Saturday, April 10, 2004

Doubling on a 6:5 Blackjack

Blackjack games aren't all the same from one casino to the next. In addition to the obvious differences of how many decks are in play, what range of bets is allowed and how deeply the deck is dealt, there are many subtle rule variations. Blackjack players look for such favorable variations as:
  • S17 - dealer Stands on a soft 17
  • DAS - player is allowed to Double down After Splitting
  • DOA - player can Double On Any two initial cards, not just 10 or 11.
  • Surrender - player may give up half his bet in lieu of playing out the hand.
All of those options help the player, if present.

But a new rule variation that really hurts the player has become fairly common in the last year. The latest thing is to have blackjack tables where being dealt a natural blackjack (an ace and a ten-valued card) pays 6:5 instead of the usual 3:2. That has a huge negative impact on the player, so much so that most advantage players and many non-advantage players won't play this game.

In thinking about whether this game might be used for cover purposes or was worth playing at all, I got to wondering whether the reduction in blackjack payoff justified any changes in strategy. In particular, it seemed to me lowering the payoff on blackjack from 150% to a mere 120% might be big enough that it would produce situations in which the right thing to do would be to double down on your blackjack. I couldn't find numbers on this, so I wrote a simulation. And now that I've got numbers, it turns out it's almost always still better to take the blackjack than to double down.

But "almost always" isn't the same as "always". So when IS the break-even point?

Here's the answer: if you're playing a single deck S17 game and you get a blackjack, the correct play is to double down if the true count is more than ten. You need a count in excess of ten to get an expected rate of return of over 120%. Against a dealer's 5 showing, the return on doubling starts at 67% and goes up by about 5 percentage points for each increment of the count until it finally catches up with and surpasses the option of taking the blackjack bonus.

Here are the actual numbers for the decision of blackjack specifically versus a dealer 5 -- here's how much money you make by doubling down on it:
count -1 wins 669002 units, 66.9002 pct.
count +0 wins 710436 units, 71.04 pct.
count +1 wins 750798 units, 75.08 pct.
count +2 wins 790728 units, 79.07 pct.
count +3 wins 836690 units, 83.67 pct.
count +4 wins 882222 units, 88.22 pct.
count +5 wins 926308 units, 92.63 pct.
count +6 wins 981836 units, 98.18 pct.
count +7 wins 1029052 units, 102.91 pct.
count +8 wins 1080914 units, 108.09 pct.
count +9 wins 1137868 units, 113.79 pct.
count +10 wins 1195290 units, 119.53 pct.
count +11 wins 1257614 units, 125.76 pct. (==there it is!
count +12 wins 1316834 units, 131.68 pct.
That's all based on one million trials per line. The numbers aren't all that different at the high end if the dealer has other upcards showing; they all show profit around +11 or so.

[UPDATE]: The numbers above aren't valid, especially for heads-up play, because I failed to account for card-removal effects. When you're the only player at the table and you take the blackjack, the dealer doesn't play out his hand. But if you double down, you take an extra card and the dealer takes as many extra cards as he needs to play out his hand. Those extra cards are all being used up at a time when the deck is hugely positive and you might rather leave them IN the deck so as to get an extra round of play.

Performancing