Axioms for Consensus Functions on the n-Cube

Abstract

A $p$ value of a sequence $\pi =(x_{\mathrm{1}},x_{\mathrm{2}},\dots ,x_k)$ of elements of a finite metric space $(X,d)$ is an element $x$ for which ${\sum }_{i=\mathrm{1}}^k{d}^p(x,x_i)$ is minimum. The ${\mathcal{l}}_p$–function with domain the set of all finite sequences on $X$ and defined by ${\mathcal{l}}_p(\pi )=\{x\text{:}\hspace{0.17em}\hspace{0.17em}x$ is a $p$ value of $\pi \}$ is called the ${\mathcal{l}}_p$–function on $(X,d)$. The ${\mathcal{l}}_{\mathrm{1}}$ and ${\mathcal{l}}_{\mathrm{2}}$ functions are the well-studied median and mean functions, respectively. In this note, simple characterizations of the ${\mathcal{l}}_p$–functions on the $n$-cube are given. In addition, the center function (using the minimax criterion) is characterized as well as new results proved for the median and antimedian functions.