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Published in: Social Choice and Welfare 3/2024

09-12-2023 | Original Paper

To be fair: claims have amounts and strengths

Author: Stefan Wintein

Published in: Social Choice and Welfare | Issue 3/2024

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Abstract

John Broome (Proc Aristot Soc 91:87–101, 1990) has developed an influential theory of fairness, which has generated a thriving debate about the nature of fairness. In its initial conception, Broomean fairness is limited to a comparative notion. More recent commentators such as Hooker (Ethical Theory Moral Pract 8:329–52, 2005), Saunders (Res Publica 16:41–55, 2010), Lazenby (Utilitas 26:331–345, 2014), Curtis (Analysis 74:47–57, 2014) have advocated, for different reasons, to also take into account non-comparative fairness. Curtis’ (Analysis 74:47-57, 2014) theory does just that. He also claims that he furthers Broome’s theory by saying precisely what one must do in order to be fair. However, Curtis departs from Broome’s (Proc Aristot Soc 91:87-101, 1990) requirement that claims are satisfied in proportion to their strength. He neglects claim-strengths altogether and identifies claims with their amount. As a result, the theory of Curtis has limited scope. I present a theory of fairness that fulfils all three desiderata: it incorporates non-comparative fairness, it recognizes that claims have both amounts and strengths, and it tells us precisely what one must do in order to be fair.

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Appendix
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Footnotes
1
See Hooker (2005), Saunders (2010), Tomlin (2012), Curtis (2014), Lazenby (2014), Henning (2015), Kirkpatrick and Eastwood (2015), Paseau and Saunders (2015), Vong (2015, 2018, 2020), Sharadin (2016), Heilmann and Wintein (2017), Piller (2017), Wintein and Heilmann (2018, 2020, 2021).
 
2
Broome contrasts claims with teleological reasons and side-constraints but does not offer a detailed account of the nature of claims. Hence, in this sense his theory of fairness is incomplete. However, as Piller (2017: 216) observes, “this incompleteness might not matter ...because we understand talk of claims pre-theoretically”.
 
3
Applying method P to Owing Money yields allocation (10, 30). Method L, which applies to Horses, is described in §2.
 
4
Curtis (2014: 55) writes that he departs from Broome in this respect for the same reason that Hooker (2005: 340–341) and Saunders (2010: 44-47) do.
 
5
Note that Anna has a claim to receive part of the £20K whereas Abram, in Owing Money, has a claim to receive all of the 20 ducats. As I will explain in Sect. 3.1, this means that Abram’s claim is absolute while Anna’s claim is notional.
 
6
This formalizes a concern for efficiency that has been raised in the literature, e.g. Hooker (2005), Saunders (2010), Lazenby (2014), and Curtis (2014).
 
7
Absolute fairness requires the satisfaction of absolute claims (cf. footnote 5) and justifies MFC(i). For Investing Time, MFC(i) is justified by the fact that, as explained in section 3.1, the group consisting of Anna and Beta has an absolute claim.
 
8
As we are assuming that the number \(\mid N \mid\) of receiving agents is greater-than-or equal to 2 and as, in this paper, claims strengths only figure in the requirements of comparative fairness, this assumption is justified for the situations that we consider in this paper.
 
9
Below, in the text following the Lottery Theorem, we elaborate on this interpretation.
 
10
Note that the alternative method is not a proper method in the sense that it does not prescribe how to allocate the estate in an arbitrary Broomean problem.
 
11
See Wintein and Heilmann (2024a) and Wintein and Heilmann (2022b)
 
12
E.g. Hooker (2005), Saunders (2010), Curtis (2014), Lazenby (2014), Vong (2018).
 
13
Vong uses ‘individual’ for what we label ‘absolute’.
 
14
We will delete the units (\(\pounds K\)) for the numbers in Investing Time from here on.
 
15
With fairness understood as having both a comparative and a non-comparative (absolute) dimension, the question arises what exactly distinguishes fairness from justice For, as Phillip Montague (1980: 131) writes “discussions of justice have traditionally focused on two very general kinds of concerns: [non-comparative and comparative ones]”. The interesting and complex question as to how fairness and justice relate I hope to address in depth on another occasion. For now, I just want to remark that my intuitions concerning this relation resemble those of Thomas Mulligan (2017: 106), who writes that “as best I can tell, justice and fairness are one and the same thing conceptually”.
 
16
For an in-depth discussion of the Fairness formula and its constitutive notions, I refer the reader to Wintein and Heilmann (2024a) and Wintein and Heilmann (2022a).
 
17
Thanks to an anonymous referee for raising this question.
 
18
See Wintein and Heilmann (2021) for a general account of how the philosophical literature on fairness relates to various economic literatures e.g. cooperative game theory, apportionment theory and that on bankruptcy problems.
 
19
Thomson (2019) presents an excellent and extensive overview of this literature, of which Thomson himself is one of the leading scholars.
 
20
For formal definitions and an in-depth comparison of these three rules see e.g. Herrero and Villar (2001) who jointly refer to the three rules as “The Three Musketeers”. The role of D’Artagnan, i.e. a fourth rule that is axiomatically related to the three rules but not part of the original three, is reserved for the “Talmud Rule” (Aumann and Maschler 1985).
 
21
When a matrix equation \(Ap = s\) is induced from a remainder problem \({\mathcal {B}}^\star\), the columns of A correspond to the integer allocations of \(\textbf{MaxSat}({\mathcal {B}}^\star )\) and both the sum of the entries of each column of A and the sum of the entries of s equal \(E^\star\). Thus, neither (4) nor (5) can be induced from a remainder problem \({\mathcal {B}}^\star\).
 
22
With A and s as in (4): A has rank 2 whereas the rank of (A|s) is 3.
 
23
It is not hard to show that the rank of the \(|N^\star | \times m\) matrix A of an instantiation of (\(\star\)) equals \(|N^\star |\) from which, as \(|N^\star | \le m\), it follows that for any \(s \in {\mathbb {R}}^n\) the rank of (A|s) equals \(|N^\star |\) as well.
 
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Metadata
Title
To be fair: claims have amounts and strengths
Author
Stefan Wintein
Publication date
09-12-2023
Publisher
Springer Berlin Heidelberg
Published in
Social Choice and Welfare / Issue 3/2024
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI
https://doi.org/10.1007/s00355-023-01494-y

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