Once a barber moved to another town and chose to open a barbershop in the vicinity of two other barbershops on the same street.
The sign above the first one read:
The best barbershop in town
The second one advertised itself as
The best barbershop in the country
The newly opened barbershop entered the competition as
The best barbershop on this street
I liked this joke a lot when I heard it many years ago. And it only occurred to me last month that all three claims can genuinely be true!
This can be merely an effect of marketing reach. A new business has to focus on a narrow customer base with the purpose of getting strong foothold before starting to grow (as described in “Crossing the Chasm” by Geoffrey Moore). Therefore businesses of different level of maturity (and success) operate in different leagues. In 2023 Manchester City won the 31st UEFA Champions League while Radcliffe FC became the winner of the Manchester Premier Cup in the same year; the latter competition involvees non-league football clubs and I couldn’t find (with minimal effort put into it) if Manchester City ever competed at that level during its history. Thus a claim of being the best on the street may only mean that the author of that claim doesn’t even (yet) dare to compete in a broader market.
In such a setting the best-in-town barbershop can be objectively better than the best-on-the-street barbershop but the former isn’t even interested in disputing the less ambitious claim.
However, all three claims can be true in a somewhat strict sense.
The rating of a barbershop (or any other similar business) is determined by its clientele while the tastes of population can vary significantly. Thus a locally best barebershop can appeal to the norms, standards and/or expectations of the neighborhood while failing to satisfy the wider base of customers from the town/country. In other words, if you surveyed the opinions of the residents of the said street the new barbershop might indeed come at the top. But once the survey is expanded to the entire town (or country) the ranking can change without any manipulation of the results.
Initially I was under the impression that this was possible only when the rating is computed as a mean/average, but after more careful thought there is no such restriction. Any way of aggregating individual customer ratings (provided only that none of the values of the input ratings is completely dismissed when computing the result) seems to allow for such a “paradox”.
Consider a function f() taking a set of real numbers and returning a single real number, such that any of the input elements may influence the output. Then it should be possible to identify two non-empty finite sets A₁ and B₁ of the same size, and two other non-empty finite sets A₂ and B₂ of equal (but, possibely, different) size such that
- f(A₁) < f(B₁)
- f(A₁ ∪ A₂) > f(B₁ ∪ B₂)
Here A₁ and B₁ represent the individual ratings of the two barbershops by a small set of customers (street), while A₂ and B₂ correspond to the ratings of the same barbershops by an additional group of customers (rest of town).