Axiomatization of an exponential similarity function

An agent 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, ys n+1, be the weighted average of all previously observed values yi, where the weight of yi is the similarity between the vector xn+1, ..., x m n+1, associated with yn+1, and the previously observed vector, xi , ..., x m i . 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.