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

2. 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

3. Implementation Visual Studio.NET –Visual Basic

4. Implementation User Input –GUI Executable –Strategies Play like dealer Recommended Counting –Betting Standard Bet Counting –Advanced Counting

5. Implementation Model Output –Text Files –CSV

6. 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

7. 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

8. Deck Randomness Shuffle –Used a Reducing Uniform Distribution First card = Uniform (1, 312) Second card = Uniform (1, 311) [after 1 st card shuffled] Cut Location –Uniform (52, 260) Cannot Cut within 1 Deck of Beginning or End of Shoe

9. Deck Randomness End of Shoe Location –Triangular (208,  26) End Of Shoe Should be at 4 Decks and within a half a deck

10. Deck Randomness Cards –Average Value

11. Analysis Single Player Results As Difference of Matched Pairs –i.e. E[Strat 1 /Bet 1 ] – E[Strat 2 /Bet 2 ] confidence interval, E[Strat 2 /Bet 2 ] – E[Strat 3 /Bet 3 ] confidence interval –Hypothesis Test H 0 : (  1 -  2 ) = 0H 1 : (  1 -  2 ) > 0 95% Confidence Interval –250 Simulations

12. Analysis TEST  1 -  2 dd Tt 0.05 Result Dealer with Standard Bet vs. Dealer With Simple Counting 1.4287.53040.0771.96 Failed to Reject Recommended with Standard Bet vs. Recommended With Simple Counting -15.2347.7754-0.6911.96 Failed to Reject Counting with Standard Bet vs. Counting With Simple Counting -80344.2587-3.674-1.96 Reject Null Hypothesis Betting Strategy Analysis –Standard Bet vs. Simple Counting Bet –Cannot determine if standard bet is better or worse than simple counting based bed

13. Analysis TEST  1 -  2 dd Tt 0.05 Result Dealer with Simple Counting vs. Dealer With Advanced Counting 67.8391.2862.741.96 Reject Null Hypothesis Recommended with Simple Counting vs. Recommended With Advanced Counting 1.128510.30930.0351.96 Failed to Reject Counting with Simple Counting vs. Counting With Advanced Counting -71.096995.8087-1.1291.96 Failed to Reject Betting Strategy Analysis –Simple Counting Bet vs. Advanced Counting Bet

14. Analysis with Discussion TEST  1 -  2 dd Tt 0.05 Result Recommended with Simple Counting vs. Dealer With Simple Counting 242.2958.843.991.96 Reject Null Hypothesis Counting with Advanced Counting vs. Recommended With Advanced Counting -71.0961214.98-2.971.96 Reject Null Hypothesis Playing Strategy Analysis Definitively, the Recommended Strategy gives better results on average than the dealer’s strategy or the counting strategy

15. 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(Strat 1, 1 /Bet 1, 1 )] – E[Batch(Strat 1,2 /Bet 1,2 )], E[Batch(Strat 2, 1 /Bet 2, 1 )] – E[Batch(Strat 2,2 /Bet 2,2 )] Determine if the Number of Players affects Results Test for Best Strategy to Use with N Players

16. 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 H 0 :  1 -  2 = 0 for H 1 :  1 -  2 > 0 Need to Increase Sample Size –Standard Deviations are too High –N needs to be larger

17. 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 H 0 :  1 = 0 H 1 :  1 > 0 –Fail to reject Null Hypothesis –However, larger n and stabilized  and  mean other tests may change

18. 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)

19. 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

20. 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