
The German Tank Problem (2010) - flashman
http://www.statisticalconsultants.co.nz/blog/the-german-tank-problem.html
======
alanh
Countermeasures* to this problem reminded me of an incident in which the
Germans were able to outsmart Allied military intelligence.

* [https://en.wikipedia.org/wiki/German_tank_problem#Countermea...](https://en.wikipedia.org/wiki/German_tank_problem#Countermeasures)

This is something I read in _The American Axis,_ which discusses how Charles
Lindbergh was duped into helping delay American involvement in WWII. The short
version: He was taken on a grand, multi-day tour of the Luftwaffe’s large air-
force… never the wiser that the planes he saw on airbase after airbase were
the same ones, flown in overnight from the last airbase! Naturally the US MI
questioned Lindbergh after his travel and grossly overestimated the power and
size of the Germain air force, just as they had intended!

(As I recall, Lindbergh wasn’t a Nazi sympathizer so much as a dupe; although
the other subject of the book – Henry Ford — certainly was. In fact, Ford
plants produced the faultiest American war planes, by far, dooming quite a
number of American pilots. There’s every reason to think this was
intentional.)

 _Update_ I read this book 8 years ago and my memory isn't 100%. It looks like
Lindbergh, per the book, was also a Nazi sympathizer to some degree, although
he comes off much more sympathetically than Ford.

~~~
hackuser
> Ford plants produced the faultiest American war planes, by far, dooming
> quite a number of American pilots. There’s every reason to think this was
> intentional.

That is quite an assertion to make about someone, even long dead. Could you
provide some evidence?

~~~
alanh
As I mentioned, I’m pulling this from a book. [http://www.amazon.com/The-
American-Axis-Charles-Lindbergh/dp...](http://www.amazon.com/The-American-
Axis-Charles-Lindbergh/dp/0312290225)

One reviewer on Amazon says:

> _This is one of the most carefully researched, extensively footnoted,
> beautifully written, books on this subject ever published. The sources
> include entries from Lindbergh 's journal, Lindbergh's speeches, FBI files,
> letters from German archives, Ford company records, and many other primary
> sources. Every assertion of fact is backed up with unimpeachable
> documentation._

IIRC — by this point it was Ford's son running the plant(s) in question, but
Henry Ford himself is not just a known anti-Semite, but quite literally ran
and anti-Jewish propaganda paper in Dearborn, MI. It hardly takes a leap to
look at the abysmal performance of planes built by Ford and suspect that
management there was particularly … lackadaisical in their pursuit of quality.

~~~
hackuser
> Henry Ford himself is ... a known anti-Semite

Agreed; he's not my favorite person at all and made an awful call by
supporting Hitler. But ...

> It hardly takes a leap to look at the abysmal performance of planes built by
> Ford and suspect that management there was particularly … lackadaisical in
> their pursuit of quality.

Things like that alarm me; it's a proven method for condemning the innocent
and unpopular. We can form hypothetical links between many events - Billy is a
mean person and disliked Joe, Joe's been murdered, so hang Billy; or, every
morning the rooster crows and then the sun rises, therfore the rooster is
doing it - but most will be false. We need evidence to differentiate truth
from imagination.

From the Amazon page, Publisher's Weekly seems to think less of the book
(though Kirkus likes it):

 _Although Wallace (Who Killed Kurt Cobain?) is a recipient of Rolling Stone
's Award for Investigative Journalism and appears to have done much primary
research, he delivers a highly speculative rehash of material handled much
better in A. Scott Berg's Lindbergh, Robert Lacey's Ford: The Man and the
Machine and such seminal studies as Charles Higham's American Swastika.
Wallace tries and fails to sensationalize well-known facts about the parochial
American fifth column of the late 1930s and early '40s, a bungling movement of
which Ford and Lindbergh were among the most public faces. Wallace sees a
conspiracy in what he presents as Ford's pro-Nazi partnership with Lindbergh:
a dark and powerful alliance designed to hinder the Allies at every turn. In
fact, the two men were far more naive than effectual in their attempts to prop
up American isolationism before Pearl Harbor. And Lindbergh, who counted Harry
Guggenheim among his closest friends, found Ford's hatred of Jews repugnant.
Once war was declared, both Lindbergh and Ford helped the Allied effort.
Lindbergh helped develop the Corsair and later, as a "civilian observer," flew
more than 25 combat missions over the South Pacific. At the same time, Ford
(with Lindbergh's help, and after a few false starts) became the leading
manufacturer of the B-24 bomber. Were Ford and Lindbergh half-witted dupes of
Nazi propaganda before the war? Undoubtedly. Were they Nazi agents either
before or after the start of hostilities? Wallace fails to make the case._

~~~
alanh
I used the word "suspect," didn’t I? The numerical output of B-24 bombers is
dramatically undercut by their failure rate.

The quote you’re posting is mostly about Lindbergh. I don’t recall the book
trying to say he was "partnered" or "allied" with Ford in a Nazi conspiracy,
mostly that he was a useful dupe. This changes nothing. We will probably never
know for sure how much of the bombers' poor construction was intentional or
not, but you can hardly say there’s no reason to seriously _suspect_ it.

~~~
hackuser
> We will probably never know for sure how much of the bombers' poor
> construction was intentional or not ...

That's the reasoning that so alarms me. That can be said about anything (and
is often used by propagandists): We will never know for sure if person X
kidnapped and killed a prostitute, or if AIDS is a conspiracy of western
doctors to kill Africans, or if the Muslim next door is a terrorist, or if
aliens kidnapped my cousin, or if the KGB is following me.

Without evidence it's just a random guess and slanders the accused.

------
Maultasche
Note to self: when manufacturing tanks, use UUIDs as identifiers, not
sequential integers.

~~~
herge
This also matters for user visible ids in applications, like user ids, etc.
It's really easy to say your competitor has only 2000 users if you sign up to
their service and your id is 1992.

~~~
brianwawok
pks are sequential. If you sign up as 1992 they have 1992 customers.

~~~
jeremysmyth
_pks are sequential. If you sign up as 1992 they have 1992 customers._

Does not follow. Only follows if:

\- PKs are sequential

\- _and_ PKs are non-zero non-negative

\- _and_ they have not been reset at any point (or the sequence generation
otherwise modified)

\- _and_ all preceding PKs are those of customers (as opposed to test values,
invalid PKs due to a sequence number that was generated but the registration
did not complete, or customers who have since left the service)

If PKs are sequential and natural and the sequence has not been modified, then
you know there are _no more than_ 1992 customers. Not exactly 1992.

------
mikeash
I always love this story.

I wonder about the potential for error due to bias in the collection process.
For example, I would expect older tanks to be more prone to breaking down due
to mechanical problems, which might change how likely they are to be captured
or destroyed. (I can't tell if that likelihood would go up, because they're
more likely to break down in combat, or if it would go down, because they're
more likely to be in the shop instead of on the battlefield.)

Evidently it didn't affect the accuracy of this technique too badly, but it
makes me wonder.

~~~
Peaker
But only the single largest serial number, and the number of samples are being
used. Not any of the serial numbers seen besides the largest one.

That means any such noise as you describe will have no effect. As long as a
high-serial number tank is ever destroyed or captured, the method will work.
It only takes one.

~~~
mikeash
You can get away with only using the count and the maximum because of the
assumption of uniformity. Once that assumption goes away, the assumption no
longer works.

If earlier tanks are much more likely to be captured, then the number of
samples will be exaggerated compared to the maximum number seen, so you'll
underestimate production. If earlier tanks are much less likely to be
captured, then the number of samples will be smaller compared to the maximum,
and you'll overestimate.

------
UnBe
Alternative application of the solution. It's an interesting bit of math, with
many theoretical and real world applications.

[http://what-if.xkcd.com/65/](http://what-if.xkcd.com/65/)

------
eggoa
Thus Seal Team "Six".

------
goodcanadian
The moral of the story is to use random serial numbers.

~~~
TearsInTheRain
no you want to just skip numbers randomly to inflate the estimates

~~~
bmmayer1
The irony here is that, at the beginning, the Nazi Party started counting
members at 500 to make themselves seem bigger than they were.

~~~
Terr_
<glances at the UIDs in his system's /etc/passwd>

------
bendykstra
This reminds me of a popular German graduation prank. Three goats are set
loose in a school building. However, before the goats are released, the
students paint the numbers 1, 2 and 3 on them to facilitate an orderly
recapture procedure.

~~~
justinsaccount
I think you mean 1,2 and 4.

~~~
AcerbicZero
You may have missed the joke...

~~~
gruez
Explain? I don't get it

~~~
alanh
The joke plays off the one we've all heard before, about pigs being released
with numbers implying one is "still at large" — the joke being that the
Germans are such industrious and efficient engineer-types that even school
pranksters still facilitate an efficient clean-up. I thought it was clever!

~~~
Mithaldu
I'm a german and i didn't get it. Which may in itself be an implicit joke
about germans.

~~~
alanh
I’m not surprised. It's based on stereotypes that an actual German would not
be as likely to share. (You may also be less likely to have heard the joke
this plays off.)

------
pinaceae
Ok, can someone _explain_ why this method performs so well?

Take the highest serial number, add a tenth of it then subtract 1. Presto.

What is the underlying logic here?

~~~
justinsaccount
It's not "add a tenth":

> For example, let’s say 10 tanks were captured

It's "add 1/n" where n is the number you collected.

Think of this related problem. Numbers are randomly chosen from 1 of 2 ranges.
One range is from 0-10, the other is 0-100. The numbers are:

4 3 1 8 5 2 7

Which range were the numbers chosen from? It could be either, but
statistically the answer would be from 0-10, since if they were chosen from
0-100 the chances of all of the numbers being lower than 10 is very low.

Now, say you didn't know the ranges ahead of time, and the numbers were:

117, 232, 122, 173, 167, 12, 168, 204, 4, 229

Now, the upper bound can't be 100 or 200, since there is a 232. So it is at
LEAST 232.

Which is more likely, that the upper bound is 256, 500 or 1000 ?

Intuitively you can assume that if it was 500+, you would expect at least one
number in the 300-500 range.

The highest number should be 232 + some amount that is inline with all the
other numbers. It could be exactly 232, but the chances that you happened to
pick the highest number itself is also low.

Basically you are solving this problem:

If you were going to select 10 numbers at random from a range of 0 to N, how
high should N be so that there is a high probability that the highest number
you select is only 232.

Since a uniform distribution is assumed, you are basically just adding one
more uniformly distributed data point.

~~~
JeremyHoward
It's actually "add m/n" where n is the number you collected and m is the
highest sampled serial number.

------
wahsd
An interesting aspect to this that seems to be overlooked is that the
conventional intelligence was probably intentionally fudged in order to garner
funds and support and raise anxiety and fear as a means for motivation.

You can't overestimate the production of tanks by that much when you are
observing the production facilities. You would either have to have been
tricked, i.e., knowing that it is being observed the same tanks were "shipped"
multiple times or you, for some reason thought there were more factories which
you didn't have eyes on. I'm not sure how to judge whether it would have been
in the Nazi interest to provide an impression of high productivity or hide the
true nature of the numbers.

Another interesting aspect to this is that it provides the background for why
the US military operates with what would otherwise be seen as abject
retardation. There is a strategic value in being aimless, illogical, and
confused in your operations for various reasons. One reason is that if you
have "random" serial numbers your enemy can't do these kind of calculations.

------
yuhong
Reminds me of [http://blog.cryptographyengineering.com/2014/10/attack-of-
we...](http://blog.cryptographyengineering.com/2014/10/attack-of-week-
unpicking-plaid.html)

