*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.

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