A Blackjack solver. The player's full hand composition is considered, not only the total sum.

Parameters

• Deck (default: standard single deck)
• Blackjack amount (default: $B=21$)
• Dealer's target (or lowest amount to stand on) (default: $T=17$).
• Ace alternate extra value. Base value is 1. (default: 10)

The dealer always stands on reaching soft target $T$.

Todo

• Add double down
• Add split
• Add option for dealer to peek for blackjack
• Add option for dealer to hit when reaching soft target

Blackjack solver - rough outline

The steps to determine an optimal policy using dynamic programming:

1. List all valid hands (all hands that have not busted), including empty and 1-card hands.

2. Find the probability $D_s(h, c, t)$ that the dealer will stand at $t = T,...,B$, for each player hand $h$ and each revealed dealer card $c$.

3. The EV (expected value) of standing at state $(h, d)$ is

$$ EV_s(h, d) = P(\text{dealer busts}) + P(t < S(h)) - P(t > S(h)) $$ $$ = \sum_{t \geq S(h)} D_s(h, d, t) - \sum_{t > S(h)} D_s(h, d, t) $$

where $S(h)$ is the score or sum value of hand $h$.

4. The optimal policy/value for each state $(h, c)$ can be determined in one sweep of all states going backwards from the largest hand size states to the lowest.

Solutions

For base parameters:

$EV = -0.04246$. "H" is hit. "S" is stand". "U" is surrender. For two-card hands not shown, the strategy is always to hit.