COWLES FOUNDATION FOR RESEARCH IN ECONOMICS
AT YALE UNIVERSITY

Box 208281
New Haven, CT 06520-8281

COWLES FOUNDATION DISCUSSION PAPER NO. 1485

Axiomatization of an Exponential Similarity Function

Antoine Billot, Itzhak Gilboa and David Schmeidler

October 2004

An agent is asked to assess a real-valued variable y based on certain characteristics x = (x¹,...,x^m), and on a database consisting of n observations of (x¹,...,x^m,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, y^s_n+1, be the weighted average of all previously observed values y_i, where the weight of y_i is the similarity between the vector x¹_n+1,...,x^m_n+1, associated with y_n+1, and the previously observed vector, x¹_i,...,x^m_i. This paper axiomatizes, in terms of the prediction y_n+1, a similarity function that is a (decreasing) exponential in a norm of the difference between the two vectors compared.

Keywords: Similarity, Exponential

JEL Classification: C8, D8