Working on a backgammon engine means asking what might happen after each legal move. The next dice matter, but so does the opponent’s choice: if a roll lets them hit, escape or make a point, the analysis has to consider which reply they will choose.
Blackjack gives us a useful comparison because its dealer follows prescribed rules. We still have to account for unknown cards, but we can calculate the dealer’s responses without predicting a freely chosen strategy. That distinction explains why two games with luck can present very different problems to a computer.
In September 1956, the Journal of the American Statistical Association published an analysis of blackjack by Roger Baldwin, Wilbert Cantey, Herbert Maisel and James McDermott, all affiliated with Aberdeen Proving Ground in Maryland. Their calculated strategy differed substantially from the advice of card experts and the play commonly seen in casinos.
The authors identified a crucial simplification near the beginning of their paper: the casino dealer’s strategy was fixed by known rules. The player faced uncertain cards, but did not have to anticipate a clever opponent changing plans.
The dealer cannot choose the most awkward reply
A blackjack dealer instead follows the table’s instructions about taking cards and standing. Rules differ between games, including whether the dealer takes another card on a soft 17, a total containing an ace valued at 11. Once a particular set of rules is specified, the player can calculate the dealer’s possible finishing totals and their probabilities.
That turns the comparison between hitting and standing into a calculation of expected gains and losses. The dealer’s rule is part of the model, so the computer can calculate the distribution of results it produces.
The best choice can still expect to lose
A common trap in discussing optimal play is to assume that finding the right move means finding a profitable one. The following simplified calculation makes the distinction visible.
A hard 16 against the dealer’s 6
You have a total of 16 with no ace counting as 11, and the dealer shows a 6. Compare standing with taking a card and then standing if you survive. A win pays one stake, a loss costs one stake and a tie returns the stake.
Stated model: cards drawn with replacement, aces worth 1 or 11, ranks 2–9 as likely as an ace, and ten-valued cards four times as likely. The dealer hits through 16 and stands on every 17, including soft 17. Surrender, doubling and splitting are outside this choice.
Compare the expected gains and losses
| Decision | Expected result |
|---|---|
| Stand | Loss of 15.4 |
| Take a card | Loss of 43.1 |
Standing loses less on average and is the better of the two choices. The best available decision can therefore still have a negative expected return. The result of one hand cannot validate the calculation.
Original calculation by enumerating the model’s possible card continuations, without simulation. Drawing with replacement keeps the probabilities constant; a real shoe has a changing composition.
Standing with 16 feels vulnerable because it cannot beat a dealer who reaches 17 or more. It can still win when the dealer goes over 21. Taking a card offers stronger finishing totals, but also introduces an immediate chance of going over 21 yourself, which loses before the dealer’s result can rescue you.
The calculation does not promise a win on the next hand; it says which of these two choices loses less on average under the stated assumptions. That is the sense in which an expected-value calculation can identify a correct decision in an unfavourable position.
The cards that have gone change the problem
Our example draws cards with replacement, keeping the probabilities constant. Actual blackjack draws from a finite deck or shoe. Once cards have been dealt, the composition of the remainder changes, so the state of the calculation can include more than the player’s total and the dealer’s visible card.
Edward Thorp’s 1961 paper, A Favorable Strategy for Twenty-One, showed how dependence between successive draws could support a favourable mathematical strategy under the conditions he studied. A known dealer policy and knowledge about remaining cards were different ingredients in the calculation.
This also separates cards from backgammon’s independent dice. Seeing several sixes does not use up any sixes in a fair dice process; seeing cards leave a shoe changes which cards remain. Calling both games random should not hide that difference.
What this adds to a comparison of games
The 1956 paper analysed specified casino rules, not every game sold under the name blackjack. Its historical calculations should not be treated as a proof of perfect strategy for every rule variation. The important structural point is the authors’ stated reason the problem was manageable: the dealer’s behaviour was known.
Bridge has random cards too, but adds partners, strategic opponents and decisions made without seeing other hands. Backgammon has public positions and random rolls, with an opponent choosing what to do after each roll. Chess has no random draw in the rules, but the opponent is free to find the reply you missed.
A ranking of computational difficulty has to specify which task it compares. Calculating a choice against fixed behaviour, proving perfect play against any opponent and learning a useful approximation are separate projects, even when each ultimately assigns a value to the available moves.
Sources: Baldwin, Cantey, Maisel and McDermott, The Optimum Strategy in Blackjack (1956), with an institutional copy of the original paper; Edward Thorp, A Favorable Strategy for Twenty-One (1961). The hard-16 calculation is original and is not copied from either paper. It enumerates card continuations in the explicitly stated with-replacement model, with the dealer standing on soft 17. Its expected returns are conditional on this position and these two actions, not a whole-game house edge or a result for every casino rule set.