Thanks Don,

I wish your wife a speedy recovery.

It is a pleasure to read your posts (and your wonderful book BJA3 which I have read multiple times).In fact , I would like to take this opportunity to let you know that that IMHO, it ranks amongst the top 3 books I would recommend to anyone on BJ. I am awaiting the next book of yours.

I am also thankful that you put my misery to rest by clarifying the 2 hands 1 hand puzzle for negative scenario. I happened to notice one thing-in making the Sd calculation from the variance, you ignored the square root of R but that should not affect the results as it would affect both the cases in equal measure.

Are there any other articles that are not published in BJA3 that I can purchase or read?

regards
waugh

> Sorry not to get back sooner. My wife broke her wrist,
> so things have been a bit hectic here for a couple of
> days.

> I suppose that, over 30+ years, no question has ever
> been asked more than the one/two hands one, and,
> frankly, I usually don't have the patience to answer
> again and again. But yours, below, has a different
> twist, because of the negative e.v., so I will do my
> best to give you a thorough answer.

> Alone, correct. With others, two will be better.

> Right.

> My intuition tells me no, but we will do the math in a
> second. It may be very close.

> It will help, but it will not be superior to playing
> one hand at table min. Again, we are assuming that you
> are alone at the table. And again, let's do the math
> later.

> So far, so good.

> Well, we haven't considered the risk yet, and since
> SCORE (for positive-e.v. situations) is a
> risk-adjusted measure, we would want to do the same
> thing here and consider e.v./s.d, or (e.v./s.d.)^2, as
> well. We'll do that in a minute.

> No, that isn't true. You can never reduce variance by
> betting more! You bet more, you have higher variance.
> Period.

> The maximum card-eating effect will be when you are
> alone. When going from two hands on the table (yours
> and the dealer's) to three hands (your two and the
> dealer's), you are adding 50% more cards per round. Of
> course, you are also betting twice as much per round,
> so you really have to consider the two together.

> So, before we do any more complicated math, just
> consider the following: suppose there are just enough
> cards left to deal six hands. Playing one hand, that
> will be three rounds, and I will bet three units,
> until the shoe ends. Playing two hands, I will play
> only two rounds, but each will have two units bet, so
> a total of four. Why do I want to risk four units
> before the shoe ends, instead of three? Well, because
> we haven't yet factored in the variance.

> Let's do some more math. Suppose the e.v. for a hand
> in this situation is -2%. You bet one unit. And
> suppose the s.d. is, say, 1.15 units. That makes the
> variance 1.15^2 = 1.32 squared units.

> For two hands (total of two units wagered), the
> variance becomes 2*[1.32 + 0.48 (the covariance)] =
> 3.60 squared units.

> But, in the second instance, we play R rounds per unit
> of time, while in the first, we play 1.5R rounds. So,
> playing one hand, the total e.v. is -2% x 1.5R =
> -0.03R and the total variance is 1.32 X 1.5R = 1.98R.
> So, the Sharpe ratio
> (akin to the DI, Desirability Index, which I reserved
> for positive-e.v. situations = e.v./s.d.) is
> -0.03R/1.41R = -0.021.

> For the two-hand situation, the e.v. is -2% x 2 x R =
> -0.04R, while the variance is 2 x 1.80R = 3.60R.
> Sharpe ratio becomes -0.04R/1.90R = -0.021!!

> Ha! Same thing. Interesting, huh?

> Don