Sunday, November 24, 2013

Benford is not kind to strippers

Over at MR, Tyler discusses inflation and strippers:
"Nobody dislikes currency inflation more than strippers." 
But is that claim true?  It depends on the margin.  Let’s say the standard tip is a dollar, and price inflation lowers the real value of that dollar.  A lot of customers won’t substitute into stuffing $1.43 into the stripper’s garments.  They might do two or three singles, but strippers will be shortchanged at various points going up the price pole.  There is something about handing out a single bill that is easier and more transparent, or so it seems.
I was a bit disappointed to see no mention of Benford's law!  I love Benford's law and cannot pass up an opportunity like this...

Benford's law says that if you go to the supermarket (say) and look at the prices, the leading digits will not be distributed uniformly like you might expect; rather, lower digits are much more common, with almost a third of the prices starting with 1!

The underlying reason for this surprising fact is that prices keep pace with inflation and thus exhibit exponential growth.  Think about simply doubling your money every year (a 100% growth rate):

1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 10242048, 4096, 8192, ...

As you can see, over 2/3rds of the numbers begin with a digit less than 5!  As soon as you get to a leading 5 or above, doubling is sure to boost you into the next order of magnitude, leading with a 1 again!

What does this have to do with strippers and indivisible tips? To find out, let's assume that:
  1. The inflation rate is constant at, say,  5% per year.
  2. The "correct" real tip for a stripper's services is $1 in 1980 dollars.
  3. Customers know the correct tip, but insist on paying only in a single bill, limiting them to payments of $1, $5, $10, $20 and so forth.
  4. In any given year, customers compensate by straightforwardly rounding their payment up or down to the bill with value closest to the correct tip for that year.
In 1980, the customer simply tips $1, and everyone is happy.  As inflation kicks in, though, the dollar loses value.  In 1981, a stripper must be paid $1.05 to be equally well off as before.  The customer realizes $1 is underpaying by $0.05, but that's better than paying $5, which overpays by $5 - $1.05 = $3.95.

So the customer underpays.  In 1982, the stripper should be paid $1.10.  Again, $1 underpays, but surely a 10 cent underpayment is better than a $3.90 overpayment with a $5 bill.

We continue in this fashion, year after year.  It is not until 2003 that the customer starts paying with a $5.  In 2003, the correct tip is $3.07, which is closer to $5 than $1.

Now the customer is overpaying.  In 2004, he overpays again.  In 2005, he overpays again.  That's good, since he's been underpaying for the last 22 years.  It will balance out, right?


Sadly, no, it won't balance out.  By 2013, the stripper's correct tip is $5.00, meaning a $5 bill in 2013 is worth the same as a $1 bill in 1980.  (By the way, the true inflation rate in the last 33 years has been much lower than 5%; a 1980 dollar is actually equivalent to about $2.83 today.  But whatever).

The stripper is paid exactly correctly in 1980 and 2013.  But in between, we have 22 years of underpaying versus only 10 years of overpaying. In fact, the sum total of the underpayments in 1980 dollars is $8.84, versus $3.20 in overpayments, for a net loss of $5.64.

What happened?  The customer followed the seemingly innocuous rule of considering the correct payment, and then rounding to the nearest acceptable unit of currency.  It seems like roughly half the time you'd be rounding up and half the time rounding down, balancing out in the long run.  But that misses a fundamental fact about exponential growth, namely that the bigger you are, the faster you grow.

Here is the complete beginning of the sequence of the stripper's "correct" tips:

1.00, 1.05, 1.10, 1.16, 1.21, 1.28, 1.34, 1.4, 1.48, 1.55, 1.63, 1.71, 1.80, 1.98, 2.08, 2.18, 2.29, 2.40, 2.53, 2.65, 2.79, 2.93, 3.07, 3.23, 3.39, 3.56, 3.73, 3.92, 4.11, 4.32, 4.53, 4.76, 5.00, 5.25, 5.52, 5.79, 6.08, 6.39, 6.70, 7.04, 7.39, 7.76, 8.15, 8.55, 8.99, 9.43, 9.91, 10.40, ...

You can observe Benford's law in full effect here. See how many prices begin with 1?

Was there anything special about picking $1 and $5 as our denominations?  Not really.  No matter what, growth accelerates.  That means that you will always traverse the first half of the distance more slowly, which means any policy of rounding to the nearest of two denominations will tend to round down more than up.


Aside: Mind you, I never said how the "correct" tip is determined.  This discussion makes it seem like strippers will systematically be shortchanged, but why should the customers be the ones who determine the "correct" wage, and decide the fair way to round?  Presumably there is some sense in which strippers set their own prices (say, by giving dirty looks to underpaying customers), and thus can correct for Benford's law.  And if strippers are being shortchanged anyway, supply should adjust accordingly.  Also, even if any individual payment is restricted to $1 or $5, there's no reason an individual or group cannot randomize their payments to achieve a correct intermediate payment on average.  But I hope these caveats will not diminish your respect for Benford's law!

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