Merging discrete measurements

A merging function synthesizes a vector of numbers (representing measurements, scores or quantitative opinions) into a single number (representing a consensus or collective measurement, score or quantitative opinion). Assuming that all the involved numbers are drawn from a discrete set, it is shown that projection functions are the only merging functions satisfying three properties satisfied by the arithmetic mean (defined for real numbers). Another projection result is obtained under alternative assumptions when merging functions are assumed to transform matrices of numbers from a discrete set to a vector of numbers from the discrete set.