Blackjack Betting and Playing Strategies:A Statistical ComparisonBy Jared Luffman

MSIM 75212/3/2007

Model Dynamics• Randomness of Deck

– Shuffling, Cutting, End of Shoe

• Designing Betting Patterns– Flat Bet– Modify Bet by Counting

• Designing Player Strategies– Dealer Rules– Recommended Strategy– Counting for Probability

• Rules/Game Play– Surrender, Even Money– No Insurance

Implementation• Visual Studio .NET

– Visual Basic

Implementation• User Input

– GUI Executable– Strategies

•Play like dealer•Recommended•Counting

– Betting•Standard Bet•Counting

– Advanced Counting

Implementation• Model Output

– Text Files– CSV

Modeling Objectives• Determine Statistical Significance of

– Betting Strategy• Counting vs. Standard Bet

– Playing Strategy• Playing Like Dealer vs. Recommended• Playing Recommended Strategy vs. Counting

– Table Strategy• Playing Alone vs. Playing with Other Players• Playing Alone with Counting vs. Playing with

Others who Play Recommended Strategy• Playing Along with Counting vs. Playing with

Other Players who Count

Analysis• Compared Results After Playing a

Complete Shoe– All Results are IID if based on

Randomly Distributed Cards

• Deck Randomness– Shuffling

•Cut Location•End of Shoe Location•Cards

Deck Randomness• Shuffle

– Used a Reducing Uniform Distribution• First card = Uniform (1, 312)• Second card = Uniform (1, 311) [after 1st card

shuffled]

• Cut Location– Uniform (52, 260)

• Cannot Cut within 1 Deck of Beginning or End of Shoe

Deck Randomness• End of Shoe Location

– Triangular (208, 26)• End Of Shoe Should be at 4 Decks and within a half

a deck

Deck Randomness• Cards

– Average Value7.307692/)( nCardValue i

Analysis• Single Player Results As Difference of

Matched Pairs– i.e. E[Strat1/Bet1] – E[Strat2/Bet2] confidence

interval,E[Strat2/Bet2] – E[Strat3/Bet3] confidence

interval

– Hypothesis Test• H0: (1 - 2) = 0 H1: (1 - 2) > 0

• 95% Confidence Interval

– 250 Simulations

Analysis

TEST 1 - 2 d T t0.05 Result

Dealer with Standard Bet vs. Dealer With Simple Counting

1.4 287.5304 0.077 1.96Failed to

RejectRecommended with Standard Bet vs. Recommended With

Simple Counting-15.2 347.7754 -0.691 1.96

Failed to Reject

Counting with Standard Bet vs. Counting With Simple

Counting-80 344.2587 -3.674 -1.96

Reject Null Hypothesi

s

• Betting Strategy Analysis– Standard Bet vs. Simple Counting

Bet

– Cannot determine if standard bet is better or worse than simple counting based bed

Analysis

TEST 1 - 2 d T t0.05 Result

Dealer with Simple Counting vs. Dealer With Advanced

Counting67.8 391.286 2.74 1.96

Reject Null

Hypothesis

Recommended with Simple Counting vs. Recommended

With Advanced Counting1.128

510.3093

0.035 1.96Failed to

Reject

Counting with Simple Counting vs. Counting With

Advanced Counting-71.096

995.8087

-1.129 1.96Failed to

Reject

• Betting Strategy Analysis– Simple Counting Bet vs. Advanced

Counting Bet

Analysis with Discussion

TEST 1 - 2 d T t0.05 Result

Recommended with Simple Counting vs. Dealer With

Simple Counting242.2 958.84 3.99 1.96

Reject Null

Hypothesis

Counting with Advanced Counting vs. Recommended

With Advanced Counting-71.096 1214.98 -2.97 1.96

Reject Null

Hypothesis

• Playing Strategy Analysis

• Definitively, the Recommended Strategy gives better results on average than the dealer’s strategy or the counting strategy

Analysis• Playing Strategy Analysis with Multiple

Players– Common Strategy w/ Normal Betting & N Additional Players– Common Strategy w/ Counting Betting & N Additional Players– Counting Strategy w/ Normal Betting & N Additional Players– Counting Strategy w/ Counting Betting & N Additional Players– i.e. E[Batch(Strat1,1/Bet1,1)] – E[Batch(Strat1,2/Bet1,2)],

E[Batch(Strat2,1/Bet2,1)] – E[Batch(Strat2,2/Bet2,2)]

• Determine if the Number of Players affects Results

• Test for Best Strategy to Use with N Players

Analysis• Confidence Intervals Driven by Standard

Deviation– Results varied by shoe (-2700, 2900)– Average difference relatively close to 0

• Tests Inconclusive– Mixture of Rejecting the Null Hypotheses

H0: 1 - 2 = 0 for H1 : 1 - 2 > 0

• Need to Increase Sample Size– Standard Deviations are too High– N needs to be larger

Future Analysis• Ran 50000 simulations for a single

player using the Recommended Strategy and a standard bet

• Statistical values of interest =0.304 =699.0835– n = 50000

• H0: 1 = 0 H1 : 1 > 0 – Fail to reject Null Hypothesis– However, larger n and stabilized and

mean other tests may change

Lessons Learned

• Lessons Learned– Handling Simulation Run Output

• Limited by Excel– Integer Values vs. Reals

• Card values, Chips, etc are Integer Values• Probabilities are Reals• Dealing with Probabilities of Card Values

• There is such a things as too much data– Originally thought 250 simulations would be sufficient

• Saved a lot of data– Needed to Max-out Excel (50000 simulations)

• Too little time to go back and remove all data outputs• Large simulation took 30 minutes to run (broken into 5

runs)

Improvements• Data Handling

– Allow user to specify data to be recorded

• Improve Counting Strategy– Counting should improve odds

• Probabilistic Betting Strategies– “Build your own” Strategy– Dealer Strategy and Recommended were easy

because they were defined

• Optimization– A search heuristic to determine the best “action”

strategy based on perfect knowledge of the system• i.e. Could you hit on a 20 to get a 21, or hit on a 20 to

bust so your following hands would be better off

Conclusions• The only concrete result was that the

Recommended Strategy provided better payoffs on average than the dealer’s strategy or the counting strategy

• Need to make modification to output to speed up simulations to rerun hypothesis tests with increased N value

• Find a known counting strategy and implement it to see how it compares to the standard recommended strategy