Blackjack basic strategy, priced: the exact chart for this table’s rules

The rules the chart is built for

Basic strategy is not one chart. It is the best decision for every hand under a particular set of rules, and the rules move the cells: whether the dealer hits soft 17, whether you may double after splitting, whether the dealer checks for blackjack, how many decks are in the shoe. A chart printed for an eight-deck game where the dealer hits soft 17 is subtly wrong for a table where the dealer stands. So the first step is to write the rules down from the source rather than assume them.

Two of those rules make the arithmetic unusually clean. The infinite shoe means every card is an independent draw — each rank one in thirteen, ten-valued cards four in thirteen — so the value of a hand depends only on its total and whether it is soft, never on which cards made it. A hard 16 made of 10 and 6 plays exactly like one made of 9 and 7, which is not quite true in a real shoe. And the peek means that by the time you are asked to act, the dealer is known not to hold blackjack; every decision below is priced on that condition, which is why the dealer’s ace looks less frightening here than in a no-peek game.

With those facts the whole game is a finite recursion. The dealer’s final total from each upcard is solved first, as in the dealer bust table; then, for each player hand and upcard, the expected value of standing, hitting (and continuing optimally), doubling and splitting is computed exactly, and the chart is simply the largest of them.

The chart

H hit, S stand, D double (with more than two cards, hit instead).

D double, otherwise hit; Ds double, otherwise stand. *A,2 hits against a 5 and A,3 doubles.

The engine allows one split, so a pair dealt to a split hand is played by its total from the hard or soft chart. Tens mix: a king and a jack are a pair of tens.

Most of the chart is what every published S17 chart says, which is the point: it is a check that the rules were read correctly. Three rules move specific cells, and switching each one off in the same recursion shows exactly which. The dealer standing on soft 17 is why 11 hits against an ace, A,7 stands against a 2, A,8 stands against a 6 and A,4 hits against a 4; against a dealer who hits soft 17, all four become doubles. Doubling after a split is why 2,2 and 3,3 split against a 2 or a 3, 4,4 splits against a 5 or a 6 and 6,6 splits against a 2; without that rule all seven cells are hits. And the peek is why 8,8 splits against a ten or an ace, A,A splits against an ace and 11 doubles against a ten; in a game where a dealer blackjack also takes the doubled or split stake, those are hits.

The close calls, priced

EV is net: −0.54 means losing 54 cents per dollar staked on average. A split or double’s EV already counts the second stake. Cost is what the runner-up gives up per hand, in original bets.

Read the cost column before the chart and the priorities of learning it change. Hard 16 against a ten is the hand people argue about, and it is worth almost nothing: six ten-thousandths of a bet, a cent every seventeen dollars. On a finite shoe the answer even depends on whether the 16 has three or more cards; on this shoe it does not, and the hit wins by a whisker. Hard 12 against a 4 is almost as flat. These are cells to get right for tidiness, not for money.

The money is in the soft hands and the pairs. Standing on soft 18 against a 9 feels safe — eighteen is a made hand — and costs 0.083 of a bet, because the dealer’s nine makes 19 more than a third of the time and a soft hand cannot bust on the next card. Failing to double soft 18 against a 6 costs almost a tenth. Splitting tens against a weak dealer card is the famous temptation and the most expensive one on the list: a pair of tens is worth +0.704 standing and +0.576 split, so the split gives up an eighth of a bet to turn one excellent hand into two good ones. Splitting eights against a ten is the opposite case: both choices lose, and splitting loses about a twentieth of a bet less.

What the chart is worth

Every row includes the 3:2 blackjack. The difference between the first and last rows is about five and a half points of every bet made.

The table is the case for learning the chart and also its limit. Playing it exactly is worth five to five and a half points over the two folk strategies — mimicking the dealer loses because the player acts first and a player bust loses even when the dealer would also have busted; never busting loses because it hands the dealer every stiff hand. Doubles and splits alone are worth almost two points, which is why a player who has only learned when to hit leaves so much on the table.

And the top row is still under 100%. The engine header states the game at about 99.5% with basic play; the exact figure for these rules, with one split allowed and no resplitting, is 99.43%, computed here. The remaining 0.57% is the house edge that no hit, stand, double or split removes. The chart makes blackjack one of the cheapest games in the building. It does not make it a winning one, and because the shoe has no memory there is nothing to count that would change that — which is the story of the MIT team told backwards.