After I wrote the above article last month, more details about Mr. Johnson’s exploit surfaced, including the following two constraints the casino imposed on him:

1. He had to lose more than $500,000 before his 20% loss-discount would be given to him. For any losses between zero and $500,000, he would not get a rebate.
2. He was required to deposit a minimum of one million dollars "in the cage" to bankroll each session of play.

In last month’s analysis, we did not factor in either of these constraints. I applied his 20% discount on all blackjack sessions where he lost any money, and assumed he played with an infinite bankroll (i.e., he played every blackjack session to the specified number of rounds.) This month, we’re going to add in the above constraints, and look at a few other possible variations in his play that can impact his edge.

These extra constraints came out in the final revisions of Mark Gruetze’s interview with Mr. Johnson. I had already completed my analysis and submitted my article, but Henry called me up wondering how these extra constraints would impact his edge. I told Henry, "I don’t know for sure, I’ll have to do more analysis and simulations. However, the $500,000 minimum loss requirement before he would get the 20% rebate would decrease his edge for sure. My guess is that it might bring down the number of rounds where the edge flips back to the casino, from 990 to maybe 500 or 600. And I think the million-dollar bankroll might actually work in his favor."

One of the reasons I love the mathematics of blackjack and other advantage gambling games is that it is filled with surprising results. When a mathematician thinks they know the answer, and the math turns out to prove them completely wrong… that is one of the most exciting things (ya, I know… a strange way to get your kicks in life). I remember my excitement 20-years ago when I first heard that blackjack was a beatable game through card counting, and I thought it was not possible, then I did my research and realized that indeed it was mathematically true. I remember when Henry challenged me to come up with an easier card counting system for average players (than the traditional High-Low), so I invented Speed Count. and proved that it provided a healthy edge over the casino while being far easier to learn and use.

Now Mr. Johnson has revealed his incredible loss-discount strategy, which turns out to be a way to get a huge potential edge over the casino, without card counting. Studying loss-discounting has provided the same thrill, with non-intuitive results from the mathematical analysis . In addition, the particular constraints that the casino imposed on Johnson turned out to be equally surprising and non-intuitive!

Let’s start with the same analysis as last month, and add only the first new constraint that the player only gets his 20% rebate on losses if he loses more than $500,000. The player still plays with no upper or lower limit on his bankroll, and we look at different number of rounds played per session.

I used the same simulation set up a as last month’s article: 6 decks, 2/3 penetration, DAS, S17, resplit aces, late surrender, $5 flat betting with perfect basic strategy. I ran an ROR simulation with one-million playing sessions for each number of rounds below.

Note: The ">-5 bet units" column in the table below is the fraction of sessions with an ending bankroll of -$500,000 or better, and the "$ win" column is the average of the ending bankrolls for these positive sessions (with the simulations $5 unit bet size.) The "< -5 bet units " column is the fraction of sessions with a negative ending bankroll, and the "$ lose" column is the average of the ending bankroll for these negative sessions (with the simulations $5 unit bet size.) The return column (win rate per round) is computed as: [(% win) * ($ win) – (% lose) * ($ lose) * 80% ] / $5 / (# rounds). It represents the profit or loss per blackjack round, in bet units.

The above results tells us that very few sessions end up in the -$500,000 to zero range. And, we’re only talking about a 20% reduction from those few 5% of sessions. The end result is that the player’s edge does not change very much at all! The casino has not insured itself any protection by setting the minimum loss limit at $500,000. If Johnson was getting his loss-discount applied every couple of hours (200 rounds), his hourly win rate dropped from $36,037.60 to $32,639.63, about a 9% decrease.

The next casino requirement imposed on Mr. Johnson was that he had to deposit at least $1,000,000 as his session bankroll before he played. From the start, I knew this requirement played in Mr. Johnson’s favor since it would give him a great excuse to play less rounds of blackjack, which we learned last month was the name of the loss-discount game! The less rounds that he played before asking for the 20% rebate on his losses, the more money he could earn. Even though a million dollars sounds like a lot, playing with a 10 bet-unit session bankroll is actually extremely low. (e.g., the 5% risk-of-ruin bankroll is 2.8 million for the 200 round game).

To study this extra bankroll requirement, I placed a combination of different lower and upper bankroll limits in the session simulations, representing different styles of play Mr. Johnson could employ.

Note: The Lower and Upper limit columns show the amount, in bet units, after which the session is stopped. For example, –10 bet units is $1,000,000 for Mr. Johnson. Because of the extra constraints, the Rounds column shows the actual average number of rounds per sessions.

Rows 1 to 3: This is the same 200 rounds example as before where he could make $32,639.63 an hour, but with the added wrinkle that if he gets to a $1,000,000 loss, he’ll stop playing and get his loss discount applied. Incredibly, his win rate improves substantially over the casino to $51,858.50 an hour, a 59% improvement! The reason is that we are only averaging 135 rounds of play per session, instead of the intended 200. Therefore, the casino-imposed million dollar bankroll requirement actually turned out to help Mr. Johnson substantially!

Asking for his loss-discount every hour or so when he is down a million, which is what will happen if he stops playing regardless after 200 rounds, may get the casino’s attention. When he loses, he plays only an hour, but when he’s winning he plays for two hours. In rows 2 and 3, we increased the intended number of rounds he’ll play to 300 and 400 respectively. In both cases, he still makes more per hour than he did without the casino’s bankroll requirement!

An interesting fact is that as the player plays more rounds, his chance of a winning session goes down, all the way to 32.61% with 400 rounds as the target. This happens because as he plays more, there is a greater chance of hitting the not very low million-dollar bankroll limit. Despite losing two-thirds of the sessions, the player is still making a killing over the casino in all cases, since the few winning sessions will be very large indeed. The fact that he would tend to lose more sessions than win, would be great camouflage for his advantage playing strategy.

Row 4: Although rows 1 to 3 are following the casino’s million-dollar bankroll guideline, they may eventually get wise if the player is playing very short one- or two-hour sessions before cashing out. In this example, we intend to play for 1200 rounds (12 hours), but will also stop when you reach 4 million dollars in profit (along with stopping at one million in losses.) This reduces the player’s edge to $21,980.96 per hour, due to the increased number of rounds per session (316.)

Row 5: Here, the player intends to play 970 rounds, the pivot point where the casino gets the edge back, but with their imposed million-dollar bankroll limit. Interestingly, the player still has a good advantage over the casino of $14,017.70/hour, since the average number of rounds is 361, far less than the intended 970. The player still has the edge, even though they only have a winning session one in five times that they sit down to play!

Rows 6 and 7: Here, we impose a lower and upper bankroll limit, and will play the full amount of rounds to reach either boundary. In row 6, we’ll stop when we lose a million, or win 6 million, and this results in an average of 464 rounds (4 to 5 hours of play.) The player’s edge has dropped substantially, to less than 1/10^th of a bet unit per hour, which is marginal at best given the big bankroll at risk. These rows highlight that a savvy player really has to manage their play carefully when they leverage a loss limit, and make sure the method they employ has them cashing out frequently.

We learned last month that loss-discounts, especially large ones around 20%, can provide a high roller with a huge advantage over the casino as long as they play as few rounds of blackjack as possible before settling up. We learned this month that casino’s might try to protect themselves from this, by requiring a minimum loss before applying the discount, and by requiring large bankroll deposits in advance.

The end result is that a minimum loss limit of five bet units is simply not enough to overcome a savvy player’s edge. Imposing a bankroll deposit requirement of 10 bet units, actually helps the player by minimizing their length of play sessions overall. Even with both of these requirements, a $100,000 basic-strategy high roller can still make over $30,000 an hour on average.

It should be noted that Mr. Johnson is not the first person to get interested in leveraging loss-discounts for profit. Peter Griffin, a well-known blackjack mathematician, has written on the subject, and possibly a few other experts. But, I think Mr. Johnson is the first Big Player to seek out such a huge edge by arranging such an incredible loss-discount deal, and use his intuition to make it payoff over 10 million!