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A model is proposed which treats rankings given bya group of judges as representing regions in an isotonic space of dimensionality r. Three possible criteria for estimating lower bound dimensionality are discussed: mutual boundary, cardinality, and the occurrence of transposition groups. Problems associated with each criterion are mentioned. The task of a psychological scaling technique is to search for some form of lawfulness, i.e., redundancy, in experimentally collected data. This redundancy, when it is present, permits a description of the items (also perhaps of the subjects) which is simpler than an exhaustive account of the response of every subject to every item, and yet tells the experimenter everything he wants to know about his data. If the scaling technique is the Guttman scale [5], for example, the experimenter hopes to find that the score of one of his subjects will tell him not simply how many items the subject passed but which items he passed, within some reasonable margin for error. Sometimes this scaling process is an end in itself; at other times the items or the people are scaled in order to "calibrate " them for application in some other context. In either case, the scaling technique chosen for a particular application must be appropriate to the task given the subjects, i.e., whether they were asked to agree with the items, pass them, rank them, or compare them in pairs, etc. This paper is concerned with a scaling technique designed for the analysis of ranked preference data. The subject is asked to rank a group of items, for example, the names of hobbies. The model states that each hobby can be characterized by its position on each of several underlying attributes (e.g., scientific-artistic, solitary-gregarious, killed-unskilled, etc.). The model states further that every subject can be characterized by his own maximum preferences on each of these attributes, and that he will ran