Axiomatization of an exponential similarity function

A. Billot, I. GILBOA, D. Schmeidler

Mathematical Social Sciences

March 2008, vol. 55, n°2, pp.107-115

Departments: Economics & Decision Sciences, GREGHEC (CNRS)

An individual is asked to assess a real-valued variable y based on certain characteristics x = (x1,', xm), and on a database consisting of n observations of (x1,', xm, y). A possible approach to combine past observations of x and y with the current values of x to generate an assessment of y is similarity-weighted averaging. It suggests that the predicted value of y, yn+1s, be the weighted average of all previously observed values yi, where the weight of yi is the similarity between the vector xn+11,', xn+1m, associated with yn+1, and the previously observed vector, xi1,', xim. This paper axiomatizes, in terms of the prediction yn+1, a similarity function that is a (decreasing) exponential in a norm of the difference between the two vectors compared. Keywords: Similarity function; Axiom; Exponential decay