Effect of Standard Deviation on Playing Strategy in BlackJack

Blackjack strategies are based upon optimizing Expected Value. That is, for each decision conventional strategy recommends the action (doubling, hitting, surrendering, insuring, etc.) which has the highest EV. Intuitively, however, in "borderline cases" standard deviation ought to be looked at. That is, a decision such as doubling, splitting, or not-surrendering involves accepting higher risk, which should clearly be taken into account in making strategy decisions.

For Kelly players, there is a simple solution to this question. Players who completely accept the Kelly principles will seek to optimize, not their EV but their Expected Logarithm (EL). Given this, they should select for each strategy decision, the choice which produces the greater EL. If we use the standard quadratic approximation to the Log, we see that the following parameter should be looked at

     EL   =  [ m -  f * SS / 2  ] * f



  where   m  =  Mean Earnings 

          SS = Sum of Squares 

          f  = Fraction of Bankroll bet on the Hand 

Now when small amounts of the bankroll have been wagered, the second term is negligible and EL is just f * EV. So maximizing EL will be the same as maximizing EV when f is small. In these circumstances Kelly strategy would agree with conventional strategy. However as f increases, then the SS term becomes more important. Indeed, if the decision of highest EV also increases Variance (such as doubling or not-surrendering) then we may compare the ELs for these decisions and obtain a formula for the critical fraction:

          f  =  2 k ( m1 - m2 ) / ( SS1-SS2 )



 Where   m = Mean Earnings for Actions 1 & 2

        SS = "Sum oF Squares" (Variance +  m2 )  

         k = 1 for Kelly System

	

If more than this amount has been wagered, then the decision of lower-risk would be preferred.

Of course, Kelly players could obtain the exact value of this critical fraction by solving the corresponding Log equation directly. However I like these equations, in part because they indicate clearly the tension between the 2 key parameters. Also, they suggest generalization to other (non-Kelly) systems, discussed later.

Some Examples:

1. The Basic Strategy for 6 decks calls for not surrendering 87 vs. 10. However this a very close play and I calculate the Critical fraction to .068% This means that a Kelly player who had only bet 1 unit on this play would not be justified in following the Basic Strategy unless the bankroll exceeded 1,470 units!
2. BS calls for doubling Ace-2 vs. 5 in <=9 decks. However in 8 decks, I calculate that more than 200,000 units are required to make this play. Even in 6 decks, I estimate that the required bank to be 576 units.
3. Several weeks ago, we discussed the hypothetical situation of knowing that your first card was an Ace. Calculations showed that Kelly players would bet approximately 40% of their bankroll. Were they to do that, they should violate BS and not double. Even a lucrative doubles, like A7 v 6, are not justified if more than 13.7% of the bank has been bet.

Unfortunately, except in unusual circumstances like our last example, this theory may have little practical value. The reason is that counters are not playing BS; they are making decisions based upon indices. The difference in expected values is very sensitive to changes in deck composition, which influences the decisions that counters make. In our first two examples, in any positive count situations the correct Kelly play will be the ordinary correct play (doubling A2v5 and surrendering 87v10). If players use index numbers that round to the more conservative choice (as many do), they will have usually compensated for the effect of risk. However there are

Other Systems

The Kelly Criterion takes as an objective function the expected logarithm. While there are some sound mathematical reasons for accepting this, it is not the only system that one could choose. For example, I have seen arguments supporting the use of square root instead of log. However, we can apply our same methodology to such systems. We would obtain a similar equation for critical fraction, except that the constant k listed above would be different (for square root, I believe that k is 2).

Players with lower risk tolerance may have an objective function far more conservative than Kelly and may have a much lower k. For these players, the study of critical fractions may be more important.

May Your Earnings Always Exceed Your Expected Value!