Preference modelling
From a purely mathematical point of view, the book presents an elementary theory of certain properties of binary relations in a finite set A. A relation can be equivalently seen as a graph or a matrix; a different look will often exhibit different properties of a relation. The three ways of presentation appear often throughout the book. The authors also consider connections between a relation and a numerical function (defined over A). A preference structure in a set A is a triplet consisting of three binary relations in A respectively named: preference, indifference and incomparability relation. The authors separately study the tournament, total order, weak order, total interval order, total semiorder, partial order, quasi-order, partial interval order and partial semiorder structures. Among known theorems presented in the book are those of Rédei (1934), Landau (1955), Fishburn (1970), Hajos (1957), Ghouila and Houri (1962), Gillmore and Hoffmann (1964), Fulkerson and Gross (1965), Scott and Suppes (1958), Roberts (1969) and Szpilrajn-Marczewski (1930). Special attention is given to relations derived from a numerical function defined over \(A^ 2\); certain such functions, understood to reflect intensity of preferences are named (which is somewhat misleading) complete valued preference structures. Structures assuming only two values 0 and 1 are studied separately. Complete valued preference structures especially ''appear in probabilistic or fuzzy contexts''. The book also contains many specific graph and matrix examples.
scientific article; zbMATH DE number 3914015