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DON JOHNSON: COULD HE HAVE WON MORE?ĖPART 3

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.

Editorís Note: In a series of articles in the BJI (issues 138 and 139), our contributing writers analyzed how Don Johnson exploited several Atlantic City casinosí loss-rebate programs to win millions. The following is Part 3 of a three-part series of articles reprinted with permission from Dr. Jacobson, showing his recent analysis of how Johnson accomplished this. Part 1 appeared in BJI #161 and Part 2 in BJI $162.

 

When Don Johnson (DJ) beat several East Coast casinos in late 2011 through early 2012 by exploiting their loss rebate programs, few understood the scope of his triumph.  After his success, there were a number of articles (for example, this article in Atlantic Magazine) that tried to explain his success as a mix of luck and experience. Several other articles produced mathematical analyses that did not directly address the main questions. As I described in this post, DJ succeeded because there was an off-the-top profit to be made simply by playing within the constraints he negotiated and strategically choosing quit points. The question I pose in this post is if DJ could have done even better?

At the World Game Protection Conference, DJ stated that there were three constraints on his play.

  • He had to place one million ($1,000,000) dollars cash in the cage.
  • He was allowed to wager a maximum of $100,000 per hand.
  • If he lost $500,000 or more in a day, he could stop playing and get a 20% rebate on the totality of his losses for the day.

The common wisdom in the aftermath of DJís success was that DJ should play until he either lost or won $500,000, then quit for the day. As I showed in my previous post, this "wisdom" about optimal stopping points is patently false. With a one million dollar starting bankroll, DJís optimal quit points were to stop when his current bankroll was either less than $200,000 (quit-loss) or after winning $2,000,000 (quit-win).

With these stopping points:

  • DJís average edge over the house was 0.63%.
  • DJís expected total win per day was $85,800.
  • DJís expected number of hands per day was 137.

Yesterday, I realized that there was a hidden assumption in my analysis: that Don Johnson must deposit exactly $1,000,000, no more, no less. As I researched this assumption online, I could find no evidence in any of the multitude of media articles about DJ that he ever deposited more than $1,000,000. Did he? I donít know. Regardless, the question is if he could have won even more by strategically sizing his deposits to reflect specific market conditions.

To be precise, suppose that in addition to the two strategic decisions (quit-win, quit-loss) DJ could make about when to stop for the day, there was a third decision he could make as well: the size of his starting bankroll.  What if the requirement for DJ was not that he place exactly one million dollars in the cage, but instead the requirement was that DJ place at least one million dollars in the cage? Was his deposit amount a variable under his control? Could DJ have made even more by depositing an optimal amount in the cage?

Here is the setup for DJís optimization problem:

  • DJís starting bankroll per day was at least $1,000,000.
  • DJ was allowed to wager $100,000 per hand.
  • DJ was given a 20% rebate on his full losses if he lost at least $500,000 in a day.

Given these constraints, what are the appropriate bankroll, quit-loss and quit-win points to maximize DJís daily net income?

To answer this question, I ran a bunch of simulations...

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