The aim of this project is to apply graph theory to explore the sample space for blackjack, a popular card game in casinos that has been widely studied and analyzed. My goal is to generate the various possible game states and how they are connected with each other to see which sequence of player actions results in the highest expected value for a given starting hand and the visible dealer card.
Blackjack is a popular card game in casinos. It is generally played with one to four players who all aim to beat the 9. All the cards have a specific value that counts towards the score of a hand (i.e. the cards that belong to a player or the dealer).
The shoe is a box that contains one or more standard decks of cards from which cards are drawn during a round of blackjack. Historically, the cards were shuffled first and then placed inside the shoe, but nowadays there are continuous shuffling machines which shuffle the cards inside the shoe.
A hand is a set of cards that is used for scoring. In American blackjack, the dealer starts with a hand of one visible card (face up) and one hole card (face-down, not visible). The dealer can never have more than one hand at a time. Players each start with a hand of two visible cards, and if the hand is splittable, they can split the hand into two and receive two more cards from the shoe, resulting in more than one hand for the player. Scoring happens independently for each hand, so it is possible for a player to have one hand that beats the dealer's and another that loses, for example.
The dealer in blackjack deals the cards and draws cards from the shoe when necessary. The dealer also has a hand and therefore plays against the players, representing the house. They collect lots bets or pay out players based on their hands and their decisions. When all the players' turns have ended, the dealer follows a fixed set of instructions until their turn ends. When all the players and the dealer have played, the round concludes, at which point any lost bets are collected and any winning hands are paid out.
The player is someone who bets against the house and plays against the dealer. They are allowed to make certain decisions during their turn, such as asking more cards from the shoe, splitting a hand or surrendering.
In blackjack, each card has a value which is used to calculate the total hand score. The face cards or court cards (Jack, Queen and King) have a value of 10, all other non-Ace cards have a value equal to their number of pips, and the Ace can count as 1 or 11, whichever yields the highest score without exceeding 21. A blackjack, which is what the game is named after, is a hand that consists of an Ace and a ten-valued card. In many rule sets, a blackjack beats non-blackjack hands that have a total score of 21.
If a hand contains an Ace and its value counts as 11, then that total is considered soft. For example, A♣6♥ is a soft 17. A hard total applies to hands that do not contain an Ace, or where the Ace can only count as a 1. For example, 8♣9♥ and A♣6♥Q♠ are both hard 17s.
If the score of a hand exceeds 21, it has gone bust.
The dealer deals the cards one by one, first for the players then for themselves, until everyone has two cards. For simplicity, let's assume there is only one player.
The player can make decisions that may change the score of their hand, hopefully to beat the dealer's hand.
| Decision | Description | Hand signal |
|---|---|---|
| Stand | End your turn. | Wave your hand. |
| Hit | Get the next card from the shoe and add it to your hand. | Tap the table. |
| Double down | Double your bet, hit one card and end your turn. | Increase your bet and point with one finger. |
| Surrender | Give up half of your bet and end your turn with no further losses. | - |
| Split | If your hand has two cards with the same value, split them into two hands and get two new cards from the shoe to bring each hand to two cards. | Increase your bet and make a "peace" sign. |
The dealer's decisions are more constrained. Their initial hand score after revealing their hole card will dictate what the dealer must do next. If the hand total is 17 or less, the dealer must hit. For a hand total of 18 or more, the dealer stands. Most casinos have the dealer hit on a soft 17. For hand scores of 18 or more, the dealer always stands, no matter if the total is soft or hard.
A player loses their entire bet when they lose, or half when they surrender. Winning means that a player get twice their original bet back, or more for a blackjack (generally 2.5x or 3x).
The game state encapsulates the necessary information about the player, the dealer and the shoe to be able to calculate all possible next states, for example because a player or the dealer make a decision. When there are no more possible states, it means that the round has come to an end.
The player has one or more hands which can be in play or which can be ended (by standing, doubling down, surrendering or busting). Each hand has a state and allowed player decisions that could lead to future states.
The dealer state is similar to the player state, with a single hand, state and allowed decisions which may lead to future states.
The shoe state simply contains the remaining cards for a given state. All the cards in play (i.e. the dealer's cards and the player's cards) and the remaining cards in the shoe should make up the starting number of full standard decks.
Two players are equivalent if their hands are equivalent, and two hands are equivalent if they contain the same number of equivalent cards and if the hand states and decisions are the same.
Two dealers are equivalent if their hands, their state and their allowed decisions are equivalent.
Two shoes are equivalent if they contain the same number of cards and if each card in a specific position of the first shoe is equivalent to the card in the same position in the other shoe.
Two game states are equivalent if their player states, dealer states and shoe states are all equivalent.
Decision trees are a way to model the sample space of a probability calculation using a tree graph structure. Each node in the tree represents a possible states of the sample space (for example, heads or tails in a coin flip, rain, snow or sunny weather for forecasting, etc.) and the weight of an edge between two nodes is the probability or likelihood of the state changing from the starting state to the end state.
If the events or states are statistically independent from one another, it is possible to calculate the probability of a certain final state (i.e. a state that has no subsequent next states) by traversing the path from an initial state all the way to that final state and multiply all the edge weights together.
In blackjack, the aim is to maximize your wins or to minimize your losses. Given all the possible outcomes and all the decisions a player can take, there are many possible end states. But using the decision tree representation, it is possible to calculate the average expected win (or loss) based on the probabilities calculated from each state to its next.