Competitive facility location models are based on the assumption that customers select a facility to patronize according to some rule involving the distance to the facility. A customer prefers facility A over facility B if the distance to facility A is shorter than some function of the distance to facility B. The authors define a consistency property of a selection rule. A selection rule is consistent if the selection of a facility remains unchanged along the shortest route to that facility: namely the rule cannot indicate a preference for a different facility while en route to the selected one. Otherwise, the rule is inconsistent. The authors prove that under mild conditions, the only consistent rule is the one based on an additive function of the distance, a function which is the sum of the distance and a constant. All other rules, such as a product of the distance and a constant (a multiplicative function), are inconsistent.
