The recent The Atlantic article “The Man Who Broke Atlantic City” tells the story of Don Johnson who won millions of dollars in private room custom rules high stakes blackjack. The method Mr. Johnson reportedly used is, surprisingly, not card counting (as made famous by professor Edward O. Thorp in *Beat the Dealer*). It is instead likely an amazingly simple process I will call a martingale money pump. Naturally the Atlantic wouldn’t want to go into the math, but we can do that here.

Blackjack Wikimedia

## The game

According to the article Don Johnson developed a reputation as a non card-counting high roller in gambling circles. This tempted revenue hungry Atlantic City casinos to invite him to private room gambling with custom rules and special considerations. Mr. Johnson who describes himself as “not naive in math” got different casinos to agree to the following two important game changes:

- Various rule changes to blackjack changing the player’s ability to split hands, dealer’s hitting rules, and other things. Mr. Johnson is quoted as estimating the rule changes brought the house advantage or edge down to about 0.25% per game. This is much less than the typical house edge of over 0.5% for multiple deck games. But it is still in the house’s favor (no doubt casinos understand the mathematics of blackjack very well).
- A per-visit refund or rebate privilege: on any visit where Mr. Johnson leaves the table down $500,000 or more he needs only to pay 80% of his deficit to settle his account. This is essentially free money- but most gamblers lack the discipline to take advantage of it.

The blackjack rule changes were not the problem. The issue was the rebate. Casinos regularly give gamblers initial stakes (such as $100 of chips just for walking in) and apparently routinely negotiate different refund and rebate programs with high rollers. Again, the casinos know what they are doing. Rebates are free money, but most gamblers lack the discipline to hold onto the money.

However, Mr. Johnson clearly is a disciplined gambler. In the article he stated his strategy was to cash in if he lost enough to trigger the rebate (so probably stop if he was $500,000 in the hole) but continue to bet and bet large if he was ahead. His delightfully disarming quote on why he bets when he is ahead is:

So my philosophy at that point was that I can afford to take an additional risk here, because I’m battling with their money, using their discount against them.

And he is very right. There is an advantage to betting with the house’s money. Here is some math I am sure Mr. Johnson knows (either formally or intuitively, probably both).

## The casino’s fallacy

Mathematically, the discount is an obvious bad bet on the part of the casino. They known this. It is simple to exploit: come in every night and make only a single $500,000 bet. If you win end the visit, if you lose pay off the 80% ($400,000). Roughly every two days the casino gives you about $100,000 (or $50,000 a day on average).

The casino likely feels confident offering such deal to a high-roller, because likely the high-roller has been losing or winning more than $50,000 a day when playing and the casino can just cancel the deal if they notice the gambler has started to accumulate profits. The hoped for benefit is: the gambler lacks discipline and loses a great deal of money every night at your casino (and not at a competitor’s casino). If the gambler never ends a night ahead, then any money collected by the casino *seems* like profit (no matter how deep a discount is offered).

## Breaking a fair game

Mr. Johnson no doubt knows how to break this scheme. He will play a very disciplined strategy that loses a bounded amount of money most days (keeping the casino happy, yet exploiting the discount), looks like the behavior of an undisciplined gambler, but happens to have a positive expected return for Mr. Johnson.

Suppose you try to exploit these rules. Call the amount of money you are up or down for the day $X ($X starts at zero). For this section everything will be only “for the day.” Let’s make the generous assumption that you negotiated so many rule changes, play so well, control bet sizes, and do just enough card counting to make the odds 50/50. That for each and every bet the casino has exactly a 50% chance of winning.

Suppose your current net-winnings for the day are $X (X starts at zero). Further suppose you gamble with the following strategy. You start what we call “a phase” by writing down your current net-winnings for the day $X and a positive integer $B. The phase continuous until your net-winnings for the day are driven to a new value that is either $(X-B) or $(X+B).

A phase is played as follows:

- If and only if your net winnings for the day are $(X-B) or $(X+B) (our interval boundaries) the phase ends.
- Otherwise if your current net-winnings for the day are $Y with $(X-B) < $Y < $(X+B) you bet any integer number of dollars between $1 and $min(Y-(X-B),X+B-Y) (i.e. any amount at least $1 that won’t jump over the interval boundaries).

For example you can start a phase by betting $B (which guarantees the phase will be exactly one bet long), or run a phase of always betting $1 until you hit the boundary. Because we assumed this was a “fair game” (neither you or the casino have any advantage on bets) then martingale theory tells us the expected value at the end of a phase must equal the value at the beginning of the phase. You started the phase with a value of $X, so the expected value at the end of the phase must also be $X. The expected value is p*(X-B) + (1-p)*(X+B) where p is the (unknown) probability of your betting exiting the interval as a winner. But if p*(X-B) + (1-p)*(X+B) = X then we must have p=0.5, no matter what variation of bets you execute.

This is a standard result in Martingale theory. And it just means: if neither you or the house have an advantage on individual games played, then neither you or the house have an advantage on any sequence of games even with you choosing the bet sizes and choosing when to stop playing. This is one of the hooks of gambling: players think there is great power in choosing bet size and when to stop, when usually there is no way to use those to any advantage beyond choosing not to gamble.

Things change when we add in the discount.

Let us always pick B such that X-B is -$500,000. So B = X+500,000 and our interval is -$500,000 to $(2X + 500,000). And our odds of exiting the interval at the left or right boundary remain 50/50. However, if you walk away when you are down -$500,000 you are only expected to pay $400,000. So the expected value of a bet of X+500,000 is: 0.5*(-400,000) + 0.5*(2X + 500,000) = $(X+50,000), not $X. It actually makes sense (in terms of expected value, not in terms of risk) for you to bet. You have an expected profit of $50,000 every time you bet $(X+500,000) due to the casino’s generous discount. In a sense casino is subsidizing every one of your large bets, not just the end of day settlement. Also notice the player betting a bit more than their daily winnings each time they win is pretty much equivalent to the house betting a bit more than their daily loss, or the doubling pattern of the “small martingale” (a famously risky system).

Let’s push this strategy forward a bit. One complete multi-phase strategy is: pick a daily winning target (say $3,500,000) and running betting phases of the form $(X+500,000) until you are at least that far ahead or until you are $500,000 down and then quit. That is: each night you either see three winning phases in a row and take home $3,500,000 or see a loss and pay $400,000. On average you would take home $3,500,000 one night in 8 and be down $500,000 each of 7 nights in 8. Since 7*$500,000 = $3,500,000 we have expected winnings match expected losses, prior to the discount. But you are only paying $400,000 each night you lose, so your expected net take home over 8 nights is $3,500,000 – 7*$400,000 = $700,000. So, in expectation, the casino is leaking about $87,500 a night to you.

## With non-fair odds

The previous section assumed “fair odds.” Mr. Johnson is quoted as saying he think his rule changes took the house’s edge down to about 0.25%. This is good for multiple deck blackjack, but still a problem.

Even if the odds were fair or near-fair you want to make large bets to exit the phase interval in a reasonable amount of time. You are not going to win or lose a million dollars quickly in $100 bets. The issue is the scheme requires virtuoso play on each hand. You have to play well almost every hand to eat into the house advantage, and that is going to be exhausting.

Once the house odds are against you, you don’t just want to make large bets- you *need* to make large bets. The only way of having a high probability of exiting on the profitable side of one of your phase intervals is to make large bets. With small bets the law of large numbers will almost always force you to lose the phase (and the day). With a large bet you don’t exit with the 50/50 odds, but you at least win with appreciable probability. Also with bad odds, you find above a certain size you no longer want to bet. The house has an expectation advantage proportional to the size of your bet, so eventually the bonus you are pulling per-bet (due to the house insuring losses) is overwhelmed and bets become unprofitable.

## Getting barred

The fair-odds phases strategy is pretty much guaranteed profit if one is allowed to play long enough. To do this you have to have enough bankroll to finance enough losing nights to have a good chance of a win, and you have continue to have access to the discount.

We know three casinos eventually stopped given Mr. Johnson a discount when he had nights where he took home $4 million, $5 million, and $6 million. Suppose the casino will immediately stop your play if you are ever ahead $3,500,000 or more (the article says casinos cut Mr. Johnson off on nights at winnings somewhat above this value), and not invite you back if you are ever net-ahead over a few nights.

When we add this possibility of getting barred: the overall betting scheme looses money is if it looses 9 nights in a row (as we assumed the casino will not allow you to win enough to recoup that loss). Suppose you now play at the casino until you have a winning night and are barred (in our case a day that wins $3,500,000), or until you lose 10 nights in a row (and are thus down $4,000,000). Some calculation shows this scheme has a 30% chance of losing money and a positive expected value of $515,847.10. This represents a high risk expected return on the stake (most money the scheme is prepared to lose) of around 13% in ten days. The Sharpe ratio is 0.18 which is very large for a 10 day investment (for example Morningstar quotes a good Sharpe ratio of over an annual term as being 0.40, and an average annual return being closer to 0.29). The idea is: Sharpe ratios on smaller time-period investments are necessarily smaller (smaller returns, and larger variances). A crude conversion to move from a monthly scale to a yearly scale would be to just look at how variance should decline (by a factor of about sqrt(12)), so a good monthly financial instrument might have Sharpe ratio of around 0.4/sqrt(12) or around 0.12 (and this is ignoring any issue of compounding of value). So this is a high-risk but also high-return scheme.

To lower your risk you would want to play this scheme at multiple casinos, as the Atlantic reported Mr. Johnson did. Mr. Johnson is reported to have won millions from three casinos. Since the strategy has about a 70% win rate it is safe to assume to see 3 wins we would have to play at about 4 casinos.

Unfortunately my calculations show there is still a good chance of losing money remain high (now around 33%, but your Sharpe ratio is now 0.35 showing an improved reward to risk ratio). This is also assuming The Atlantic doesn’t write an article about you and get you barred from some casino private rooms before you finish your betting sequence.

## My interest

I don’t actually play blackjack. I do however, love thinking about martingales.

Categories: math programming Opinion Quantitative Finance Tutorials

### jmount

Data Scientist and trainer at Win Vector LLC. One of the authors of Practical Data Science with R.

For more on the Sharpe ratio see

http://www.win-vector.com/blog/2015/06/what-is-a-good-sharpe-ratio/

http://www.win-vector.com/blog/2008/09/a-quick-appreciation-of-the-sharpe-ratio/