Playing it safe

Proper doubling is an advantage bet at blackjack

The extra chips you risk on sanctioned doubles at blackjack are advantage bets. Advantage, of course, is why they're sanctioned.

Here's what this means. Make believe you bet $10 at the start of a round. Pretend further that, according to the way you play and the largesse of the casino bosses, the house's edge in the game is half a percent. When you make the bet, prior to the deal, your $10 has an expected value of $9.95. That is, on the average, it'll be worth $9.95. This figure accounts for the chances of receiving various sets of initial hands and dealer upcards, as well as the likelihoods of payoffs or loses on each combination.

Once the cards are dealt, however, you're in a "conditional" mode. Sometimes the news is bad. As an illustration, if you get 10-6 against 10-up, you're a serious conditional underdog. Your best shot is to hit, in which case your $10 has an expected value of only $4.65. Standing would be slightly worse, at $4.59.

The news is good often enough to make blackjack a popular game. An example is 10-9 against seven. In this situation, standing, your $10 has an expected value of $16.15. Hitting or doubling well, you don't need the numbers to know you shouldn't.

Conditionally advantageous bets like these are highly desirable. But solid citizens can't do much about them except hope they'll come along regularly and play them properly when they do.

Proper doubles, however, offer opportunities to place wagers that are advantageous at the moment they're made. Consider, for instance, 6-5 versus a dealer's four. If you hit this total, the $10 you bet on the round would have an expected value of $12.94. You can double down by betting another $10, subject to the constraint that you draw one and only one additional card. To be sure, this isn't much of a restriction here because it's what you'd do anyway. (One more card would put you in the range from 12 through 21, on which you'd stand against a four.) So the extra $10 you bet is worth the same as the original $10, a theoretical $12.94. Not on a hope, but when you slide it onto the layout.

Circumstances differ somewhat when doubling on hands on which you might want take another card if given the option. Picture a 6-4 versus seven-up. This is conditionally advantageous; hitting has an expected value of $12.62, accounting for the possibility that you'll be dealt a card from two through six such that you'd hit again were it allowed. Doubling with an additional $10 but being unable to keep drawing is still an advantage bet. When the effect on the initial wager along with the money at risk on the double is tallied, the auxiliary $10 has an expected value of $11.44. And the total $20 up for grabs has an expected value of $24.06.

An interesting condition occurs with A-2 versus four-up. It's conditionally advantageous when hit $10 jumps in theoretical value from the initial $9.95 to $11.04. Blackjack buffs recognize this as a hand that Basic Strategy says not to double. It turns out that if you do, the extra $10 put in the ring is worth only a theoretical $9.64. Still, the double wins more often than it loses. The total $20 has an expected value of $20.68. And, at times, you may prefer being slightly favored to earn $20 over a better shot at gaining $10, given the relative chances of losing the two amounts. A state when utility may eclipse probability.

This is by no means a general guideline for soft doubles. A-2 versus three-up is conditionally advantageous when hit; the $10 bet has an expected value of $10.74. But doubling loses more than it wins, such that the extra $10 is only worth a theoretical $9.26 and the combined $20 has an expected value of $19.99.

When in doubt, it's prudent to play "by the book." But particular personal priorities may justify an alternate strategy. Such as the pleasure of a bigger profit outweighing the penalty of an added loss, or a greater prospect of success appealing more than a higher win. Even these decisions ought not be made on a whim. Rather, by grasp and accepting what you're sacrificing in pursuit of a goal. As the bettors' bard, Sumner A Ingmark, wrote:

When there's a rule that you're defying, Know what's the cost of not complying.