Postcard from the Edge-case

"It's obvious," said the Professor, sketching rapidly on the table cloth, "that the CAMRA locALE scheme is bound to produce some anomalies, particularly in branches not completely surrounded by other branch areas, as will be the case with, for example, those which lie on the coast."

"Firstly," the professor outlined one of the square demarcations, "here we have a branch area, with numbers indicating individual breweries in the branch - I've shown each branch having 3 breweries for simplicity"

"OK professor, and the circles would be showing the breweries considered LocALE?"

 "Right," the professor beamed "My diagram isn't an exact representation of any particular area, but it it suggestive of how different branch policies may interact and lead to sub-optimal outcomes."

He continued, "Let's compare the strategies of branch A and branch B."

I could see that Branch A had a bigger circle around it than B, and I told the professor so.

"Yes" he smiled, "A larger area, but the same number of breweries! As it happens, seven. This branch has chosen a broader definition of local so as not to penalise their members for a geographical accident.  Being at the edge would have given them only 4, or possibly 5, locALE breweries if they'd used the same criterion as their neighbours in branch B"

 I told the professor that this seemed fair enough.

 "Oh yes" he continued "perfectly fair for the members, and after all, it's a membership organisation. But look at it from a brewery point of view." He regarded me expectantly.

I shook my head, I couldn't see what he was getting at.

"It's simple," He wagged a finger at me, "How many circles is each brewery in? That is to say, how many branches consider each brewery local?"

I must have looked more than usually puzzled, because he continued, "How many circles is brewery 3 in?"

That was easy, "Just one"

"Yes" he chuckled, only one branch considers brewery 1 a locALE brewery.  But what about brewery 11?"

I studied the tablecloth.  Then it struck me, "Ah, brewery 11 is contained in 3 circles!"

"Yes" said the professor, "Brewery 11 has been made locALE by branch A, but of course branch B hasn't reciprocated by offering this courtesy to brewery 1.  Why would they?"

"But what this means" he continued sadly, "is that in order to make sure that their members have a good choice of locALE, branch A has handed a competitive advantage to breweries not based in their branch area, which is denied to breweries that actually are!"