Re: Twenty Beautiful Women

[mpd@netcom.com (Mike Duvos)]

tcmay@got.net (Timothy C. May) writes:

 > You're both reading far too much into this problem. David
 > S. specified "beauty," the personal judgment of the chooser.
 > No deep philosophical meaning.

Which means that given two candidates, we can order them with
regard to beauty.  No other information is implied.

 > Perhaps an equivalent formulation will make this clearer:

 > One is passing through a town with 20 gas stations, with
 > gas at various prices. The stipulation is that one cannot
 > turn around. Once a gas station has been passed, there's no
 > turning back. So, what is the best strategy for finding the
 > lowest gas price (or shortest lines, or cleanest
 > appearance, or brightest sign, or whatever one wants to
 > analyze). Or even by the most beautiful girl standing in
 > front, to return us to the original statement.

 > So, you see, the problem is well-defined, with an elegant
 > solution.

Ahem.  I think the sticking point here lies in the translation of
the phase "lowest gas price" into the appropriate function to be
minimized.

Suppose we have 20 gas stations with prices p[1] through p[20]
which we have an equal chance of encountering in any of the 20!
possible orders while driving through town.  We have a
deterministic strategy for picking a station to buy gas at which
tells us whether or not to buy at the current station as a
function only of the rankings of the prices of gas at stations so
far encountered, including the current one.

This strategy maps every one of the 20! permutations of the gas
stations into one of the 20 prices, namely the price at which we
purchase gas by applying the specified strategy when stations are
encountered in the given order.

Finding the "best" strategy implies that we have some function
whose domain is the set of such strategies, and whose output is a
real number, such that the "best strategy" is one for which this
number is minimized.  Calling this function the "lowest gas
price" is somewhat misleading, since there are a variety of
different notions of "average" we may use to condense the 20! gas
prices a given strategy generates into a single number.

If we were concerned about our wallets, we would probably want
the strategy such that the arithmetic mean of gas prices was
minimized over all orderings of stations, but this is a
parametric notion, and requires that we know specific numeric
values of prices, and not simply whether one price is bigger than
another.

Non-parametrically speaking, the only obvious way of ordering
strategies is lexographically, where a strategy which yields more
occurrences of the lowest ranked price is better than one which
yields less, and if two strategies are equal in their yields of
the N lowest ranked prices, we then compare them on the price
ranked N+1.

This is a function only of the relative rankings, but may not
necessarily choose the strategy which on average results in the
expenditure of the least amount of money.

So while the solution may be elegant, I would argue that the
problem, as given, is far from "well-defined", unless some
explicit metric which admits arithmetic means is introduced.

--
     Mike Duvos         $    PGP 2.6 Public Key available     $
     mpd@netcom.com     $    via Finger.                      $