# An axiomatic characterization of a class of rank mobility measures

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## Abstract

In this paper we provide an axiomatic characterization of a total preorder for assessing the mobility associated with any pair of rankings relative to a same group by means of a new class of rank-mobility measures. These measures constitute a strong generalization of Spearman’s \(\rho \) index. The problem of dealing with mobility of subgroups, which is a pressing issue in several and various applicative contexts ranging from the use of socio-economic indicators to information retrieval, is addressed in terms of *partial permutations* which include standard permutations as a special case. Literature is lacking in a theoretical discussion for comparing mobility of variable-size subgroups: we show that our axiomatic approach fills this gap.

## Notes

### Acknowledgements

We would like to thank the anonymous referees for their careful reading and valuable comments that have been helpful for providing a much improved version of the present work.

## References

- Bossert W, Can B, D’Ambrosio C (2016) Measuring rank mobility with variable population size. Soc Choice Welf 46(4):917–931CrossRefGoogle Scholar
- Chakravarty SR, Dutta B, Weymark JA (1985) Ethical indices of income mobility. Soc Choice Welf 2:1–21CrossRefGoogle Scholar
- Cook WD, Seiford LM (1978) Priority ranking and consensus formation. Manag Sci 24(16):1721–1732CrossRefGoogle Scholar
- Critchlow DE (2012) Metric methods for analyzing partially ranked data, vol 34. Springer Science & Business Media, BerlinGoogle Scholar
- Dardanoni V (1993) Measuring social mobility. J Econ Theory 61:372–94CrossRefGoogle Scholar
- D’Agostino M, Dardanoni V (2009) The measurement of rank mobility. J Econ Theory 144:1783–1803CrossRefGoogle Scholar
- Debreu G (1954) Representation of a preference ordering by a numerical function. In: Coombs CH, Thrall RM, Davis RL (eds) Decision processes. Wiley, New YorkGoogle Scholar
- Diaconis P (1988) Group representations in probability and statistics. Institute of Mathematical Statistics, HaywardGoogle Scholar
- Dowrick S, Dunlop Y, Quiggin J (2003) Social indicators and comparisons of living standards. J Dev Econ 70(2):501–529CrossRefGoogle Scholar
- Emond EJ, Mason DW (2002) A new rank correlation coefficient with application to the consensus ranking problem. J Multi-Criteria Decis Anal 11(1):17–28CrossRefGoogle Scholar
- Fagin R, Kumar R, Sivakumar D (2003) Comparing top k lists. SIAM J Discrete Math 17(1):134–160CrossRefGoogle Scholar
- Fields GS, Ok E (1996) The meaning and measurement of income mobility. J Econ Theory 71:349–77CrossRefGoogle Scholar
- Fields GS, Ok E (1999) The measurement of income mobility. In: Silber J (ed) Handbook of income inequality measurement. Kluwer Academic Publishers, DordrechtGoogle Scholar
- Foster JE, Shorrocks AF (1991) Subgroup consistent poverty indices. Econometrica 59:687–709CrossRefGoogle Scholar
- Jäntti M, Jenkins SP (1997) Income mobility. In: Atkinson AB, Bourguignon F (eds) Handbook on income distribution. Elsevier, AmsterdamGoogle Scholar
- Kemeny JG (1959) Mathematics without numbers. Daedalus 88(4):577–591Google Scholar
- Kemeny JG, Snell JL (1962) Mathematical models in the social sciences. Ginn Publishing Inc., BostonGoogle Scholar
- Kendall M, Gibbons JD (1990) Rank correlation methods. Edward Arnold, LondonGoogle Scholar
- Maasoumi E (1998) On mobility. In: Giles D, Ullah A (eds) The handbook of economic statistics. Marcel Dekker, New YorkGoogle Scholar
- Mitra T, Ok E (1998) The measurement of income mobility: a partial ordering approach. Econ Theory 12:77–102CrossRefGoogle Scholar
- Permanyer I (2012) Uncertainty and robustness in composite indices rankings. Oxf Econ Pap 64:57–79CrossRefGoogle Scholar
- Pečarić JE, Proschan F, Tong YL (1992) Convex functions, partial orderings and statistical applications. Academic Press Inc., New YorkGoogle Scholar
- Ruiz-Castillo J (2004) The measurement of structural and exchange mobility. J Econ Inequal 2:219–228CrossRefGoogle Scholar