rukus said:
youre variance of 1x100 and 2x75 are about equal. youre variance would be lower if you split your 100 bet into 50 and 50, but not if you split it into 75 and 75. the 75% "rule" was determined by setting the variance for the two scenarios (1 hand vs 2 hands) equal to each other. same thing for the 3 hands. but in your case of 2x75 vs 3x50, yes your variance would be lower in the 3 hand scenario.
Well, the way I understand it, your variance of 1*100 vs 2*75 would not be equal.
Assume, to keep it simple, an optimal bet defined by roll*%advantage divided by the sum of variance and co-variance if any.
Let's take Wong's benchmark numbers from Table 85. Variance=1.28 and co-variance=.47. 1/1.28=.78. 1/(1.28+.47)=.57. 1/(1.28+.47+.47)=.45. Which explains his factors in Table 86.
So now, let's assume a 1.28% adv with a $10K roll and determine an optimal bet for 1 hand. It's 78% of your edge * your roll or $100. For 2 hands it's 57% of your edge *$10K= $73 on each of 2 hands and a $146 bet in total. For 3 hands it's .0128*.45*$10K=$57 for each of 3 hands or $171 in total.
Hence Mr. Renzey's numbers of 2 * 73 or 3 times 57 if one chooses to express as a percent of your original optimal bet. The same %'s, btw, Don uses in Table 2.4 when analyzing the "eating up good cards" stuff.
In each case, one is betting more money in 2 hands than one hand and more in 3 hands than 2 hands. One is "getting more money on the table" to get a higher $win rate. At the same time, one is increasing variance. Since we're optimal betting here, giving rise, I think to a Kelly-risk, it's the ratio one's win rate has increased compared to the ratio one's variance has increased. When done right, the risk to one's bankroll remains unchanged in effect enjoying a higher win rate with the same overall risk to roll. The trade off for more money on the table is more risk.
So, in PrimeTime's case, if his orig $100 bet was an optimal one, betting 2 @75 might be a slight overbet but fine. Betting 3 @ $50 maybe makes little sense in a way since, yes, he has decreased variance but his win rate remains the same. In effect he is playing with a lower? risk to his roll than he was with his original $100 bet. Which is fine but then why not play with that risk all the time kind-of thing if you like it so much? It's probably not that big of a deal in reality lol.
Also, note how variance and co-variance will be different with different rules in place. Although that 2*73 and 3 * 57 always seems to be about right from what I can tell lol.
Also I'd guess co-variance would also change at different TC's and also change with use of indexes, just as variance does, and that his 0.47 (and 1.28) in this case is probably just an average over all hands? Any comments on this?
And, given the above, and since in real-life a fixed-spread guy can't always make the optimal bet, it's just one more reason to let a sim figure it out for you anyway lol.
Auto Monk - not sure I quite get your mnemonic lol. Something about it doesn't seem quite right to me but I'm not sure what lol. Not that that means anything anyway lol. Maybe something to do with stan dev? Like for $100 at three hands I'd go $100* square root of 3 times the stan dev for the 3 hands? Whatever, so what if it 2*71 may be a slight underbet or not lol.
There's alot more I don't get about spreading to multiple hands than I think I might get lol.