Why "Doubling for Less" Isn't Always as Dumb as They Say

This, for two reasons. First, players have an edge on the extra bet, so expected profit is most with the biggest possible outlay. Second, since doubling means drawing once and only once, cases may arise where inability to take again outweighs the benefit of an added wager smaller than the maximum allowed. An illustration of this effect might be a total of nine versus three-up. Bettors who drew twos or threes, for totals of 11 or 12, would keep going if they could. Statistically expected profit on ordinary hits would be almost $0.11 per $1 bet; on full doubles it's roughly $0.13. Doubling for less than about 65 percent of the flat wager reduces expected profit below $0.11 per dollar, turning a windfall into a woe. In the other direction, if a casino offered tripling or quadrupling as a promotion, all else being equal, it could end up as an underdog at its own tables.

That's the gospel according to expectation. But other considerations may sometimes suggest betting less than the original wager for the double, or holding back altogether.

The emotive value or happiness solid citizens associate with various amounts of money (the classy word is "utility") is the main mitigating factor. This is why some folks intuitively double for less. Whatever their reasons, they enter rounds betting big, either in absolute dollars or relative to their current bankrolls. The opportunity pops up to double. Say they do it, matching what's already bet. Joy if they win, dejection if they lose. It's the same figure plus or minus, but the magnitude of the potential pain may outweigh that of the foreseeable pleasure.

Here's one way expected utility pans out for a possible double on 11 versus five-up. This is not only a strong hand, but a single card would be drawn doubling or hitting, and players' chances in both cases are 61.9, 31.1, and 7.0 percent to win, lose, and push, respectively. Assume a player's utility function equals the square root of the money at issue. This is a good model and fits instances where losses disturb more than equal wins delight.

For the cited situation, with base wagers below 30 percent of a current bankroll, expected utility peaks with a full double. The optimum auxiliary bet plummets as initial wagers grow. Starting at 40 percent of a current bankroll calls for a double of half the nominal value. A round begun with half a bankroll on the layout is optimized by doubling at the 20 percent level. Players having utility functions reflecting a "go for broke" attitude maximize expected happiness by doubling aggressively on greater relative original wagers. People more sensitive to losses would pull back earlier, and might not double at all if the bets they made before the deal exceed 10 or 15 percent of their reserves.

The "Kelly criterion" offers yet another reason to double for less under some conditions. This standard tells blackjack card counters, as well as racing and sports handicappers who believe they have an edge, the fraction of a current stake to bet for the fastest projected rate of bankroll growth, given the odds and payoffs. The principle, adapted to the 11 versus five-up double, leads to supplementary bets of the full amount when beginning wagers are under 16 percent of a starting bankroll, no doubles at all for openers over 33 percent, and a sliding scale downward between these levels. For example, double for 65 percent of an initial wager equal to 20 percent of a bankroll, and 10 percent of an initial wager equal to 30 percent of a bankroll.

Nobody may ever calculate double reductions for all conceivable contingencies. But, understanding these concepts may help you when you're tempted to hold back. You may want to choose between doubling for less and simply hitting (or standing with soft 18). Either way, you'll know why the butterflies in your belly now and then rebut Basic Strategy. The immortal muse, Sumner A Ingmark, envisioned the essence of such experience in his veritable verse:

The lessons seasoning oft teaches,
Extend beyond the rule book's reaches.