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" John Nash meets Jorge Hirsch : Scale Invariant Citation Indices "

by Josep Freixas, Roger Hoerl, and William S Zwicker

A number of citation indices have been proposed for measuring and ranking the research publication records of scholars. Some of the best-known indices, such as those proposed by Hirsch and Woeginger, are designed to reward most highly those records that strike some balance between productivity (number of papers published) and impact (frequency with which those papers are cited). A large number of rarely cited publications will not score well, nor will a very small number of heavily cited papers. We propose several new citation indices, each resting on the notion of scale invariance, as introduced by John Nash in his solution of the two-person bargaining problem. Our main focus is on one of these—a scale invariant version of the Hirsch index, which has been independently proposed by Levene et al as the \Chi-index. We argue that it has advantages over the original ; it produces fairer rankings within subdisciplines, is more decisive (discriminates more finely, yielding fewer ties) and more dynamic (growing over time via more frequent, smaller increments). Simulations with Poisson noise suggest that scale invariance reduces the number of "accidental" reversals, wherein random irregularities cause researcher A to receive a lower index value than B, although A’s productivity and impact are both slightly higher than B’s. Moreover, we provide an axiomatic characterization of the scale invariant Hirsch index, via axioms that bear a close relationship, in discrete analogue, to those used by Nash. This argues for the mathematical naturality of the new index.