Opinion

Felix Salmon

Guest post: Michael Geismar’s blackjack strategy

June 5, 2012

When mathematician and blackjack expert Jonathan Adler saw my post about hedge fund manager Michael Geismar’s antics at the Vegas blackjack tables, he offered to explain just how silly Geismar was being. I jumped at the chance. So enjoy:

Like most stories dealing with probability, this one starts with a coin flip. If you take a fair coin and flip it, you would expect it to have a 50/50 chance of landing heads. But let’s say that you flipped the same coin ten times, and on a wild streak of luck each of those times it happened to end up being heads. What are the chances that the eleventh flip will also be heads? While it may feel like you’re “owed” a tails, the odds of it being a heads are still 50/50, since the coin didn’t change in any way. Each flip is what mathematicians call an independent event: the outcome of each flip has no impact on the outcome of any other flips. The idea that after seeing a bunch of one side of the coin on past flips you are more likely to see the other on future flips is called the gambler’s fallacy. The fallacy comes from the confusion between the long run outcome (with a large enough sample size, I expect half of my coin flips to be heads and half to be tails) and the outcome on any one flip (since I have seen a bunch of heads before, I need to start getting tails to balance things out in the long run).

While there are many different rule sets for blackjack depending on the casino, the core game is generally the same: the payout, if you win, is the same as your wager, unless the player has a blackjack — a face card and an ace — in which case the payout is one and a half times the wager.

While in each round the player has several choices on what to do, once the player sees their hand and the dealer’s card there is generally a single best action for them to maximize their potential payout. This set of best actions is called “Basic Strategy” and is well known. The player really doesn’t have much choice in terms of what they do on a single round; any decent player will just take the optimal move based on what’s showing. Assuming the player always takes the best possible action, for every dollar they bet in a round they should lose around half a cent.

Blackjack is a game where it is easy to fall prey to the gambler’s fallacy. As a player, if you receive several losing hands in a row it is easy to think that you’re “due” for a winning hand. However since each hand is essentially an independent event (and I’ll get back to this later), the number of losses you have had in a row doesn’t change chance of you getting a win on your next hand. Even if you get a run of bad hands in a row, your next hand is still just about as likely to lose as the previous one, similar to the situation with flipping a coin.

If you go to the casino with the goal of burning a little time and money in exchange for the atmosphere and free drinks, then the fact that the house has an edge isn’t too distressing. But if you go to Vegas and want to try and win as much money as you can, the expected loss on each hand seems like a problem. There are two main ways to legally attempt to overcome the fact that each hand on average loses you a bit of money. You can either change the odds to be in your favor, or you can try and change your bet amounts to make it less likely you will lose. Only one of these methods actually works.

By changing the structure of the game, you can make it that your average hand has a positive return. This was famously done by a group of MIT students using a method called card counting. The students exploited the fact that unlike our coin tosses from earlier, hands of blackjack aren’t truly independent events. That’s because each round of blackjack comes from the same shoe of cards, so if you keep track of what cards have been played in earlier rounds, you will have a small amount of knowledge on what cards you are likely to see in future hands. When there are mostly face cards and aces remaining in shoe then the player is actually at a slight advantage to the dealer. If you only place bets when the deck is to your advantage then you can make yourself money. The MIT students counted the number of face cards that had been seen already to estimate what proportion of remaining cards were face cards. When there were a high proportion of face cards left in the shoe they would make large bets. Card counting is completely legal since you aren’t technically altering the game nor are you using any mechanical devices to aid you. All that is involved in card counting is exploiting a weakness in the design of the game, although in practice this is extremely difficult to do.

Another way to try and overcome the expected loss on each hand by having the casino change the rules for you. If you’re a high enough roller, sometimes casinos will entice you to play by giving you discounts on your losses. When they offer these discounts on losses, they attempt to run the math to ensure that you should still be expected to lose money on your trip, however as described in the article it’s not clear they always get it right.

Most people don’t have the skill and manpower to count cards, they don’t have enough money to warrant a discount, nor do they have any other way to get the odds on each hand in their favor. So to try and overcome the house edge, they will try to cleverly alter the amount they are betting on each hand. A betting strategy, or a martingale, is a set of rules to determine how much a player should bet on each hand to try and compensate for previous wins or loses. This is different from counting cards because it doesn’t take into account what cards are left in the shoe; it only uses how many times the player has won or lost.

For example, let’s say you and your spouse go to a blackjack table with \$1,024 \$1,023 and hope to win an additional dollar. Your spouse suggests you just play one hand and if you lose then walk away, but you have a better idea in mind. On your first hand you bet a single dollar. If you win you do walk away, but if you lose you bet two dollars. If you lose twice in a row you bet four dollars, if you lose three times in a row you bet eight dollars, and you continue to double your bet until you get a win. Any time you win a hand you will wipe out all of your previous losses and you’ll get a dollar in winnings. The only way not make of money is to lose 10 straight hands in a row, and since losing 10 straight hands in a row is extremely unlikely, you expect to almost always make the dollar you were hoping for. Or in terms of the coins from before, instead of betting a dollar that a coin will flip heads, you bet \$1,024 that out of ten flipped coins at least one will be heads. If you win you get an extra dollar, otherwise you lose all of your \$1,024 \$1,023.

If you followed your spouse’s advice, you would have slightly less than a 50% chance of winning a dollar, and slightly greater than 50% chance of losing a dollar. By not following their advice, you have around a 99.9% chance of winning the dollar, and a 0.01% chance of losing all the money you walked in with. In fact because the amount you would lose when you get ten bad hands in a row is so catastrophically high, the expected amount you win overall is still negative. Your clever betting strategy didn’t actually change the house’s advantage over you; all it did was push the risk out so that you lose very rarely and when you lose you lose big. You can mathematically prove that any betting strategy you use, no matter how hard you try and optimize it, will fail to change the fact that the house has an advantage – you’ll still lose money by playing.

The two methods of trying to adjust the outcome of the game have parallels in investing. When a large investment firm develops a new method of trying to predict the stock market, say by trying to incorporate people’s emotions from twitter, they are using new information to increase the chance that they can call which way a stock will move. This is analogous to how the MIT team was trying to predict how a hand of blackjack will play out before it gets dealt. In both cases they are using special knowledge of the situation to increase the underlying probability of success.

Alternatively, when a bank sets up a hedge against one of their investments, they are trying to decrease the number of possible outcomes in which they lose money. For example the bank may hedge its investment in Microsoft by shorting Google. If they both drop in price, the short on Google will cancel out the losses on Microsoft. But at this point, to get the same level of return as investing in just Microsoft, they will have to increase their leverage.

Once the bank has increased their leverage, this becomes similar to the betting strategy in blackjack. Most of the time, the bank’s pair of investments will yield a decent return. Every once in a while, Microsoft will decrease in value while Google increases, and the bank will lose much more money than if they hadn’t hedged at all. Just like the person using a betting strategy, they have pushed their risk to the tail events: only when the market moves in a particular way will they lose money, but when it does, they’ll lose big.

As Wall Street has created more and more complicated financial products, it has become nearly impossible for a buyer to determine how much of the product’s return is due to shifting risk to the tails. In terms of blackjack, consider a person who tells you they can get an average return of five cents for every dollar you give them to play, but doesn’t tell you how they do it. Unless you watch them play, there is really no way for you to know if they are actually changing the game like the MIT students, or if they are just employing a betting strategy and at some point will lose all of your money. This lack of information is a problem for clients trying to get a good return from a bank, and also a problem for banks CEOs trying to ensure their company has a good return.

The JP Morgan case is a good example of how investing in Wall Street is actually worse than Vegas. Bruno “the London Whale” Iksil made a series of hedges to try and ensure that the case where he would lose money was unlikely to happen. Unlike at a blackjack table where the dealer has a fixed set of actions she has to follow, on Wall Street there are other investors looking to exploit other people’s mistakes. Once other investors saw that the Whale left a chance for his investment to go sour, they were able to take actions to exploit this, and caused the event that seemed unlikely to come to pass.

Lawrence Delevingne’s story on Michael Geismar’s time in Vegas is a great anecdote showing that people in charge of billions of dollars on Wall Street don’t understand the idea of shifting risk. After hearing Ben Mizrech speak, Geismar was seen using a betting strategy to try and improve his winnings at the blackjack table. After every winning hand, he would increase his bet by \$1,000. After a losing hand he would lower his bet. The article doesn’t say by how much, but let’s assume after losing a hand he would reset his bet to \$1,000.

This betting strategy has the opposite effect the one described before; instead of having a single win wipe out previous losses, a single loss will wipe out much of the earlier winnings. On most sequences of hands Geismar would lose money, but occasionally he will have an unlikely winning streak and make a very large amount. Instead of shifting the downside risk to the tail events, Geismar shifted the upside risk to tail events. Over time this betting strategy is expected to lose Geismar money, just like all other betting strategies. But Geismar fell victim to the gambler’s fallacy: he thought that a run of winnings changed the chance of getting another winning hand.

Just to be clear, despite having perhaps been inspired by Mizrech, this betting strategy is not at all the same as card counting. The amount Geismar was betting was unrelated to the proportion of face cards remaining the deck; it was only changed by the numbers of wins and losses he had seen. It may be that he would get a losing hand and reset his bet to \$1,000 even while the deck is still hot, or similarly he may have increased his bet when the remaining shoe was mostly low cards.

This misunderstanding of how probability works didn’t stop Geismar from winning several hundred thousand dollars on his trip. But if he were to keep going to Vegas he would lose money in the long run. His lack of understanding is distressing since he is the co-founder and president of a \$4.6 billion hedge fund, and the mistakes he made in Vegas could easily be made in other forms of risk management.

The lesson here is that whether on Wall Street or the strip in Las Vegas, it’s easy to confuse increasing the chances of winning with shifting risk. Increasing the chances of winning improves the amount you should expect as payout. Shifting the risk makes it so that most of the time you get a good payout, but every once and a while you lose catastrophically. As a culture, we should be trying to ensure that the people making financial decisions are looking to do more of the former and less of the latter, especially given the systemic consequences of recent catastrophic market collapses.

I find this analysis a bit silly. Once you acknowledge that the optimal strategy was for him to abstain from playing, all that’s left is to analyze his revealed preference over these (suboptimal) outcomes. As you point out, his method of playing (which aligns with the reverse of the Gambler’s fallacy outcome) skews his outcomes to likely small losses and unlikely large wins. As far as gambling goes, that sounds like a fairly fun way to skew the risk.

This doesn’t mean he was stupid (unless you consider gambling with negative EV stupid at the outset). It’s not prima facie evidence of ‘misunderstanding of how probability works’. It’s gambling, which he can afford to do for pleasure.

Posted by absinthe | Report as abusive

It’s also worth noting that this distribution is similar to Taleb’s strategy. Of course, Taleb believes that he has an edge.

Posted by absinthe | Report as abusive

I believe when I played with him Mike used to ask that double down cards always be dealt down. I’m glad the original article didn’t mention this, because I can only imagine the nonsense I’d be reading now – “Geismar believes that he can change the value of cards by not looking at them! Don’t let him near your money!”

Posted by Felagund | Report as abusive

@absinthe

Taleb’s strategy tends to work because 1) it’s specifically NOT looking for randomness, the kind you get in Vegas, but for chaos, ‘chaos’ as in ‘Mandelbrot’, complex systems that can exhibit enormous excursions from seemingly unimportant events aka, strongly non-linear, stateful systems, and 2) the strategy’s counter-parties are not as smart as casinos and tend to severely misprice tail events.

Getting 2) is pretty easy. It’s called Wall Street. Plus it’s really nice. They can come back loss after loss after loss thanks to all their pals in DC and to Uncle Ben. Going to the casino is awesome when you’re playing with somebody’s else money.

Getting 1) is very difficult and I have no idea if Taleb is really that good at it. But at least, having a vague idea of what you’re looking for is a better starting point than most purported investment strategies out there.

By the way, Felix, the total amount at play in the blackjack bet is not \$1024 but 1+2+4+8+16+32+64+128+256+512, \$1023.

That’s how you can make \$1 with the spousal strategy … unless you go 10 down in a row, of course.

Posted by Frwip | Report as abusive

Having spent a lot of years in a gambling town, there are so many angles to this it’s hard to know where to begin. In BJ you know only one thing for sure – you are going to lose more hands than you win – count on that. If you lose 5 hands in a row you want to lose your minimum bet x5; when you win 5 in a row you must win more than 5x the minimum. Still, stick around long enough and you come away with nothing but a few “free” drinks.

What The Whale was doing seems more like poker than BJ – he was playing against other individuals, not against the math. He and Macris seem to have been attempting to “buy the pot” by raising the stakes on a losing hand until the other players were forced to throw in their cards. Didn’t work then, but strangely – it could work now and get JPM out of this whole mess for damn near nothing.

In an ultra-high-stakes game, the counterparties can’t play straight-up against JPM.

Posted by MrRFox | Report as abusive

Contrary to the previous commentators, I think this is an excellent post that casts the CIOs actions in a light I had not previously considered. Thanks are due to Adler for writing it and to you for posting it, Felix.

Posted by Greycap | Report as abusive

You summarise card counting fairly well. Essentially it’s key to increase the bet size when the count is high and bet the minimum when the count is low. The MIT students could still lose but increasing the bet size when the probability has significantly improved is the key to winning over a long set of sessions.

The double bet strategy is mitigated by the casinos maximum bet limit on blackjack tables. So a \$2 min bet table will have a \$100 max bet, only allowing \$2, \$4, \$8,\$16,\$32,\$64 – so that’s just 6 bets before that strategy fails not 10. Even if it was \$200 then you would have one more bet (7 bets) \$128 bet. Generally the max bet is 100 times the size of the min bet which would equate to 7 bets. Casino’s also restrict the strategy with roulette, simply by adopting a max bet limit.

If you take away the bet maximum limit on the tables then you could make small returns for free at no risk at blackjack. The google / Microsoft parallel doesn’t really match the blackjack betting strategy of doubling your bet. They are only comparable as poor investment strategies. The max bet limit is the issue for blackjack players using that strategy. Adopting the double bet strategy without the limit is a superb idea! If you can invest without a restricted (low) limit then you can make lots of money in the investing world. In a world without limits the large hedge funds, can set there own limits. Using the blackjack anaolgy, the amount of bets can increase from 7 to however high they wish by moving the goalpost limit.

Posted by VinnyMC | Report as abusive

I had the pleasure of travelling with VinnyMC to Napoleon’s Casino in London one evening, it is a high class establishment with a tasteful decor. He wanted to show me his skill at blackjack however the burly staff refused to let him in, simply saying he was “too good”, and that we must leave. We then went to the bookies and rinsed the lot on the 2.30 at Kempton.

Posted by super_pete | Report as abusive

The differences between Geismar placing bets for a hedge fund and gambling are these:

- Running the hedge fund he has opportunities to play on insider information. This comes out over and over for consistent winners on Wall Street. Any mathematician who knows probability theory will tell you that consistently winning is near perfect proof of cheating (i.e. insider information.)

- To the extent that your analysis is correct, and he is not playing insider information, he is still optimizing his income. It’s a good strategy for his income to place those kind of large tail bets. If he wins, which will be most of the time, he is a hero and makes huge amounts of money for himself. Odds are that he can keep that going for quite a while. But if he loses? If he’s smart he won’t have all his money in one basket. He will still have his personal money. At worst he will get called before Congress for a flapdoodle session of no consequence. He will be able to raise more money to manage once again and odds are he won’t crash for a while once again.

————-
I will also note that the MIT “kids” didn’t card count. That wasn’t good enough to give them their consistent, large wins. They learned how to cut decks accurately and other things in addition. This is described in “Burning Down the House”.

Posted by BrPH | Report as abusive

Maybe the tax rate on gambling winnings should be cut to boost the economy?
Also, the original article is a case study on just about the perfect targeted ad.

Posted by thispaceforsale | Report as abusive

Excellent post, Felix; nice use of a reader’s expertise.

Posted by Sunset_Shazz | Report as abusive

When I read the account of Michael Geismar’s play I thought there were three possibilities. First, that he was in fact counting, but disguising this by talking about a dumb betting strategy. Second, I thought he is describing “variance” as skill and I hope he doesn’t believe that. The third possibility is that he is very very ignorant of statistics.

With regard to BrPH’s comment, the MIT team did count. A little time with google will convince you that “Burning down the house” is far more fiction than fact (as are other titles by the same author), and the only way you can know *how* to ‘cut the deck’ is to do calculations based on the count. As far as I know JP invented the technique – and the simple data logging methods – and worked out the odds and betting strategy – circa 1982. JP called it the ‘non-random shuffle.’

I was there.

Posted by AlchemistGeorge | Report as abusive

Wow. There is just an unbelievable amount of nonsense getting posted. I am going to attempt to address it all, but it will happen over several posts. Right now, I just want to make one point that I have made several times before, and I want you guys to read it, understand it, and believe it. Mike Geismar in no way believes now, believed when he won his 710 grand in Vegas, or believed fifteen years ago when I used to play with him, that any betting strategy will improve his (negative) expectation at blackjack. This is a fiction that Delevingne carelessly created in his article in a (failed) attempt to make a lucky run sound like the result of skilled play.

Posted by Felagund | Report as abusive

On the MIT team – of course they counted. But yes, they also used other techniques. Exploiting a non-random shuffle is frequently known as “shuffle tracking” and the MIT team did do it, but JP Massar did not invent the method. It is true that one cannot shuffle track without also counting. However, AlchemistGeorge, I believe when BrPH refers to “cutting decks accurately” he is talking about another technique, known in the biz as “bottom steering” or just “steering.” While this requires a tremendous amount of skill and preparation, it does not involve counting cards and is, in fact, very difficult to do while counting at the same time. This is in no way a fiction that was created for Burning Down the House, and most casinos have now adopted policies to thwart steerers. I learned bottom steering from JP himself, and have watched him do it. As far as I know he CAN be accurately credited with inventing this method.

Posted by Felagund | Report as abusive

Felagund – both of your posts sound good to me.

Posted by AlchemistGeorge | Report as abusive

Okay, now let’s talk about the most egregious stuff – the comparisons of hedge funds in general and QIM specifically to a Martingale strategy at blackjack.

In the case of QIM in particular, a lot of these comments are way off base. They have had, what, like 15 winning years in a row? Sure, I guess that could be a Martingale-type system at work (although, how many losing months have they even had). But I have pretty good info that it isn’t. I work with a guy who taught machine learning at UVA. That is a pretty serious field and a pretty serious university. He knows QIM’s CEO and tells me that guy used to live in artificial intelligence chatrooms. He says you could tell he was on to something by the questions he was asking. And he has no reason to build up QIM – he is jealous as hell that it wasn’t him. For that matter, I am jealous too – the guy is on the short list of people in the world who are smarter than I am (though he is not better at basketball). If you go to the QIM compound and see the library of books on neural networks, etc. – well, they are putting on quite a show if all they are really doing is pressing on losses. I just don’t believe it.

Posted by Felagund | Report as abusive

Oh, a note on the term “Martingale.” The article makes it sound as though that term refers to any betting strategy. It does not – it refers specifically to the doubling on losses strategy or a similar group of strategies not considered “tame.” For simplicity we’ll say that non-”tame” strategies can be defeated by a betting limit – any limit at all. This should not be confused with the more general martingales of probability theory, which refer only to fair games, where future outcomes are independent of the current sigma algebra of information.

Posted by Felagund | Report as abusive

“But Geismar fell victim to the gambler’s fallacy: he thought that a run of winnings changed the chance of getting another winning hand.”

Its not clear to me how you get to this conclusion. I find it much more reasonable to conclude that Geismar reduces his bets after losing and increases his bets after winning, because he has less or more “gambling money” left. This is in fact the optimal strategy to maximize the total amount of fun he has gambling (with certain assumptions about the relationship of his fun with winning, this is a mathematical theorem), and having fun is the justification for gambling with negative monetary expectation.

It’s not very intelligent to bet the same amount of money regardless of how much money you have. In the investing world, that would be like recommending that you should invest \$50k in the stockmarket, regardless of whether you have a total of \$50k to invest, or a total of \$1mm to invest. A sensible investment strategy would be something like “keep 50% of your portfolio in stocks”. The result of that is: if you start with \$100k, you put \$50k in the stock market. If the market goes down 10%, you have \$95k left, so you put \$47.5k into the stock market, which means you bet less after losing.

Posted by niveditas | Report as abusive

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