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DON JOHNSON: HOW HE BEAT BLACKJACK–PART 1
by Eliot Jacobson, Ph.D.
For 25 years, Dr. Jacobson has held positions as professor of mathematics and professor of computer science. With dozens of research articles, publications, and media appearances, Dr. Jacobson is widely recognized as one of the world's top experts on casino table games and casino game mathematics. His web site is
www.apheat.net, which contains the latest information on beatable casino table games, side bets, and promotions.Editor’s Note: In a series of articles in the BJI (issues 138 and 139), our contributing writers analyzed how Don Johnson exploited several Atlantic City casinos’ loss-rebate programs to win millions. The following three-part article, reprinted with permission from Dr. Jacobson, is his recent analysis of how Johnson accomplished this.
When I heard Don Johnson (DJ) speak at the
World Game Protection Conference last Tuesday, I expected him to dismiss his blackjack winnings as the result of a little bit of advantage play and a lot of luck. That’s how he portrayed events in this article in Atlantic Magazine. However, after his explanation of how he beat horse racing (see this post), there was little doubt about his skill as one of the world’s top advantage players. When DJ smoothly quoted the exact house advantage of one of the blackjack games he played (0.263%), he removed all doubt.As DJ explained, the deal he negotiated in Atlantic City had the following terms:
- DJ placed $1,000,000 in credit with the casino.
- Any time DJ lost $500,000 in a day, he could stop playing and receive a 20% rebate on his loss.
- He could place a maximum wager of $100,000. Some places, this maximum was $50,000.
In the casino-world, it is often the case that people state something "obvious" which is in no way "obvious" and is often just wrong. Such was the case in the aftermath of DJ’s slaughter. Almost everyone took it for granted that the best strategy was for DJ to play until he either won $500,000 or lost $500,000, then leave for the day. This "strategy" appears logically sound (if you’re not using any logic). But, strangely, it didn’t represent the actual course of events in DJ’s play. In fact, DJ played much longer on the days he was winning than this "strategy" dictated. On losing days, DJ often endured a loss of more than $500,000. I decided to model the circumstances of DJ’s play to determine the optimal stopping points. What I found showed that DJ was right to continue playing well beyond the mythical "stopping points" of +/- $500,000.
First, I fixed the game played. For top players, the following rules are standard on a shoe game:
- Six decks
- Dealer stands on soft 17
- Player can double on any first two cards
- Player can double after split
- Player can re-split aces
- Player can re-split to four hands
- Late surrender
DJ certainly played this game. He may have played against other rule-sets, but he did not disclose those.
Next, I wrote a computer program that allowed me to do a Monte Carlo simulation of possible stopping strategies used by DJ. I then ran 16 simulations for various stopping strategies (from a $500,000 win up to a $2,000,000 win) to determine the optimal stopping point (rounded to $100,000). Each of the 16 simulations consisted of modeling one hundred million (100,000,000) players using that stopping strategy, and averaging their results. The program’s output included the player’s edge over the house, the expected win for the player, and the average number of hands played by the player.
The following table gives the results of the analysis of the stopping points for DJ with a $100,000 flat wager:
These results show that the optimal stopping point for DJ was a win of $1,600,000 (not $500,000, as is popularly believed). DJ was playing with an overall edge of 0.93% over the house. DJ’s expected win was $61,786. The expected number of hands played was 66.1.
The following table gives the results of the analysis of the stopping points for DJ with a $50,000 flat wager:
In this case, the optimal stopping point for DJ was a win of $1,000,000. DJ was playing with an overall edge of 0.58% over the house. DJ’s expected win was $46,852. The expected number of hands played was 160.2.
At this point you may also wonder at the credibility of stopping with a loss of $500,000, and you would be right. Once again, popular belief is misguided. After DJ lost $500,000, he is essentially playing with a discount of 20% on every bet he makes, since he no longer has to qualify for his loss discount. The only constraint is that DJ should keep enough to make a double down or split ($200,000 with a $100,000 bet and $100,000 with a $50,000 bet). DJ clearly stated in the talk that on several occasions he lost more than $500,000 before quitting for the day. Do you think DJ knew what he was doing?
With two degrees of freedom (high and low stopping points), I re-did the analysis given in the tables above. Essentially, I fiddled with a lot of simulations. Here are the optimal stopping points and other pertinent data when wagering $100,000 per hand...
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