Blackjack Betting and Playing Strategies: A Statistical Comparison By Jared Luffman MSIM 752 12/3/2007.
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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
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