# How to rank rankings? Group performance in multiple-prize contests

- 86 Downloads

## Abstract

When groups of individuals compete in several multiple-prize contests, the performance of a group is a vector of ordered categories. As the prizes are aimed at ranking the participants, group performances are not trivially comparable. This note provides a theoretical discussion on how to rank group performances. In order to do so, I draw from the parallel that this problem has with the formally similar problem of measuring inequality. I describe three alternatives that generate partial orders for group performances. I define partial orders based on the first- and second-order dominance, two classes of performance measures, and two sets of basic transformations, and I prove equivalence theorems between them. I apply these theoretical results to discuss several sports ranking problems.

## Notes

### Acknowledgements

I would like to thank Juan Pablo Torres for his valuable feedback. I acknowledge financial support from the Institute for Research in Market Imperfections and Public Policy, ICM IS130002, Ministerio de Economía, Fomento y Turismo.

## References

- Abul Naga R, Yalcin T (2008) Inequality measurement for ordered response health data. J Health Econ 27(6):1614–1625CrossRefGoogle Scholar
- Allison R, Foster J (2004) Measuring health inequality using qualitative data. J Health Econ 23:505–524CrossRefGoogle Scholar
- Atkinson A (1970) On the measurement of inequality. J Econ Theory 2:244–263CrossRefGoogle Scholar
- Bazen S, Moyes P (2012) Elitism and stochastic dominance. Soc Choice Welfare 39(1):207–251CrossRefGoogle Scholar
- Chakravarty S, Zoli C (2012) Stochastic dominance relations for integer variables. J Econ Theory 147(4):1331–1341CrossRefGoogle Scholar
- Dalton H (1920) The measurement of the inequality of incomes. Econ J 119(30):348–361CrossRefGoogle Scholar
- Dasgupta P, Sen A, Starrett D (1973) Notes on the measurement of inequality. J Econ Theory 2(6):180–187CrossRefGoogle Scholar
- Dutta I, Foster J (2013) Inequality of hapiness in the US: 1972–2010. Rev Income Wealth 59(3):393–415CrossRefGoogle Scholar
- Fishburn PC, Lavalle IH (1995) Stochastic dominance on unidimensional grids. Math Oper Res 20(3):513–525CrossRefGoogle Scholar
- Gravel N, Magdalou B, Moyes P (2014) Ranking distributions of an ordinal attribute. In: AMSE working paper , vol 50. Aix-Marseille School of EconomicsGoogle Scholar
- Hardy G, Littlewood J, Polya G (1934) Inequalities. Cambridge University Press, CambridgeGoogle Scholar
- International Olympic Committee (2015) Olympic Charter. IOC, Lausanne, SwitzerlandGoogle Scholar
- Kolm S-C (1976) Unequal inequalities II. J Econ Theory 13(1):82–111CrossRefGoogle Scholar
- Konrad K A (2009) Strategy and dynamics in contests. Oxford University Press, OxfordGoogle Scholar
- Lins MPE, Gomes EG, de Mello JCCS, de Mello AJRS (2003) Olympic ranking based on a zero sum gains DEA model. Eur J Oper Res 148(2):312–322CrossRefGoogle Scholar
- Lorenz MO (1905) Methods of measuring the concentration of wealth. Publ Am Stat Assoc 70(9):209–219Google Scholar
- Moldovanu B, Sela A (2001) The optimal allocation of prizes in contests. Am Econ Rev 91(3):542–558CrossRefGoogle Scholar
- Rothschild M, Stiglitz JE (1973) Some further results on the measurement of inequality. J Econ Theory 2(6):188–204CrossRefGoogle Scholar
- Saaty TL (2008) Who won the 2008 Olympics? A multicriteria decision of measuring intangibles. J Syst Sci Syst Eng 17(4):473–486CrossRefGoogle Scholar
- Shorrocks A, Foster JE (1987) Transfer sensitive inequality measures. Rev Econ Stud 54(3):485–497CrossRefGoogle Scholar
- Sisak D (2009) Multiple-prize contests: the optimal allocation of prizes. J Econ Surv 23(1):82–114CrossRefGoogle Scholar
- Sitarz S (2013) The medal points’ incenter for rankings in sport. Appl Math Lett 26(4):408–412CrossRefGoogle Scholar