I recently made some additional changes to my blackjack combinatorial analysis software, in response to some interesting questions about card-counting systems. In the process, I encountered some even more interesting– and mostly unrelated– results that I do not fully understand, since they seem to contradict generally accepted ideas about the game.
There are many different variations on the rules of blackjack: the dealer may stand or hit on a soft 17, the player may or may not be able to split pairs more than once, etc. The game may also be played with different numbers of 52-card decks, from just a single deck to as many as 8 decks (416 cards) shuffled together. The question here is: how does a player’s achievable expected return vary with the number of decks used… and why does it vary?
The first question is relatively easy to answer; the following plot shows the optimal expected value of a round of blackjack, in percent of initial wager, as a function of the number of decks used.
There are a couple of interesting observations. First, in the game with a single deck, the player actually has the advantage over the house! Of course, this is why these particular rule variations, despite being the “simplest” and most common, are not found in single-deck casino games.
More importantly, the trend is clear: fewer decks are better for the player. As the number of decks increases, a player’s expected value decreases asymptotically, approaching the “infinite shoe” expected house edge of about half a percent.
Which brings us to the question motivating this post: why are fewer decks better for the player? The answer that I see and hear most commonly is similar to that on the above-referenced Wikipedia page:
“All things being equal, using fewer decks decreases the house edge. This mainly reflects an increased likelihood of player blackjack [my emphasis], since if the players draws a ten on their first card, the subsequent probability of drawing an ace is higher with fewer decks. It also reflects a decreased likelihood of blackjack-blackjack push in a game with fewer decks.”
I have never bought this. Although it is true that player blackjacks are indeed more likely with fewer decks (and pushed blackjacks are less likely), I thought that this was a secondary effect, and the real reason for the trend in overall expected return was something more information-theoretic, so to speak. We’ll get to this shortly.
But first, it occurred to me that we could verify the claim by considering a modified form of the game: blackjack, but “without the blackjack.” That is, suppose that we get rid of the 3:2 bonus of player blackjack, as well as the penalty of dealer blackjack, so that an initial ten-ace hand is just another hand totaling 21, no better or worse than, say, 9-7-5. Of course, the absolute expected return in such a game will be miserable. But the point is that, if the above claim is correct, then the trend in expected return vs. number of decks should disappear or at least diminish.
No such luck; following is the same plot as above, but with the addition of the red points indicating the optimal expected return for the player in the game without blackjacks.
At least from this perspective, the blackjack bonuses and penalties do not seem to be the primary reason that “fewer decks are better.” So what is the reason?
My view has been that the main reason is that, with fewer decks, each individual card dealt from the shoe provides more information to the player than a card dealt from a larger shoe. That is, with fewer decks, each new card seen causes a larger change in the distribution of card ranks not yet seen (i.e., still in the shoe).
This additional information manifests itself in the player’s strategy being “more composition-dependent” in games with fewer decks. For example, consider the extreme case of a game played with 1000 decks, essentially an “infinite shoe” as far as we are concerned. Optimal strategy in this case is “total-dependent,” so that, for example, you should always hit a hard 16 against a dealer 10, no matter how that hard 16 is made up (e.g., 10-6, 9-7, 8-4-4, etc.). But as the number of decks decreases, the “composition” of those hard 16s begins to matter, to the point where, for example, optimal strategy may be to hit 10-6 against dealer 10, but to stand on 8-4-4.
With the recent additions to the blackjack analysis software, we can measure the extent of this “composition-dependence,” as shown in the following interesting plot. For each number of decks, we count how many exceptions there are to total-dependent strategy, where each exception consists of a particular player hand composition and dealer up card.
This is not quite what I expected to see. The really complex single-deck strategy, with over 300 composition-dependent special cases to remember, makes sense, as does the eventual decrease to zero composition-dependent exceptions with the absurdly large 125-deck shoe. But the initial increase in strategy complexity, to a local maximum at 54 decks is, well, weird. Maybe someone can provide an explanation?
Anyway, things seem to get curiouser still; recall the earlier exercise where we modified the game to eliminate blackjacks, to test the claim that increased likelihood of blackjack was the main reason that fewer decks are better for the player. To test the speculation that the cause is actually greater opportunity to take advantage of more per-card information– via more composition-dependence in strategy– suppose that instead we modify the playing strategy used, so that no matter how many decks in the shoe, we always use the same total-dependent strategy. In this way, we do not “know” whether we are playing with fewer decks. If my speculation is correct, then as before, the trend in expected return should flatten out.
Once again, no such luck. The following plot compares expected value for the player using optimal composition-dependent strategy (the blue points) with the expected value using a fixed “infinite shoe” total-dependent strategy (the red points).
At this point, I am stumped. Hopefully someone may be able to shed some light on this, by either pointing out errors in my calculations, suggesting additional factors that I am missing, or even just suggesting an appropriate way to compare the relative contribution of these two factors.