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Hit or Stand: The Hard 16 Dilemma in Blackjack11 November 2003
So, 16 against a high upcard being between a rock and a hard place, just how bad is it to stand? Further, assuming you're tempted to stand, are you better off doing it with some dealer upcards than others? And, if so, which are which?
The answers are in the numbers. After all, besides luck, that's what gambling
is about. The numbers for the two-card 16s are shown in the accompanying table.
Entries give expected loss per dollar bet when standing or hitting, and the
penalty for the latter, against each upcard. Data are for eight-deck shoes with
"playable" 10- or ace-up. Playable means the dealer doesn't have a
blackjack, since the decision would be then irrelevant. For the cynically inclined,
these figures are not made up based on what certain oracles think, or what the
bosses say, or what some guru concludes after watching lots of games and jotting
down results or tabulating millions of simulated virtual hands; they're found
by a "combinatorial analysis" using a computer to increment through
and count each and every possible outcome.
Expected losses and penalties per dollar bet, for standing
The data are revealing in several respects. On a practical level, they show
that the penalty for standing is most severe when the dealer has ace-up, nearly
$0.15 on the dollar for either two-card hard 16. Next worst, less than half
as bad, occurs against seven. From there, it drops with rising up-cards to under
a cent on the dollar, nearly nothing, with a dealer 10. So, if you're to stand
at all, do it against a 10 where the mathematical misery is the least; this
will happen on about half of your 16s because the shoe has as many 10s as it
does sevens, eights, nines, and aces together. And, for reasons that the numbers
should make obvious, don't even think about standing on that 16 against an ace.
Losses on hitting also increase as upcards go from seven to eight, nine, ace and 10. This phenomenon tracks the impact of the levels at which dealers are apt to finish when they don't bust, lowest when showing a seven and highest a 10. As completed dealer totals get higher, they become more difficult to beat.
The theoretical costs of flouting basic strategy don't follow either the chance of the dealer busting or finishing at a high level, though. They depend on the differences between the two. Other than against an ace, the penalties fall as the actual expected losses rise. Trade-offs like these make blackjack tough to second-guess and show the danger of applying qualitative logic to quantitative problems. They also explain why so many players wonder whether Basic Strategy really makes sense. The poet, Sumner A Ingmark, envisioned the essence of the enigma like this:
Most systems for gambling are artful delusions,
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