Nash's bargaining problem and the scale-invariant Hirsch citation index

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 discuss three new citation indices, one of which was independently proposed in Fenner et al. (PLOS ONE 13(7): e0200098, 2018). Each rests on the notion of scale invariance, fundamental to John Nash's solution of the two-person bargaining problem. Our main focus is on one of these-a scale-invariant version of the Hirsch 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), and exhibits enhanced centrality and tail balancedness. Simulations suggest that scale invariance improves robustness under Poisson noise, with increased decisiveness having no cost in terms of 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 (Econometrica 18(2):155-162, 1950). This argues for the mathematical naturality of the new index.