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Is it always best to double down in blackjack when Basic Strategy says to?

19 November 2012

Experienced blackjack buffs know that the correct Basic Strategy choice for certain combinations of two-card totals and dealer up-cards is to double-down. That is, to match their original bets and draw one and only one additional card.

Few players, however, have a clue about the rationale for this dictum – except, perhaps, in vague terms such as they win more by doing so. Don’t embarrass anyone by asking whether winning more refers to an amount of money or the frequency of successful hands, though. And, let’s not discuss the substantial proportion of solid citizens who believe bettors must adhere to the gospel of the game or they’ll change the flow of the cards and ruin the table for everyone else. Anyway, lots of folks figure “the book” was written by experts so it must be right.

When Basic Strategy decrees a double, the reason is that the move maximizes the “expectation” for the hand. Statistically speaking, expectation isn’t just a fancy way of saying “hope.” It refers to the theoretical earnings per dollar wagered at the start of the round, as predicted by the laws of probability. On a more intuitive level, it’s equivalent to the average profit obtained by players, relative to their bets, after numerous instances of the hand.

Overall, the joint expectation associated with receiving and then resolving all possible blackjack hands is about minus 0.5 percent,. This means the loss of half a cent per dollar bet. Some hands, once dealt, have positive expectation. How positive depends on the manner in which the round is executed. Situations calling for doubles have positive expectation whether players double or choose the next best alternative – usually hitting but occasionally standing (as with A-7 vs 6-up).

A two-card 10 vs 9-up illustrates the way this all works. The nearby table gives the probabilities of bets winning, pushing, or losing by hitting or doubling. The figures show that players actually have greater chance to win and smaller chance to lose by hitting than doubling. Chance notwithstanding, expectation is greater by doubling. This, because more money is involved. For hitting, per dollar bet at the start of the round, players have expectation of earning an average of 0.501 x \$1 - 0.386 x \$1 = 11.5 cents. For doubling, it’s 0.493 x \$2 - 0.420 x \$2 = 14.6 cents.

Probabilities of wins, pushes, and losses when hitting or doubling on a 10 vs 9-up

```action          resolution
win    push    lose
hit       50.1%  11.4%   38.6%
double    49.3%   8.7%   42.0%
```

There’s yet another way to view the benefit of doubling. As indicated, before a round is dealt, players’ expectation is negative. After the deal, conditioned on the hand being a proper double, expectation is positive. For 10 vs 9-up, the projected profit by hitting is 11.5 cents. In allowing doubles, the bosses let players bet another dollar, already knowing they have an edge worth 14.6 - 11.5 = 3.1 cents on the money being added.

There’s a downside, too. Since the likelihood of success drops and that of failure rises as more money goes up for grabs, doubling may have a pernicious impact on individual with low cash reserves. An argument could also be made that 11.5 cents is the theoretical profit on a \$1 investment while 14.6 cents is the long-term average return on a \$2 outlay. On this basis, per dollar at risk, hitting comes in at 11.5 cents while doubling yields only 14.6/2 = 7.3 cents. The 14.6 cents indicates how occurrences of this hand impact players’ outlook for the game as a whole. But it’s not unreasonable to evaluate the options based on gain relative to potential loss.

No discussion of doubling is complete without mentioning players who “double for less” – accepting a smaller gain from their position in the catbird seat by not hazarding as much. In situations when, after drawing one card, a player might want to be able to hit again (for instance, doubling on an 11 vs 10-up and pulling a deuce), doubling for less does more than reduce risk and reward in tandem. Pretend a player with 10 vs 9-up augments the original bet by only 50 percent. Expectation is then 0.493 x \$1.50 - 0.420 x \$1.50 = 10.95 cents, which is lower than it would be by simply hitting. Other hands and discounts on the auxiliary bet would yield different results. In general, doubling for less imposes a rate of return penalty on soft hands and on hard two-card totals when the dealer has 7-up or above.

Doubling when Basic Strategy says to do so improves expectation in blackjack. This is the usual criterion for players. Usual. Not exclusive. Understanding the pros and cons of this and other options is the essence of good gambling. And, as for anyone who criticizes you about changing the flow of the cards, well, here’s what the punters’ poet, Sumner A Ingmark, wrote:

Bettors losing in a game
Look for someone else to blame.