"You always double down on 11, baby." Sage advice from Vince Vaughn's character in the 1996 film Swingers. At one point in the film, Trent (played by Vaughn) and Mike (played by Jon Favreau) make an impromptu trip to Las Vegas, and Mike ends up completely out of his depths at a high-stakes blackjack table.
Due to the success of films like 21 and The Hangover, blackjack is undoubtedly one of the most well-known casino games. But for those who need a refresher, here are the rules. Each player (including the dealer) is dealt two cards. Every card has an associated point value: numbered cards are worth their number, face cards are worth ten, aces are worth either one or eleven, whichever is more favorable. Each player is then given the opportunity to request more cards (called "hits"), one at a time, until she is satisfied with her total. If her total exceeds 21, she "busts" and her bet is forfeit. At the end of the round, a player wins if her total is higher than the dealer's, and no larger than 21.
Those are the basics, but there are other rules that the more advanced player can exploit. One is called doubling down: instead of hitting as many times as you like, you're allowed to double your bet in exchange for exactly one hit. This is sometimes a wise investment, though there's also risk involved. For example, if your total is ten, you double down, and your third card is a two, you're stuck with a total of twelve and can only win if the dealer busts.
This brings us to the clip above. Mike receives a six and a five, giving him a total of eleven. Trent urges him to double down, and indeed, this seems like good advice. After all, in a deck of 52 cards, 16 of them have a value of 10 - that's over 30%! Always doubling down on eleven is also consistent with the basic blackjack strategy popularized by Edward O. Thorp in his book Beat the Dealer. From a mathematical standpoint, Trent is right. You should always double down on eleven.
This brings little consolation to Mike when he loses $200, however. Afterwards, Mike and Trent have the following exchange:
Trent: I'm telling you baby, you always double down on an eleven.
Mike: Yeah, well obviously not always.
Trent: Always, baby.
Mike: I'm just saying, not in this particular case.
Trent: You always double d--
Mike: I lost, okay? How could you say always?
Mike is clearly upset, as any of us would be after losing $200. But is he correct? Knowing what he now knows about that hand, was doubling down the right decision?
From a finanicial perspective, yes. Sorry Mikey. Hindsight is always 20/20, but the whole point of blackjack strategy is that the players have incomplete information. While players can lose any individual hand by doubling down on an eleven, basic strategy tells us that in the long run, we win more by doubling down on eleven than we do by not doubling down on eleven. Doubling down in this case was perhaps an unlucky move, but not an unwise one. (For more mathematical musings on the word "luck," I highly encourage you to check out a delightful exchange between Chris Lusto and Chris Danielson.)
One way to explain why doubling down is smart is to use expected value. Roughly speaking, the expected value tells us how much Mike can expect to earn on average in that same situation if he plays many hands. Based on these numbers, Mike has a 58.3% probability of winning $200 and a 34.0% probability of losing $200 when he has a six and a five, and the dealer's visible card is a two. The remainder of the probability (7.7%) corresponds to the case of a "push", when Mike and the dealer end up with the same total - in this case, Mike neither wins nor loses any money.
Given these probabilities, how much do we expect Mike to earn? We calculate by multiplying each possible amount Mike could win or lose by its corresponding probability. Using the numbers above, the expected value of doubling down is
(0.583)($200) + (0.077)($0) + (0.340)(-$200) = $48.60.
The probability of winning if Mike doesn't double down is actually a bit smaller, and his probability of losing is a bit larger1, so doubling down more than doubles the amount of money he can expect to win.
But maybe Mike wasn't interested in winning money. Maybe he just wanted to have a nice time. Instead of trying to maximize his earnings, what if he was trying to maximize his happiness? Mike seems pretty high strung, so I imagine the remorse he feels if he loses is stronger than the glee he feels if he wins. Let's suppose that losing feels twice as bad as winning feels good, and that his happiness (or unhappiness) is proportional to the amount of money he wins (this probably isn't true, but for amounts of money within a small range it may not be too far off the mark). If we define our units so that winning $100 is the worth one "Mikey Happiness Unit" (or MHU for short), then with these assumptions, we get the following table of values:
What is the expected value of MHU when Mike decides to double down? Replacing dollar amounts with MHU values in our earlier calculation, we get
(0.583)(2) + (0.077)(0) + (0.340)(-4) = -0.194.
While the expected value of money is positive, the expected value of Mike's happiness is negative. Perhaps this explains why Trent and Mike have such a lively argument at the lower stakes table: they didn't agree on a choice of units first! Of course, by this reasoning, Mike probably shouldn't have sat down in the first place. A little math could've saved him from a big headache.
1: If Mike hadn't doubled down but had otherwise followed the basic strategy, he'll take one hit unless his first hit is an ace, in which case he'll hit twice. We can use this information to compute the probabilities for each possible final total for Mike. On the other hand, these tables provides probabilities for each final possible total for the dealer, given that his visible card is a two (though the fact that we know Mike has a 5 and a 6 changes these values somewhat). With these numbers we can estimate the probability that Mike will win, push, or lose by not doubling down. It turns out these numbers are 56.47%, 7.49%, and 36.04%, respectively. In the interest of space, details on the calculation may have to wait for another time; the main takeaway is that Mike's probability of winning increases slightly when he doubles down, while his probability of losing decreases.