From: hall@kronos.arc.nasa.gov (Michael Hall)
Newsgroups: rec.gambling
Subject: BJ: Peanuts! Kelly Criterion Strategy Adjustments! Popcorn!
Date: 28 Jul 92 09:30:20 GMT
Organization: NASA/ARC Information Sciences Division

I hacked up Steve Markowitz's strategy adjustment derivation program
to optimize expected log(bankroll) rather than expected value, as Steve
Jacobs and others have suggested.  (I still have reservations about this,
but never mind for now.)  I have very, very little faith in the Kelly
strategy adjustment numbers I have generated, mostly because I had to
modify someone else's program.  I have the least faith in the dealer 10
and ace columns (due to handling the special case of dealer blackjacks.)

But just in case they are correct, I've included them here.  Once I am
more sure of the validity, I'll repost Markowitz's program with the
modifications, if Markowitz does not object, and then everyone can
generate risk-adverse strategy adjustments for their favorite counting
systems.  Until then, be patient.

The adjustments below are for Hi-Opt I, downtown Vegas single deck.
I did not compute splitting numbers (well, I did, but they were clearly
bogus). Each situation appears in triplicate:

1) first is the Kelly adjustment number for a 6 unit bet of a 300 unit bank
2) second is the Kelly adjustment number for a 1 unit bet of a 300 unit bank
3) third is the original expected value maximizing adjustment number

What you'll see is that there is very little difference between the 1/300
strategy and the expected value maximizing strategy.  That's pretty much
what one would expect.  There are just a few places where the 1/300 strategy
is slightly more conservative, such as doubling 8 vs 4 at +6 instead of +5.
The 6/300, on the other hand, has quite a few deviations in the doubling;
however, the deviations are not striking, e.g., 8 vs 4 at +7, so I'm afraid
the nay-sayers may be correct that the Kelly indicies are not much
better than the already published expected value maximizing indicies.
(But hey, I'll take every free ounce of advantage I can get!)

Of course, the 1/300 through 6/300 tables could be combined into one
table, taking into account the different bets at each count level.
(But you'd have to pick one favorite betting spread.) Thus, the
compiled Kelly table would be exactly like the strategy adjustment
tables we all know by heart, just with slightly more conservative
numbers especially at high counts where we will have big bets out.

Again, I have little faith in these numbers, and I am very suspicious of 
the 10 and ace columns.  Also, Markowitz's program uses a lot of
approximations, so even the expected value maximizing numbers are just
a bit off what they should be.

Drumroll please...
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                     Blackjack Count Strategy Adjustments

                Cards:    2   3   4   5   6   7   8   9  10 Ace
                Values:   0   1   1   1   1   0   0   0  -1   0

               1 deck, no double on splits, dealer hits soft 17


                              Hit/Stand Strategy

                                Dealer's Upcard

Player's Total     2     3     4     5     6     7     8     9    10    Ace

   Hard 18        ...   ...   ...   ...   ...   ...   ...   ...   ...   ...   
   Hard 18        ...   ...   ...   ...   ...   ...   ...   ...   ...   ...   
   Hard 18        ...   ...   ...   ...   ...   ...   ...   ...   ...   ...   

   Hard 17        ...   ...   ...   ...   ...   ...   -16   -17   -20    -6   
   Hard 17        ...   ...   ...   ...   ...   ...   -16   -17   -20    -6   
   Hard 17        ...   ...   ...   ...   ...   ...   -16   -17   -19    -4   

   Hard 16         -9    -9   -10   -12   -11     9     8     4     0     1   
   Hard 16         -9    -9   -10   -12   -11     9     8     4     0     1   
   Hard 16         -9    -9   -10   -12   -11     9     8     4     0     3   

   Hard 15         -6    -5    -7    -8   -10    10     9     7     3     3   
   Hard 15         -6    -5    -7    -8   -10    10     9     7     3     3   
   Hard 15         -6    -5    -7    -8   -10    10     9     7     3     5   

   Hard 14         -4    -4    -5    -7    -8    12    19   Hit    11     5   
   Hard 14         -4    -4    -5    -7    -8    12    19   Hit    11     5   
   Hard 14         -4    -4    -5    -7    -8    12    19   Hit    13     8   

   Hard 13         -1    -1    -3    -4    -6   Hit   Hit   Hit   Hit     9   
   Hard 13         -1    -1    -3    -4    -6   Hit   Hit   Hit   Hit     9   
   Hard 13         -1    -1    -3    -4    -6   Hit   Hit   Hit   Hit    13   

   Hard 12          2     2     0    -1    -2   Hit   Hit   Hit   Hit    13   
   Hard 12          2     2     0    -1    -2   Hit   Hit   Hit   Hit    13   
   Hard 12          2     2     0    -1    -2   Hit   Hit   Hit   Hit   Hit   

   Soft 20        ...   ...   ...   ...   ...   ...   ...   ...   ...   ...   
   Soft 20        ...   ...   ...   ...   ...   ...   ...   ...   ...   ...   
   Soft 20        ...   ...   ...   ...   ...   ...   ...   ...   ...   ...   

   Soft 19        ...   ...   ...   ...   ...   ...   ...   ...    19*  ...   
   Soft 19        ...   ...   ...   ...   ...   ...   ...   ...    19*  ...   
   Soft 19        ...   ...   ...   ...   ...   ...   ...   ...    18*  ...   

   Soft 18        -13   -13   -11   -12   -12   -20   -10   Hit   Hit     1   
   Soft 18        -13   -13   -11   -12   -12   -20   -10   Hit   Hit     1   
   Soft 18        -13   -13   -11   -12   -12   -20   -10   Hit   Hit   Hit   

   Soft 17        Hit   Hit   Hit   Hit   Hit   Hit   Hit   Hit   Hit   Hit   
   Soft 17        Hit   Hit   Hit   Hit   Hit   Hit   Hit   Hit   Hit   Hit   
   Soft 17        Hit   Hit   Hit   Hit   Hit   Hit   Hit   Hit   Hit   Hit   

                             Double Down Strategy

                                Dealer's Upcard

Player's Total     2     3     4     5     6     7     8     9    10    Ace

   Hard 11         -9    -9   -10   -11   -12    -6    -4    -2     2   ...   
   Hard 11        -10   -10   -11   -12   -12    -7    -5    -3     0   ...   
   Hard 11        -10   -10   -11   -12   -13    -7    -5    -3    -4    -1   

   Hard 10         -8    -7    -9   -10   -11    -5    -2     0   ...   ...   
   Hard 10         -8    -8    -9   -11   -12    -6    -4    -1   ...   ...   
   Hard 10         -9    -8    -9   -11   -12    -6    -4    -1     5     4   

   Hard  9          2     1    -1    -3    -4     5    10   ...   ...   ...   
   Hard  9          0     0    -2    -4    -5     3     9   ...   ...   ...   
   Hard  9          0     0    -2    -4    -5     3     8   ...   ...   ...   

   Hard  8         14    10     7     4     4    17   ...   ...   ...   ...   
   Hard  8         12     9     6     3     3    16   ...   ...   ...   ...   
   Hard  8         12     8     5     3     2    15   ...   ...   ...   ...   

   Hard  7        ...    16    11    10    10   ...   ...   ...   ...   ...   
   Hard  7         20    15    10     9     9   ...   ...   ...   ...   ...   
   Hard  7         20    15    10     8     9   ...   ...   ...   ...   ...   

   Hard  6        ...    18    14    12    14   ...   ...   ...   ...   ...   
   Hard  6        ...    17    13    11    13   ...   ...   ...   ...   ...   
   Hard  6        ...    17    13    11    12   ...   ...   ...   ...   ...   

   Hard  5        ...    19    14    13    14   ...   ...   ...   ...   ...   
   Hard  5        ...    18    14    12    13   ...   ...   ...   ...   ...   
   Hard  5        ...    18    13    12    13   ...   ...   ...   ...   ...   

   Hard  4        ...    19    15    14    15   ...   ...   ...   ...   ...   
   Hard  4        ...    19    14    13    14   ...   ...   ...   ...   ...   
   Hard  4        ...    18    14    13    14   ...   ...   ...   ...   ...   

   Soft 20         11    10     8     6     6    17   ...   ...   ...   ...   
   Soft 20         10    10     7     6     5    16   ...   ...   ...   ...   
   Soft 20         10     9     7     6     5    16   ...   ...   ...   ...   

   Soft 19          8     5     4     2     1    14   ...   ...   ...   ...   
   Soft 19          7     4     3     2     0    13    20   ...   ...   ...   
   Soft 19          7     4     3     1     0    13    20   ...   ...   ...   

   Soft 18          3     2    -5    -6    -8    16   ...   ...   ...   ...   
   Soft 18          1     0    -6    -8   -10    15   ...   ...   ...   ...   
   Soft 18          0     0    -7    -8   -11    14   ...   ...   ...   ...   

   Soft 17          4     1    -3    -8    -9   ...   ...   ...   ...   ...   
   Soft 17          0    -2    -5   -10   -11    11   ...   ...   ...   ...   
   Soft 17          0    -2    -5   -10   -11     9   ...   ...   ...   ...   

   Soft 16         15     7     2    -2    -8   ...   ...   ...   ...   ...   
   Soft 16         11     5    -1    -4   -10   ...   ...   ...   ...   ...   
   Soft 16         10     4    -1    -5   -11   ...   ...   ...   ...   ...   

   Soft 15         18     8     2    -2    -5   ...   ...   ...   ...   ...   
   Soft 15         14     5    -1    -4    -7   ...   ...   ...   ...   ...   
   Soft 15         13     4    -2    -5    -8   ...   ...   ...   ...   ...   

   Soft 14         13     8     2    -1    -3   ...   ...   ...   ...   ...   
   Soft 14         10     6     0    -2    -4   ...   ...   ...   ...   ...   
   Soft 14          9     5     0    -3    -5   ...   ...   ...   ...   ...   

   Soft 13         11     7     3     0    -2   ...   ...   ...   ...   ...   
   Soft 13          8     5     1    -2    -3   ...   ...   ...   ...   ...   
   Soft 13          8     5     1    -2    -3   ...   ...   ...   ...   ...   

       * indicates adjustment in the opposite of normal direction

-- 
Michael Hall            \ The Jelly criterion says, to a first approximation,
NASA Ames Research Center\ to spread proportional to your peanut butter and
hall@ptolemy.arc.nasa.gov \ your bread.  It is the optimal way to make a PBJ.
Newsgroups: rec.gambling