Playing two hands and Kelly (not short)

From Stanford Wong's BJ21

Posted by Grimy Fellow on 1 Jul 1998, 11:13 p.m.

I. Background:
A. Overall Kelly
As we already know, when playing "all hands" or even any hands in negative expectation situations there is an "overall" Mean and Standard Deviation that determines the "required" or "associated" Kelly bankroll size for the system being employed. This overall Kelly bank size is usually larger than that which would be calculated for the bets made at any individual positive count. (This is one of the disadvantages of playing in negative and/or zero counts. Fortunately, the size of this overall Kelly bankroll is trivially estimated once one knows the overall Standard Deviation and "advantage".)

B. Kelly Effect of Covariance When Playing Two Hands
When the Kelly calculation is applied to bet sizes played at a single, positive expectation count a quick, general rule of thumb is: the correct Kelly bet is 80% of one's advantage times one's total stake (see Professional Blackjack, 1994 edition p 204), while the correct Kelly bet for each of two hands is roughly 60% of one's advantage times one's total stake. Thus, if it is correct Kelly to bet $100 on a hand it would be approximately correct to bet $75 on each of two hands, according to Wong.

II. Motivation: Angst
When I have the opportunity I play the game, and I usually like to know that the amount of money I'm betting falls withing Kelly guidelines that I have determined by first simulating the system I've decided to play. But all my simulations have been of the "single hand" variety. That is, my own RYO simulator doesn't support multiple hands for a single player. Furthermore, when I play it sometimes happens that I have the opportunity to play two hands in a positive count (my local casino is usually very crowded, so this isn't as common as it might be for you). When I get this opportunity I always calculate what 75% of my single hand bet would be, and then bet that amount on each of my two hands - but I do this without having the benefit of having simmed this play to see what effect this has on my "overall" Kelly bank requirement. I always say to myself, "well, I sure hope Wong knows what he's talking about".

III. Hypothesis
In positive counts betting two hands at 75% of the amount one would bet at one hand has little or no effect on the overall associated Kelly bankroll size.

IV. Test
A. General Idea
Run two simulations using Wong's BCA. Both sims play the same game except that one sim bets $20 at all counts of +2 and over while the other sim bets 2 hands of $15 at these counts. Determine required Kelly bank for each sim and compare.

B. Details
6 decks, S17, DAS, DOA. Bet $5 in any negative or 0 count. Bet $10 at +1.
Sim "1x20" bets $20 at all counts of +2 or greater.
Sim "2x15" bets two hands of $15 at all counts of +2 or greater.

V. Preliminary Results
(BCA provides SD and edge for two ranges of data, per 100 rounds and per shoe. Thank Goodness that the estimated Kelly Bank is roughly the same for each! I give both of them here)

A. "1x20"
35.7 million hands
SD per 100 rounds: 122.74
Est. Kelly bank: $4875
Edge per 100 rounds: 3.09
SD per shoe: 45.91
Edge per shoe: .43
Est. Kelly bank: $4902
SE = .03

(as a check, my own sim calls for a Kelly bank of $4812 for the game above.)

B. "2x15"
25 million rounds
SD per 100 rounds: 138.07
Edge per 100 rounds: 4.81
Est. Kelly bank: $3963
SD per shoe: 51.04
Edge per shoe: .66
Est. Kelly bank: $3947
SE = .04

VI. Preliminary Conclusion
The hypothesis is not true for this game. It is actually significantly "safer", as well as more productive, to bet two hands of 75% of the originally intended bet size.

VII. Theoretical Speculation
The 80% factor mentioned in Professional Blackjack is based on a single count, or "simple", Kelly calculation. When one's system requires that one bets in counts with negative expectations the 80% factor is reduced for each count. But the greater the ratio of the total amount bet in positive expectations to the total amount bet in negative expectations the less the reduction of the Kelly size for each positive count. Betting two hands at 75% increases this ratio, which has the effect of reducing the overall required Kelly bank to being closer to that which would be expected if we only bet Kelly-sized bets at the positive counts.

VIII. Conclusion
No need for angst when betting two hands at 75% each instead of betting the originally planned amount on a single hand. To the contrary - bet this way whenever possible.

IX. Misc. Comments
I am NOT recommending that anyone play all hands at a six deck shoe and limit the spread to 4 to 1. This was only a test betting system using the first numbers that came to mind. I believe that in a more realistic system the difference in required Kelly stake numbers would have been much smaller.

I sincerely hope that by now every would-be Kelly bettor recognizes the value of determining Kelly stake requirements for entire systems of play and knows how to quickly estimate Kelly stake requirement from sim results, or from the tables in chapter 10 of Blackjack Attack (D. Schlesinger).

Good cards to everyone!

Grimy