Merging discrete evaluations

A merging function synthesizes a vector of numbers (representing any form of quantitative evaluation) into a single number (representing a consensus or collective evaluation). 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 when 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 into a vector of numbers from the discrete set.