanation in BJA3 (though it may well be in there, evading my gaze ). Could anyone clarify this for me?
Well, I'll give it my shot lol. And, I'm not sure I understand it completely either.

Fundamentally, like you say, it is meant ( I think) as a way to make the best use of one's time. Assuming an hour means 100/rds seen or played. Also, as I'm sure you know it always also assumes a \$10K roll and an "optimal" bet ramp. It strives to always bet in such a way that one's ROR will always be close to the "Kelly Holy Grail" of 13.53% or so. (one is maximizing growth to roll).

I think it is also true that, for an optimal, fixed bet-ramp Kelly guy, \$Win rate per 100 rds seen or played will also equal the SCORE. Just look at his Tables to confirm how close the \$win/100 is to SCORE.

I guess "SCORE" has sometimes come to more generically mean "win-rate".

Not sure, but I'm not even sure I like your def of Score as EV^2/Var in the first place lol.

But I'm not sure I'm not lmao.

I just use Don's definition on page 203 for DI and square it to get SCORE.
It usually seems to agree with his stated SCORE in his tables. Although, like I say, what it REALLY may mean, not so sure lol.

The key to me is ROR is always the same by assumption. Basically, answering the question, how much and when should I bet my \$10K roll keeping risk at 13.53% or so, maximizing the growth to my roll, when pen, rules, index use etc change.

I guess as opposed to a guy who might bet exactly the same \$ramp at the same points in time when even though the pen has changed from 4.5/6 to 5.5/6 not realizing, maybe, how much lower his risk to roll has probably become by doing that.

If I did this stuff, likely just me, the point of SCORE to me was it showed to me it is posible to always bet with an equal risk, the one I chose to begin with, to my roll and let win rate fall where it may.

Probably no help London.

Oh yeah - and also that that underlying 13.53% ROR assumed in the SCORE assumption would be INSANE to actually play at.

#### Nynefingers

##### Well-Known Member
London Colin,

I think I understand your question. I'm going to take a stab at it, coming from a completely different direction than the previous replies. I've wondered the same thing before and your post has caused me to dig into this a little more. I want to say up front that I have not read anywhere close to all of the info in the references I'm listing. I'm using them because they had the specific pieces of the puzzle I happened to be looking for with that particular Google search, and they both come from http://www.bjmath.com (Archive copy), which seems to be a good source of information.

First, the SCORE is built around a 13.5% risk of ruin. According to page 7 of this link (Archive copy), a 13.5% risk of ruin corresponds to full Kelly betting based on the initial bankroll, with no bet resizing. So I'm starting with the assumption that we are betting full Kelly.

As sagefr0g stated, "SCORE is DI squared and DI is 1000[ (winrate/hr divided by 100)/(sd/hr divided by 10)]." Another way to say this is DI = 1000 * (EV per hand) / (standard deviation per hand). I'm going to drop the "per hand" from here on, but it is assumed. Continuing, we can say that SCORE = (1,000 * EV / SD)^2 = 1,000,000 * EV^2 / SD^2. And since SD^2 = Var, we get:

SCORE = 1,000,000 * EV^2 / Var

Now, this link (Archive copy) states that "The correct 'Kelly' bet at any level is Bankroll*ev/var, where ev is the unit gain per hand, and var is the unit variance per hand."

Since our bankroll is \$10,000, our correct full Kelly bet would be 10,000 * EV / Var. For correct Kelly betting, our bet size is directly proportional to our EV for each individual bet, so our average bet size should also be proportional to our average EV. Our average per hand winrate would be our average bet times our average EV, or (10,000 * EV / Var) * EV. Our hourly winrate at 100 hands per hour would be 100 * (10,000 * EV / Var) * EV = 1,000,000 * EV^2 / Var. Since this is the same as the SCORE, we can say that the SCORE equals our hourly winrate with a \$10,000 bankroll and a 13.5% risk of ruin.

More generally, hourly winrate at a 13.5% risk of ruin is equal to (hands per hour) * bankroll * EV^2 / Var. I haven't read BJA, so I don't know why SCORE was defined the way it was. I assume the formula isn't nearly so simple for hourly winrate at other risk levels, which probably makes 13.5% the only reasonable risk level to use for generating a simple to use formula. Regardless, our primary goal is to maximize our winrate for a given RoR, so creating a good way of comparing winrates is really what matters. We don't really care to understand how it affects us if our standard deviation changes. We don't care to fully grasp the impact if our %EV changes. With a predetermined acceptable RoR, those things will cause us to change our bet sizing to match our risk tolerance. We care how those factors interact to affect our hourly rate, give our risk tolerance. That's why I feel Don Schlesinger based SCORE off of the winrate formula above. It automatically rolls the EV and SD for a given situation into one number that allows direct comparison of one game or one situation to another for the purpose of determining which opportunity will give us the best winrate for a given risk level.

All that said, the constants we use for hands per hour and for bankroll are really irrelavant, although for the sake of getting numbers that have some meaning to us, it helps if we use a large enough constant to bring the formula's output up to a range we can easily think about. If the constant were one, we would be talking about how much difference there is in a SCORE of 0.000028 and 0.000032. Percentage-wise, that's no different than 28 vs. 32, and since no one will likely be playing at 13.5% RoR, the absolute numbers are not particularly useful either, but for many people it isn't as easy to think about the smaller numbers. That's one good argument for picking a large constant when defining SCORE. If we use the standard assumption of 100 hands per hour, and choose a \$10k bankroll, then that gives a constant of 100 * 10,000 = 1,000,000, which puts the numbers in the "right" range and makes it easy to explain to people what the SCORE really means in meaningful (if not particularly useful) terms (i.e. winrate on a \$10k bankroll with a 13.5% RoR). The fact that this makes the SCORE equal to DI^2 is an added bonus, but not really important. It is my view that the \$10k bankroll assumption was probably chosen simply because it made for a convenient constant for the formula.

I have no idea if this was the thought process Don went through in developing SCORE, but it does make good sense to me. Hopefully I've answered your question, but if not, then I hope I've at least given you some food for thought.

#### London Colin

##### Well-Known Member
Eureka (I think)

Thanks very much everyone. I think I now see the source of my confusion. While I had a conceptual understanding of what SCORE represents, I didn't really know the mechanism by which it is calculated. I stumbled upon the definition EV^2/Var (in BJA3, but only mentioned tangentially) and took that as my starting point, leading me to the question 'why should that equal win-rate?'

Really, win-rate should be taken as the definition, and how to calculate it should be the first question. (But to qualify as a SCORE, rather than just a generic win-rate, there must also be the fixed parameters of \$10K BR, Kelly-optimal betting without resizing, etc.)

Having digested the last few posts, I'll try and state what my current understanding is. Any corrections or additional thoughts would be welcome...

To calculate win-rate you just need to sum all the different betsize * EV * frequency, for each true count. And you can do that with any parameters you like, in terms of bankroll, bet ramp, etc., in order to generate a personalised score. That score then represents the win-rate that you can actually antcipate if you play in that manner. To get a meaningful comparison of games, however, you ought to ensure that you keep the RoR constant, though you are free to choose your own value, rather than 13.5%.

For the standardised parameters of SCORE, we get a slight simplification. Each betsize will be 10,000 * EV/Var [EV and Var being specific to each TC], and so the win-rate for 100 hands will be 100 *10,000 * EV/Var * EV [which is 1,000,000 * EV^2 / Var, as Nynefingers says]. And that also happens to be DI^2.

So, to summarise:

Win-rate is good way to compare games, if you structure your bet ramps to maintain a fixed RoR.

SCORE is an idealised win-rate, which coincidentally equals DI^2.

Is that all OK?

#### blackjack avenger

##### Well-Known Member

London Colin said:
blackjack avenger, I give up. It seems you are being deliberately obtuse.

I still live in hope that someone might respond to the question I actually asked, but I've given up any hope that it will be you.
I take my time and try to answer your question, which was not worded very well. I was the only one to try to help you for several days!

Many who have followed with responses repeated basically what I stated.