DI vs. SCORE

Can anyone tell me the benefits or drawbacks form one or ther other? And why not just use a return on investment figure like in business? Thanx

Under the correct assumptions, SCORE is merely the square of your DI. Why prefer one to the other. Because the dimensions of SCORE involve the reciprocal of time. If you put dimensions on DI, it would involve the square root of time.

Here is an example that may help. Suppose that we compare game A with DI of 7, and game B with DI of 8. In terms of DI's, B is 14% better. But if we look at their SCOREs, we see that A is 48, B is 64, and B is over 30% better. This SCORE tells us that game B is 30% faster than game A. That means that 100 hours of play at Game B is worth 130 hours at Game A, assuming you are betting optimally in both.

Here is an example that may help. Suppose that we compare game A with DI of 7, and game B with DI of 8. In terms of DI's, B is 14% better. But if we look at their SCOREs, we see that A is 48, B is 64, and B is over 30% better. This SCORE tells us that game B is 30% faster than game A. That means that 100 hours of play at Game B is worth 130 hours at Game A, assuming you are betting optimally in both.

I think you meant to say "game B has 30% faster bankroll growth". Although this provides a good approximation, it is worth pointing out it is not strictly accurate because the game B player will have a slightly large bankroll to bet optimally from at each and every stage. This "compound interest" effect is quite minor over short periods of time but becomes quite significant over the course of a professional, or even recreational gambling career.

Also, when discussing games with very large advantages, such as exposed hole-card games, this effect becomes very important indeed, and a completely different method of comparison is required to give a true picture.

With most risk/reward measures, as with counting systems, a trade-off exists between simplicity and accuracy.

What I said above :

That means that 100 hours of play at Game B is worth 130 hours at Game A, assuming you are betting optimally in both.

Let me make the statement more precise. Suppose Player A plays Game, and player B plays B. Suppose the SCORE of B is 30% higher than A. Let us assume that A and B have the same initial Bankroll, and that they have the same betting strategy. That betting strategy might be to bet a fixed level with an initial RoR of 5%. Or it may be resizing to Half-Kelly with 10% swing in Bank. Or something similar.

Now let WA be A's bankroll after 130 hours. Let WB be B's bankroll after 100 hours. Now these are random variables; they don't have a single value, they have a probability distribution. What I am saying is that the

Probability Distribution of WA is the same as the Prod Dist of WB. Their Means are the same, their Variances are the Same, their Medians are the Same, etc. etc. That is what I mean when I say 100 hours at B is worth 100 hours at A.

This does assume that A and B are "normal", i.e. that their outcomes have a Normal Distribution. This is obviously not quite exact in the real world, but it is a good approximation for blackjack.

I do agree that isn't a good method for games with "large edges", and that other measures are needed to capture the gain in those games. However, I don't think I agree about the example you mentioned: hole-carding. I believe that in BJ the gain available to hole-carding is only a little over 10%, and less with reasonable cover measures. (I think Morgan has posted some exact numbers, but I don't remember the precise values.) I don't think that is enough to change the normal approximation.

However, I do agree that there are other situations with big edge where our approximation breaks down. But I would rather not get into a discussion of those here.

I do agree that isn't a good method for games with "large edges", and that other measures are needed to capture the gain in those games. However, I don't think I agree about the example you mentioned: hole-carding. I believe that in BJ the gain available to hole-carding is only a little over 10%, and less with reasonable cover measures. (I think Morgan has posted some exact numbers, but I don't remember the precise values.) I don't think that is enough to change the normal approximation.

What I was commenting on with regard to the hole-carding was that, after a relatively short space of time your bankroll is significantly larger than it was to begin with causing the optimal bettor to resize bets. The gain compounds quite quickly.

I think you are talking about the skew you get with games like video poker. I would not comment on that here since its not a beginners topic, and a bit out of my depth.

Regarding the gain from hole-carding, there is some dispute over the exact figure. Braun's figure was 9.9%. I have heard of a figure of 13% from Morgan. I am unsure which figure to believe in the absence of independent verification, though I am virtually certain that Braun did not take insurance into consideration in the initial study which would explain some, if not all, of the discrepancy. I think there was a good reason for this, but I'm speculating.

...the 13% figure was including perfect insurance decisions.

ANS