TLDR: No card counting. He negotiated a 20% discount if his losses were to reach $500K, and negotiated small tweaks to the rules (eligible splitting, doubling hands) that brought the house edge down to 0.25%. At that point he was basically playing a 50/50 game with some downside risk protection, and got lucky with the cards. EDIT: He could do this because Atlantic City is a dump and is desperate for customers now that Pennsylvania has casinos all over the place.
He won’t say what all the adjustments were in the final e-mailed agreement with the Trop, but they included playing with a hand-shuffled six-deck shoe; the right to split and double down on up to four hands at once; and a “soft 17” (the player can draw another card on a hand totaling six plus an ace, counting the ace as either a one or an 11, while the dealer must stand, counting the ace as an 11)
The blackjack rules (excluding the discount agreement) mentioned don't seem out-of-the-ordinary. Most Las Vegas casinos have these exact rules for their higher minimum tables http://wizardofvegas.com/guides/blackjack-survey/.
The house edge of this game (using http://wizardofodds.com/games/blackjack/calculator/) is 0.33457. There's no additional player-favorable rule that I can imagine to which all 3 casinos agreed (like late surrender, a rule I've never seen in a Las Vegas strip casino) that would account for the ~0.08 difference in house edge. Like another comment suggested, I think there's a good chance he intended to count cards when calculating his odds. A simplistic card-counting strategy (http://wizardofodds.com/games/blackjack/card-counting/high-l...) could account for the difference.
If you have 20% downside protection in a nearly even odds game, you can figure out the optimal strategy. Also, he was betting $100k a hand, so he didn't have to play too many hands before deciding to stop as a winner or take his 20% loss rebate. (If you have the 20% loss protection, the optimal strategy is to bet big.)
For example, his downside protection was 20% of a loss of at least $500k. If he was at -400k, he had no reason to quit; he should keep playing until he was ahead or at -500k.
Also, casinos previously negotiated 20% loss rebates on a LIFETIME basis. He negotiated the 20% loss rebate on a PER-TRIP basic, correctly pointing out that if he was ahead at a casino, he had no reason to return if another casino also offered the rebate.
The lt;dr summary is: He didn't cheat or count cards. The 20% loss rebate combined with blackjack being a nearly even game gave him an edge.
You could do the same thing at craps. At craps, there's an obscure rule that lets you make a bet at true odds after establishing a point, sometimes for 10x or more of your original bet. If you're playing at a place that offers the 10x backup bet and the 20% loss rebate, you could do the same thing at craps.
The article implies that the 0.25 house edge was calculated before the rebates were considered, still leaving the ~0.08 house edge as a mystery. The quote below follows the quote from my original comment. The author of the article may have made a mistake, though.
When Johnson and the Trop finally agreed, he had whittled the house edge down to one-fourth of 1 percent, by his figuring. In effect, he was playing a 50-50 game against the house, and with the discount, he was risking only 80 cents of every dollar he played.
Charming story -- but there's plenty of controversy that comes with it. Some Canadian mathematicians ran their own analysis in 2014 and published it in Chance magazine, arguing that Johnson's winning prospects weren't nearly good enough under the terms described for him to have been likely to have achieved such scores. The full article is behind a $48 paywall, but here's a link to the start of it: http://www.tandfonline.com/doi/abs/10.1080/09332480.2014.890...
Meanwhile, Elliot Jacobson, a guy with his own academic pedigree (U of Arizona Ph.D. in math; former Ohio University asst/assocmath professor) has published a defense of Johnson's claims and a counter-rebuttal to the Canadians. Jacobson now runs a gambling-consulting firm. His rebuttal is here:
http://apheat.net/2014/03/25/fact-checking-an-article-on-don...
Jacobson's rebuttal is perhaps more fascinating than the Atlantic article, though I think the context of the article is necessary. Details such as: Johnson would coax two or three dealer errors per day in play, valued at +$200,000 per day; or that the Canadian mathematicians took some strange shortcuts in modelling or running simulations.
Johnson was very aggressive and belligerent at the table. He made the dealers and pit nervous and agitated. For example, he would sometimes half-way signal his intention and if the result was not to his liking, dispute that he had asked for a hit card. He would pound the table and shout. And the casinos were willing to give him the benefit of the doubt because he was such a high-roller.
Fascinating! Echoes of Phil Ivey arranging to play baccarat with a deck and dealer of his own choosing ... and the casino agreeing to it until learning that Ivey thus had a bigger edge than expected.
Seems fairly simple to me. Tightening up the odds and the trip-to-trip 20% discount means he would be seeing a nearly 10% return on every visit. Those odds don't sound like good business.
Re: the card counting thing, is it really that clear cut? He sounds like a sharp guy. Couldn't he just be pattern matching and getting a small advantage without actively trying to do card counting? I'm kind of talking out of my ass here, but it seems like you could get some mileage out of simply tracking whether you've seen a lot of high cards or not: https://en.wikipedia.org/wiki/Card_counting#Basics
A couple other related thoughts: I'm taking a course on Markov chains this summer, and it's nice to be able to relate it to a real world example (i.e. "Gambler's ruin"). In short, this guy would be guaranteed to lose his money if he played indefinitely EVEN if the odds were 50-50 https://en.wikipedia.org/wiki/Gambler's_ruin
>Another common meaning is that a gambler with finite wealth, playing a fair game (that is, each bet has expected value zero to both sides) will eventually go broke against an opponent with infinite wealth. Such a situation can be modeled by a random walk on the real number line. In that context it is provable that the agent will return to his point of origin or go broke and is ruined an infinite number of times if the random walk continues forever.
And if you like these sort of pieces on gambling, maybe you'd like this article I wrote recently about winning money in an indie gambling game: http://www.jere.in/13
Re Re card counting: The vast, vast majority of advantage when counting is the choice to change your bet size depending on the count. When a deck starts you are at some initial count and then it goes up or down as the deck goes along. While the count is at or below a certain point (low counts are considered bad and high counts are considered good), you usually just bet the table minimum as you do not have an advantage over the house. Once the count reaches a certain level you start betting more aggresively. There is usually a spread to counts where for example if the count is 1 you bet 2x table minimum if it is 3 you bet 4x table minimum etc. This is how you take advantage of the fact that the odds are in your favor - if you just bet the same amount all the time you are not really doing anything different (assuming you are following Basic Strategy at all times).
The rub here is that this type of betting can be seen by the house. If you are of enough interest to the casino (something you can get playing $100 hands let along $100K hands) they will watch you bet and try and look for this pattern. Most casinos employ former counters or have people on staff you can, after a little bit of time, determine if you are counting or not and politely ask you to leave. In fact most counters will specifically give away some of their advantage by going against the recommended bet at a given count to try and make it look less like they are counting. One such practice would be to never go from a minimum bet to a max bet, as it is most likely not something a regular counter would do (aka bet $100 one hand then bet $1K the next hand).
You are correct however that some advantage can be gained by just "tracking whether you've seen a lot of high cards". Players use indexes which will change their play depending on the count, and use this skill to augment their changes in betting amounts to better take advantage of their current position. You can see the effect of using indexes vs. no indexes here: https://www.qfit.com/cardcountingindexes.htm
I am not sure I understand this. He never actually gained an edge on the house and the discount never actually came into play from what I can tell. I did not think that there was a way to predict variance, though admittedly I know little about this.
Thanks, couldn't read that one earlier(blocked at work). I think I am starting to get it, But I still feel like all the facts were never presented. Basically, it was a 50/50 game except he had a 20 percent cushion on losses. You would think a cool million would at least get you single deck black jack opposed to 6 deck.
His expected outcome was in his favor because he whittled the house edge down to nearly even odds, and he had loss protection of 20%. But you're right that that didn't come into play because the actual outcome happened to be in his favor and ballooned out of control thanks to a random walk.
The discount is only downside protection, it does not change the odds. The changes that mattered were playing with a hand-shuffled six-deck shoe and the right to split and double down on up to four hands at once; and double a "soft 17". These changes essentially made the game 50/50. Still, an incredible amount of luck, though he was playing $100,000 a hand, so when you have an amazing split, split, double, double hand it adds up. The key factor here was once he was up, he basically could keep going, because statistically odds were even.
The fact that he insisted on a 6 deck shoe seems to suggest that he was, in fact, counting cards. I was under the impression that standard casino blackjack shoes were 8 decks (machine shuffled).
He may not have played as a standard card counter would (as that would make him more detectable), but he still could have made riskier plays when the shoe was "hot" to give him a slight edge over 49.75-50.25
And whether a game is machine-shuffled or hand-shuffled will usually depend on the min bet. The higher the minimum, the more likely the table is to be hand-shuffled, and the lower tier the casino, the lower that minimum is likely to be. For example: the Wynn Las Vegas' $100 min table is machine-shuffled, while Harrah's Lake Tahoe $100 min table is hand-shuffled.
My understanding is that the 6-deck shoe confers a mathematical advantage, not simply a card-counting advantage, over an 8-deck shoe. If you use the calculator here you will see that it improves odds even under basic strategy play:
http://wizardofodds.com/games/blackjack/calculator/
