Merging relative scores

This paper considers the problem of scoring alternative technologies based on their performance on different benchmarks, normalizing the scores relative to those of a base technology, and then finding a merged or combined score based on the normalized or relative scores. Similar problems occur if we score alternative job candidates or students on the basis of different tests or criteria and seek to merge relative scores after normalization. They also occur when we obtain index numbers such as price indices. Here, we measure different items at different times, normalize relative to some base time, and merge the relative measures of the different items obtained at a fixed time. We attempt to identify merging methods which will guarantee that certain types of conclusions based on them will be invariant if the base technology (job candidate, student, or time) is changed.