Would some statistically-knowledgeable folk be good enough to explain
the mathematical/statistical reasoning (or, if applicable, the
principles which contrave) the following statement by Nobel Physicist
Steven Weinberg (writing about books on war, not about physics) in the
most recent (Nov. 6, 2003) issue of the "New York Review of Books"?  

  "It should have been obvious that the solution to
   the U-boat threat was to require merchant ships
   to sail in convoy.  As Churchill later explained in
   The World Crisis,
         The size of the sea is so vast that
          the difference between the size of
          a convoy and the size of a single ship
          shrinks in comparison almost to in-
          significance.  There was in fact
          nearly as good a chance of a convoy
          of forty ships in close order slipping
          unperceived between the patrolling
          U-boats as there was for a single ship;
          and each time this happened, forty
          ships escaped instead of one.
  (This is also the reason that fish of many species
  swim in schools.)"

Putting aside the at best highly questionable (and "scientific"?)
parenthetical throw-away remark about "the reason" ascribed to what
"many" fish do, and also disregarding for the moment the variable of
the role of spying/intelligence, does the Churchill quotation really
(accurately) "explain" what Weinberg characterizes as "obvious"?  

Does Churchill's (and, implicitly, Weinberg's) use of "vast" conflate
that word with "infinite" and, conversely, is "vast" itself helpful
bearing in mind that, even if a particular shipping route with respect
to one specific ship (or one convoy) at one time is not known in
advance, the geographical parameters of British (or of U.S. or other
"allied") shipping routes was (more or less) reasonably ("probably"?)
predictable?

(And, BTW, what might Weinberg have been referring to by his
parenthetical reference to "many" fish?)

Thanks.

        Consider the square mileage of the Atlantic. Now, consider the
number of sea-worthy U-Boats patrolling at any given time, and their
ability (range) to detect a single ship or a convoy.
        HOW MUCH undetectable area is left IF detection of a single ship
is approximately the same probability as detecting a convoy.

        What is the probability of detecting a single ship versus a
small closely grouped convoy???

        WHERE would be the best places to locate patrolling U-Boats???
Remember, the U-Boats must , also, AVOID detection!!!

        A good question would be WHAT is the optimal area of a convey to
MINIMIZE detection by the patrolling U-Boats??? What numbers of ships
fit into this area??? What types of ships should be included and how
many of each???

        IMHO...

        MJR
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I'm guessing here, filling in the blanks:  

If the lone predator-fish finds one fish, he will always eat it;
but he will never eat up the whole school.  So
the individual members of a school *some*  chance of
getting  away, while having the same chance of being spotted.

If the U-boat finds one ship, he will always sink it;
presumably, some of the convoy can get away.
 ... Then we get into having guards for the convoy.

This works pretty well for me.  Nature has this habit of
discovering what works, by the original 'genetic algorithms'.

[snip, rest]

--
Rich Ulrich, wpi...@pitt.edu
http://www.pitt.edu/~wpilib/index.html
"Taxes are the price we pay for civilization."

  If the U-boat finds a convoy, why can't he sink several ships?

The limiting factor is detection of convoys by U-boats.

Assume

A) detecting convoys is the rate-limiting step (true)

B) one U-boat meeting one merchant ship on open water
   can sink it 100% of the time.  {likely}

C) detecting a convoy is not much easier than detecting
   a single merchant ship  {the assertion}

If, by contrast, one U-boat is unable to sink all 40 ships
in a convoy, as seems rather likely, especially if there's
a destroyer chasing it back, then there's a win to
to convoys.

Churchill is right.

Imagine that the U-boat however were able to call in quick, massive
air support which would obliterate a whole convoy.  Then each
detection would lead to a massive loss, and thus, in order to reduce
variance, it would be better to send them in one at a time and make
the enemy spread resources.

Guerilla warfare against an enemy with complete air superiority works
like that.  There, detection of a large formation by air is easy,
and obliteration of the whole formation is likely.

U-boat surveillance could still only see a small fraction of
the ocean at a time.

prey fish swim in schools.   The idea being that if a predator
happens upon the school, the predator might be able to catch
one or two before its hunger was sated, and the rest swam away.

Whereas, if the predator saw each such fish individually, it could
pick them all off.

With smart enough, and hungry enough predators, this doesn't work.

I saw an Imax film about a pod of dolphins which were able to "trap" a
large school of fish into swimming around and trying instinctively to
escape, but there was always a dolphin to thwart them.

The dolphins massacred (and ate) the entire school.

  But what if it is able sink some number greater than one but
less than all?   Clearly the fraction of the typical convoy that
is sunk has some bearing on the original question.  

It seems to me that there is an optimization argument concealed
here, and that there is a host of various factors, upon which
the size and very existence of the optimum depend.  It simply
does not follow from any of the back-of-the-envelope arguments
put forth above, that the question has an answer -- without
holding those various factors fixed to some stipulated values.

Two numbers have been suggested (somewhere I no longer remember) about
the visibility of convoys. The idea is that a single ship can be spotted
from about 10 miles whereas a 40-ship convoy can be spotted from about
12 miles.

Of course, a single warship can guard more than one ship in a convoy.

Convoys were also used during the 1700s and 1800s due to visibility
constraints.

If the fraction is less than one then it's still a win.

not quite mathematically, but militarily it's true.  you can put a
destroyer or two hidden in a convoy which would disrupt any attack and
reduce the number of ships lost per attack.

of course the counter strategy would be to hunt in sufficient
numbers to destroy many ships in a convoy and to radio
in positions of found ships for later destruction.  Which is also
what happened.

statistically it is a bias versus variance problem.  :)

Convoys may reduce the proportion of sunk overall, but increase the
variance in the number of ships which get through in any period.

the optimization problem is in the size of the convoy.  which
empirically was not one, and it was not the maximum (send all
available ships over in a single convoy).

There is also the issue of espionage; if a convoy is known to
be in any location at a time, a waiting pack of u-boats could
in fact eliminate the whole thing.  That is a downward force
on the optimal size of convoys.