Is Doubling Down at Blackjack Worth the Extra Money at Risk?

29 March 2006

By Alan Krigman

One of the most perplexing predicaments baffling blackjack buffs is whether to double down every time their wallet-sized card ordains it as correct for the situation at hand. Why the enigma? After all, everyone wants to earn twice as much. And occasions to do so under favorable circumstances should be tempting.

The problem is that nothing's guaranteed. So players have to balance the prospects of the supplementary return against the two fold loss. The downside may be quite daunting, for instance, if an individual's bankroll is ebbing, an especially big bet is already on the table, or luck seems to be running bad that day.

In gambling, bets and decisions are often rated according to "expectation." This is typically stated as a percentage, but you can picture it as cents won or lost per dollar at risk, averaged over many trials. Expectation is the statistical foundation of Basic Strategy in blackjack. As an illustration of how it works with doubles, assume a player starts a round by betting $10 and receives a 6-3 versus the dealer's six-up. Expectation by hitting is 20.0 percent, $2.00 profit on the $10; doubling, it's 32.5 percent on that initial $10, $3.25 profit. Since $3.25 exceeds $2.00, Basic Strategy decrees a double.

Note that in this example, twice as much at stake doesn't promise a commensurate return. Expectation is $2.00 with $10 at risk, but $3.25 and not $4.00 with $20 up for grabs. The predicted average marginal profit on the extra $10 is only $1.25. Marginal profit on the auxiliary bet offers a means by which solid citizens who are uncomfortable sliding out the simoleons can identify when the benefit they hope to gain may not justify what they can lose.

For reference, the strongest possible marginal gain is on 11 versus six-up. Hitting with $1 at risk, the expected profit is $0.34. Doubling, with $2 at risk, it's $0.68. The added dollar, bet under favorable conditions when the 11 versus six is known, is worth the same $0.34 as the buck shoved onto the table before any cards were dealt. The second strongest case is 11 versus five-up, which comes in at $0.32 per marginal dollar wagered.

The weakest double dictated by Basic Strategy based strictly on the laws of probability, when measured by marginal expectation for the additional wager, is soft 13 versus five-up. With these cards, the gain promised by the incremental dollar is three thousandths of a cent. A mere $0.03 on an initial bet of $1,000.

The next-weakest Basic Strategy double is soft 15 versus four-up. Here, the theoretical return on that extra dollar is two tenths of cent so you risk $10 to average a $0.02 increase in earnings.

Back home on planet earth, though, you may well wonder how much it's worth risking, even when the odds are on your side, to reap a relatively small return. If you're feeling your oats in a casino and $1 or even $10 seem like chump change, would you think the same way about a $1,000 bet to average $0.03 profit?

Basic Strategy for multi-deck blackjack with standard rules includes 39 doubles. The accompanying list gives these in descending order of expected marginal profit on the additional dollar, in $0.05 steps.

Marginal theoretical profit by doubling, per additional $1.00 bet

Marginal profit is a utility-based alternative to expectation that you shouldn't be ashamed to let guide your blackjack decisions. And while this criterion may seem to flout the laws of probability, it's far from following hunches, superstitions, and butterflies in the belly. Not that the latter aren't also factors, as the rhymer, Sumner A Ingmark, recognized in writing: