## Abstract

We axiomatize the class of mixed utilitarian–maximin social welfare orderings. These orderings are convex combinations of utilitarianism and the maximin rule. Our first step is to show that the conjunction of the weak Suppes–Sen principle, the Pigou–Dalton transfer principle, continuity and the composite transfer principle is equivalent to the existence of a continuous and monotone ordering of pairs of average and minimum utilities that can be used to rank utility vectors. Using this observation, the main result of the paper establishes that the utilitarian–maximin social welfare orderings are characterized by adding the axiom of cardinal full comparability. In addition, we examine the consequences of replacing cardinal full comparability with ratio-scale full comparability and translation-scale full comparability, respectively. We also discuss the classes of normative inequality measures corresponding to our social welfare orderings.

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## Author information

## Additional information

We are grateful to Geir B. Asheim, Kaname Miyagishima, Paolo G. Piacquadio, Marcus Pivato, Stéphane Zuber and three referees for their comments and suggestions. The paper was presented at the 14th Meeting of the Society for Social Choice and Welfare in Seoul, the 2018 Workshop of the Central European Program in Economic Theory in Udine, the Development Bank of Japan, Hitotsubashi University, Fukuoka University, Tohoku University and the University of Luxembourg. A preliminary version was prepared while Kamaga visited CREA at the University of Luxembourg. We acknowledge the financial support from the Fonds de Recherche sur la Société et la Culture of Québec and a Grant-in-Aid for Young Scientists (B) (No. 16K17090) from the Japan Society for the Promotion of Science.

## Appendix

### Appendix

To prove that the axioms used in Theorems , 3, 4 and 5 are independent, consider the following examples.

First, the social welfare ordering \(R=D\times D\) satisfies the axioms of Theorems , 3, 4 and 5 except for weak Suppes–Sen.

Second, define the ordering *R* on *D* as follows. For all \(x,y\in D\),

This social welfare ordering satisfies all of the required axioms except for the Pigou–Dalton transfer principle.

Third, the leximin social welfare ordering *R* on *D* satisfies the requisite axioms except for continuity.

Fourth, assume that \(n\ge 3\) and consider the generalized Gini \(R^{G}_\beta \) on *D* with \(\beta _{i}>\beta _{i+1}\) for all \(i \in \{1,\ldots ,n-1\}\). This social welfare ordering satisfies the axioms of Theorems , 3, 4 and 5 except for the composite transfer principle. (Note that the axiom is vacuous for \(n=2\) and, therefore, it is redundant in the two-agent case.)

Fifth, consider the restriction of \(R_{\psi }\) to \({\mathbb {R}}^{n}_{++}\) associated with \(\psi \in {\Psi }\) given by \(\psi (z)=\ln (z+1)\). This social welfare ordering on \({\mathbb {R}}^{n}_{++}\) satisfies the axioms in Theorem 4 except for ratio-scale full comparability.

Sixth, define the ordering *R* on \({\mathbb {R}}^{n}\) as follows. For all \(x,y\in {\mathbb {R}}^{n}\),

where the function \(g :{\mathbb {R}}\rightarrow {\mathbb {R}}\) is given by

This social welfare ordering satisfies the axioms in Theorem 5 except for translation-scale full comparability.

From Theorems 4 and 5, cardinal full comparability is independent of the other axioms in Theorem 3. Finally, the axioms of Theorem are independent because they constitute a subset of those in Theorem 3.

We conclude this appendix by showing that it is impossible to replace the composite transfer principle with its weak version defined by adding the restriction \(\delta \ge \varepsilon \) in Theorems , 3, 4 and 5. Suppose \(n=3\), and consider \(R^{G}_{\beta }\) on *D* associated with \((\beta _{1},\beta _{2},\beta _{3})=(2/3,1/3,0)\). To show that \(R^{G}_{\beta }\) satisfies the weak version of the composite transfer principle, let \(x,y\in {\mathbb {R}}^{3}\) and suppose that there exist \(i,j,k \in N\) and \(\delta ,\varepsilon \in {\mathbb {R}}_{++}\) with \(\delta \ge \varepsilon \) such that \(x_i = y_i + \delta \), \(x_j = y_j - \delta - \varepsilon \), \(x_k = y_k + \varepsilon \), \(x_i \le x_j < x_k\), \(y_i < y_j \le y_k\) and \(x_\ell = y_\ell \) for all \(\ell \in N {\setminus } \{i,j,k\}\). Then, we obtain \( \sum _{i=1}^{3}\beta _{i}x_{(i)}-\sum _{i=1}^{3}\beta _{i}y_{(i)}=2\delta /3-(\delta +\varepsilon )/3=(\delta -\varepsilon )/3\ge 0. \) Thus, \(xR^{G}_{\beta }y\) holds. Note that \(R^{G}_{\beta }\) satisfies all the axioms in Theorem 3 except for the composite transfer principle. It can be verified that \(R^{G}_{\beta }\) violates the composite transfer principle as follows. Consider \(x=(2,2,7),y=(1,5,5)\in {\mathbb {R}}^{3}_{++}\). Then, we obtain \(yP^{G}_{\beta }x\) since \(\sum _{i=1}^{3}\beta _{i}x_{(i)}-\sum _{i=1}^{3}\beta _{i}y_{(i)}=-1/3\). We note that it is straightforward to extend this example to higher values of *n*.

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Bossert, W., Kamaga, K. An axiomatization of the mixed utilitarian–maximin social welfare orderings.
*Econ Theory* **69, **451–473 (2020). https://doi.org/10.1007/s00199-018-1168-y

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### Keywords

- Social welfare ordering
- Utilitarianism
- Maximin principle
- Normative inequality index

### JEL Classification

- D63