I usually try hard to avoid writing here about anything not directly related to mathematics and physics, but on rare occasions I can’t resist. Many readers may want to skip this posting as unserious, but maybe some will find it entertaining.

My travels last week took me to Las Vegas, where I stayed overnight with my old housemate John Chang and his family. During my graduate student days at Princeton, one year I lived with two fellow physics graduate students who were part of a card-counting team which had started up to take advantage of the recent opening of casinos in Atlantic City. Many of the other members of the team were based at MIT, and my roommates often mentioned one of them, “John”. A few years later I was looking for a place to live in Cambridge, answered an ad, and ended up going to meet the owner of a house who was looking for a housemate. After we talked for a while, I realized that he was the “John” my Princeton roommates had been telling me about.

Anyway, you can read more about John in a story just put up at the Xconomy web-site. He’s the model for the character “Mickey Rosa” in the book “Bringing Down the House”, which has just been made into the movie “21”, opening this weekend (Kevin Spacey plays “Mickey Rosa”). I’m looking forward to seeing the movie on Saturday.

**Update**: For more about John and the card-counting business, see this article in Men’s Vogue.

**Update**: If you’re interested in this, you should definitely check out John’s blog, which he has started updating again with new postings.

This would be great jump-off point for “The String Kings”. The String Theorists start out as card counters at Atlantic City, but are kicked out once the management realises what they are up to. Unperturbed, the String Theorists hire their own muscle, forcing the casinos to readmit them, and the cycle of supersymmetric, eleven-dimensional violence starts …

Gambling theory and “statistical logic” are among my interests. The optimum strategy for blackjack was worked out analytically and verified on computer sims as far back as the late 50s and published in American statistical journals. It minimises the house edge but can’t overcome it. MIT math professsor Ed Thorpe worked out that the player can gain a 1% edge by counting cards so you can estimate how rich the deck is in aces and tens. There are some amusing stories about him liked how in Las Vegas he doubled $10,000 of mobster money. Thorpe went into finance and later used his abilities to exploit statistical anomalies in various types of securities and make a lot of money. I think he now runs a hedge fund.

However, according to blackjack folklore card counting was actually worked out first by a character in Las Vegas called “Greasy John”, an obese and somewhat obnoxious individual who always had a large basket of fried chicken. But when he played blackjack he won regularly. He seemed to know exactly when to up his bets. However, he took his secret with him since he died of a massive heart attack while playing, but Ed Thorpe gets the credit. There are also various amusing BJ stories like people playing with computers strapped to their backs under their clothes. Card counting is banned in Vegas now I think or else security “asks you to leave”. They also introduced multiple-deck games and other distractions. I also heard some of the MIT team got seriously physically threatened in Europe.

When casinos opened in Atlantic City in the late 70s they did so with a unique rule for BJ–early surrender–which actually gave BJ players an advantage over the house. However, the New Jersey Casino Control Commission who mandated the rule did’nt actually know the math and by the time they got rid of it about a year later a lot of “college kids” and pros had already milked it for millions.

As author of “String Kings”, I do recall they had set up rigged slot machines with only a 1 in 10^500 chance of paying out;) I suppose they could also introduce “supersymmetric 11-dimensional quantum blackjack”.

On what planet can a player gain an advantage over the house in blackjack? John Scarne has debunked this idea many years ago.

But let’s review a critical fact: the house advantage in blackjack is based 100% (as a minimum) on the way the game is played. The advantage is due to the fact that players go first, and that ties don’t pay off. Any player going over 21 instantly loses. Also, after the first two cards are dealt, if the dealer gets 21, players either lose, or ‘tie’.

Just these rules give the house a 8.27% advantage, but the payoff when a player wins with a 21, which pays 3 to 2, reduces this to 5.9%.

Card counting is useless in blackjack as long as the last 1/5 or so of the cards are discarded. With 208 cards, at least 40 are never played.

Certain other rules can bring down the house advantage, maybe to 1-2%, but it is impossible to overcome the advantage the house has from playing last.

The main advantage that players have is knowing the dealers up card, and also knowing that the hole card can’t add up to 21.

Maybe the best proof is that the MIT group supposedly made $10 million, but John was still seeking a roommate.

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stevem,

I am not sure quite how one classifies the skill of being able to write insightful reviews of non-existent films, but whatever it is, you clearly have it in spades. My suggestion was actually for the prequel, as some kind of explanation is required as to how a bunch of Princetonian rough-diamonds with bad hearts end up ruling the clubs.

Bob Mondello reviewed ’21’ this evening on NPR’s

All Things Considered. He wasn’t especially complimentary:(Actually, he covered two films in one review—usually not a good sign.)

tomj,

The information stevem gives agrees exactly with what I remember from my days hanging out with card counters, at least one of whom was an extremely talented physicist, expert in the field of Monte-Carlo calculations, which he performed not just on physics problems.

When I was living with John, he was still in his twenties, had only been doing this for a few years. At the time he definitely didn’t have $10 million, but he did own the Cambridge house we were living in ….

Peter,

I read the thesis of your friend. Compared to the analysis of Scarne, who never even graduated high school, your friend is an idiot. I’m not even sure that your friend understands basic probability theory.

For instance, your friend assumes an ‘infinite deck’. That means that each card always has 1/52 change to be dealt.

But the theory of counting cards assumes some non-normal distribution of cards. It is obvious that if you could detect a non-regular distribution of cards, you could gain an advantage, on average, how big is this advantage?

If there was not a huge structural advantage from going last, the game of blackjack would not exist. This is the secret of gambling: somehow promote the idea that more work, more analysis will overcome the structural deficit.

It isn’t going to happen. This is the huge con. The idea is that you can beat the house. You just need to memorize some tables, memorize some exceptions.

Maybe you have some additional information, but what I have read is that you have attested to _missing_ money, not actual money.

I think it is important. Someone who wishes to make money via gambling, and, maybe worse, by organizing a gang of gamblers, just to eek out a 1-2% advantage (at most), who would do this? I mean, we are talking about commodities trading. Only an MIT geek would think this is something worth reporting.

This is pretty close to the string theory bs. Some moron over-exaggerating, and all his friends being amazed. It is just bs.

tomj is correct that the house edge arises from the double bust probability, less the 3:2 blackjack payoff to the player. But he seems not to realize that the house has a fixed strategy, whereas the player has the option to double, split, hit, stand, or even surrender. Depending on the exact rules and number of decks, proper play reduces the house edge to a small amount, generally around 1/2%.

Ed Thorp published the first counting system in 1961. He determined the effect of removing one card of each denomination on the expectation of the game. The sum of the effects is close to linear. It turns out that an excess of small cards are bad for the player, or conversely, an excess of tens and aces helps the player. This makes intuitive sense, since more tens and aces increases the probability of blackjack, which pays 3:2 to the player; it increases the chance the dealer will bust with a small card showing; and it helps the player with his doubles.

Anyway, if you can determine there is an excess of 10s and Aces, you can also determine when you have a positive expectation bet. Depending on exactly how many cards are dealt, the rules, how much you bet at various “counts” and so forth, your average expectation is generally 1-2% of the money bet.

As for Scarne, he was no mathematician. He was a great sleight-of-hand magician, but his blackjack writings have long been debunked.

And as for making money, if your average bet is $1000 a round, you will bet about $1 million in a weekend of play. At an advantage of 1-2%, your expectation would be $10-20K. Of course, winning is by no means guaranteed, but that’s why you form a team, to allow the law of large numbers to work. Even going on one trip a month, 10-20K is a nice income for a hobby.

The method is simple, but the discipline to carry it out is uncommon. As for being a moron, I guess that depends on what company you keep.

Perhaps the best proof of the profitability of blackjack is that casinos bars counters. And have you ever had 80K in cash to misplace?

First, the notion that there is no winning strategy for counting cards in blackjack is ridiculous. If the deck is rich enough in ten cards and aces, the gave favors the player. If reserved almost all your betting until that happens, you will expect to win. That may not be a realistic or interesting strategy, but it works.

Also, the attitude of casinos towards card counters varies a great deal by place and, more importantly, how much money is involved. Back in the 1960s, my dad and two other MIT students took a trip to the Bahamas, and they spent an evening counting cards in a casino. They weren’t very professional about it. One guy counted the ten cards, my dad counted the total number of cards, and the third guy actually played. They did this quite openly, and the casino didn’t care, because they were playing at a $5 table. They won less than $100 total. I’ve heard similar things from other people about more recent trips to Vegas and Atlantic City. If you’re playing for peanuts, (some) casinos really don’t care if you count.

Tomj, everything John says makes perfect sense but I don’t get your reasoning. It is mathematically very well established that card counting works, although putting it into into practice is difficult and hard work. It is also intuitively obvious that tracking the fluctuations in the composition of the deck(s), and varying the wagers accordingly, is advantagous to the player. The player’s edge comes from the increased probability of blackjack and the ability to perform operations like doubling and splitting that are not available to the dealer/house. It does’nt matter that the player “goes first”.

My own experience of blackjack is limited to the online game where the cards are “electronically shuffled” after each hand so card counting is useless. An interesting question therefore–and one that remains unanswered as far as I can tell–is whether an edge could be gained in the computer-generated online version of the game. That is, is the online game perhaps more “pseudo-random” than random? Could one data mine a large statistical sample of computer-generated hands and look for repetitive patterns, trends, predictive signals, correlations–kind of like what Jim Simons does with financial data–and make bets or alter the magnitude of the wager in accordance? For example, could you reasonably predict when the negative fluctuations or variance/volatility is going to hit at which point you would start betting small? It is pretty much an open problem in cryptography as to whether one can effectively distinguish between a large sample of purely random objects and a large sample of pseudo-random objects. Some modern rngs are pretty sophisticated however, but computers still essentially remain deterministic machines and must generate random objects (card hands, dice throws) from a formula or iteration algorithm. However, in the online game successive bets can be laid down very fast and even with a slight edge the law of large numbers would quickly do the rest if one could even get a small edge. The hassle associated with real casinos is avoided and you can play anywhere at anytime. Sorry to rant on but I’ve never had a response to this query;)

Chris, as a spinoff or prequel to “String Kings”, one could perhaps base it on Scorcese’s film “Casino”: string mobsters set up a casino in Vegas called “The Only Game in Town” (with flashing neon) and featuring rows and rows of the 1-in-10^500-payout slot machines, skimming the profits for the bosses back east in Princeton and Harvard; and with a “bagman” taking the cash back east on regular trips. (Guess who?;)). I can imagine a lot of violence though with cheaters and counters being taken to a back room and getting their hands and heads busted with heavy bound archival copies of Nuclear Physics B.

stevem,

I have no idea how good the random number generators being used are. If you did manage to figure out how to take advantage of their pseudo-randomness, presumably whoever operates them would notice this and change to a different one.

Card counters over the years have put a lot of effort into trying to exploit non-randomness in casino shuffles. I don’t know what the current state of this is, although I did hear that at least one casino hired mathematician Persi Diaconis to check out their shuffle.

It’s not against the law to do what card-counters do, although casinos (at least in Las Vegas) can kick you out if they think you are doing this too well. Using a hidden computer to play in a Las Vegas casino is now against the law there, and you could end up in jail if you do this. Most card-counters are careful to not do anything illegal, so this is not one of their tactics in Las Vegas.

tomj,

Please stop insulting people. I should have deleted your comment when first posted, will just leave it because of the informative response from John that it generated.

So how did you like the movie? Lauren and I went on Friday night. The theater was completely full. But we were quite unimpressed by the movie. And its depiction of math was awful. It was fun to see MIT and Cambridge in a movie, though.

Deane,

I thought the movie was kind of a disappointment. Not that good (especially the performance of the lead actor, who was vapid), and it could have been a lot better. Having so much of it set in Planet Hollywood was both ridiculous and boring, and the movie was too full of product placement and a bit too much of a bland product itself. But it was entertaining enough, and I just read that it’s number one at the box office this weekend, so the filmmaker is doing something right, at least as far as making a product with as wide appeal as possible.

Check out John’s blog for his take on it, which is interesting.

stevem,

I had not thought about the psedo randomness of the cards in online Black Jack before, but if you know what generator is being used and if you could find the seed value used the game would become totally deterministic. However as far as I know it is extremely difficult (if at all possible) to determine the seed value from observing the random numbers (and you cannot really observe the numbers only the cards) even if you know what random number generator is being used.

If the casino uses an old generator with a short period then eventually you will get the same sequences of cards again, but a modern generator such as the “Kiss-Monster” repeats itself after 10^8859 numbers so then you will have to play for awhile ðŸ™‚

Peter have you seen the talks at Lindefest (which has talks on string theory and anthropics)

Thanks Shantanu,

I hadn’t seen that. Especially glad to see the notes from Witten’s lecture, which was the same one I saw at Stony Brook last week and hope to get a chance to write about.

No problem, Peter. Also for some reason my post about Coleman’s 1975 lectures

and a very interesting paper by Carlip is no longer there. I was hoping you would

blog about it (since both of these are related indirectly to string theory)

Did you take the same class by Coleman at Harvard?

Thanks

Shantanu,

As the comment policy says “off-topic comments better be interesting…” otherwise I’ll delete them. I wasn’t much interested in the Carlip paper, just because my interest in that kind of debate about quantizing gravity is limited. There are lots of things worth reading on the arXiv that aren’t discussed here.

As for the Coleman lectures, this was mentioned on several other blogs already and I think I’ve written about Coleman and his QFT course here in the past. I actually attended two quantum field theory courses at Harvard, and during the first, which wasn’t taught by Coleman, I watched some of those lectures on videotape in the science center. The second time around I took the course from him. He was a great teacher and it was a great course. But trying to follow a course of blackboard lectures by watching low resolution video on the internet seems to me to be a painful idea.

I just got home after seeing “21”.

My idea (doing the movie as a prequel to “String Kings”) is better.

In particular, I’d like to have a scene where the young String Kings abduct a security guard from an Atlantic City casino who has previously thrown them out for card counting, dragging him to basement and tying him to a chair.

THE KID

How many space-time dimensions are there, punk?

SECURITY GUARD

Three space, one time.

THE KID (punching him in the face)

Wrong, totally moronic crackpot! The answer is TEN space, one time! Let’s try again, shall we? How many spacetime dimensions are there?

SECURITY GUARD

Three space … one time.

THE PLUMBER (A string theorist)

WRONG! Get this wrong one more time and you’re going to go glub, glub, glub to the bottom of the sea.

Etc.

On the topic of gambling and card counting in the movies, I was reminded of “Rain Man,” where Tom Cruise’s character takes Rain Man to a casino and has him count the cards at blackjack.

There was a famous poker champion named Stu Unger, who had a total-recall memory and could memorize every card in a 6 deck blackjack shoe, yet Unger’s game was poker and he supposedly won over $30 million in poker tournaments. For more info see:

http://en.wikipedia.org/wiki/Stu_Ungar

Another excellent gambling movie is “Rounders,” starring Matt Damon as a poker player.

As for me, I like all of the casino games, and the only counting I do is my money, Rule number 1 of my ‘winning formula’ is to quit after winning 10 percent of what I came in with; Rule number 2 is don’t bring an ATM card; Rule number 3 (of 3 Rules) is don’t be a gambling addict.

Just a quick one to throw into the pot, which involves adding a few variables other than the numbers themselves into the equation for beating house edge

To really beat house edge, you have to think outside the box. That is, you have to think of gaining benefits other than the montry win itself from your play. This means factoring in comps

The easiest way to do this is to play slots and the first thing to do is find a slot that pays 99% or over. Slot play now attracts the best perks in casinos. Better still, combine this with entering a slot tournament where you can get free rooms in hotels when you enter and will often get a free lunch too. In doing this you automatically have an edge worth hundreds of dollars. Now, when playing the machine, you must use your points card in the machine. This will monitor your play and you will then begin to acquire morefreebies such as complimentary meals, rooms, shows, limos etc

Of course, the edge you get is only of value if you actually desire these freebies and you will never get rich on them…

Similar edges can be gained on blackjack. If you are good enough at blackjack and can get the floor house edge down to less than 1%, that is a good start. You must then get your play ‘rated’ by the floor manager. He can then pass that info along to the casino system and you can then accumulate comps in the same way that will be greater than the 1% you have been losing on blackjack over time.