Hold Percentage: The Emperor of All Numbers

The Emperor’s New Clothes is a short tale by Hans Christian Andersen about two rogue weavers who promise an emperor a new suit of clothes that are invisible to those unfit for their positions, stupid or incompetent. When the emperor parades before his subjects in his “new clothes,” a child cries out, “But he isn’t wearing anything at all!” The tale has been translated into more than 100 languages. (I got that figure from Wikipedia, so it must be true.)

Now let’s look at another story, the story of the pre-financial revue analysis. It goes something like this:

Boy, we just had a bad month. Hold percentage in blackjack was only 4 percent. Baccarat didn’t fare much better at 6.5 percent. Carnival games were OK with the usual 24 percent, but that only accounted for 5.5 percent of the total drop. Year-to-date blackjack hold is down more than 1 percent compared to last year—last quarter we only held 9.5 percent. Our monthly promotion culminated in the Mercedes giveaway. The drawing night produced great headcounts, but we only held 8 percent that night. Also, Willy the Whale hit us hard this month. … OK, so take out Willy’s play, forget the drawing night … Let’s see, that’s back to 11 percent. Ehmm, still no good. What’s going on?


Who amongst us has not been through the above, albeit abridged, analysis prior to the financial revue? Perhaps it’s time for an analysis of the analysis.

What is hold percentage? Drop divided by win. Two numbers that are quantifiable and, many years ago, were the only two numbers we had. There was no player rating, no average bet calculation, no theoretical win, no comp dollars—it was management by the seat-of-the-pants. “Well,” we said, “let’s divide one number by the other and see what turns up. OK, looks like it’s hovering around the 14 percent mark, so that’s it then. Fourteen percent it is.” (Feel free to insert whatever number you are happy with if 14 is not for you.)

So, how is it we hold 14 percent when our definitive house advantage is only, let’s say, 2 percent? Player A arrives at the casino with $100. He sits himself down at a $10 blackjack game, buying in (drop) for his $100. He bets $10 per hand, win or lose. Out of 10 hands, he wins four and loses six, netting $80. If he leaves now, we hold 20 percent. But, in theory, he has $98. Now he stays for eight more hands, winning five and losing three, netting $100. But, in theory, he has $95 and change, given the theoretical $98 he had prior to playing a further eight hands.

Are you following me so far?

Player A carries on for another 10 hands, winning seven and losing three, netting $140. But he is down to $93 and change, in theory. Player A carries on for a while—let’s hope for an average three or four hours, in which case we should hold our 14 percent—and eventually leaves with $500. What happened there? Drop $100, lose $400. In the words of Charles Dickens’ Mr. Micawber, “result misery.” Hold percentage: minus a lot.

Never mind, there are lots of other Player A’s we can average out to come up with our magical 14 percent hold. On average, they turn over their money enough times to average out to 14 percent. Or so we hope.

But you can’t teach your customers how to be customers, though it would be nice if we could. The conversation might go something like this: “Excuse me Player A, you can’t leave now, you need to turn your money over another 12 times in order for us to meet our ‘expected gain.’ Oh, and by the way, I would prefer it if you did not cash out your winnings until you are done playing, as that creates what we call a ‘false drop’ and it’s going to mess up my numbers.”

So all these players are out there just doing as they please, with nary a thought for our back-of-house calculations. How dare they?!

There are a couple of mathematical issues at odds here. First, averages alone can often be misleading. Second, the averaging of averages will seldom produce a result from which an intelligent conclusion can be drawn. To illustrate, allow me to draw a couple of sporting analogies.

Baseball Team A and Baseball Team B produced batting averages last year (2010) of .267 and .274, respectively.

Naturally, one might assume, Team B was the more successful of the two. Except that Team A was the New York Yankees (95–67), and Team B was the Kansas City Royals (67–95). Remember, these are actual averages.

Now, consider the averaging of averages.

Let’s take Mark Teixeira of the New York Yankees, for example. (I confess, at this point, I know more about cricket than baseball, so maybe I can learn something here, too.) Here are some of Teixeira’s “averages”:

Career Batting Average: .286

Regular Season (2010) Average: .256

Post Season (2010) Average: .148

If my math is correct, on average, Teixeira bats … 0.23? Last I heard, Mark was making $20 million a year. Maybe it’s time we checked his file?

Clearly the above, albeit facetious, analysis is absurd. And yet we perform the same analyses of our own hold percentage repeatedly, puzzling at the results.

Over and above the problem of averaging lies another mathematical axiom that cannot be ignored: The Law of Large Numbers.

There is no principle that a small number of observations will converge to the expected value or that a streak of one value will immediately be “balanced” by the others. So says The Law of Large Numbers. One night’s observations (Drawing Night) or Willy the Whale’s play are not large numbers.

Further, you cannot “take out” the activities of certain players or certain nights to satisfy your expectations. If you do, you arrive at another number, which is neither your “hold percentage” nor any other number I am familiar with. One day, one week, one month are not “large numbers” in terms of assessing hold percentage. Given the (uncontrollable) variables that go into the equation they are, well, just numbers.

So, if 14 percent is your number, then I suggest you collect a “large number” of observations to support it. In the meantime, I would concentrate on a far more important number—let’s call it “volume.” It’s not really a number, but when you have more of it, you make more revenue, because, last time I looked, the customers bust before we do.

I know of two other groups that are not interested in your hold percentage: thieves and bank managers.

Perhaps, like me, they are wondering if the emperor is wearing any clothes.

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