The Blackjack Strategy Solver and Its Limits
Abstract. DiceGem offers many blackjack variants, each with its own deck and rules. One solver computes the expected value of every decision for all of them: it powers the in-game Hint button, the bots and the published strategy charts. This paper describes the model, the dealer and player calculations, and the approximations that keep the main bet of these games from carrying an exact published figure.
Summary
- One solver covers every blackjack variant: deck makeup, dealer rules, doubling, splitting, surrender and bonus payouts all come from the variant’s rule set.
- It assumes an infinite deck: the chance of each card value comes from the variant’s deck and does not change as cards are dealt.
- It does not model re-splitting, and it leaves out the 6-7-8 and 7-7-7 bonuses. Its advice is very close to best play, but not exact. That is why our fairness page lists the main bet of these games as depending on your decisions.
1. Card probabilities
The chance of drawing each value (ace, 2 to 9, and ten-valued cards) is taken from one deck of the variant. In a standard deck a ten-valued card has a 16 / 52 chance. In a deck without tens, as in Spanish-style variants, the chances change and so does the best play. This model is called an infinite deck, because the chances never change during the hand.
2. The dealer
The dealer’s play is fixed by the rules, so the solver computes the probability of every way the dealer’s hand can finish: 17, 18, 19, 20, 21, bust, blackjack, or a push on 22 where a variant has that rule. It follows each possible next card from the visible up card, with repeated states remembered so that each is solved only once.
When the dealer checks for blackjack before you act, the dealer’s hidden card is conditioned on not making a blackjack. Whether the dealer hits soft 17 changes the result as well.
3. The player
For your hand (its total, whether it is soft and its number of cards) the solver computes the expected result of each legal decision, in units of the original bet:
- Stand: compare your total with every dealer outcome from section 2.
- Hit: average over every next card, then play the best decision from the new hand.
- Double (and triple or quadruple down where a variant allows): one more card, then stand, with the stake multiplied.
- Split: the value of one hand that starts from one of the paired cards, counted for each of the two hands.
- Surrender: always −0.5.
- Redraw (where a variant allows it): two new cards, then the best decision.
The best decision is the one with the highest expected value. Bonuses for winning with five or more cards are part of the calculation.
4. What the model leaves out
- Card removal. In a real shoe, every card dealt changes the chances of the next one. In single-deck and few-deck variants this changes some close decisions. The infinite-deck model ignores it.
- Re-splitting. A split hand that receives another card of the same value is valued as if it could not be split again.
- 6-7-8 and 7-7-7 bonuses. These pay extra for specific three-card hands. The solver does not count them, so where a variant offers them its advice can differ from best play on a few hands.
Each of these effects is small, but together they mean the solver cannot certify an exact return for the main bet to six decimal places, which our exact return-to-player method requires. Side bets, where a blackjack game offers them, depend only on the dealt cards, so they do have exact published figures.
5. Where it is used
- The Hint button in our blackjack games shows the solver’s choice for your current hand.
- The bots at multi-seat tables play the same strategy.
- The blackjack strategy charts print the solver’s first decision for every two-card hand against every dealer up card, for each variant.
Tests check the solver against well-known basic strategy decisions, such as doubling 11 against a 6, standing on hard 17 against an ace and splitting eights against a ten.