Hotelling’s principle of minimum differentiation
PAPER IX — SOCIOLOGY · QUESTION 31 OF 31
In plain terms: The free standard reference supplying the machinery the proof requires, and the setting in which competitors cluster.
The question as set
Prove Hotelling’s principle of minimum differentiation for two sellers on a line, and exhibit the failure at three.
The question in Paper IX · Its section on the reference page
The question in context
This is question 31 of the 31 set in Paper IX — Sociology of The Truly Hardest Exam in the World, the examination of The College of All Minds. In the College’s subject index it is filed under Algorithms and complexity and Games and strategy. Like every question of the College, it carries three references of record, verified live at publication and described twice — technically, and in plain terms — so the ground can be judged before it is walked. The response required is an argument, not a survey, of at most three thousand words, as technical as the question demands and no more.
The references of record
Algorithmic Game Theory (open copy)
The equilibrium apparatus in full: existence, computation, and the location games in which sellers converge on a line.
In plain terms: The free standard reference supplying the machinery the proof requires, and the setting in which competitors cluster.
Hotelling games — equilibria and their failure
The three-player case treated: non-existence of pure equilibrium on the line, and the modifications that restore one.
In plain terms: A free modern treatment giving exactly the counterexample the second half of the question demands.
The equilibrium concepts and their existence conditions, against which the three-seller failure can be diagnosed rather than merely noted.
In plain terms: The free scholarly entry supplying the standards by which the breakdown is to be read.
Background: Hotelling’s law · Nash equilibrium