Profitability of Card Counting on Single Deck Versus Double Deck Versus Shoes On Aug 24, 1998, in rec.gambling.blackjack.moderated Abdul Jalib wrote: In this article, I'll compare the profitability of card counting on single deck versus double deck versus shoes. I'm going to first define some mathematical terms, but if you are allergic to even simple math, don't panic... I'll get to the qualitative conclusions by the end of the article. Let's define r to be the win rate or mean result or expected value, V to the be variance of this win rate, and B to be our bankroll. I am assuming Kelly/logarithmic utility, so our goal is to maximize log(B). The Kelly bankroll B is then V/r. William Chen noted over on rec.gambling.poker that the "expected rate of growth would simply be: U = r/B - V/2B^2..." I've been using just r/B in the past, but I think his adjusted figure is more precise. You can rewrite it as U = r/(V/r) -V/2(V/r)^2 = r^2/V - r^2/2V if your bankroll is the proper Kelly bankroll. See also old posts on "certainty equivalent". Next I'll show some simulation results. I've tried to hold the percent bankroll growth per 100 rounds nearly constant (between .40% and .49%), so what we are left with are a number of comparably profitable games, with different numbers of decks, rules, penetrations, and betting spreads. The counting system is Wong Halves adjusted for aces, and the betting is optimal, with an optimal number of units being bet at each count level, a minimum of 1. No cover was used for the betting; we could talk about adding that into the simulations if you want, though determining the optimal betting pattern within additional restrictions is difficult. No preferential shuffles were simulated; if you experience these, you should merely move on to another table or another casino. .025 units per 100 rounds have been taken out for tipping and other sources of lost EV. SIMULATION RESULTS     WIN RATE   BANK ROLL % BANK GROWTH / 100 ROUNDS  MEAN  VARIANCE DECKS RULES   PENET.  SPREAD (R) (V) (B) 1 H17 DOA 55%* 1-3 .030 10.04 335 0.45% 1 H17 D10/11   70% 1-5 .032 7.53  301 0.45% 2 S17 DOA DAS  75%  1-5   .026 6.92 265  0.49%  2 S17 DOA DAS 66% 1-10  .039 18.66 483 0.41% 6 S17 DOA DAS LS RSA  83%  1-40**  .107  141.43  1326  0.40% 6 S17 DOA DAS LS RSA  83%  1-100  .247  642.52  2716 0.47%      * Actually, the single deck penetration is 4 rounds to 2 spots, with you playing both hands. This game does exist. ** Includes abandoning negative counts.  Now for the qualitative discussion of these results... Look in particular at the first and last lines. A shoe with great rules, great penetration, and a 1-100 spread produces about the same bankroll growth as a single deck game with much worse rules, much worse penetration, and a mere 1-3 spread. If this comes as a shock to you, it may be because you are accustomed to looking at "advantage" as a metric to judge games. Advantage doesn't do a good job of taking risk into account. I don't have the advantage figures handy for these simulations, but I'm sure the advantage for that 1-100 spread shoe game would be much higher than the advantage for the 1-3 spread single deck game. Unfortunately, a higher advantage does not imply a better game. Fortunately, higher bankroll growth, as defined above, does imply a better game, if you have Kelly/logarithmic utility. Such a utility of money results in the largest expected bankroll in the limit. Another observation is that changing the single deck to have better penetration but worse rules still results in your having to use a considerably more aggressive spread (1-5 compared to 1-3). For double deck, penetration is *really* important: 75% is very nice, while 66% forces you to a dangerously large spread if you wish to have a bankroll growth per 100 rounds. I think the spreads of 1-5 on single deck, 1-10 on double deck, and 1-100 on shoes are very likely to get you barred. The 1-40 on shoes while abandoning negative counts could also get you barred, though this is about what I do on the rare occasions when I do frontal assaults on shoes. The sweet spots are the 1-3 spread on downtown Vegas single deck getting 4 rounds to 2 spots, and the 1-5 spread on the 75% penetration double deck. These spreads are less likely to trigger heat and barrings, though you should still add in some cover. Double decks with 75% penetration are an endangered species, but getting 4 rounds to 2 spots in downtown Vegas rule single deck can still be found, even if it takes some hunting. Does that mean casinos should deal shoes to protect themselves? Well, no, I don't believe so, for several reasons. First, what matters to a casino is not a counter's bankroll growth but his expected value. The counter is winning .107 to .247 units per round on the shoe game examples compared to just .030 units for the counter on the single deck example. With smaller spreads, shoes are a lose-lose situation for counters and casinos: the counters don't make as much as they would on single or double deck, and the casinos lose more than they would if the counter were playing single or double deck. Again, this is because there are two bankrolls involved: the counter's finite bankroll versus the casino's effectively infinite bankroll (though most pit critters sweat a $500 bet like it will put the casino out of business if they lose it.) Second, shoes open up casinos to a number of holes to be exploited by highly skilled teams with huge bankrolls. One such hole is the Ken Uston style call-in. There are several other shoe-vulnerabilities I'd rather most casinos remain ignorant about. Finally, again for teams with huge, say multimillion dollar, bankrolls, such teams don't care about the variance, since the casino won't accept the kind of bets their bankroll could support, so the teams effectively just care about maximizing expected value for whatever action they can get down. Since larger expected values (in absolute dollars) are possible on shoes than on single or double deck, such teams can make the casinos pay dearly for their paranoid replacement of single and double deck with shoes. Abdul Jalib wearing the hat of Professional Degenerate Gambler AbdulJ@PosEV.com (Abdul Jalib) ©1998 Abdul Jalib