A simple axiomatization of the median procedure on median graphs

Martyn Mulder and Beth Novick

No EI2011-25, Econometric Institute Research Papers from Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute

Abstract: A profile = (x1, ..., xk), of length k, in a finite connected graph G is a sequence of vertices of G, with repetitions allowed. A median x of is a vertex for which the sum of the distances from x to the vertices in the profile is minimum. The median function finds the set of all medians of a profile. Medians are important in location theory and consensus theory. A median graph is a graph for which every profile of length 3 has a unique median. Median graphs are well studied. They arise in many arenas, and have many applications. We establish a succinct axiomatic characterization of the median procedure on median graphs. This is a simplification of the characterization given by McMorris, Mulder and Roberts [17] in 1998. We show that the median procedure can be characterized on the class of all median graphs with only three simple and intuitively appealing axioms: anonymity, betweenness and consistency. We also extend a key result of the same paper, characterizing the median function for profiles of even length on median graphs.

Keywords: consensus axiom; consensus function; location function; median; median function; median graph (search for similar items in EconPapers)
Date: 2011-08-01
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