Blackjack numbers
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).
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