AbstractCompetitive 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 facilityAover facilityBif the distance to facilityAis shorter than some function of the distance to facilityB. We 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 whileen routeto the selected one. Otherwise, the rule is inconsistent. We 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.