Haze this! Posted with prior typos on bjmath.com counting board and on yahoo.com card counters cafe. Here it is corrected, and proofed a bit.
Risk Adverse indexes are Inferior to Delta Force tables.
equation 1: normal algebraic indexes in general: -m*ssp/ip
ssp=sum of squares of point count ip =inner product/dot product EORs and point count values.
equation 2: ssp/ip gives delta TC to change edge by 1%
equation 3: ip/ssp = delta %/delta TC; TC=true count
label 1: equation 3 becomes delta %
label 2: x=50/50 algebraic index
label 3: y=extra TC needed to convert x to risk adverse index
label 4: delta%(bet) is with betting EORs
label 5: delta%(play) is with playing EORs
equation 4: y/delta%(play)=(x+y)/delta%(bet)
problem 1: equation 4 ignores the starting advantage and how it changes a risk adverse index
equation 5: (modifying equ 4 by problem 1): y/delta%(play)=SA+(x+y)/delta%(bet)
problem 2: linear solutions of equ 5 result in delta%(play)/delta SA>2
problem 3: practical solutions of equ 5 require new risk adverse tables for every change in SA
Proof of problem 2 is found in mathprof's posting on bjmath.com approximating risk adverse indexes using SBA to find an index with a bet at twice Kelly criteria fraction. This is instead of using a recursive method and SBA. But it also can demonstrate a tendency for small moves in SA percentage to require more percent "edge" in an index to make it risk-adverse.
Proof of problem 3 can be derived from problem 2 or by inspection of where small changes in SA result in changing risk adverse indexes by more than 3 TC points. This is well beyond the rounding done by hi-lo lite (discussed in 2nd editon Blackbelt in Blackjack) or hi-lo express (discussed on the card counters cafe). This can be taken as a minimum standard to require new risk-adverse tables be drawn-up for your given game, in terms of starting advantage.
Problem 4 comes from the limits to Jalib's true count theorem. in general profits in games are proportional to the volitility of the true count. All potential normal risk adverse indexes are a fraction of those that can be obtained by adding delta% tables to normal 50/50 edge indexes to DIRECTLY only add money where delta%(play)*(TC-50/50 index TC) is greater than the edge diagnosed at the time the initial bet was made.
An exercise for the reader:
Simulate the differences between the returns. I have found that the bankroll required for a near zero edge, in a marginal 8 deck shoe game, was 99.87% of the Kelly bankroll required for normal risk adverse indexes. In a deep one deck game, with parameters sized to get approximately .03 units per hand and an SD of 2.3 units per hand, this fraction dropped to 94.7%
Only 2 tables are needed: 50/50 indexes and delta%(play) tables. You also can make playing decisions with risk-adverse goals no matter what cover bets or less than optimal spreads you use.
or you can go ahead and follow the herd and still need new risk adverse tables every time the SA changes. Or am I just "casting pearls before______,"(clones)?