How basic strategy is derived

Basic strategy is not a set of opinions about blackjack. It is the output of a calculation: for every player hand against every dealer upcard, compute the expected value of each legal action by enumerating everything that can happen next, and keep the action with the highest value. The first such calculation was published in 1956 and done on desk calculators; today a strategy engine repeats it in a fraction of a second for whatever rules you enter.

The expected-value comparison

Take hard 12 against a dealer 4 under six-deck S17 rules. The Wizard of Odds expected-value table lists, for the composition 10+2, stand −0.211115, hit −0.210364 and double −0.420729 per unit staked. The best available option is the least negative one. Hit wins by 0.00075 here, but for 9+3 the same table gives stand −0.208039 and hit −0.215390, so stand wins. A total-dependent chart averages across all compositions of 12 and, for six decks, stands.

Each of those numbers is a sum over outcomes. For “stand”, the engine takes the dealer’s final-hand distribution for a 4 upcard with the player’s cards removed from the shoe, and adds +1 for every outcome that busts or finishes below 12 (only busts, since any completed dealer hand is 17 or more), −1 for every outcome of 17 or more. For “hit”, it considers each of the 13 possible draw ranks weighted by how many remain, and for each resulting hand asks the same question recursively: stand or hit again? The recursion ends when the hand busts or reaches 21. “Double” is one forced draw at double stakes. “Split” is the hardest, because it creates two hands whose results are correlated through the shoe.

The dealer’s distribution is itself computed the same way. The dealer bust probability table is a by-product of the strategy calculation, and the player bust probabilities are the simplest term in the hit branch.

1956: four desk calculators

Roger Baldwin, Wilbert Cantey, Herbert Maisel and James McDermott, mathematicians doing US Army service at the Aberdeen Proving Ground, published “The Optimum Strategy in Blackjack” in the Journal of the American Statistical Association in September 1956. Wikipedia describes it as “the first mathematically sound optimal blackjack strategy”. They had no computer; the Four Horsemen page records that they “used only the desk calculators available at the military base”, and their work was expanded into the 1957 book Playing Blackjack to Win.

Their method was the one described above, with approximations where exact enumeration was impractical by hand. The remarkable fact, recorded on the same page, is how little those approximations cost: when Thorp later recomputed the strategy by machine he found it accurate “within a couple of hundredths of a percentage point”, and the gambling writer Arnold Snyder put the cost of using the 1956 chart today at “a few hundredths of a percent” against perfect basic strategy. The Four Horsemen article tells the story in full.

Thorp and the IBM 704

Edward Thorp, then at MIT, took Baldwin’s hand calculations and re-ran them on an IBM 704, which Wikipedia records he “used as a research tool in order to investigate the probabilities of winning”. He confirmed basic strategy, published “A favorable strategy for twenty-one” in the Proceedings of the National Academy of Sciences in 1961, and wrote Beat the Dealer in 1962, substantially updated in 1966.

Thorp’s larger contribution was the next step: having a program that could evaluate any deck composition, he asked what happened to the player’s expectation when particular cards were removed, and found that a deck short of tens favoured the house while a deck rich in tens favoured the player. That observation is card counting, and the ten-count system in Beat the Dealer was its first published form. Basic strategy is the fixed-composition special case of the same calculation; the Thorp article and the card counting guide continue from there.

Total-dependent and composition-dependent

A total-dependent strategy makes one decision for each (total, upcard) pair. That is what every printed chart does, including the site’s strategy chart: one cell for hard 16 against 10, whatever cards make the 16.

A composition-dependent strategy makes a decision for each specific set of cards. Because removing cards from the shoe changes both what you can draw and what the dealer can draw, some near-tie decisions flip for particular compositions. The Wizard of Odds appendix 15 prices the difference for six decks, S17, double after split, no surrender: house edge 0.4062% total-dependent versus 0.4031% composition-dependent, a gain of 0.0031%. In a single deck the gain is 0.0387%. The composition-dependent exceptions article lists the hands that matter.

Both are “basic strategy” in the sense that neither uses information from previously dealt cards. The moment you use cards no longer in your hand, you are counting.

Reading a strategy engine’s output

Modern engines (the Wizard of Odds strategy calculator and the blackjackinfo.com Basic Strategy Engine, among others) take a rule set and return a chart and a house edge. Reading them well means checking four things:

1. The rules assumed. Deck count, soft-17 rule, doubling restrictions, double after split, surrender, peek or no-hole-card. The blackjackinfo engine, for example, shows 0.44% for six decks, S17, double any two, DAS, no surrender, peek, and 0.66% for the same game with H17.
2. Which table a cell comes from. Hard totals, soft totals and pairs are separate tables; a pair of 8s is looked up as a pair, not as hard 16. Where no pair row exists (5s and 10s on some charts) the hand is played as its hard total.
3. Fallback actions. Cells such as “double, otherwise hit” or “surrender, otherwise hit” encode what to do when the preferred action is not available, for instance a three-card 11 that cannot be doubled.
4. What the engine does not know. Insurance and even money: the Wizard’s strategy page states that its advice “should not be taken on insurance or even money”, meaning never take them without a count.

Expected-value tables such as the Wizard’s six-deck S17 table go one layer deeper and show the numbers behind each cell, which is the only way to see how close a decision is. A cell that wins by 0.0007 (12 vs 4) and one that wins by 0.2 (double 11 vs 6) look identical on a chart.

Why rules change the chart

The expected values depend on everything in the rule set, so changing a rule changes some winners. The Wizard of Odds rule-variation table prices the overall effect; the chart effect is where it shows up:

None of these is a matter of style. Each is the result of re-running the comparison with a different payoff structure. That is why the site’s chart states its profile (six decks, S17, double any two, DAS, no surrender, US peek) and why the rule variations article matters before you sit down at a table whose rules differ.

Related reading

• Composition-dependent exceptions
• How the house edge is calculated
• The Four Horsemen of Aberdeen
• Basic strategy guide