An antimedian of a profile $\\pi = (x_1, x_2, \\ldots , x_k)$ of vertices of a graph $G$ is a vertex maximizing the sum of the distances to the elements of the profile. The antimedian function is defined on the set of all profiles on $G$ and has as output the set of antimedians of a profile. It is a typical location function for finding a location for an obnoxious facility. The `converse' of the antimedian function is the median function, where the distance sum is minimized. The median function is well studied. For instance it has been characterized axiomatically by three simple axioms on median graphs. The median function behaves nicely on many classes of graphs. In contrast the antimedian function does not have a nice behavior on most classes. So a nice axiomatic characterization may not be expected. In this paper such a characterization is obtained for the two classes of graphs on which the antimedian is well-behaved: paths and hypercubes.

Antimedian, consensus axiom, consensus function, consistency, hypercube, path
Erasmus School of Economics
hdl.handle.net/1765/22803
Econometric Institute Research Papers
Report / Econometric Institute, Erasmus University Rotterdam
Erasmus School of Economics

Balakrishnan, K, Changat, M, Mulder, H.M, & Subhamathi, A.R. (2011). Axiomatic Characterization of the Antimedian Function on Paths and Hypercubes (No. EI 2011-08). Report / Econometric Institute, Erasmus University Rotterdam (pp. 1–17). Erasmus School of Economics. Retrieved from http://hdl.handle.net/1765/22803