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. $