Not MathProf ...
... so, you'll just have to settle for me! :-)
"I was wondering, why not just take the ratio of win rate per hand seen to SD per hand seen and then convert this to SD per Hr to WR per hour? Wouldn't that be more meaningful than DI in and of itself?"
Perhaps ... to you! If you reread the pertinent BJA3 sections very carefully, you'll understand that DI mimics the Sharpe ratio, one of the most universally well-known and accepted measures of the "desirabilty" of an investment. DIs of, say, 1 to 10 also mimic the univerally accepted practice of rating the "goodness" or quality of something on a scale of 1 to 10, that just about the whole world seems comfortable with -- except you.
"In BJA3, the author says"
You keep writing "the author," like you aren't sure who he is. :-) Maybe you could write "Don."
"DI = [win rate per hand seen/SD per hand seen] x 1000. Wouldn't the ratio be more meaningful without multiplying by 1000?"
Well, again, maybe to you. If you'd like to talk about DIs of 0.006, be my guest. I just assumed that most people would prefer "normal" numbers like 1, 2, 3, 4. You seem to be an exception. :-)
"For example, if the DI for a game is 5, this does not really tell us anything without us doing some extra work. Just read on."
Neither does the Sharpe ratio. It's a number, on a relative scale, designed to be compared to other such numbers.
"Let's work backward. Divide 5 by 1000 = .005. Now we know the ratio of WR/Hand to SD/Hand is 1 to 200. Assuming 100 hands dealt per hour, then WR = $100 and SD = $2000. Thus, on an hourly basis, the ratio of SD to WR = $2000/$100 = 20 to 1. The smaller this ratio the stronger the game. This seems like a simple way to compare games with different rules, spreads, etc."
And Brett Harris thought that way a long time ago. His "N0" (N-zero)
-- see BJA3, bottom p. 212, top p. 213 -- is exactly the reciprocal of the SCORE, rather than the DI, without the 1,000^2, or one million, multiplier. The value represents the number of hands needed to be played such that the e.v. equals one s.d., or such that e.v. -s.d. = 0. And so, smaller is better.
"What do you guys think?"
"On another note, I do realize that DI^2 = SCORE."
It's hard not to realize that, if you've read the chapter carefully.
"But why would squaring this number give the hourly WR which is optimized for kelly and a 10k BR?"
Explained in BJA3, pp. 152-154. Average (optimal) bet is (edge/variance) times (Kelly) bank. Multiply that by your edge and you get edge squared over variance, which is SCORE. In turn, square root of SCORE is simply edge/s.d., which is DI, or Sharpe ratio.
REREAD the explanations!