An axiomatization of the mixed utilitarian-maximin social welfare orderings

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1 An axiomatization of the mixed utilitarian-maximin social welfare orderings Walter Bossert and Kohei Kamaga November 16, 2018 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 ratioscale full comparability and translation-scale full comparability, respectively. We also discuss the classes of normative inequality measures corresponding to our social welfare orderings. Journal of Economic Literature Classification No.: D63. Keywords: Social welfare ordering, Utilitarianism, Maximin principle, Normative inequality index 1 Introduction The utilitarian social welfare ordering and the maximin principle are firmly rooted in moral philosophy. Utilitarianism is a way of formalizing Bentham s (1789) principle of the greatest happiness of the greatest number, and it ranks utility vectors by comparing total utilities or, equivalently, average utilities in a fixed population framework. An early axiomatic characterization of utilitarianism is presented by d Aspremont and Gevers (1977). The maximin social welfare ordering is the 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 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. Department of Economics and CIREQ, University of Montreal, Canada. walter.bossert@videotron.ca Faculty of Economics, Sophia University, Tokyo, Japan. kohei.kamaga@sophia.ac.jp 1

2 utility analogue of Rawls s (1971) maximin principle who advocates this criterion in the context of primary goods rather than well-being. Maximin ranks utility distributions by comparing their respective minimum utilities, and the first axiomatization of this social welfare ordering can be found in Strasnick (1976). There is a stark contrast between utilitarianism and maximin when it comes to their attitude towards distributional equity. While the utilitarian social welfare ordering is neutral with respect to utility inequality (see, for instance, Blackorby, Bossert and Donaldson, 2002), the maximin principle is extremely inequality-averse by paying attention to the worst-off individual only (see Bosmans and Ooghe, 2013, Miyagishima, 2010, and Miyagishima, Bosmans and Ooghe, 2014, for example). The two orderings do, however, share an invariance property with regard to the measurability and interpersonal comparability of utilities specifically, both are compatible with cardinally measurable and interpersonally fully comparable utilities; see Hammond (1976) and Maskin (1978). More on utilitarianism and maximin in the context of information-invariance assumptions can be found in Blackorby, Bossert and Donaldson (2005), Bossert and Weymark (2004) and d Aspremont and Gevers (2002). See also a recent contribution by Ou-Yang (2018). In his analysis of social welfare orderings that are compatible with cardinally measurable and interpersonally fully comparable utilities, Roberts (1980) suggests an ordering that is a possible compromise between utilitarianism and maximin. The social welfare ordering proposed by Roberts ranks utility vectors by comparing convex combinations of average and minimum utilities. Alvarez- Cuadrad and Long (2009) refer to such an ordering as a mixed Bentham-Rawls social welfare ordering. They use an infinite-horizon extension of Roberts s ordering their mixed Bentham-Rawls criterion, which is formulated as a convex combination of discounted utilitarianism and the infimum criterion. On this latter point, see also Figuières, Long and Tidball (2017), Long (2007), Long and Martinet (2018) and Tol (2013). Each member of this class is identified by the relative weights used in the requisite convex combination. Each mixed utilitarian-maximin social welfare ordering is a special case of the generalized Gini social welfare orderings introduced by Mehran (1976) and Weymark (1981). A generalized Gini ranks utility vectors by comparing weighted utility averages, where the weights are rank-dependent and larger weights are given to lower utility levels in order to be compatible with the well-known Pigou-Dalton transfer principle. Among the generalized Gini social welfare orderings, the mixed utilitarian-maximin orderings have the distinguishing feature that they are monotone with respect to the evaluations of the utilitarian and the maximin criteria. The main purpose of this paper is to present a characterization of the mixed utilitarian-maximin social welfare orderings. We use the weak Suppes-Sen principle, the Pigou-Dalton transfer principle, continuity and the composite transfer principle, in addition to the information-invariance property of cardinal full comparability. Weak Suppes-Sen guarantees that the social evaluation respects efficiency and impartiality. The Pigou-Dalton transfer principle is the well-established distributional-equity property that is based on rank-preserving progressive utility transfers. Continuity ensures that small changes in individual utilities do not lead to large changes in the social 2

3 ranking. The composite transfer principle is another distributional equity property and it refers to the consequences of a composition of rank-preserving progressive and regressive utility transfers. Except for the composite transfer principle, these properties are well-established and do not require much discussion. The axiom that is crucial in distinguishing the class of mixed utilitarian-maximin orderings from other social welfare orderings is the composite transfer principle, and we provide a detailed explanation of this property after introducing its formal definition. Deschamps and Gevers (1978) provide a joint characterization of utilitarianism and leximin the lexicographic extension of maximin. Our approach differs from theirs in that we impose the weak variant of the Suppes-Sen principle as compared to their use of anonymity and strong Pareto. Moreover, their minimal-equity axiom is absent from our setting and, instead of their separability property, we employ the composite transfer principle. Ours and their contribution share the information-invariance assumption of cardinal full comparability. Our first result shows that the conjunction of the above axioms except for cardinal full comparability is equivalent to the existence of a continuous and monotone ordering defined on pairs of average and minimum utilities that can be used to rank any two utility vectors. In other words, these four axioms together restrict the informational basis of evaluation to average and minimum utilities. Using this result, we then proceed to our axiomatization of the mixed utilitarian-maximin social welfare orderings by adding cardinal full comparability to the list of axioms. In addition, we examine ratio-scale invariance and translation-scale invariance as alternative informational assumptions, and we discuss the normative inequality measures that correspond to our classes of social welfare orderings. Section 2 presents the basic notation and definitions employed in the paper. In Section 3, we provide a characterization of the mixed utilitarian-maximin social welfare orderings. Section 4 explores ratio-scale invariant and translation-scale invariant generalizations of our orderings, and Section 5 concludes. The independence of the axioms used in our axiomatizations is established in the appendix, which also contains an example showing that one of our axioms cannot be weakened in a particular fashion. 2 Notation and definitions 2.1 Preliminaries Let R be the set of all real numbers and R + (respectively R ++ ) be the set of all non-negative (respectively positive) real numbers. The set of all positive integers is denoted by N. Our notation for vector inequalities is given by the symbols, > and. The symbols and are used for weak and strict set inclusion. Let N = {1,..., n} with n 2 be the set of all individuals. The set of all possible utility vectors for N is the n-dimensional Euclidean space R n. A typical element of R n is denoted by x = (x 1,..., x n ) where x i is the utility level of individual i N. For each x R n, x ( ) = (x (1),..., x (n) ) 3

4 denotes a non-decreasing rearrangement of x, ties being broken arbitrarily. The arithmetic mean of x R n is denoted by µ(x). The origin of R n is 0 n, and 1 n denotes the n-dimensional vector that consists of n ones. For any x R n, k N and c R, the vector y = (x k, c) is defined by letting y i = x i for all i N \ {k} and y k = c. A binary relation R on D R n is a subset of D D. For simplicity, we write xry for (x, y) R. The asymmetric and symmetric parts of R are denoted by P and I. A social welfare ordering on D is a complete and transitive binary relation. We consider the two domains D = R n and D = R n + \ {0 n }. Note that µ(x) > 0 for all x D in the latter case. 2.2 Mixed utilitarian-maximin social welfare orderings A mixed utilitarian-maximin social welfare ordering is a convex combination of utilitarianism and the maximin principle, first introduced by Roberts (1980, pp ). Given a weight α [0, 1], the mixed utilitarian-maximin social welfare ordering on D associated with α is defined as the following binary relation R UM α. For all x, y D, xr UM α y αµ(x) + (1 α)x (1) αµ(y) + (1 α)y (1). is repre- For any α [0, 1], the associated mixed utilitarian-maximin social welfare ordering R UM α sentable by the continuous social welfare function Ξ UM α : D R given by Ξ UM α (x) = αµ(x) + (1 α)x (1) for all x D. Ξ UM α (x) coincides with the representative utility corresponding to x according to the ordering R UM α whenever this specific utility level is well-defined. That is, except for the cases in which Ξ UM α (x) D, Ξ UM α (x) is implicitly defined by ( Ξ UM α (x),..., Ξ UM α (x) ) Iα UM x. Note that the only circumstances in which this representative utility is not defined are those where α = 0, D = R n + \ {0 n } and x i = 0 for some i N. The mixed utilitarian-maximin social welfare orderings constitute a class of orderings, one for each α [0, 1]. The special cases of utilitarianism and of maximin are obtained for α = 1 and for α = 0. Moreover, any mixed utilitarian-maximin social welfare ordering is a generalized Gini; see Mehran (1976) and Weymark (1981). A social welfare ordering R is a generalized Gini ordering if there exists a vector β = (β 1,..., β n ) of parameters in the n-dimensional unit simplex with β 1 > 0 and β i β i+1 for all i N \ {n} such that xry n β i x (i) i=1 n β i y (i). i=1 4

5 The generalized Gini social welfare ordering associated with the parameter vector β is denoted by R G β. It is straightforward to verify that, for β = (α/n + (1 α), α/n,..., α/n), it follows that R UM α = R G β. A feature that distinguishes the mixed utilitarian-maximin social welfare orderings from the other generalized Ginis is that they possess a monotonicity property with respect to the evaluations generated by the utilitarian and the maximin orderings. Let R U and R M denote the utilitarian and the maximin social welfare orderings, that is, for all x, y D, xr U y µ(x) µ(y) and xr M y x (1) y (1). A social welfare ordering R is UM-monotone if R U R M R and P U P M P. The following result shows that the mixed utilitarian-maximin social welfare orderings are the only generalized Ginis that respect the agreement of utilitarianism and maximin as defined in our UM-monotonicity property. Theorem 1. A generalized Gini social welfare ordering R G β exists α [0, 1] such that R UM α = R G β. is UM-monotone if and only if there Proof. If. We show that R UM α R U R M R UM α y (1) ) 0, it follows that xr UM α is UM-monotone for any choice of α [0, 1]. We first show that. Suppose that x, y R n, xr U y and xr M y. Because α(µ(x) µ(y)) + (1 α)(x (1) y. To prove that P U P M P UM α, suppose that x, y R n, xp U y and xp M y. Because α(µ(x) µ(y)) + (1 α)(x (1) y (1) ) > 0, we obtain xp UM α y. Only if. By way of contraposition, suppose there is no α [0, 1] such that R G β = RUM α. Then there exist i, j N with 1 < i < j such that the weights (β 1,..., β n ) satisfy β 1 > β i > β j 0. To show that R G β is not UM-monotone, consider x, y Rn + \ {0 n } such that x (k) = y (k) for all k N \ {1, i, j} and x (1) = 1 + (β i β j )/β 1, x (i) = 2, x ( j) = 5, y (1) = 1, y (i) = 3, y ( j) = 4. It follows that 0 < (β i β j )/β 1 < 1 and, thus, xp U y and xp M y. But we also have xiβ G y because i N β i (x (i) y (i) ) = 0. Thus, R G β is not UM-monotone. 5

6 2.3 Normative inequality indices Inequality indices can be derived from social welfare functions by using average utility and the representative-utility function. The prevalent method of obtaining a relative measure of inequality is that of Atkinson (1970), Kolm (1969) and Sen (1973). In the absolute case, the approach of Kolm (1969) and of Blackorby and Donaldson (1980) can be employed. A convex combination of the two appears in Bossert and Pfingsten (1990) but our focus here is on the relative and absolute variants. The Atkinson-Kolm-Sen inequality index measures utility inequality as the percentage shortfall of representative utility from average utility. For the mixed utilitarian-maximin social welfare ordering R UM α, the corresponding relative inequality index Jα RUM : R n + \ {0 n } R + is defined by Jα RUM (x) = µ(x) ΞUM α (x) = (1 α) µ(x) x (1) µ(x) µ(x) for all x R n + \ {0 n }. This index can be interpreted as the relative variant of the absolute meanmin index mentioned in Chakravarty (2010, p. 34), adjusted by the multiplicative constant (1 α). Chakravarty (2010) refers to this measure as the absolute maximin index but we prefer to use the term mean-min because the maximum level of utility does not appear in its definition. The Kolm-Blackorby-Donaldson normative inequality index is defined as the absolute shortfall of representative utility from average utility. In the case of the mixed utilitarian-maximin social welfare ordering R UM α, the absolute inequality index Jα AUM : R n R + is given by Jα AUM (x) = µ(x) Ξ UM α (x) = (1 α)(µ(x) x (1) ) for all x R n. Both of these normative indices can also be viewed as special cases of the relative and absolute measures corresponding to the generalized Ginis. The relative inequality measures defined above are ordinally equivalent across different values of the parameter α, and the same comment applies to the absolute class. As to possible characterizations of these classes of inequality measures, we note that this may be achieved following the method employed by Kolm (1976) in his characterizations of two alternative classes of relative and absolute indices. In each case, Kolm (1976, p. 426) defines a separability property that implicitly applies to the social welfare ordering corresponding to the requisite class, which permits him to obtain a characterization of the inequality measures which are not separable themselves. An analogous procedure seems to be applicable in our setting as well. 3 An axiomatization of mixed utilitarianism-maximin In this section, we prove our main result. As a preliminary observation, we characterize the class of social welfare orderings that satisfy the following four conditions. Our first axiom is a weak variant of the well-known Suppes-Sen grading principle (see Suppes, 6

7 δ δ+ε ε y i x i x j y j y k x k Figure 1: Utility transfers in the composite transfer principle 1966, and Sen, 1970) which appears in d Aspremont and Gevers (2002, p. 504). It encompasses an efficiency property and an impartiality requirement but is weaker than the conjunction of weak Pareto and anonymity. Weak Suppes-Sen: For all x, y D, if x ( ) y ( ), then xpy. The Suppes-Sen grading principle is obtained if the inequality is replaced with > in this definition. A standard axiom is the (weak) Pigou-Dalton transfer principle. It demands that a rank-preserving progressive utility transfer does not decrease social welfare; see Pigou (1912) and Dalton (1920). Pigou-Dalton transfer principle: For all x, y D, if there exist i, j N and δ R ++ such that x i = y i δ y j + δ = x j and x k = y k for all k N \ {i, j}, then xry. The third axiom is another well-established property. Continuity requires that small changes in utilities do not lead to large changes in social welfare. Continuity: For all x D, the sets {y D : yrx} and {y D : xry} are closed in D. All of the above properties are well-established in the literature. They are widely considered to be rather innocuous and uncontroversial. The following composite transfer principle is introduced in Kamaga (2018). In analogy to the transfer sensitivity axiom in Shorrocks and Foster (1987), it asserts that a composition of progressive and regressive utility transfers involving three individuals does not decrease social welfare as long as the relative ranking of the three individuals is preserved and the progressive transfer involves lower utilities than the regressive transfer. See Figure 1 for a diagrammatic illustration of the property. Composite transfer principle: For all x, y D, if there exist i, j, k N and δ, ε R ++ such that x i = y i + δ, x j = y j δ ε, x k = y k + ε, x i x j < x k, y i < y j y k and x l = y l for all l N \ {i, j, k}, then xry. This axiom is, of course, the property that sets the class of mixed utilitarian-maximin orderings apart from the (quite large) class of social welfare orderings that meet the remaining requirements. The use of this principle can be motivated quite naturally because it is implied by the conjunction of strong Pareto and Hammond equity; see Hammond (1979, p. 1132). These two properties are defined as follows. 7

8 Strong Pareto: For all x, y D, if x y, then xry and if x > y, then xpy. Hammond equity: For all x, y D, if there exist i, j N such that y i < x i < x j < y j and x k = y k for all k N \ {i, j}, then xry. To show that the composite transfer principle is implied by the conjunction of these two properties, suppose that x, y D, i, j, k N and δ, ε R ++ are such that x i = y i + δ, x j = y j δ ε, x k = y k + ε, x i x j < x k, y i < y j y k and x l = y l for all l N \ {i, j, k}. By strong Pareto, xp(x k, y k ) and by Hammond equity, (x k, y k )Ry so that, by transitivity, xpy and hence xry. This observation provides a strong case in favor of the composite transfer principle because Hammond equity is a widely accepted property. Note that, even though we do not employ strong Pareto in our main result (Theorem 3 below), we discuss the consequences of strengthening weak Suppes-Sen to the conjunction of strong Pareto and anonymity after the statement of our axiomatization. Thus, the implication just established is of even more relevance for the resulting characterization of a subclass of the mixed utilitarian-maximin orderings. Further, it should be noted that, as the resulting characterization will show, the composite transfer principle is compatible with the conjunction of strong Pareto and continuity. This contrasts with the incompatibility between Hammond equity and the two properties. To show their incompatibility, suppose that x, y D and i, j N are such that y i = x i < x j < y j and x k = y k for all k N \ {i, j}. Letting x m m N be the sequence in D defined by x m = (x i, xi m) and xm i = x i +(x j x i )/(2m) for all m N, it follows from Hammond equity that x m Ry for all m N, so that xry by continuity, whereas ypx by strong Pareto. Therefore, the composite transfer principle is an important weakening of Hammond equity when we explore strongly Paretian and continuous (and thus, representable) social welfare orderings. To provide a further illustration of the principle, observe that there is a close analogy between a composite transfer involving three individuals and a combination of a rich-to-poorer transfer and a poor-to-richer transfer as alluded to by Cowell (1985). As Cowell (1985, p. 568) notes, to go beyond the implications of the Pigou-Dalton transfer principle, some kind of weighting must be imposed on top-end transfers as against bottom-end transfers. The composite transfer principle proposes to always resolve the conflicts arising in such situations by ensuring that the (progressive) rich-to-poorer transfer does not outweigh the (regressive) poor-to-richer transfer. Interpreted in this way, our axiom is akin to a consistency property: changes in the details involving the composite transfers do not change the direction of the effect the combination of the two transfers has on the social welfare ordering. One may want to consider a weaker version of the composite transfer principle that is obtained by adding the restriction that the amount of a progressive utility transfer δ must be equal to or greater than that of a regressive transfer ε. However, as we show in the appendix, it is impossible to use this weaker axiom instead of the composite transfer principle to obtain our axiomatization results. In conjunction with our other axioms, the composite transfer principle is equivalent to the requirement that a regressive transfer that leaves minimum utility unchanged does not decrease social welfare. We choose the above-defined variant because it allows to draw parallels with the con- 8

9 tribution of Kamaga (2018) but note that the alternative formulation just described could be used instead. The composite transfer principle focuses on what Cowell (1985, p. 569) refers to as type 1 questions because it addresses the relative effect of transfers that leave total utility unchanged. Axioms such as Hammond equity, on the other hand, deal with type 2 questions because they involve transfers that may change total utility. It may be of interest to note that our axiomatization employs an axiom (the composite transfer principle) that is of the type 1 variety, which contrasts with the characterizations of maximin or leximin that rely on type 2 considerations. To characterize the class of social welfare orderings satisfying these four axioms, we need some additional notation and definitions. Let X = {(x 1, x 2 ) R 2 : x 1 x 2 } and, analogously, X + = {(x 1, x 2 ) X : x 1 > 0 and x 2 0}. Note that, for any x R n, we have (µ(x), x (1) ) X. Furthermore, for any x X, there exists x R n such that (µ(x), x (1) ) = x. The same relationship holds between vectors in R n + \ {0 n } and X +. Let D = X or D = X +. Clearly, D = X corresponds to the domain D = R n, and D = X + applies to the case D = R n + \ {0 n }. A binary relation R on D is monotone if for all x, y D, x y implies xp y and x y implies xr y. The relation R is continuous if, for any x D, the sets {y D : yr x} and {y D : xr y} are closed in D. The following theorem shows that a social welfare ordering satisfying the four axioms presented above can be represented by an ordering on pairs of average and minimum utilities. In other words, the information we can utilize for comparing utility vectors is limited to average and minimum utilities. Theorem 2. A social welfare ordering R on D satisfies weak Suppes-Sen, the Pigou-Dalton transfer principle, continuity and the composite transfer principle if and only if there exists a continuous and monotone ordering R on D such that, for all x, y D, xry ( µ(x), x (1) )R (µ(y), y (1) ). (1) Proof. We only present the proof for the case in which D = R n and hence D = X; the proof for the domain D = R n + \ {0 n } and the associated set D = X + is analogous. If. Since R is a monotone ordering satisfying (1), it is immediate that R satisfies weak Suppes- Sen, the Pigou-Dalton transfer principle and the composite transfer principle. To show that R satisfies continuity, let x R n and consider any sequence y m m N in {y R n : yrx} that converges to y R n. Because the functions f (y) = µ(y) and g(y) = y (1) defined on R n are continuous, the sequence (µ(y m ), y m (1) ) m N converges to (µ(y ), y (1) ). By (1), (µ(ym ), y m (1) )R (µ(x), x (1) ) for all m N. Since R is continuous, we obtain (µ(y ), y (1) )R (µ(x), x (1) ) and, using (1) again, y Rx follows. Thus, 9

10 {y R n : yrx} is closed. The proof that the set {y R n : xry} is closed is identical. Only if. Step 1. We show that, for all x, y R n, if µ(x) > µ(y) and x (1) > y (1), then xpy. Let x, y R n and suppose that µ(x) > µ(y) and x (1) > y (1). By weak Suppes-Sen and continuity, it follows that xiy for all x, y R n such that x (i) = y (i) for all i N. Thus, we can without loss of generality assume that x i = x (i) and y i = y (i) for all i N. We now distinguish two cases. (i) n = 2. If x 2 > y 2, we obtain xpy by weak Suppes-Sen. Now suppose that x 2 y 2. Let δ = x 1 y 1 and ε = y 2 x 2. Note that δ > ε since µ(x) > µ(y). Consider z = (y 1 +(δ ε)/2, y 2 +(δ ε)/2). By the Pigou-Dalton transfer principle, we obtain xrz. Weak Suppes-Sen implies zpy and, because R is transitive, it follows that xpy. (ii) n 3. Let ε = (x 1 y 1 )/(n 1) and δ (0, ε). We define the vectors z 1,..., z n 1 R n as follows. Let z 1 = y and, for all t {2,..., n 1}, define z t by z t i = zt 1 i z t 1 for all i N \ {1, t, n}, z t 1 = 1 if x t > y t z t δ if x t y t, z t 1 z t t = t (= y t ) if x t > y t x t ε if x t y t, z t z t 1 n if x t > y t n = z t 1 n + y t x t + ε δ if x t y t. By the composite transfer principle and the reflexivity of R, we obtain that, for all t {2,..., n 1}, z t Rz t 1 if x t y t and z t Iz t 1 if x t > y t. Since R is transitive, z n 1 Ry follows. Note that, by the definition of the vectors z 1,..., z n 1, we obtain x i > z n 1 i If x n > z n 1 n for all i n and n i=1 z n 1 i = n i=1 y i., then xpz n 1 by weak Suppes-Sen. Since R is transitive, we obtain xpy. Now suppose that x n z n 1 n. Let = z n 1 n x n and r i = (x i z n 1 i )/ n 1 j=1 (x j z n 1 j ) for all i n. Note that r i > 0 for all i n and n 1 i=1 r i = 1. We define the vectors w n, w n 1,..., w 1 R n as follows. Let w n = z n 1 and, for all t {1,..., n 1}, define w t by w t i = wt+1 i for all i N\{t, n}, w t t = wt+1 t + r t, w t n = w t+1 n r t. Note that, for all t {1,..., n 1}, it follows that w t t < w t n since w t t < x t x n w t n. For all t {1,..., n 1}, we obtain w t Rw t+1 by the Pigou-Dalton transfer principle if > 0 and w t Iw t+1 by the reflexivity of R if = 0. Since R is transitive, we obtain w 1 Rz n 1. Finally, we define w R n by w i = w 1 i for all i N\{n 1, n}, w n 1 = w 1 n 1 + (x n 1 w 1 n 1 )/2, w n = w 1 n (x n 1 w 1 n 1 )/2. 10

11 By definition of the vectors w n, w n 1,..., w 1, it follows that w 1 n 1 < x n 1 x n = w 1 n. Furthermore, x i > w i for all i N since w 1 i < x i for all i n. Thus, we obtain wrw 1 by the Pigou-Dalton transfer principle and xpw by weak Suppes-Sen. Recalling that z n 1 Ry, we obtain xpy since R is transitive. Step 2. Since R is continuous, it follows from Step 1 that, for all x, y R n, [µ(x) = µ(y) and x (1) = y (1) ] xiy. (2) We define the binary relation R on X as follows: for all x, ȳ X, xr ȳ if and only if there exist x, y R n such that x = (µ(x), x (1) ), ȳ = (µ(y), y (1) ), and xry. By (2) and the transitivity of R, R satisfies (1) for all x, y R n. We first show that R is an ordering on X. To prove that R is complete, suppose that x, ȳ X. Let x, y R n be such that x = (µ(x), x (1) ) and ȳ = (µ(y), y (1) ). Since R is complete, we obtain xry or yrx. By (1), we obtain xr ȳ or ȳr x. Next, we show that R is transitive. Let x, ȳ, z X and suppose that xr ȳ and ȳr z. Let x, y, z R n be such that x = (µ(x), x (1) ), ȳ = (µ(y), y (1) ) and z = (µ(z), z (1) ). By (1), xry and yrz. Because R is transitive, we obtain xrz and, by (1), it follows that xr z. The next property to be established is the continuity of R. Let x X and consider any sequence ȳ m m N in {ȳ X : ȳr x} that converges to ȳ X. We define x, y R n by x n = x 2, y n = ȳ 2 and, for all i n, x i = (n x 1 x 2 )/(n 1) and yi = (nȳ 1 ȳ 2 )/(n 1). Furthermore, define the sequence y m m N in R n as follows: for all m N, y m n = ȳ m 2 and ym i = (nȳ m 1 ȳm 2 )/(n 1) for all i n. It follows that (µ(x), x (1) ) = x, (µ(y ), y (1) ) = ȳ, and (µ(y m ), y m (1) ) = ȳm for all m N. By (1), y m m N is a sequence in {y R n : yrx}. Since ȳ m m N converges (coordinate-by-coordinate) to ȳ, y m m N converges to y. Because R is continuous, it follows that y Rx. By (1), we obtain ȳ R x, which means that {ȳ X : ȳr x} is closed. The proof that {ȳ X : xr ȳ} is closed is analogous. Finally, we show that R is monotone. Since R satisfies (1), it follows from Step 1 that, for all x, ȳ X, xp ȳ if x i > ȳ i for all i {1, 2}. To complete the proof, let x, ȳ X and suppose that x i ȳ i for all i {1, 2}. Suppose, by way of contradiction, that ȳp x. Since R is complete and continuous, the set L { z X : ȳp z} is open and it follows that x L. Thus, there exists ε R ++ such that B ε ( x) L, where B ε ( x) is the open ball with center at x and radius ε. Define z X by z i = x i + ε/2 for all i {1, 2}. By definition, z i > ȳ i for all i {1, 2} and, thus, it follows that zp ȳ. This is a contradiction since z B ε ( x) L. The final axiom to be used in our main result is cardinal full comparability. This well-established information-invariance property postulates that utilities are cardinally measurable and interpersonally fully comparable. Cardinal full comparability: For all x, y, x, ȳ D, if there exist a R ++ and b R such that 11

12 x i = ax i + b and ȳ i = ay i + b for all i N, then xry xrȳ. Our axiomatization establishes that the conjunction of the five axioms introduced in this section characterizes the utilitarian-maximin social welfare orderings. Thus, adding cardinal full comparability to the axioms of Theorem 2 implies that the level sets of R must be linear. Theorem 3. A social welfare ordering R on D satisfies weak Suppes-Sen, the Pigou-Dalton transfer principle, continuity, the composite transfer principle and cardinal full comparability if and only if there exists α [0, 1] such that R = R UM α. Proof. If. This part of the proof is straightforward and we do not state it explicitly. Only if. Assume first that D = R n + \ {0 n }. By Theorem 2, there exists a continuous and monotone ordering R on X + that satisfies (1). Since R satisfies cardinal full comparability, R is invariant with respect to common increasing affine transformations. That is, for all x, y, x, ȳ X +, if there exist a R ++ and b R such that x i = ax i + b and ȳ i = ay i + b for all i {1, 2}, then xr y xr ȳ. (3) Let z {x X + \ {1 2 } : x 1 1 and x 2 1} be such that zi 1 2 ; such a point exists because R is continuous and monotone. Furthermore, let t (0, 1). Now define ẑ = tz + (1 t)1 2. We show that ẑi 1 2. By (3), zi 1 2 implies tzi t1 2. By definition, ẑ i = tz i + 1 t for all i {1, 2}. Furthermore, tz i = ẑ i (1 t) and t1 2 i = 1 2 i (1 t) for all i {1, 2}. Thus, by (3), tzi t1 2 implies ẑi 1 2. Next, let ž be such that z = tž + (1 t)1 2. We show that ži 1 2. By (3), zi 1 2 implies (1/t)zI (1/t)1 2. Note that (1/t)z i = ž i + (1 t)/t and (1/t)1 2 i = 1 2 i + (1 t)/t for all i {1, 2}. Thus, by (3), (1/t)zI (1/t)1 2 implies ži 1 2. By the transitivity of R, we obtain ẑi ž. Since z {x X + \ {1 2 } : x 1 1 and x 2 1} and t (0, 1) was arbitrarily chosen, there exists α [0, 1] such that xi y for all x, y X + with αx 1 + (1 α)x 2 = αy 1 + (1 α)y 2 = 1. By (3), it follows that, for all a R ++, xi y for all x, y X + with αx 1 + (1 α)x 2 = αy 1 + (1 α)y 2 = a. Since R is monotone, it follows that, for all x, y X +, xr y αx 1 + (1 α)x 2 αy 1 + (1 α)y 2. (4) By (1), this completes the proof for the case of D = R n + \ {0 n }. Now consider D = R n. Applying the same argument as in the case of D = R n + \ {0 n }, there exists a continuous and monotone ordering R on X that satisfies (1) and (3). Furthermore, since R n + \ {0 n } R n and X + X, there exists α [0, 1] such that R satisfies (4) for all x, y X +. For any x, y X with αx 1 + (1 α)x 2 αy 1 + (1 α)y 2, there exist x, ȳ X + such that x i x i = ȳ j y j for all i, j N. Thus, (4) extends to X by (3). An alternative proof that directly shows that R satisfies (4) on X can be found in Blackorby, 12

13 Donaldson and Weymark (1984, pp ) and in Bossert and Weymark (2004, pp ). The mixed utilitarian-maximin social welfare orderings are special cases of the generalized Ginis. Two axiomatizations of the class of generalized Ginis are presented by d Aspremont and Gevers (2002, pp and p. 517). In particular, d Aspremont and Gevers (2002) characterize (i) the generalized Gini social welfare orderings using the axiom of cardinal unit-comparability defined for rank-ordered utility vectors instead of the composite transfer principle and cardinal full comparability, and (ii) the generalized Ginis associated with positive weights using the separability axiom defined for rank-ordered utility vectors instead of the composite transfer principle. Their weakened cardinal unit-comparability axiom implies their weakened separability axiom. As shown in Theorems 2 and 3, our results do not rely on the results of d Aspremont and Gevers (2002), but rather show how the composite transfer principle in conjunction with other axioms restricts the informational basis that can be utilized. We conclude this section with a brief discussion of possible variations in the set of axioms that are employed in Theorems 2 and 3. If weak Suppes-Sen and continuity are replaced with the conjunction of the strong Pareto principle and anonymity in Theorem 3, permissible social welfare orderings include leximin. Consequently, the informational basis of evaluation is not restricted to average and minimum utilities. Examples of some other permissible social welfare orderings, including utilitarianism and the lexicographic composition of utilitarianism and leximin that applies utilitarianism first, are presented in Kamaga (2018). On the other hand, if we retain continuity and replace weak Suppes-Sen with the conjunction of the strong Pareto principle and anonymity, then the ordering R on D identified in Theorem 2 becomes sensitive to average utility in the sense that, for all x, y D, if x 1 > y 1 and x 2 = y 2, then xp y. In addition, the range of permissible values of the parameter α in Theorem 3 becomes the half-open interval (0, 1]. The strict Pigou-Dalton transfer principle demands that xpy rather than merely xry results if x is obtained from y via a progressive transfer. It is not possible to strengthen the Pigou-Dalton transfer principle to its strict counterpart in our theorems there exists no social welfare ordering satisfying the resulting set of axioms. This observation follows from Proposition 1 in Kamaga (2018), which shows that there is no upper semicontinuous binary relation on R n that satisfies the strict Pigou- Dalton transfer principle and the composite transfer principle; besides, this impossibility extends to R n + \ {0 n } because his proof uses only positive utility vectors. In analogy to the previous modification, it is possible to strengthen the composite transfer principle by, again, replacing the requirement that xry in the consequent with the relational statement xpy. For n = 3, this has the effect that the ordering R on D in Theorem 2 becomes sensitive to minimum utility so that, for all x, y D, if x 1 = y 1 and x 2 > y 2, then xp y. Moreover, the range of permissible parameter values α in Theorem 3 now becomes the half-open interval [0, 1). For n 4, however, we obtain an impossibility. Defining x = (2, 3, 3, 6, 7..., 7) and y = (2, 2, 5, 5, 7..., 7), we obtain xiy from Theorem 2, whereas the strong composite transfer principle as defined above 13

14 Figure 2: Non-linear functions in Φ requires xpy. 4 Ratio-scale and translation-scale invariance 4.1 Definitions In this section, we examine information-invariance properties that differ from cardinal full comparability. The natural candidates for this task are ratio-scale full comparability and translation-scale full comparability. Clearly, replacing cardinal full comparability with ratio-scale full comparability or translation-scale full comparability extends the class of social welfare orderings compatible with the requisite information-invariance assumption. Although it is possible to study ratio-scale invariance while allowing for non-positive utility values (see, for instance, Tsui and Weymark, 1997), we restrict attention to the positive orthant of R n. As Tsui and Weymark (1997) prove, considering domains that include negative utilities leads to dictatorships or even impossibility theorems once a Pareto condition is imposed. We begin with the generalizations that correspond to ratio-scale full comparability. Define Φ as the set of all continuous and non-decreasing functions φ: (0, 1] (0, 1] with the following two properties. z φ(z) 1 for all z (0, 1], φ(az) aφ(z) for all a, z (0, 1]. (5a) (5b) Geometrically, the first property means that, for any z (0, 1], the point (z, φ(z)) lies on or above the 45-degree line in (0, 1] 2. Note that property (5a) implies that φ(1) = 1. Property (5b) means that, for any z (0, 1] and any z (0, z), the point (z, φ(z )) lies on or above the line segment connecting (z, φ(z)) and the origin 0 2. An example of φ Φ is given by a linear function φ(z) = (1 α)z + α 14

15 with α [0, 1]. Further, examples of a non-linear function in Φ are given by φ(z) = z and 3z if z (0, 0.3], φ(z) = 0.9 if z [0.3, 0.6], 0.25z if z [0.6, 1]. These non-linear functions are illustrated in Figure 2. From the latter example of a non-linear function φ, a permissible function in Φ is not necessarily concave. For a function φ Φ, we define the social welfare ordering R φ associated with φ by letting, for all x, y R n ++, ( ) ( ) x(1) y(1) xr φ y φ µ(x) φ µ(y). (6) µ(x) µ(y) If φ is given by a linear function φ(z) = (1 α)z + α with α [0, 1], it follows that R φ = R UM α. For any φ Φ, R φ is representable by the function Ξ φ : R n ++ R ++ such that ( ) x(1) Ξ φ (x) = φ µ(x) µ(x) for all x R n ++. As before, Ξ φ (x) is the representative utility of x according to R φ since Ξ φ (Ξ φ (x),..., Ξ φ (x)) = φ(1)ξ φ (x) = Ξ φ (x). The normative relative inequality index J R φ : R n ++ [0, 1) corresponding to Ξ φ is given by J R φ(x) = µ(x) Ξ φ(x) µ(x) ( ) x(1) = 1 φ µ(x) for all x R n ++. By (5a), 0 Jφ(x) R < 1. We now consider the social welfare orderings that correspond to the assumption of translationscale full comparability. Let Ψ be the set of all continuous and non-decreasing functions ψ: R + R + satisfying the following two properties. 0 ψ(z) z for all z R +, (7a) ψ(z + b) ψ(z) + b for all b, z R +. (7b) Property (7a) means that, for any z R +, the point (z, ψ(z)) lies on or below the 45-degree line. It also implies that ψ(0) = 0. Property (7b) means that, for any z R + and any z [z, ), the point (z, ψ(z )) lies on or below the line with slope one passing through (z, ψ(z)). An example of ψ Ψ is given by a linear function ψ(z) = αz with α [0, 1]. Examples of a non-linear function in Ψ are 15

16 Figure 3: Non-linear functions in Ψ given by ψ(z) = ln(z + 1) and 0.25z if z [0, 0.4], ψ(z) = z 0.3 if z [0.4, 0.7], 0.5z if z [0.7, ). These non-linear functions are illustrated in Figure 3. As with permissible functions φ Φ, a function ψ Ψ is not necessarily concave. For ψ Ψ, we define the social welfare ordering R ψ associated with ψ by letting, for all x, y R n, xr ψ y ψ(µ(x) x (1) ) + x (1) ψ(µ(y) y (1) ) + y (1). (8) If ψ Ψ is given by a linear function ψ(z) = αz with α [0, 1], it follows that R ψ = R UM α. For any ψ Ψ, a specific representation of R ψ is given by the function Ξ ψ : R n R such that Ξ ψ (x) = ψ(µ(x) x (1) ) + x (1) for all x R n. Since ψ(0) = 0, it follows that Ξ ψ (x) is the representative utility of x according to R ψ. The normative absolute inequality index Jψ A corresponding to ψ is defined as J A ψ (x) = µ(x) Ξ ψ(x) = µ(x) x (1) ψ(µ(x) x (1) ) for all x R n. By (7a), it follows that Jψ A (x) 0. 16

17 4.2 Axiomatizations We now provide characterizations of the social welfare orderings that are obtained if cardinal full comparability is replaced with ratio-scale full comparability and translation-scale full comparability, respectively. These well-known information-invariance properties are defined as follows. Ratio-scale full comparability: For all x, y, x, ȳ R n ++, if there exists a R ++ such that x i = ax i and ȳ i = ay i for all i N, then xry xrȳ. Translation-scale full comparability: For all x, y, x, ȳ R n, if there exists b R such that x i = x i + b and ȳ i = y i + b for all i N, then xry xrȳ. Some further definitions are required. Let X ++ = {(x 1, x 2 ) X : x i > 0 for all i {1, 2}} denote the subset of X that contains positive components only. Depending on the class of orderings considered, the set D denotes either X or X ++. A function f : D R is monotone if, for all x, y D, x y implies f (x) > f (y) and x y implies f (x) f (y). It is well-known that any continuous and monotone ordering R on D is representable by the representative-utility function Ξ; see, for example, Theorem 4.1 in Blackorby, Bossert, and Donaldson (2005). The following lemma states the properties of Ξ that will be used in the characterization results of this section. Lemma 1. If an ordering R on D is continuous and monotone, then there exists a continuous and monotone function Ξ: D R such that x 2 Ξ(x) x 1 for all x D, (9a) (Ξ(x), Ξ(x))I x for all x D, (9b) xr y Ξ(x) Ξ(y) for all x, y D, (9c) Ξ(ξ, ξ) = ξ for all ξ R (resp. ξ R ++ ). (9d) The following theorem characterizes generalizations of the utilitarian-maximin social welfare orderings that satisfy ratio-scale full comparability. Theorem 4. A social welfare ordering R on R n ++ satisfies weak Suppes-Sen, the Pigou-Dalton transfer principle, continuity, the composite transfer principle and ratio-scale full comparability if and only if there exists φ Φ such that R = R φ. 17

18 Proof. If. Let φ Φ. To show that R φ satisfies weak Suppes-Sen, let x, y R n ++ and suppose that x ( ) y ( ), which implies µ(x) > µ(y) and x (1) > y (1). Let a = µ(x)/µ(y) and a = x (1) /y (1). It follows that a, a > 1 and ( ) ( x(1) a ) y (1) φ µ(x) = φ aµ(y). µ(x) aµ(y) If a /a 1, we obtain xp φ y since φ is non-decreasing and a > 1. Now suppose that a /a < 1. By (5b), we obtain Since a > 1, it follows that xp φ y. φ ( a ) ( ) y (1) y(1) aµ(y) φ a µ(y). aµ(y) µ(y) Because φ is non-decreasing and continuous, R φ satisfies the Pigou-Dalton transfer principle, the composite transfer principle and continuity. By (6), R φ satisfies ratio-scale full comparability. Only if. Suppose that R satisfies the axioms of the theorem statement. By Theorem 2, there exists a continuous and monotone ordering R on X ++ satisfying (1). Lemma 1 implies that there exists a continuous and monotone function Ξ: X ++ R ++ satisfying the four properties (9a) through (9d). Since R satisfies ratio-scale full comparability, R satisfies the corresponding property defined as follows. For all x, y, x, ȳ X ++, if there exists a R ++ such that x i = ax i and ȳ i = ay i for all i {1, 2}, xr y xr ȳ. Thus, from (9b), it follows that, for all x X ++ and for all a R ++, (Ξ(x), Ξ(x))I x implies (aξ(x), aξ(x))i ax, which in turn implies Ξ(ax) = aξ(x) (10) since R is monotone and transitive. The solution of the functional equation in (10) is given by Ξ(x) = φ ( x2 x 1 ) x 1 (11) for all x X ++, where φ: (0, 1] R is defined by φ(z) = Ξ(1, z) for all z (0, 1]; see Aczél (1966, p. 229). Next, we show that φ Φ. Since the domain of Ξ is X ++, the domain of φ is (0, 1]. From (9a), it follows that, for all x X ++, x 2 /x 1 φ(x 2 /x 1 ) 1. Thus, φ is a function that maps into the interval (0, 1] satisfying (5a). Furthermore, φ is continuous since it satisfies (11) and Ξ is continuous. To show that φ is non-decreasing, suppose that z, z (0, 1] are such that z < z. Let x, x X ++ with x 1 = x 1 be such that z = x 2/x 1 and z = x 2 /x 1. Since Ξ is monotone, we obtain Ξ(x ) Ξ(x). From (11), it follows that φ(z ) φ(z). To show that φ satisfies (5b), suppose that a, z (0, 1]. Let x X ++ be such that z = x 2 /x 1. 18

19 Because Ξ is monotone, (11) implies ( Ξ(x 1 /a, x 2 ) Ξ(x) φ a x ) 2 x1 x 1 a φ ( x2 x 1 ) x 1 φ(az) aφ(z) and, therefore, φ Φ. From (1) and (9d), it follows that, for all x, y R n ++, xry (µ(x), x (1) )R (µ(y), y (1) ) Ξ(µ(x), x (1) ) Ξ(µ(y), y (1) ) and, by (11), it follows that R = R φ. Translation-scale full comparability leads to the following axiomatization. Theorem 5. A social welfare ordering R on R n satisfies weak Suppes-Sen, the Pigou-Dalton transfer principle, continuity, the composite transfer principle and translation-scale full comparability if and only if there exists ψ Ψ such that R = R ψ. Proof. If. Let ψ Ψ. To show that R ψ satisfies weak Suppes-Sen, let x, y R n be such that x ( ) y ( ). Let b = µ(x) µ(y) and b = x (1) y (1). It follows that b, b > 0 and ψ(µ(x) x (1) ) + x (1) = ψ(µ(y) y (1) + b b ) + y (1) + b. If b b, we obtain xp ψ y since ψ is non-decreasing and b > 0. Now suppose that b < b. From (7b), it follows that ψ(µ(y) y (1) ) ψ(µ(y) y (1) (b b)) + b b. Thus, we obtain ψ(µ(y) y (1) + b b ) + y (1) + b ψ(µ(y) y (1) ) + y (1) + b. Since b > 0, we obtain xp ψ y. Thus, R ψ satisfies weak Suppes-Sen. By (7b), R ψ satisfies the Pigou-Dalton transfer principle and the composite transfer principle. Since ψ is continuous, R ψ satisfies continuity. By (8), R ψ satisfies translation-scale full comparability. Only if. Suppose that R satisfies the axioms of the theorem statement. The same argument as that employed in the proof of Theorem 4 establishes that there exists a continuous and monotone ordering R satisfying (1) and a continuous and monotone function Ξ: X R satisfying (9a) to (9d). Because R satisfies translation-scale full comparability and R satisfies the corresponding property, it follows that, for all x X and for all b R, Ξ(x 1 + b, x 2 + b) = Ξ(x) + b. The solution to this equation is given by Ξ(x) = ψ(x 1 x 2 ) + x 2 (12) for all x X, where ψ: R + R is defined by ψ(z) = Ξ(z, 0); see Aczél (1966, p. 231). 19

20 Since the domain of Ξ is X and Ξ is continuous and satisfies (9a), ψ is a continuous function that maps into R + satisfying (7a). To show that ψ is non-decreasing, suppose that z, z R + are such that z < z. Let x, x X with x 2 = x 2 be such that z = x 1 x 2 and z = x 1 x 2. Since Ξ is monotone, (12) implies that ψ(z ) ψ(z). To show that ψ satisfies (7b), suppose that b, z R +. Let x X be such that z = x 1 x 2. Since Ξ is monotone, it follows from (12) that Ξ(x 1, x 2 b) Ξ(x) ψ(z + b) + x 2 b ψ(z) + x 2 ψ(z + b) ψ(z) + b. Thus, ψ satisfies (7b). Finally, from (1), (9d) and (12), it follows that R = R ψ. 5 Concluding remarks The mixed utilitarian-maximin social welfare orderings provide an intuitively plausible method of finding a compromise between utilitarianism and the maximin principle. These orderings were first proposed by Roberts (1980) but no characterization was provided in the literature prior to this contribution. The primary objective of this paper was to fill this gap, which we achieved by means of Theorem 3. In our results, we allow for all possible relative weights to be assigned to the extremes when defining a convex combination of the two. An interesting issue to be addressed in future work is the possibility of characterizing specific members or subclasses on the basis of additional axioms that may shed light on normatively appealing parameter choices. Appendix To prove that the axioms used in Theorems 2, 3, 4, and 5 are independent, consider the following examples. First, the social welfare ordering R = D D satisfies the axioms of Theorems 2, 3, 4, and 5 except for weak Suppes-Sen. Second, define the ordering R on D as follows. For all x, y D, xry µ(x) + x (n) µ(y) + y (n). 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 3 and consider the generalized Gini R G β on D with β i > β i+1 for all i {1,..., n 1}. This social welfare ordering satisfies the axioms of Theorems 2, 3, 4, and 5 except 20

21 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 ψ to R n ++ associated with ψ Ψ given by ψ(z) = ln(z + 1). This social welfare ordering on R n ++ satisfies the axioms in Theorem 4 except for ratio-scale full comparability. Sixth, define the ordering R on R n as follows. For all x, y R n, where the function g: R R is given by xry g(µ(x)) + g(x (1) ) g(µ(y)) + g(y (1) ) g(z) = z if z 0, 2z if z < 0. 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 2 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 δ ε in Theorems 2, 3, 4 and 5. Suppose n = 3 and consider R G β on D associated with (β 1, β 2, β 3 ) = (2/3, 1/3, 0). To show that R G β satisfies the weak version of the composite transfer principle, let x, y R3 and suppose that there exist i, j, k N and δ, ε R ++ with δ ε such that x i = y i + δ, x j = y j δ ε, x k = y k + ε, x i x j < x k, y i < y j y k and x l = y l for all l N \ {i, j, k}. Then, we obtain 3i=1 β i x (i) 3 i=1 β i y (i) = 2δ/3 (δ + ε)/3 = (δ ε)/3 0. Thus, xr G β y holds. Note that RG β satisfies all the axioms in Theorem 3 except for the composite transfer principle. It can be verified that R G β violates the composite transfer principle as follows. Consider x = (2, 2, 7), y = (1, 5, 5) R Then, we obtain yp G β x since 3 i=1 β i x (i) 3 i=1 β i y (i) = 1/3. We note that it is straightforward to extend this example to higher values of n. References Aczél, J. (1966), Lectures on Functional Equations and Their Applications. Academic Press, New York. Alvarez-Cuadrad, F. and N.V. Long (2009), A mixed Bentham Rawls criterion for intergenerational equity: Theory and implications. Journal of Environmental Economics and Management 58: