Cacarulo: Optimal way of playing two hands

[Cacarulo]

Hi to All,

The answer to this problem is very interesting. Let's see what we get when playing solo and
what when playing two hands all the time. Remember that there are no other players
at the table and speed is not considered.

Conditions: 6D,S17,DOA,DAS,SPA1,SPL3,NS,5/6,Hi-Lo,C22 floored, 10000 million rounds.

1) Playing Alone

 play-all | rounds played = 100.00% 

 spread   ev/h   sd/h   ror%      n0    di  score      ekb   avb    unit  kelly 

 ------------------------------------------------------------------------------ 

 1 -  8  1.841 36.238  13.53   38735  5.08  25.82   713.21 2.147  14.021  1.000 

 1 - 12  2.752 46.665  13.53   28749  5.90  34.78   791.22 2.507  12.639  1.000 

 1 - 16  3.551 55.899  13.53   24785  6.35  40.35   880.02 2.821  11.363  1.000

Note that for 1:16 the betting pattern should be:

TC <=  -1 ==>   1.00 u | $  11.36 

TC  =   0 ==>   1.00 u | $  11.36 

TC  =   1 ==>   2.90 u | $  32.96 

TC  =   2 ==>   6.48 u | $  73.62 

TC  =   3 ==>  10.07 u | $ 114.41 

TC  =   4 ==>  13.75 u | $ 156.22 

TC >=   5 ==>  16.00 u | $ 181.81

2) Playing two hands all the time

 play-all | rounds played = 100.00% 

 spread   ev/h   sd/h   ror%      n0    di  score      ekb   avb    unit  kelly 

 ------------------------------------------------------------------------------ 

 1 -  8  3.730 60.143  13.53   26001  6.20  38.46   969.78 2.144  10.312  1.000 

 1 - 12  5.570 77.596  13.53   19408  7.18  51.53  1081.00 2.508   9.251  1.000 

 1 - 16  7.191 93.167  13.53   16784  7.72  59.58  1207.00 2.827   8.285  1.000

Note that for 1:16 the betting pattern on each hand should be:

TC <=  -1 ==>   1.00 u | $   8.28 

TC  =   0 ==>   1.00 u | $   8.28 

TC  =   1 ==>   2.91 u | $  24.08 

TC  =   2 ==>   6.47 u | $  53.62 

TC  =   3 ==>  10.10 u | $  83.65 

TC  =   4 ==>  13.79 u | $ 114.28 

TC >=   5 ==>  16.00 u | $ 132.56

It's obvious that under these considerations it's better to play two hands all the time.
But, what if we restrict our second hand to a particular TC? Say that our second bet will
be placed only if the TC is greater or equal to a certain TCX. The problem is now reduced to
determine what this TCX should be. The answer is not straightforward since we
need to determine this value by a trial and error process. Besides, it's
SPREAD related.
So here we go. I'm not going to post every single TCX, just the ones that matter for
optimal spreads lower or equal 1:16.

3) For TCX = -3

 play-all | rounds played = 100.00% 

 spread   ev/h   sd/h   ror%      n0    di  score      ekb   avb    unit  kelly 

 ------------------------------------------------------------------------------ 

 1 -  8  3.573 56.525  13.53   25032  6.32  39.95   894.31 2.026  11.182  1.000 

 1 - 12  5.304 73.555  13.53   19231  7.21  52.00  1020.01 2.374   9.804  1.000 

 1 - 16  6.825 88.590  13.53   16848  7.70  59.36  1149.86 2.674   8.697  1.000

Note that for 1:16 the betting pattern on each hand should be:

TC <=  -1 ==>   1.00 u | $   8.70 

TC  =   0 ==>   1.00 u | $   8.70 

TC  =   1 ==>   2.77 u | $  24.11 

TC  =   2 ==>   6.16 u | $  53.61 

TC  =   3 ==>   9.61 u | $  83.59 

TC  =   4 ==>  13.12 u | $ 114.14 

TC >=   5 ==>  16.00 u | $ 139.15

4) For TCX = -4

 play-all | rounds played = 100.00% 

 spread   ev/h   sd/h   ror%      n0    di  score      ekb   avb    unit  kelly 

 ------------------------------------------------------------------------------ 

 1 -  8  3.644 57.784  13.53   25147  6.31  39.77   916.33 2.066  10.913  1.000 

 1 - 12  5.431 75.193  13.53   19168  7.22  52.17  1041.02 2.425   9.606  1.000 

 1 - 16  6.964 90.086  13.53   16733  7.73  59.77  1165.28 2.725   8.582  1.000

Note that for 1:16 the betting pattern on each hand should be:

TC <=  -1 ==>   1.00 u | $   8.58 

TC  =   0 ==>   1.00 u | $   8.58 

TC  =   1 ==>   2.81 u | $  24.08 

TC  =   2 ==>   6.25 u | $  53.60 

TC  =   3 ==>   9.75 u | $  83.69 

TC  =   4 ==>  13.30 u | $ 114.13 

TC >=   5 ==>  16.00 u | $ 137.31

5) For TCX = -5

 play-all | rounds played = 100.00% 

 spread   ev/h   sd/h   ror%      n0    di  score      ekb   avb    unit  kelly 

 ------------------------------------------------------------------------------ 

 1 -  8  3.686 58.622  13.53   25293  6.29  39.54   932.29 2.093  10.726  1.000 

 1 - 12  5.501 76.162  13.53   19167  7.22  52.17  1054.42 2.457   9.484  1.000 

 1 - 16  7.048 91.052  13.53   16689  7.74  59.92  1176.25 2.757   8.502  1.000

Note that for 1:16 the betting pattern on each hand should be:

TC <=  -1 ==>   1.00 u | $   8.50 

TC  =   0 ==>   1.00 u | $   8.50 

TC  =   1 ==>   2.83 u | $  24.08 

TC  =   2 ==>   6.31 u | $  53.65 

TC  =   3 ==>   9.84 u | $  83.63 

TC  =   4 ==>  13.43 u | $ 114.15 

TC >=   5 ==>  16.00 u | $ 136.03

6) For TCX = -6

 play-all | rounds played = 100.00% 

 spread   ev/h   sd/h   ror%      n0    di  score      ekb   avb    unit  kelly 

 ------------------------------------------------------------------------------ 

 1 -  8  3.714 59.239  13.53   25439  6.27  39.31   944.83 2.113  10.584  1.000 

 1 - 12  5.533 76.653  13.53   19195  7.22  52.10  1061.99 2.474   9.416  1.000 

 1 - 16  7.105 91.768  13.53   16682  7.74  59.95  1185.24 2.781   8.437  1.000

Note that for 1:16 the betting pattern on each hand should be:

TC <=  -1 ==>   1.00 u | $   8.44 

TC  =   0 ==>   1.00 u | $   8.44 

TC  =   1 ==>   2.86 u | $  24.09 

TC  =   2 ==>   6.36 u | $  53.64 

TC  =   3 ==>   9.91 u | $  83.65 

TC  =   4 ==>  13.54 u | $ 114.24 

TC >=   5 ==>  16.00 u | $ 134.99

Conclusion: If we use a 1:12 spread we should go for a second hand at a TCX
of -4 and above. For 1:16 the TCX should be -6. Note that the
more we spread the lower the TCX is so, for a higher spread it's better to play
two hands all the time. Lower spreads also get TCXs higher.

Sincerely,
Cac

[Don Schlesinger]

Thank you for some very fine research.

I do, of course, have a couple of quibbles. I think the study implies, as you mention, that there is no penalty for time. As if by magic, we manage to play 100 rounds of two hands, per hour, just as quickly as we play 100 rounds of one round per hour. You must admit, this isn't a very realistic assumption.

Not only does it take time to play the second hand (it is debatable as to whether it takes twice as long as playing one hand -- after all, the dealer plays his hand only once, in either scenario), but, there are more shuffles per hour when we play two hands, and that is pure "down time," when we can't be earning any money.

Since the total optimal bet for two hands is, roughly, 150% of the one-hand amount, and since we use, playing alone, 150% more cards per round, I think that a real-world experiment, involving time in the equation, would show the results of the two approaches to be MUCH closer to each other than your simulations.

And, as I've stated in BJA3, as you add players to the table, it becomes clearer to me that there is definite value from playing the second hand.

Don

[Norm Wattenberger]

I've always been curious about this question. Fascinating stuff.

[Cacarulo]

> Thank you for some very fine research.

> I do, of course, have a couple of quibbles. I think
> the study implies, as you mention, that there is no
> penalty for time. As if by magic, we manage to play
> 100 rounds of two hands, per hour, just as quickly
> as we play 100 rounds of one round per hour. You
> must admit, this isn't a very realistic assumption.

> Not only does it take time to play the second hand (it
> is debatable as to whether it takes twice as long as
> playing one hand -- after all, the dealer plays his
> hand only once, in either scenario), but, there are
> more shuffles per hour when we play two hands, and
> that is pure "down time," when we can't be
> earning any money.

> Since the total optimal bet for two hands is, roughly,
> 150% of the one-hand amount, and since we use, playing
> alone, 150% more cards per round, I think that a
> real-world experiment, involving time in the equation,
> would show the results of the two approaches to be
> MUCH closer to each other than your simulations.

> And, as I've stated in BJA3, as you add players to the
> table, it becomes clearer to me that there is definite
> value from playing the second hand.

I'll play with including another player to the table.

Sincerely,
Cac

[Cacarulo]

> I've always been curious about this question.
> Fascinating stuff.

Thanks.

Sincerely,
Cac

[GeoC]

[Cacarulo]