An impossibility result in axiomatic location theory

We show that certain reasonable axioms for an optimal solution to the problem of locating a facility on a network, i.e., axioms of distance determination, Pareto optimality, and anonymity, and a weak Lipschitz condition, can be self-contradictory. In particular, we show that they fail for any network that has a cycle. It follows that under the axioms of distance determination, Pareto optimality, and anonymity, choice of optimal location may be very sensitive to changes in the locations of the users.