Word Stand or Hit: Simulation of Blackjack by Programming

Yiqi Huang

School of Natural Sciences, University of Manchester, Manchester, M13 9PL, U.K.

Keywords: Blackjack, Game Theory, Optimal Strategy.

Abstract: Blackjack is a global gambling game with great influence and popularity where players aim to have a hand

total closer to 21 than the dealer’s hand without exceeding it. It is a game played against the dealer or the

casino. In Blackjack, the player's subjective decision almost determines the outcome of the game, while the

dealer follows a fixed rule resulting in less subjective impact on the game. As a result, players strive to find

the optimal strategy to gain profit. This paper designs a computer program to discuss whether choosing to

continue or stop drawing cards is statistically wiser for different initial cards. The paper also emphasises how

probability principles can be applied to Blackjack, providing guidance for practical decision-making in

casinos. The optimal player actions for each combination of initial dealer and player hand types calculated by

the program are summarised in a table to provide players with effective references.

1. INTRODUCTION

Blackjack, also known as 21, is one of the most

popular gambling games nowadays. The player is

aiming to have a hand total that is closer than the

dealer’s hand but without going over 21. Each player

is dealt two cards initially and can choose to take

more cards or stop at the current total, which is called

“hit” or “stand” respectively. Face cards such as

Kings, Queens, and Jacks are counted as 10 points,

Aces can be counted as either 1 or 11 points, while

all other cards are worth their face value. The name

“Blackjack” of the game comes from the best

possible hand, which is an Ace and a 10-point card.

If the player's hand exceeds 21 points, they

automatically lose regardless of the dealer's hand,

which is called “bust”. On the contrary, if the dealer

busts cards, then all players who did not bust their

cards are considered winners.

Blackjack is ancient and simple. The game could

be traced to the 15th century when Gutenberg's

printing press was invented (Snyder, 2013). It

requires very few props: only chips and playing

cards, and the rules are simple and uniform. As a

result, the game has a unique charm that attracts

people all over the world and can be found in almost

any casino. According to statistics, as the second

most profitable gambling game in America, 31% of

table game action is made up by Blackjack (Lindner,

2023). Another important reason why this casino

game is popular among players is that using

reasonable strategies can effectively increase the

winning rate. In general, the games operated in the

casino are games with negative player expectations,

because the casino needs to make a profit. However,

the disadvantages of blackjack are smaller than other

popular casino games, such as American Roulette

(Baldwin et al., 1956). With the development of

game theory and the standardization of the game,

more and more Blackjack optimal strategies have

been proposed. The concept of the Game Theory

Optimal (GTO) strategy, which is central to game

theory, was first developed by John von Neumann

(Neumann and Morgenstern, 2007). Culbertson

proposed a strategy in 1952 that brought the player's

expectations to -0.036 (Culbertson, 1952). This

makes blackjack almost a fair game and even means

there is a chance of making money from the casino.

Manson proposed a strategy for four decks used in

1975 (Manson, Barr and Goodnight, 1975). In

addition to the analysis of decisions for action, there

is also a lot of analysis of decisions for betting based

on card counting, which was first proposed by Thorp

(Thorp, 2016). Later in 1963, Dubner proposed the

Hi-Lo system at a conference in Las Vegas, which is

a simplification of the complex strategies that had

been previously developed by Thorp, which made

card counting more accessible to the general public

(Snyder, 2013). The Hi-Lo card counting method is

widely recognized and its practical application has

Huang, Y.

Word Stand or Hit: Simulation of Blackjack by Programming.

DOI: 10.5220/0013055800004601

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 1st International Conference on Innovations in Applied Mathematics, Physics and Astronomy (IAMPA 2024), pages 253-257

ISBN: 978-989-758-722-1

253

been extensively studied (Wong, 1994). Generally,

Game theory gives ideal frameworks that can be used

to provide guidance on the allocation of limited

resources and can be applicable such as solving

transportation problems (Hughes and Chen, 2021). It

can also be used to develop pricing models aimed at

balancing loads in the grid (Ghosh et al, 2004).

Therefore, game theory and optimal strategy

research are considered broadly applicable and

meaningful.

This paper will introduce a method of using an

open-source programming language and software

environment: R language to solve the Blackjack

strategy problem. That is, R language will be used to

give expectations for selecting hit or stand under

different player versus dealer cards situations to give

players a reference for decision-making.

2. METHODOLOGY

2.1. Game Description

Specifically, the process of the Blackjack is as

follows: the dealer first deals two cards to each

player, and then deals one card to himself. These

cards are all shown publicly, which means that

players also have the information of the two cards of

other players. After that, players make decisions in

order from right to left. The player can choose to

“stand”, for which the sum of the two cards in hand

will become the final points; or choose to “hit”, for

which the dealer will deal one more card to the player

to increase the sum, and players are allowed to hit

multiple times. Mentionable, players aim to not

exceed 21 points, or they will be eliminated

immediately. The dealer will take action after all

players have ended their action round. Unlike the

players, the dealer's actions do not depend on

subjective will, but on a fixed rule: the dealer must

hit if the dealer’s card sum is less than or equal to 16

points and must stand if greater than or equal to 17

points. At this moment, all surviving players will

compare their points with the dealer and whose

points are not as high as the dealer's will lose all bets.

With a bigger card sum, the player will win the game,

and the payout varies according to the odds specified

by the table and casino. If the dealer's hand exceeds

21, then all players who survived, which means those

who did not lose at the player action, win. If the

points are equal, it is a tie, with no profit or loss for

the players.

There will be different rules according to

different casino regulations. For example, if the

player believes that his two initial cards are very

favourable compared to the dealer's one initial card,

then the player can choose to double in his round,

that is, double his original bet. In this case, the dealer

gives the player one card only, which means there are

no multiple hits. Therefore, players need to consider

whether their winning odds are worth doubling down

on, as either gains or losses are doubled. Some

casinos allow betting on flushes, straights, or triple

7s. Although the probability might be extremely low,

the payment could be very impressive, with some

odds can reach 1: 500.

2.2. Simulate Process and Convert to

Code

In summary, it is crucial for players to correctly

analyse both sides' initial cards and take action. This

article will use R to produce an answer of whether

the player should choose to hit or stand in terms of

probability by various player points and dealer points

in the situation of one deck of cards and one player

playing. First, an environment with one deck of cards

is needed to be created. The code shown in Table 1

will create a vector: numbers from 2 to 10 represent

the number cards, three 10s represent the face cards,

and an 11 represents the ace card. This vector is then

repeated 4 times to form an environment of 52

standard playing cards. The user could simply

change the variable in the repeat code to a multiple

of 4 in simulation of multiple decks of cards. In

theory, using multiple decks will slightly increase the

dealer's winning rate, which is tiny enough to be

ignorable.

Table 1. R code Function call for one standard deck.

Algorithm 1

Step 1 Import the dplyr librar

Step 2 Create an array 'deck'

- Initialize 'deck' with card values:

2,3,4,5,6,7,8,9,10,10,10,10,11.

Step 3 Create an array 'full_deck' by

licatin

'deck' four times.

Next, the simulation for the dealer’s hand is

created as shown in Table 2 by setting a parameter to

initialise it and using an infinite loop to simulate the

dealer continuously drawing cards until they meet

the standing condition or bust. After updating points,

a check action is needed to be proposed for the ace

card because of that it can be either 1 or 11.

Logically, the ace card could be considered that it

only represents 1 point after busting. This allows the

R language to subtract 10 points from the hand card

IAMPA 2024 - International Conference on Innovations in Applied Mathematics, Physics and Astronomy

254

satisfying two conditions: busted cards and ace cards

included, to achieve changing the aces from

representing 11 points to representing 1 point.

Finally, the code checks if the card is busted and

returns the outcomes. The initialization of the

player's hand is the same as the dealer's, but the

player's actions need to be defined further.

Table 2. R code Function call for dealer and player

simulation.

orithm 2

Step 1 Define a function that takes the

dealer’s initial card value input by

user.

Step 2 Start an infinite loop

- If dealer’s hand is less or equal to

16, draw a new card from the deck

and update dealer’s hand by adding

the value of the card drawn.

- If dealer’s hand is greater or equal

to 17 with an Ace, minus dealer’s

hand by 10.

- If dealer’s hand is greater or equal

to 17, exit the loo

Step 3 Return the final value of dealer’s

hand once the loop is exited (the

dealer either stands or busts

)

Step 4 Define a function that stores the

dealer’s initial card value input by

user.

Step 5 Initialize a data frame named

outcomes

- Create an empty data frame called

outcomes with three columns:

- Strategy: to store different

playing strategies the player might

use, expected as character data.

- Result: to store numerical results

related to the strategies, expected as

double-precision numbers.

- Win: to store the win or loss

outcome for each strategy, also

expected as double-precision

numbers.

In this next step, the algorithm shown in Table 3

simulates the outcome of choosing to stand or hit by

the given initial card which is input by the users. If

the dealer busts or the player's hand is higher, the

player wins and is recorded as (1). If the player's

hand is smaller, the player loses (-1). If the hands are

equal, the result is a draw (0). Finally, to record to

result a vector is created. The programming for a hit

will be significantly more complicated. Firstly, an

order is given to randomly draw a card from the deck

set before, then add its points to the player's initial

hand, with the same special design of the aces. If the

player does not bust, repeat the previous recording

method. If the player busts the card, it will be directly

recorded as (-1). Similarly, the results are recorded

in the result vector as well.

Table 3. R code Function call for hit or stand simulation.

orithm 3

Step 1

Store the result of dealer’s hand after

simulation.

Step 2

For player choosing to stand, define

an if-else function.

- If dealer’s hand exceeds 21,

output (1).

-Otherwise:

- If player’s hand is greater

than dealer’s hand, output (1).

- If player’s hand is less than

dealer’s hand, output (-1).

- If player’s hand is equal to

dealer’s hand, output (0).

Step 3

For player choosing to hit, draw a

new card from the deck and update

player’s hand by adding the value of

the card drawn.

Step 4

Define an if-else function to check

Ace card.

- If player’s hand exceeds 21 with

Ace card, minus 10 and output.

-Otherwise, out

ut directl

Step 5

Define a function for comparison.

- If player’s hand exceeds 21,

output (-1).

- If player’s hand is less than 21:

- If dealer’s hand exceeds 21,

output (1).

- Otherwise:

- If player’s hand is greater

than dealer’s hand, output (1).

- If player’s hand is less than

dealer’s hand, output (-1).

- If player’s hand is equal to

dealer’s hand, output (0).

Step 6

Return outcomes for both stand or

hit.

It is not enough to so far since the aim is to design

code that can output probability which could be

achieved by simply simulating a large number of

times and calculating expectations. In Table 4, the

program is ordered to simulate 10000 times with the

input player's card is 15 points and the input dealer's

card is 7 points. These three parameters can be input

freely by users to meet the needs of real-time

simulation. The output gives two probabilities as a

reference, which are the expectations of selecting hit

and stand for action in the situation of the input.

Players should choose the action with the larger

output to ensure a higher winning rate.

Word Stand or Hit: Simulation of Blackjack by Programming

255

Table 4. R code Function call for expectation.

orithm 4

Step 1 Use the replicate function to run the

hit and stand simulation function

10,000 times with initial player hand

and dealer card input by user.

Combine all individual simulation

results into a single data frame

Step 2 Combine all individual simulation

results into a single data frame.

Step 3 Group and summarize the data by

strategy

- Group results by the 'Strategy'

column.

- Calculate the mean of the 'Win'

column for each strate

rou

Step 4 Print the summary data frame

3. RESULTS AND DISCUSSION

3.1. Example of player 15 versus dealer 9

Suppose in a game, the player gets a King and a 5,

and the dealer gets a 9. This brings the player's points

to 15 and the dealer's points to 9. The question

interested in is whether a hit or a stand has a

statistically higher chance of winning in this case

(Table 5).

Table 5. An example of running the algorithms.

Input Out put

Replicate (10000,

simulate_player_actions (15,

9), simplify = FALSE)

Strategy

AverageWin

1 Hit Once -0.480

2 Stand -0.547

According to the results, the winning rate of hit is

-0.480, and the winning rate of stand is -0.547. This

demonstrates that this is a favourable situation for the

dealer because both actions have negative

expectations for the player, which means that in the

long run, the player expects to lose. However, the

player should choose to hit in this situation as it is

less likely to lose by comparison. According to the

rules, a player can hit multiple times. Therefore, if

there is no bust after a hit, the player can use the

updated points as input to calculate the expectation

again to decide whether a second hit is needed.

3.2. Solution on other simulations

Similarly, it is reasonable to list every possible hand

situation and the corresponding winning rates of hit

and stand. This is summarised in Table 6, with the

dealer’s hand in the row and the player’s hand in the

column.

The coloured options are the actions that are more

likely to win. It is worth noting that most of the

optimal choices are still negative. Only in very lucky

cases, one can get a large positive expectation close

to 1 with a clear advantage. This proves that

blackjack is a game that favours the dealer, even if it

is not obvious. Situations where the player's hand is

equal to or less than 10 are not recorded. While

expectations can be calculated in these situations,

they are meaningless. Assuming the player's hand is

equal to or less than 10, then a hit will increase the

points without any possibility of resulting in a bust.

Therefore, when the player encounters these

situations, they should choose to hit the card no

matter what the expectation is.

The code in this article gives the optimal solution

considering only hit and stand. In most casinos,

double or split is allowed to improve the player's

winning rate, which is not considered in the

probability calculation.

Table 6. Expected value in various situations.

Expected value table

2 3 4 5 6

Hit Stand Hit Stand Hit Stand Hit Stand Hit Stand

-0.22 -0.26 -0.22 -0.22 -0.19 -0.18 -0.17 -0.11 -0.17 -0.15

-0.28 -0.24 -0.27 -0.21 -0.27 -0.17 -0.24 -0.13 -0.21 -0.15

-0.34 -0.24 -0.34 -0.22 -0.31 -0.17 -0.30 -0.12 -0.30 -0.16

-0.41 -0.25 -0.39 -0.19 -0.39 -0.19 -0.37 -0.11 -0.37 -0.16

-0.47 -0.25 -0.46 -0.20 -0.45 -0.15 -0.44 -0.12 -0.44 -0.16

-0.53 -0.11 -0.53 -0.08 -0.51 -0.05 -0.52 -0.01 -0.52 0.01

-0.61 0.14 -0.61 0.19 -0.73 0.44 -0.62 0.23 -0.62 0.27

IAMPA 2024 - International Conference on Innovations in Applied Mathematics, Physics and Astronomy

256

-0.71 0.41 -0.73 0.43 -0.63 0.20 -0.73 0.45 -0.73 0.49

-0.85 0.65 -0.86 0.65 -0.86 0.68 -0.85 0.69 -0.86 0.70

-1.00 0.88 -1.00 0.89 -1.00 0.90 -1.00 0.90 -1.00 0.90

7 8 9 10 Ace

Hit Stand Hit Stand Hit Stand Hit Stand Hit Stand

-0.24 -0.47 -0.30 -0.51 -0.37 -0.54 -0.43 -0.58 -0.48 -0.57

-0.27 -0.47 -0.33 -0.51 -0.40 -0.53 -0.47 -0.59 -0.49 -0.57

-0.34 -0.48 -0.37 -0.52 -0.44 -0.55 -0.51 -0.59 -0.55 -0.58

-0.37 -0.50 -0.43 -0.51 -0.48 -0.55 -0.55 -0.57 -0.58 -0.58

-0.42 -0.48 -0.48 -0.52 -0.51 -0.56 -0.55 -0.57 -0.61 -0.57

-0.48 -0.11 -0.50 -0.38 -0.57 -0.41 -0.61 -0.46 -0.64 -0.47

-0.59 0.40 -0.59 0.12 -0.62 -0.18 -0.67 -0.25 -0.71 -0.24

-0.72 0.61 -0.71 0.59 -0.72 0.28 -0.74 -0.02 -0.78 -0.02

-0.87 0.79 -0.85 0.80 -0.85 0.77 -0.86 0.42 -0.87 0.21

-1.00 0.93 -1.00 0.93 -1.00 0.94 -1.00 0.89 -1.00 0.65

4. CONCLUSION

Due to its high subjectivity, people always want to

improve their odds in Blackjack through personal

decisions. However, in some seemingly even

situations such as 16 versus 10, decisions are often

hard to make. But from a probabilistic perspective,

each hand will have an optimal decision that gives

the player a higher odds of winning. In other words,

there is a set of optimal strategies that will make the

player win the most or lose the least if the player

follows it for a long term. This paper proposed a

probability analysis of strategies used in Blackjack.

Although some minor part of the rule is not

considered, the simulation is still close to real-life

games. By using the R program, every situation a

player would encounter was simulated and the

optimal strategy was given. This provides an

important reference for real casino decision-

making. At the same time, it proves the minor bias

of the game. Without considering additional

strategies, it is difficult for 21 points to become a

profitable game for players.

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