I don't know the answer to all the questions posted here but I'd like to point a couple of unsolicited things out.
> What do you know about a player's method to
> calculate TC. If a TC correctly calculated
> should be 2.8, he will probably round it up
> to 3 and you consider this as 2.You're
> rewarding imprecision.
> I told you right from the start that I
> calculated both TC and Index to the tenth
> and this has to be more precise. It's a
> mathematical law.
No one said it's not more precise. What they said is that the effect is negligible. I'm not sure if they're correct or not but the way I can think of testing this is simple. Simply generate indices with your count as integers. Figure out the advantage by simulating billions as always. Multiply your count by 10 and then do the same thing but using the same betting ramp at the corresponding points and then see what the advantage is.
> Not linear?Are you kidding. When I run my
> CA-program with a loop over all TCs from -10
> to +10 with steps of 0.1, I can see with my
> own eyes that the results are very close to
> linear to say the least.
I don't see why he'd be kidding. Even you admit it's not linear since "very close to linear" and "linear" are not the same thing.
I'm still working on my full CA to get exact values (and I mean exact given an assumption that deck comps probs are based solely on removals and not effected by playing strategies - so they should be closer than any sim), but my CA does do insurance calcs exactly in this manner.
Below are the exact values for insurance expectations for 2D based on the exact Hi-Lo TC's. I used 2 decks because there are fluctuations between positive and negative EV's as seen below. In 1 deck there are also up/down fluctuations but not between positive and negative. It's important to note for this discussion though that even within any given exact TC, there can be many subsets that are pooled and these subsets do not all have the same expectation. So even within a given exact TC results are not linear. Anyways - here are the values:
Count EV Prob EV*Prob Sum(EV*Pr)
2 -0.008307116 0.002162763 -1.79663E-05 -1.79663E-05
2.025974026 -0.009294555 0.00057123 -5.30933E-06 -2.32757E-05
2.039215686 -0.008645806 0.000700784 -6.05884E-06 -2.93345E-05
2.052631579 -0.003271964 0.0005512 -1.80351E-06 -3.1138E-05
2.08 -0.006155593 0.00217214 -1.33708E-05 -4.45088E-05
2.108108108 -0.005095736 0.00056175 -2.86253E-06 -4.73713E-05
2.12244898 -0.01098195 0.00076957 -8.45138E-06 -5.58227E-05
2.136986301 -0.006029999 0.000610501 -3.68132E-06 -5.9504E-05
2.144329897 -0.010309278 0.00010723 -1.10547E-06 -6.06095E-05
2.166666667 -0.006833983 0.002371784 -1.62087E-05 -7.68182E-05
2.189473684 -0.00837717 0.000201088 -1.68454E-06 -7.85028E-05
2.197183099 -0.000248163 0.000546889 -1.35717E-07 -7.86385E-05
2.212765957 -0.005231034 0.000876093 -4.58287E-06 -8.32214E-05
2.228571429 -0.00262123 0.000606111 -1.58876E-06 -8.48101E-05
2.23655914 -0.008040041 0.00026114 -2.09957E-06 -8.69097E-05
2.260869565 -0.002290383 0.00248055 -5.68141E-06 -9.25911E-05
2.285714286 -0.004705672 0.000325025 -1.52946E-06 -9.41206E-05
2.294117647 -0.002896043 0.000604702 -1.75124E-06 -9.58718E-05
2.311111111 0.000508239 0.001015 5.15863E-07 -9.53559E-05
2.328358209 0.001355228 0.000598426 8.11004E-07 -9.45449E-05
2.337078652 -0.003807173 0.000359713 -1.36949E-06 -9.59144E-05
2.363636364 0.002761779 0.002474197 6.83319E-06 -7.11149E-05
2.390804598 0.001484744 0.000325971 4.83983E-07 -6.53216E-05
2.4 0.00100324 0.000602508 6.0446E-07 -5.86583E-05
2.418604651 0.000661438 0.00111267 7.35963E-07 -5.61188E-05
2.4375 0.000387156 0.00054501 2.11004E-07 -4.2537E-05
2.447058824 0.003715623 0.000386717 1.43689E-06 -3.82376E-05
2.476190476 0.002616516 0.002709996 7.09075E-06 -2.26955E-05
2.506024096 0.003274295 0.000412141 1.34947E-06 -1.76647E-05
2.516129032 0.005470656 0.000597707 3.26985E-06 -1.32894E-05
2.536585366 0.003442248 0.001107006 3.81059E-06 6.72995E-06
2.557377049 0.006329472 0.000540509 3.42113E-06 1.18356E-05
2.567901235 0.002619531 0.000423812 1.11019E-06 1.30815E-05
2.6 0.00442608 0.002701857 1.19586E-05 2.9623E-05
2.632911392 0.001543417 0.000422358 6.51874E-07 3.18637E-05
2.644067797 0.010571792 0.00059039 6.24148E-06 4.02047E-05
2.666666667 0.008050885 0.001092398 8.79477E-06 5.46809E-05
2.680412371 -0.010309278 2.60629E-05 -2.6869E-07 5.59417E-05
2.689655172 0.004505934 0.000596071 2.68586E-06 6.03788E-05
2.701298701 0.013258835 0.000407963 5.40912E-06 6.5272E-05
2.708333333 0 4.51399E-05 0 6.4461E-05
2.736842105 0.011655827 0.002785261 3.24645E-05 9.8295E-05
.
When the index is floored you get an index of 2. If you want the EV maximizing exact index you get 2.363636. If you take the weighted EV of all values from 2 up to but not including 2.363636 you get:
-9.59144E-05 or -0.01%,
or if you use 2.3 as the cutoff which would be the floored TC to 0.1 accuracy we get:
-9.58718E-05 or -0.01%
which represents the lost EV by flooring with a unit bet. So in the case of insurance at least with Hi-Lo in 2D, the potential EV lost is small by rounding.
It's not clear if the aggregate effect over the whole game may be bigger or smaller, but since insurance pays 2-1 which is greater than any other bet, it's probably safe to assume the effect is less for any given play. Since there are 550 different plays - if we assumed the effect was about the same for each play and that we could gain 0.01% on every play - there is a potential to gain of about 5%.
However, it's obvious that not every play has an index that appears during game play - if we assume only 22 plays actually matter - then we could potentially gain 0.2% if the bet were the same at each affected TC and more if you include the effects of the betting ramp. So it may be possible to get 0.3% when including the betting ramp.
It would be interesting to see the results of the index*10 experiment I suggested above.
Sincerely,
MGP
Bookmarks