Extending the Intersection Approach

ABSTRACT

The intersection approach is a common method of overcoming a conflict among multiple values. Under this approach, a state is more desirable than another if it is so for all criteria in question. A fundamental difficulty is that judgment under the intersection approach lacks completeness in too many cases. We propose alternative methods that extend the intersection approach: the union and union-intersection approaches. Our methods generate a (quasi-)coherence judgment which is more completed and can be applied to most problems of ethical indices.

KEYWORDS:

• Intersection approach
• Union approach
• Union-intersection approach
• Human development
• Ethical index
• Coherency

JEL classification:

• C65
• D63
• D71
• D82

Acknowledgements

I thank Takenori Konaka, Kotaro Suzumura, an associate editor, and two anonymous reviewers forhelpful comments and suggestions. This work was financially supported by JSPS KAKENHI (18K01501).We would like to thank Editage (www.editage.jp) for English language editing.

Disclosure statement

No potential conflict of interest was reported by the author.

About the Author

Susumu Cato is Associate Professor of Economics at the Institute of Social Science, the University of Tokyo. His areas of research include welfare economics, social choice, political philosophy, organizational economics, and history of economic thought. He was born in 1981. He is a visiting scholar of the Research Institute of Capital Formation, Development Bank of Japan.

Notes

1 See, for example, Blackorby and Donaldson (1977) and Kamaga (2018).

2 See also Fine (1975) and Blackorby (1975).

3 A function is a monotonic transformation if it is a composition of an increasing function and the individual utility function.

4 The limitation of the intersection approach in this direction has been discussed by Brun and Tungodden (2004) and Fleurbaey and Blanchet (2013). Herreroetal (1998) and Echavarri and Permanyer (2008) provide an analysis of ranking over capability profiles. They employ the common capability set, which is the intersection of individual capabilities. Their framework yields a way of constructing social evaluations based on the intersection.

5 See, for example, Fishburn (1979).

6 Sen (1970b) shows that acyclicity is a necessary and sufficient condition for the existence of a maximal element. Since transitivity is logically stronger than acyclicity, transitivity is not necessary for the non-emptiness of the set of maximal elements.

7 Plott (1973) provides an analysis of path-independent choices. Quasi-transitivity, which is weaker than transitivity, is sufficient for path-independence. See Cato (2016).

8 The existence of a compatible ordering is guaranteed by a result known as Szpilrajn's extension lemma. See Szpilrajn (1930) and Suzumura (1983).

9 Here a preferred alternative is put in the left. That is, “$R_1:x,y,z$” means that x is strictly better than y and y is strictly better than z for $R_1$.

10 See Cato (2018) for this observation.

11 See Sen (1970b) and Suzumura (1983).

12 For example, consider the following two criteria:(1) $R_1=\left\{\right(x,y),(y,z),(x,z\left)\right\}\cup \mathrm{\Delta };$(1) (2) $R_2=\left\{\right(y,z),(z,x),(y,x\left)\right\}\cup \mathrm{\Delta }.$(2) The first criterion says x is better than y, which is better than z; the second one says that y is better than z, which is better than x. The union suggests that x and y are indifferent, x and z are indifferent, but y is better than z. Thus, the union is not transitive.

13 We often employ indices with discrete outputs, such as counting-number procedures or yes-no questionnaires. The strict-union-intersection approach does not work for such indices and, thus, we need a general approach for criteria, including indifferences.

14 Suzumura consistency is introduced by Suzumura (1976). It is substantially developed by Bossert and Suzumura (2010).

15 Another weakening of transitivity is quasi-transitivity. As shown before, quasi-transitivity is guaranteed by the union approach. In general, Suzumura consistency is independent of quasi-transitivity.

16 As we will explain later in Section 5, each indicator comes from some original source.

17 To prove (i), consider the following:$\begin{array}{rl}& R_0=X\times X\\ & R_1=\left\{\right(x,y),(y,z),(x,z\left)\right\}\cup \mathrm{\Delta };\\ & R_2=\left\{\right(y,z),(z,x),(y,x\left)\right\}\cup \mathrm{\Delta }.\end{array}$Note that $\bigcup _{i=1,2}(R_0\cap R_i)$ is not Suzumura-consistent. To prove (ii), we add $R_3=X\times X$ to the previous example. It is easy to see that $\bigcup _{i=1,2,3}(R_0\cap R_i)$ is not compatible with $\bigcap _{i=0,1,2,3}{R}_i$.

18 We here note why our generalisation is needed. There are two fixes to the naive application of the union-intersection approach. First, $R_0$ is replaced by $P\left(R_0\right)$, and second, $\bigcap _{i\in S}{R}_i$ is added. The former is needed for Theorem 3 and the latter is needed for Theorem 4.

19 We ignore the diagonal Δ. That is, rigorously speaking, a resolution given by (3(1) $R_1=\left\{\right(x,y),(y,z),(x,z\left)\right\}\cup \mathrm{\Delta };$(1) ) is the same as that given by (4(2) $R_2=\left\{\right(y,z),(z,x),(y,x\left)\right\}\cup \mathrm{\Delta }.$(2) ) if we add Δ to the relation generated by (3(1) $R_1=\left\{\right(x,y),(y,z),(x,z\left)\right\}\cup \mathrm{\Delta };$(1) ).

20 We note that $R_S^{\text{β}}$ may not be an extension of $R_S^{\gamma \ast }$. That is, it might be the case where $P\left(R_S^{\gamma \ast }\right)⊈P\left(R_S^{\text{β}}\right)$. Then, a judgment under the union-intersection approach may not be compatible with the union approach.

21 Human Development Data Bank (http://hdr.undp.org/en/data) collects various indicators from other original sources. Most of the original data are available on websites. In the next three footnotes, we put information about the original sources of indicators that we use in this section.

22 For details and their original sources, see the Human development Report 2018 and Human Development Data Bank.

23 This process can be generalised for the case where there must be multiple reference criteria. First, we fix the number of reference criteria. Second, given the number of criteria, we identify the combination of criteria that maximises the number of comparable pairs of alternatives.