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Fairness and Aggregation

  • A. C. PASEAU (a1) and BEN SAUNDERS (a2)


Sometimes, two unfair distributions cancel out in aggregate. Paradoxically, two distributions each of which is fair in isolation may give rise to aggregate unfairness. When assessing the fairness of distributions, it therefore matters whether we assess transactions piecemeal or focus only on the overall result. This piece illustrates these difficulties for two leading theories of fairness (proportionality and shortfall minimization) before offering a formal proof that no non-trivial theory guarantees aggregativity. This is not intended as a criticism of any particular theory, but as a datum that must be taken into account in constructing a theory of fairness.



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1 E.g. Geoffrey Cupit, ‘Fairness, by Nicholas Rescher [book review]’, Mind 113 (2002), pp. 387–90, at 389.

2 E.g. Brams, Steven J. and Taylor, Alan D., Fair Division: From Cake-cutting to Dispute Resolution (Cambridge, 1996).

3 A seminal paper on the bankruptcy (or conflicting claim) problem is Aumann, R. and Maschler, M., ‘Game Theoretic Analysis of a Bankruptcy Problem from the Talmud’, Journal of Economic Theory 36 (1985), pp. 195213. For a survey of the ensuing literature, see Thomson, W., ‘Axiomatic and Game-theoretic Analysis of Bankruptcy and Taxation Problems: A Survey’, Mathematical Social Sciences 45 (2003), pp. 249–97.

4 E.g. Reiff, Mark, ‘Proportionality, Winner-take-all and Distributive Justice’, Politics, Philosophy & Economics 8 (2009), pp. 542, at 11; Brown, Alexander, ‘Principles of Stakes Fairness in Sport’, Politics, Philosophy & Economics 14 (2015), pp. 152–86.

5 Broome, John, ‘Fairness’, Proceedings of the Aristotelian Society 91 (1990), pp. 87101.

6 E.g. Hooker, Brad, ‘Fairness’, Ethical Theory and Moral Practice 8 (2005), pp. 329–52; Saunders, Ben, ‘Fairness between Competing Claims’, Res Publica 16 (2010), pp. 4155; Tomlin, Patrick, ‘On Fairness and Claims’, Utilitas 24 (2012), pp. 200–13; Lazenby, Hugh, ‘Broome on Fairness and Lotteries’, Utilitas 26 (2014), pp. 331–45; and Kirkpatrick, James R. and Eastwood, Nick, ‘Broome's Theory of Fairness and the Problem of Quantifying the Strengths of Claims’, Utilitas 27 (2015), pp. 8291.

7 Broome, ‘Fairness’, p. 92.

8 Broome, ‘Fairness’, p. 93.

9 Shortfall minimization is discussed in Nicholas Rescher, Fairness (New Brunswick, NJ, 2002), pp. 52–3, and endorsed by Cupit, ‘Fairness [review]’.

10 Broome, ‘Fairness’, p. 95.

11 Stone, Peter, ‘Lotteries, Justice and Probability’, Journal of Theoretical Politics 21 (2009), pp. 395409, at 398.

12 Rawls, John, Political Liberalism (New York, 1993), pp. 265–71.

13 The authors share authorship and credit equally for this work. A predecessor of this article was presented at the Universities of Stirling (September 2012), St Andrews (September 2012) and Oxford (January 2013). The authors thank these audiences, along with John Broome, Hugo Dixon, Hugh Lazenby, Patrick Tomlin and two anonymous referees for the journal for their comments.

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Fairness and Aggregation

  • A. C. PASEAU (a1) and BEN SAUNDERS (a2)


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