What Advantage Players and Casinos Get Wrong About the “Long Run”

Many years ago I was asked to help edit a book by a low-level card counter who was recounting his Las Vegas adventures.  His book told the stories of the various trips he made, what he wagered and what he won or lost.  It was really more of a journal than a book, but there was entertainment value in these struggles.  For each trip the author also discussed his wager size, the risk he was taking and most importantly his expected value (EV) for the trip.

EV to an advantage player (AP) is like t-win or theo from the casino-side.  It is the amount that theory predicts the person (or casino) should win.  EV and t-win are both described by the same simple formula:

EV = (total amount wagered) x (edge over the house)

T-Win = (total amount wagered) x (edge over the player)

Back to the story.  One of the “lessons” the card counter told in his book was a story about flipping a coin.  Several times he stated that if you flip a fair coin enough times then eventually the number of heads and tails will be equal.  His point was that whether he won or lost on any particular trip, he expected that in the long run his total winnings would eventually match his EV, claiming that this had to be the case due to “reversion to the mean” or “the law of large numbers” or something else from statistics he didn’t understand.  As I said to the author when I edited his work, his conception of the long run was absolutely 100% wrong.

Coins and cards do not remember you.  They do not have some primal memory source for what your previous results were and attempt to recoup your losses or punish your wins.  This is the essence of the so-called gambler’s fallacy, that outcomes must eventually become in-line with expectation.

We see this all the time in baccarat strategies.  Players know how many “Banker”, “Player” and “Ties” are normal in a shoe and they expect the shoe to try and revert to this normal state.  If a shoe has too many Bankers, it will start favoring Player.  If there aren’t enough Ties then they believe Tie becomes a more likely outcome.  Roulette players do this as well, which is the whole purpose of the “Totem” that accompanies most roulette tables.

I read a statement by a well known card counter on a message board recently.  This counter had lost a substantial amount in his recent visits to casinos, but felt comforted by his accumulation of EV and that somehow the universe “owed” this counter what he had lost:

I subscribe to the Law of Large numbers theory that given a large enough number of trials or sample size, results will come pretty close to expectation, which is what the math says it should be. … When you start talking about “owe” this or that, you are talking short-term and in the short term anything can and will occur, but long-term, the math dictates what will happen. … There are many ways to get back to the means, to expectation. You are never “due” for anything, but the law of large numbers says you will get there.

This statement demonstrates this very common misunderstanding of the long run.  There is nothing in “the math” that says that the results will come “pretty close to expectation.”  There is nothing in “the math” that says you will get there. The gambler’s fallacy rears its head again. I agree with the poster that the AP’s goal should be to accumulate EV.  But this accumulation gives only a rough guide to the center point of what might happen.  What’s past is past.  Going forward, the goal should always be to accumulate EV, but there should never be an expectation of making up past losses by exceeding EV.  That may or may not happen, however, the universe isn’t keeping track.

This phenomenon is not restricted to the player-side.  I was delivering a seminar at one of the top international casinos a few years ago. I was taken aback when their director of marketing challenged me and stated unequivocally that people who lose more than expected were due for a big win.  At another property, one of the top executives insisted that they should be able to beat every (ordinary) player, as if winning players owed the casino.  The idea of “our money” vs. “their money” is a synonym for misunderstanding the long run.

So what is true?  Something gets closer to something in the long run, it is just not “results” that get close to “expectation.”  The correct formulation of the long run is one of the limiting value of a ratio:

(results)/(expectation) –> 1.

However, even as this ratio gets closer and closer to 1, the player’s actual “results” and “expectation” can get further and further apart.

For example, results could follow the sequence 9, 98, 997, 9996, 99995, and so on, while expectation follows the sequence 10, 100, 1000, 10000, 100000, and so on.  Then we see that these two sequences are getting further apart over time even as their ratio is getting closer and closer to 1.

9/10 = 0.9
98/100 = 0.98
997/1000 = 0.997
9996/10000 = 0.9996
99995/100000 = 0.99995

The long run does not say that the top sequence has to eventually match the bottom, only that the limiting value of the ratio is 1.

The truth is that a player’s (or casino’s) forward looking EV (or t-win) is always given by the formulas above.  Nothing about what happened in the past has any impact whatsoever on these results.  There is not a third factor in these equations that accounts for recent results or any other historical pattern.  There is no such thing as being owed EV (or t-win).  The number of heads and tails are under absolutely no obligation to ever be the same.  What’s done is done, tomorrow is a new day.

The gambler’s fallacy is insidious.  It gets inside the head of ordinary gamblers, advantage players and casino professionals.  On both sides of the table the idea that actual results must eventually equal  expected results blinds otherwise intelligent individuals from rational dialogue and fact-based decisions-making.

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