Dynamic Insurance

[k_c]

Let me try to understand your results. Firstly, we fix the dealing depth at 26 of 52 cards (thus 0.5 deck remaining). Secondly, we calculate the deficit number of aces per deck in the remaining half deck:

(4-2)/0.5=4.

Thirdly, we find the insurance index drop because of this much deficit of aces:

1.4-(-2/0.5)=5.4.

Finally, we find that, for every one deficit of aces per remaining deck, the insurance index drops:
5.4/4=1.4.

The only thing I'm doing is providing below data for HiLo single deck where the high card group consists of (0 aces + tens):

Cards

47-48     insure if RC >= -4
46        insure if RC >= -3
45        insure if RC >= -4
36-44     insure if RC >= -3
24-35     insure if RC >= -2
13-23     insure if RC >= -1
4-12      insure if RC >= 0
3         insure if RC >= 1
2         insure if RC >= 0
1         insure if RC >= 1 (can't be > 1)

Cacarulo is out to provide the statistically best single true count (= 52*RC/(cards remaining) for each data point) and to further adapt that to aces removed as a reasonable simplification that would be easy to remember.

k_c

[Cacarulo]

The only thing I'm doing is providing below data for HiLo single deck where the high card group consists of (0 aces + tens):

Cards

47-48     insure if RC >= -4
46        insure if RC >= -3
45        insure if RC >= -4
36-44     insure if RC >= -3
24-35     insure if RC >= -2
13-23     insure if RC >= -1
4-12      insure if RC >= 0
3         insure if RC >= 1
2         insure if RC >= 0
1         insure if RC >= 1 (can't be > 1)

Cacarulo is out to provide the statistically best single true count (= 52*RC/(cards remaining) for each data point) and to further adapt that to aces removed as a reasonable simplification that would be easy to remember.

k_c

This table is much clearer than the previous one. The important thing is that our results agree 100%. The only detail is that I only take EV > 0 and you take EV >= 0. There are only two results where EV = 0: -4/48 and -2/24.
What we still do not agree on is the way of presenting the data. It is very complicated to carry out a control that depends on each depth. For me, the ideal is, given a penetration, to calculate the minimum TC from which the expected value is maximum.
In this particular case and with a penetration of 32/52 (20 cards left) the best index is -4.425532. In other words, we will buy insurance when the TC is greater than or equal to -4.425532.
The choice of this index arises from a table ordered by TC from lowest to highest. In it we look for the index that maximizes the expected value.
Here is an extract of the table:

| RC |  -2 | CR |  22 | FQ | 0.13152983652263245 | EV |  -0.01178311361355877 | TC |  -4.727273 | TCF |      -4.73 |
| RC |  -3 | CR |  33 | FQ | 0.14053631374905032 | EV |  -0.00891545881695388 | TC |  -4.727273 | TCF |      -4.73 |
| RC |  -4 | CR |  44 | FQ | 0.22826086956521735 | EV |  -0.01151330938564987 | TC |  -4.727273 | TCF |      -4.73 |
| RC |  -4 | CR |  45 | FQ | 0.23473635522664191 | EV |   0.00361247947454846 | TC |  -4.622222 | TCF |      -4.63 |
| RC |  -3 | CR |  34 | FQ | 0.14352121975417773 | EV |  -0.00482779202185191 | TC |  -4.588235 | TCF |      -4.59 |
| RC |  -2 | CR |  23 | FQ | 0.13133542597315229 | EV |  -0.00563441233306816 | TC |  -4.521739 | TCF |      -4.53 |
| RC |  -4 | CR |  46 | FQ | 0.34219858156028365 | EV |  -0.01058797026357294 | TC |  -4.521739 | TCF |      -4.53 |
| RC |  -3 | CR |  35 | FQ | 0.14687994217230230 | EV |  -0.00098852793468573 | TC |  -4.457143 | TCF |      -4.46 |
| RC |  -4 | CR |  47 | FQ | 0.25000000000000006 | EV |   0.02127659574468077 | TC |  -4.425532 | TCF |      -4.43 |
| RC |  -2 | CR |  24 | FQ | 0.13127110336382861 | EV |   0.00000000000000000 | TC |  -4.333333 | TCF |      -4.34 |
| RC |  -3 | CR |  36 | FQ | 0.15067512012645540 | EV |   0.00266662480199220 | TC |  -4.333333 | TCF |      -4.34 |
| RC |  -4 | CR |  48 | FQ | 1.00000000000000000 | EV |   0.00000000000000000 | TC |  -4.333333 | TCF |      -4.34 |
| RC |  -3 | CR |  37 | FQ | 0.15501466784779966 | EV |   0.00610036826646621 | TC |  -4.216216 | TCF |      -4.22 |
| RC |  -2 | CR |  25 | FQ | 0.13133542597315226 | EV |   0.00518365934642273 | TC |  -4.160000 | TCF |      -4.16 |
| RC |  -3 | CR |  38 | FQ | 0.15996246536863803 | EV |   0.00930548216808247 | TC |  -4.105263 | TCF |      -4.11 |
| RC |  -2 | CR |  26 | FQ | 0.13152983652263245 | EV |   0.00997032690378052 | TC |  -4.000000 | TCF |      -4.00 |
| RC |  -3 | CR |  39 | FQ | 0.16565633536577018 | EV |   0.01250001308246951 | TC |  -4.000000 | TCF |      -4.00 |
| RC |  -3 | CR |  40 | FQ | 0.17246082813195468 | EV |   0.01528887217046893 | TC |  -3.900000 | TCF |      -3.90 |
| RC |  -2 | CR |  27 | FQ | 0.13185563157124838 | EV |   0.01439701560539941 | TC |  -3.851852 | TCF |      -3.86 |
| RC |  -3 | CR |  41 | FQ | 0.18016361797959371 | EV |   0.01792862967709130 | TC |  -3.804878 | TCF |      -3.81 |
| RC |  -2 | CR |  28 | FQ | 0.13231356309311726 | EV |   0.01850395693034845 | TC |  -3.714286 | TCF |      -3.72 |
| RC |  -3 | CR |  42 | FQ | 0.18956425255502335 | EV |   0.02150865160700777 | TC |  -3.714286 | TCF |      -3.72 |
| RC |  -3 | CR |  43 | FQ | 0.20284949401508143 | EV |   0.02102222740000070 | TC |  -3.627907 | TCF |      -3.63 |
| RC |  -2 | CR |  29 | FQ | 0.13290768032282205 | EV |   0.02232653847771005 | TC |  -3.586207 | TCF |      -3.59 |
| RC |  -3 | CR |  44 | FQ | 0.21009353479288723 | EV |   0.03006582458637275 | TC |  -3.545455 | TCF |      -3.55 |
| RC |  -2 | CR |  30 | FQ | 0.13364183949005423 | EV |   0.02588394140253358 | TC |  -3.466667 | TCF |      -3.47 |
| RC |  -3 | CR |  45 | FQ | 0.25208140610545793 | EV |   0.02018348623853217 | TC |  -3.466667 | TCF |      -3.47 |
| RC |  -3 | CR |  46 | FQ | 0.21276595744680851 | EV |   0.04347826086956519 | TC |  -3.391304 | TCF |      -3.40 |

If you look closely, -4.622222 would be the first index candidate since the expected value is positive. The problem is that between this candidate and the next one (-4.425532) we find four that are negative. Therefore the best index is -4.425532 since the sum of all the products between the expected value and the frequency from the index upwards is the best.

Sincerely,
Cac

[k_c]

What we still do not agree on is the way of presenting the data. It is very complicated to carry out a control that depends on each depth.

I kind of see what you may be doing but I likely don't know all of the details.

A card counter is aware of only 2 things - present running count and some sort of estimate of penetration. The data we agree on eliminates true count to find a strategy. The purpose of true count is to simplify a lot of data to a single value, although it may not be perfect. I have not given up on sticking with running count and avoiding true count, which wouldn't be perfect either.

Consider this solution for generic single deck insurance, no side counting.

let dR = decks remaining = (cards remaining)/52
let RCi = running count index

RCi = 52/36*dR - 1/18

if RCi turns out to be >0 and <1, round to 1
if RCi turns out to be >1 and <2, round to 2
etc.

if RCi=0 or RCi=1 then that is the index

if present RC>=RCi, buy insurance

I think the above approach yields the correct decision for all cards remaining except 1.

Instead of having to compute a true count, RCi is computed from decks remaining and constant values.

Just an idea. I can explain how I get the constant values. Also I think I can use this approach for RCi estimate when side counting aces.

k_c

[Cacarulo]

I kind of see what you may be doing but I likely don't know all of the details.

A card counter is aware of only 2 things - present running count and some sort of estimate of penetration. The data we agree on eliminates true count to find a strategy. The purpose of true count is to simplify a lot of data to a single value, although it may not be perfect. I have not given up on sticking with running count and avoiding true count, which wouldn't be perfect either.

Consider this solution for generic single deck insurance, no side counting.

let dR = decks remaining = (cards remaining)/52
let RCi = running count index

RCi = 52/36*dR - 1/18

if RCi turns out to be >0 and <1, round to 1
if RCi turns out to be >1 and <2, round to 2
etc.

if RCi=0 or RCi=1 then that is the index

if present RC>=RCi, buy insurance

I think the above approach yields the correct decision for all cards remaining except 1.

Instead of having to compute a true count, RCi is computed from decks remaining and constant values.

Just an idea. I can explain how I get the constant values. Also I think I can use this approach for RCi estimate when side counting aces.

k_c

If what you are interested in is finding an index that avoids the conversion to TC, you have to go for the RC side. In SD it is interesting to know that for insurance it would not be necessary to convert to TC. The trick is this: We shall buy insurance only when the RC >= +1 and the total remaining cards is less than or equal to 38. This captures ALL the positive EV available. For any other option the expected value is negative or zero. The only exception that will obviously never happen is when the RC is equal to zero and there are only two cards left. In that case there is also positive EV.
What is simpler than the above? Coincidentally +1 with 38 cards remaining matches the perfect index whose TC = +1/38 * 52 = +1.368421 that we had discussed earlier.

Sincerely,
Cac

[k_c]

We shall buy insurance only when the RC >= +1 and the total remaining cards is less than or equal to 38.

Sincerely,
Cac

As you stated if there are 2 cards left and RC >= 0 buy insurance, but nothing is perfect and this is certainly good enough.

Single deck is very simple to remember by just only considering RC but the method could hopefully apply to more complicated data with minimal inconsistencies. Still just an idea.

k_c

[aceside]

As you stated if there are 2 cards left and RC >= 0 buy insurance, but nothing is perfect and this is certainly good enough.

Single deck is very simple to remember by just only considering RC but the method could hopefully apply to more complicated data with minimal inconsistencies. Still just an idea.

k_c

Earlier I calculated this: insurance contributes a tiny portion to player's edge. The probability of a dealer ace/face/ten up card as a function of TC can be written as,


P(TC)=0.001966xTC + 0.07674.


At TC=+6.5, when the dealer shows an ace up card, the player edge gain from insurance is 7.4%, and therefore, the average player edge gain from insurance is 0.66%.


However, the player edge gain from additional blackjacks at TC=+6.5 is 0.5x(6.4%-4.7%)=0.85%.


This means player's blackjacks are about three times more important than dealer's ones, if the insurance bet is limited to 0.5 of the main bet.

[DSchles]

This means player's blackjacks are about three times more important than dealer's ones, if the insurance bet is limited to 0.5 of the main bet.

And this assumes a player's bet spread of???

Don

[Norm]

insurance contributes a tiny portion to player's edge.

Well, lots of little things have small effects. It's the sum of lots of little things that matter.