I will study this part soon, but I speculate that if you do the same research on a 5.5/6 penetration game, your results will be different. Of course, your insurance index will be different too. Just do not fix on this one game of 4.5/6 penetration.
by: aceside
No, to your first premise, and no to your second, as well. But, yes, it is true that Gronbog's chart does not directly address your concerns.
We're talking at cross-purposes here, and you're even changing your own hypothesis from your previous one. Let me explain:
1. You originally speculated that, for Hi-Lo, a true-counted system, at different levels of penetration, we ought to be using different indices for the insurance decision, as if there were a sort of "floating" insurance index, which changes as penetration changes. That is simply not true.
2. Above, you submit a different hypothesis, namely, if a 5.5/6 game is played, instead of a 4.5/6 one, the single value for taking insurance will be different from the single value of a 4.5/6 game. That also is not true, but try to understand that it is a different assumption. When you devise an index for a game with x penetration, that is the single value that maximizes EV for the play across the entire spectrum of all penetrations. We do NOT devise "floating indices" for all different penetrations of the same game! Imagine playing a 6/8 game for which you know 100 indices. You would propose learning 1,300 indices, corresponding to the 13 penetration levels??? Fortunately, it doesn't matter what you propose, because they don't change, in any event.
3. Finally, I already explained to you that the principle you're espousing DOES exist for K-O, a RC system, and is one of the reasons I don't much care for unbalanced counts. There, since the RC increases by 4 per deck, for the imbalance, it stands to reason that indices must change as penetration progresses. That we nonetheless often just propose a single index number for each play, regardless of pen, attests to the inaccuracy of using K-O indices in such a fashion.
Clear?
Don