Essentially lexicographic aggregation

Essentially lexicographic aggregationAuthor(s): Ulrich KrauseSource: Social Choice and Welfare, Vol. 12, No. 3 (June 1995), pp. 233-244Published by: SpringerStable URL: http://www.jstor.org/stable/41106129 .

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Soc Choice Welfare (1995) 12: 233-244 ΊΖ ΓΎΖΖ · Social Chwee ·

© Springer- Verlag 1995

Essentially lexicographic aggregation Ulrich Krause1

University of Bremen, Department of Mathematics, 28334 Bremen, Germany (Fax: 421-218-4856)

Received: 24 February 1992/Accepted: 14 November 1994

Abstract In the paper a new approach to lexicography is developed by which in the general framework of ordered blocks with a monotonie basis it is shown that a nontrivial ordering is translation-invariant if and only if it is essentially lexicographic of degree n. Here, the latter means that the ordering can be represented by an ordinary lexicographic ordering in η dimensions. As an application it is shown that a nontrivial social welfare ordering on Euclidean space possesses a useful invariance property (cardinality and non comparability) if and only if the ordering is essentially lexicographic of a strong kind in that it can be obtained from ordinary lexicography by permutation, cutting-off and order reversal with respect to components. This result generalizes the characterization of lexical individual dictatorship obtained by Gevers and d'Aspremont and it provides, within the social welfare approach, a strong version of Arrow's impossibility theorem by not invoking any Pareto principle at all.

1. Introduction

A few years after the original demonstration by Arrow [1] it was suggested by Luce and Raiffa [12] that Arrow's impossibility theorem could be viewed - and proved too - as a theorem about the lexicographic ordering among strings of numbers. Formally, this involved moving away from Arrow's concept of a social welfare function towards Sen's concept of social welfare functional and further to what got called a social welfare ordering, viz. an ordering among strings of numbers interpreted as individual utility levels [14,15]. "Serial dictatorship" or "lexical individual dictatorship" was formally characterized by Gevers [7] and the topic was taken up again by d'Aspremont [5].

Impossibility theorems have been obtained also within the single-profile approach to the Bergson-Samuelson welfare function (cf. [5]; see also [9]) and can be found, beside in social choice theory, also in the field of multicriteria decision making (cf. [2] for connections between the two fields). Here too, lexicography

1 The author thanks K. Arrow, W. Gaertner, L. Gevers, and two anonymous referees for helpful hints and suggestions.



234 U. Krause

turned out to be a quite natural point of view (cf. [6,10]), though in a different manner as hinted at in [12]. Concerning the mathematics, Luce and Raiffa suggested a proof of Arrow's impossibility theorem by a simple modification of a theorem due to Blackwell and Girshick [4]. Related, but different, there is an earlier and purely mathematical theorem which characterizes lexicography in terms of ordered vector spaces [3,8]; actually, valuable results on lexicography were obtained by the mathematician H. Hahn as early as 1907. What is beautiful in viewing impossibility results via lexicography is that one sees more clearly why certain impossibilities come about. It seems to be worthwhile to explore in more detail the relationship between the two different strands of impossibility theorems in economics and of characterizations of lexicography in mathematics. This, however, is not the aim of the present paper and would be a task for the future.

In this paper an alternative approach to lexicography is developed which is based on the concept of an ordered block having a monotonie basis. This concept is more general and flexible than that of an ordered vector space and is presented in detail in Sect. 2. There it is shown that a nontrivial ordering for an ordered block having a monotonie basis is translation-invariant if and only if the ordering is essentially lexicographic of degree n. Here, an ordering is essentially lexicographic of degree η if there exists a mapping into η-dimensional Euclidean space by which the ordering is represented as the ordinary lexicographic ordering. Exploiting the concept of an ordered block with monotonie basis the proof given for this result is simple and does not use the supporting hyperplane theorem (Hahn-^Banach theorem) as it is the case, e.g., with the theorem of Blackwell and Girshick. The result of Sect. 2 is applied in Sect. 3 by treating a social welfare ordering on Euclidean space as a special ordered block. It is shown that a nontrivial social welfare ordering on Euclidean space is invariant for positive affine transformations (invariance property (II), which is equivalent to "cardinality and noncomparabil- ity" in [5] and to the "ordinal and non-comparable" - property in [7]) if and only if the ordering is essentially lexicographic of a strong kind in that it can be obtained from ordinary lexicography by permutation, cutting-off, and order reversal with respect to components. By requiring the strong Pareto principle in addition to the invariance condition the processes of cutting-off and order reversal disappear which leads one to the characterization of lexical individual dictatorship as obtained by Gevers [7] and d'Aspremont [5]. This characterization by Gevers and d'Aspremont is generalized by the theorem presented in Sect. 3 which at the same time provides, within the social welfare ordering approach, a strong version of Arrow's impossibility theorem by not invoking any Pareto principle at all. Other authors have studied the consequences for Arrow's impossibility conclusion of abandoning the Pareto principle while maintaining his other assumptions. Recent results in this tradition were obtained by Malawski and Zhou [13], The social welfare ordering approach adopted in the present paper is different and is based on an implicit axiom of neutrality between social outcome descriptions which pro- hibits the use of non-utility information. Whereas neutrality is not used in [13], the social welfare ordering approach is broad enough to cover Bergson-Samuelson welfare functions as well as multicriteria decision making.

2. An alternative approach to lexicography

Let R" denote the η-dimensional Euclidean space (n ̂ 1) and denote by χ = (xi9 ... ,xn) a point in this space. The ordinary lexicographic ordering <, is



Essentially lexicographic aggregation 235

defined by χ < z y iff there exists ; € { 0, . . . , η } such that xt = yt for all 1 < i < j and Xj+ χ < yJ + 1 . (For) = 0 the condition means xx < y ι , for j = η it means χ,· = yi for all 1 <i<n) (When applied to real numbers, < or < mean the common ordering or strict ordering for real numbers.)

An ordering on a set A is a binary relation ̂ which is reflexive, transitive, and complete. An ordering is trivial if a^b holds for all a,b e A. A non-trivial ordering ^ on a set A is called essentially lexicographic of degree n, if there exists an η ̂ 1 and a mapping y : A -» R" such that

a=<fc iff y{a) <iy(b) foralla,Ò€A

(Here, η is choosen as small as possible.) Obviously, the ordinary lexicographic ordering on R" is essentially lexico-

graphic of degree n, but not conversely. For example, the ordering ^ defined on R" for a fixed index i by

x^y iff Xi < y( is not lexicographic, but essentially lexicographic of degree 1 as can be seen from

y:R"->R' y(x) = x,.

(The notion of the degree takes care of the "cutting off" - case mentioned in the previous section.)

Also, the ordering ̂ defined on R2 by x^y iff Xi > y' or (xi = yx and x2 < yi) is different from the ordinary lexicographic ordering on R2 but it is essentially lexicographic of degree 2 as can be seen from y:R2 -+ R2, y(x) = (~X!,x2).

Let Β be a nonempty subset of some real vector space V and let Β be equipped with an ordering <. Denote by -< the strict preference and by ~ the indifference relation induced by ̂ on B. The pair (B, =<) is said to be an ordered block if the following two conditions are satisfied:

(i) There exists a finite set Ε α Β and a set Κ of real numbers with 0, 1 e Κ and -KcKand such that

LeeE J

(ii) If Tee ε kee ~ £e6£ /** with keJeE Κ then for all e e £, ke = le or e - 0. Further- more, e ~ 0 and /ceK imply that fee ~ 0.

Condition (i) requires Β to be generated by some finite subset £, with coefficients taken from some set K. Condition (ii) requires a kind of uniqueness for the representation made possible by condition (i). Therefore, we shall refer to a set Ε as in conditions (i) and (ii) as a basis of the ordered block.

A basis Ε of an ordered block (B, <) is said to be a monotonie basis, if £ has the following property: Suppose L is an arbitrary choosen subset of £ and e0 is an element of £ not belonging to L and such that 0-<aee-<aoô for all eeL, where αο, aeeX with |ao| = | a. | = 1. Then it holds that

ko(xoeo + Σ fceaee</oaoeo + Σ Ά* eeL eeL

for all fcOi ke, /0, le in Κ with fe0 < lo and /e < fce.



236 U. Krause

For the special case L = 0, the above property amounts to the monotonicity property

fcoaoô^'o«oô provided that 0<0Loeo and k0 < /o. In the general case, it is allowed to add on both sides of the inequality linear

combinations of elements dominated by e0 whereas the coefficients on the left hand side dominate those of the right hand side.

For an ordered block (B, =^) the ordering is said to be translation-invariant on Β if for any four elements x, y, x' y' of Β with χ + y' = χ' + y it holds that

x^y iff x'^y'. The following theorem provides a powerful characterization of essential lexicography by translation-invariance.

Theorem 1. Let (B, ̂ )be an ordered block with a monotonie basis. J/=^ is nontrivial and translation-invariant then ̂ is essentially lexicographic of degree η ̂ 1. The represenation mapping γ: Β -► IR" can be choosen to be additive, i.e., γ(χ + y) = y (χ) + y (y), provided the set of coefficients Κ is closed under addition.

Conversely, if*$ is essentially lexicographic of degree η with an additive mapping y : Β -+ R" then ̂ is nontrivial and translation invariant.

Proof (1) Consider first the case where =^ is assumed to be essentially lexicographic of degree η with an additive mapping γ : Β -► R". Then < is not trivial and for x,y e Β x^y iff y(x) <iy(y).

For x,yyx'9yf e Β with χ + y' = χ' -H y, additivity of y yields y(x) -h y(y') = y(x + y') = y(x' + y) = y(x') + y(j>). From the definition of the ordinary lexicographic ordering it follows immediately that

y(x) <My) iff y(x) + y(x') - y(x) <r/(y) + y(/) - y(y) and hence

y(x)^,y(>0 iff y(x') </?(>>').

Therefore, x^y iff x'=</, which shows that ̂ is translation-invariant. (2) Suppose now that =^ is nontrivial and translation-invariant. Let £ be a mono- tonic basis and define

F+ = {xe£|(Kx} F. = {-x|xe£,x<0}, F* = F+uF. F = F,u{ee£|e~0}.

Then F* *0 by the nontriviality of <, and 0< /for all /eF by translation- invariance. Since (B, *Q is an ordered block, we can represent any χ 6 Β as

x = Σ k/(x)f with coefficients kf(x)eK.

The coefficients kf(x) for / e F„, are uniquely determined because by the uniqueness property for an ordered block

X kff= X lff implies that fcr = // for/eF„.

By property (ii) of an ordered block there cannot be indifference between any two different elements in F*. Therefore, we can order F* by f'>fi> ··· >/*, where η ̂ 1 is the cardinality of F* = {/lf/2 » - >Λ}· Define V :B -♦ Rw by defining the




ith component of y(x) as

yi(x) = kft(x) for /= l,...,n. We first show that for any x,yeB

y (x)^ /?(>>) implies x<y

( <tt the ordinary lexicographic ordering on R"). We may write

X = X' + X" With X' = X *,(*)/, X" = X *,(%)/

and

y = y' + y" with y = £ M?)/, y" = Σ MjO/ /«F· /«F'F·

(where F'F, = {/e F |/*F,}). By property (ii) of an ordered block we have fc/~0 for keK, feF'F+,

which together with translation-invariance implies that x" ~ 0 and /' ^ 0. By translation-invariance again, χ = χ' + χ" ~ χ' and y = / + y" ~ /. Assuming y(x) <iy(y) it is therefore enough to show that x'^/. Since this holds trivially for x' = /, assume χ' Φ y'. Because of y(x) <iy(y), there exists / 6 F* such that

kr(x)<kf(y) and kr(x) = kf(y) ïor?<feF*. Defining

L = {/6F,|/^/, kf(y)<kf(x)}, J = {/eF,|/#/ /^L} we obtain the disjunct decomposition F^ = {/}uLuJ. We have that f<J for f € L. Setting

χ = m*)/+ Σ Μ*)/. * = */ω/+ Σ kr(y)f feL feL

we therefore obtain from the assumption of a monotonie basis that

*<y- For / e J we have that kf(x) < kf(y) and 0</. By the monotonie basis again we have therefore that kf(x)f^kf(y)ffor all fe J. Setting

and using that < is translation-invariant and transitive, we obtain

x<y-

Putting together, we arrive at

*' = * + *<y + ̂ <^ + £ = / that is, x'</.

Next we show for x>yeB that, conversely,

x=<)> implies y(x)£ty(y).

Suppose that x^y but not y(x) <ty(y). Then by the definition of ̂ i we must have y(y) <ty(x). In particular, there exists some /e F# such that kj(y) < k/(x).



238 U. Krause

Furthermore, by the above it follows from y(y) <jy(x) that y^x. Together with x^y this means that χ ~ y. The uniqueness property (ii) of an ordered block implies, then, that fc/(x) = kj(y' which contradicts kj(y) < kj(x). Thus, x^y implies that y(x) <ty(y). It remains to show that γ : Β -> R" is additive provided that the set of coefficients Κ is closed under addition. By property (i) of an ordered block we have for x,yeB that

* + y = Σ Μ*)/+ Σ k/(y)f= Σ (Μ*) + kf{y))f. /€F feF feF

If /C is closed under addition, then kf(x) + kf(y) e Κ and, hence, x + j/efl. Thus, we also have that

x + y = Σ M* + JO/

and the uniqueness property (ii) of an ordered block yields that kf(x + y) = fc/(x) + fc/(y) for all x, yeB, all / e F*. By the definition of y therefore

v,(x + y) = M* + y) = */.(*) + MjO = ν/Μ + yi(y) for a11 *·

This shows y(x + y) = y(x) + y(y). Π

Some general remarks concerning the alternative approach to lexicography as given by Theorem 1 may be in order. Dealing with ordered blocks the approach presented is different from approaches concerning an ordering on a whole vector space (see, e.g., [3-5,7,8, 11,12]) as well as from combinatorial approaches (see, e.g., [1,6, 10, 14]). In the next section it will be shown that the general Theorem 1 can be specialized to vector spaces in such a way as to improve a known result in the vector space and combinatorial setting. The concept of an ordered block may be looked at as a minimal framework which allows one to pin down in a precise manner the link between translation-invariance and lexicography. To make the link as in Theorem 1 the monotonicity of a basis of the ordered block plays an important role. To illustrate this consider the following example: Let Κ = { - 1,0, 1}, Β = Κ χ Κ, Ε = {(1,0),(0, 1)}. Define a nontrivial ordering < on the nine elements of Β by

( - 1, - 1)<( - l,0K(0, - 1K( - 1, 1K(0, 0)<(l, - 1K(0, 1)<(1,O)<(1, 1).

Since for xyy e Β, χ ~> y holds only for χ = y, (Β, ̂ ) is an ordered block with basis E. It is easily verified that ̂ is translation-invariant. From the definition of <, however, it is immediate that < is not essentially lexicographic. In the light of Theorem 1 this is due to the fact that Ε is not a monotonie base. The latter can be seen by checking the definition of a monotonie basis for the following case:

Let L = {e} with e = (0, 1), e0 = (1,0) and put a0 = ae = 1, k0 = 0, /0 = 1, ke = 1, l€ = - 1. Monotonicity requires that

0·1·*ο + 1·1·*^1·1·*ο + (- 1)·1·*

that is eê0 - e. By the definition of ̂ , however, e0 - e = (1, - l)-<(0, 1) = e. Ordered blocks of the kind just discussed appear also in the field of multicriteria

decision making (cf. [2, 10]). There, to find an ordering -^ on a block K2 = KxK (or, more generally, on Km) amounts to resolve "conflicts between components" which arise from assessments according to different criteria. In the above example, e.g., conflicts are resolved in a non-lexicographic manner. The conflict resolution




would be lexicographic, viz. the ordinary lexicographic ordering with the first component as the leading one, if the ordering between (0, 1) and (-1,1) and, a fortiori, between (1,-1) and (0, 1) would be reversed.

3. An application to social welfare orderings

After the introduction of social welfare functional by Sen [14] the notion of a social welfare ordering has been examined, among others, by d'Aspremont [5] and Gevers [7]. (For the background in social choice theory as well as the relevant literature see Sen [15], especially Ch. 6). Social welfare orderings refer to "orderings on EN which satisfy certain conditions and express social welfare judgements" [7, p. 76]. Here EN denotes, for Ν = {1,2,... ,n}, the η-dimensional Euclidean space, which we denote by R". In what follows we shall generalize results obtained in [5, 7] by characterizing a strong kind of essential lexicography without invoking the strong Pareto principle. The following theorem will be obtained by specializing Theorem 1 to orderings on Euclidean space. Theorem 2· For a nontrivial ordering ^ on Rm the following two statements (I) and (II) are equivalent: (I) ^ is essentially lexicographic of degree η with a representing function

y : Rm -+ R" given by y(x) = (ει*«(ΐ), ··· >z*x*(n)) far all χ e Rw, α being a permutation of {1, ... ,n} and ε,· = +1 or ε,= - 1 for all i. In other words, for any x,ye Rm one has

X^y iff (6ιΧβα,ί...,Ε,ιΧβ((1,^,(Ει}'β(ΐ) É^,).

Thereby η is given by the number of unit vectors in Rm which are not indifferent (for ̂) to zero.

(II) For any x, y e Rm and any a, b e Rm with a( > 0 and bf = r, r € R, for all i the following conditions are satisfied: (1) ifx^y then χ -h b^y + b. (2) Ifx^y then axây

(where (ax)t· = α&ι). Proof It is easily verified that statement (I) implies statement (II). Thus we assume (II) and show that (I) holds. (A) In a first step we show that Theorem 1 is applicable under assumption (II). From conditions (1) and (2) it follows that (1) is not only true for b = (r, ... ,r) but for arbitrary b = (bx, ... ,bm) e Rm. For this it suffices to observe that

Γ Xi if bi = 0, Xi + bi-1 Wl + fcf1)* if bt>0,

lí-Wí-l+t-feir1*) if*i<0. Thus ̂ is translation-invariant on Rm.

Let Κ = R and Ε = {ex, ... ,ew}, e{ being the ith unit vector in Rm. Obviously, for the pair (Rm, =<) condition (i) in the definition of an ordered block is satisfied. Concerning condition (ii), suppose that

m (X!,...,Xm)= ^ Χ&~0

i=l



240 U. Krause

and fix an index). From (2) it follows that (Xi,... ,2xj9... ,xm) ~0 and, hence, xfij ̂ 0 by translation-invariance. Using (2) again it follows that Xj = 0 or e} ~ 0. This shows that

Σ kee ~0 implies for all e e Ε that ke = 0 or e ~ 0. eeE

Because of translation-invariance this shows condition (ii) of an ordered block. To see that £ is a monotonie basis let L be an arbitrary non-empty subset of Ε and let eoeE not belonging to L and such that 0-<aee-<aoeo for all eel, where a0, oie e R with |oto| = |ae| = 1. It follows by translation-invariance and transitivity of < that

(Kpaoeo+ £(-ae)e, (♦) eeL

where ρ ̂ 1 is the cardinality of L. Let fc0, /0 and ke, le for e e L be real numbers such that kQ < l0 and le < ke for all eeL.

Since e0 and e e L are unit vectors and eo$L a vector a = (alf ... ,am) can be choosen such that a( = (/0 - feo)/P if ̂o is the zth unit vector and such that a} = ke - /e if e is theyth unit vector. Applying condition (2) to inequality ( * ) yields

0 ̂ il^£paoeo +£(*.- /.)( - <x€)e.

By translation-invariance it follows that

fcoaoeo + X fceaee < /oaoeo -f X /eaee. eeL eeL

If L is the empty set this inequality is obvious from (2) and translation- invariance. Thus, (Rm, ^ ) is an ordered block with a monotonie basis. Further- more, ̂ is nontrivial and translation-invariant. Therefore we can apply Theorem 1. (B) Next we show that (I) follows from the conclusions of Theorem 1. Since Κ = R is closed under addition Theorem 1 yields a number η < m and an additive mapping y : Rm -> R" such that for all xjelR*

x^y iff y(x)^My). By the definition of y in the proof of Theorem 1 we have for the ith component

of y that yi(x) = kfi(x) where the real numbers kf(x) are the uniquely determined coefficients in the representation χ = £/€f kf(x)f. For λ 6 R it follows that

Σ ikf[x)f- λχ=Σ kf{ix)f fcF feF

and, by the uniqueness of the coefficients, kf{?jc) = kkf(x) for all feF. Hence y.(Ax) = kyi{x) for all I y being additive we have that y is a linear mapping. From

m

X = (X!,...,Xm)= Σ Xiei>

therefore we obtain m

yiW= Σ XiVite)· i=l




Since =< is not trivial there is at least one / such that ?i(e,) Φ 0. We shall show that there is exactly one index 'γ such that γχ(βίι)Φ 0. Suppose that, on the contrary, there exists / φ i with 7Χ(^) Φ 0. Define y e Rm by yt = yi(et)~ l and yk = 0 for fc 9e i and define ζ e Rm by z} = 2y1(e,)~

* and zk = 0 for fc ̂j. The linearity of yx implies yi{y) = 1 < 2 = yjz) and, therefore, y (y) <iy(z) which in turn implies y ̂ z. Choose α € R"1 such that a{ = 3 and ak = 1 for fc Φ i. By condition (2) we have ay ̂ az and, hence, y{ay) <iy(az). Especially, yi(ay) <, yi(az) which implies

3 = öiy4y ι (βί) = yday) £ yâz) = ajZjyêj) = 2.

Since this is a contradiction we conclude that there is exactly one index it such that yiieij) φ 0. Defining α(1) = ix and rx == yi(efl) we obtain

Vi(x) = îê(i) for all χ e Rm.

The procedure for obtaining this simple expression for yx (x) can be applied also to y2(x), . . . , yn(*)> whereby, however, the lexicographic structure has to be taken into account. More precisely, calculating y2{x), vectors y and ζ are choosen as before and such that >>β(1) = ζβ(1) holds in addition. As before it follows that there exists exactly one index i2 different from ot(l) such that y2(ei2) Φ 0. Defining ot(2) = i2 and r2 = ?2(*ί2) we obtain

72(x) = 5ιΧβ(1) + Γ2χβ(2) for all χ 6 R"'

By continuation we obtain for every j, 1 <j < n, an expression

7Á*) = 0M + Γ^«ϋ) for a11 x 6 Rm^

where r j Φ 0 and t/(x) is a, possibly proper, linear combination of the χΛ(ί) with lî<j. Obviously, (a(l), ... ,a(n)) is a permutation of (l,...,n). From the expressions of the y;(x) just obtained it follows that y(x) <>iy(y) is equivalent to y'(x) <iY(y) where the representation / is defined by 7'(x) = (r'Xî)9... ,ΓΗχβ(η)). Since Tj Φ 0 the sign ε^ of r,· is either + 1 or - 1 and y'(x) <ty'(y) is equivalent to y"{x) <iY'(y) where y" is defined by y"(x) = (ειχβ(1), ... ,εηχΛ{Η)). Thus we arrive at

x<y iffyíxí^/VÍy)

and

yWîy(y)iff?"(x)îy"(y)

which proves statement (Ι). Π

In what follows we want to relate Theorem 2 to other results obtained in the literature on characterizations of lexicography. By Theorem 2, the invariance property (II) characterizes a strong kind of essential lexicography where a representing function y can be choosen which is linear and multiplicative in the sense that y{ax) = a(fl)y(x) for all α, χ e Rm where a(a)f = au{i) for 1 < ί ̂ n. In more detail, Theorem 2 states that a nontrivial ordering on Euclidean space satisfies the invariance condition (II) if and only if it can be obtained from ordinary lexicography by the processes of "cutting coordinates" (i.e. η < m is possible), "permuting coordinates" (by permutation a) and "reversing order" (i.e. cf = -1 is possible), respectively. A different description of lexicography is given by a theorem of Birkhoff [3, p. 240] which, roughly, is as follows (for an extension to infinite dimensions see Hausner and Wendel [8]):



242 U. Krause

Let ̂ be an ordering on lRm for which Rm is a vector lattice (e.g., ̂ a simple ordering) and which satisfies the following conditions:

(Γ) If x^y then χ + b^y + b for all b e Rm. (2') If x^y then AxÂy for all positive real numbers λ.

Then there exists some vector space basis G such that

x^y iff ({i,...,£m)£ifai> - .f*) where £,·, η{ are the coordinates of χ and y, respectively with respect to the basis G. Compared with the invariance-condition (II) in Theorem 2, condition (Γ) is stronger than (1) and condition (2') is weaker than (2).

Moreover, different from Theorem 2, BirkhofFs theorem does not say that =< is lexicographic with respect to the given canonical basis of Rm - on the contrary, this is false in general under assumptions (Γ) and (2'). Lexicography with resepct to the canonical basis, however, is what is needed in social choice theory and in multicriteria decision making.

Concerning social choice theory, characterizations of lexicography have been obtained by employing the (strong) Pareto principle. There are different versions of the Pareto principle for an ordering =< on Rm (x = (xl9 ... ,xm) e Rm being the canonical representation): Weak Pareto principle (WP) If χ, < yi for all i then x<y Pareto principle (P) If x, < yi for all i then x^y

Strong Pareto principle (SP) The Pareto principle, and if xf < y ι for all i but χ Φ y then x<y.

Since in Theorem 2(1) either ex = + 1 or εχ = - 1 it follows for an ordering < on Rm possessing property (II) of Theorem 2 that it is either trivial, or satisfies WP, or satisfies the anti-WP; the latter means that xf < yt for all / implies ^-<x. (In a different framework such a trichotomy is obtained in [13].) It is easily seen, however, that a corresponding trichotomy does not apply to Ρ or SP.

From Theorem 2 the following Corollary is immediately obtained.

Corollary. Let ̂ be a nontrivial ordering on Rw and let (II) as in Theorem 2.

(i) ^ satisfies (II) and {WP) if and only if there exist η < m, a permutation α of {1,... ,h} and ε,·€{ - l,l}/or 2<i <n such that for all x,j/eRm x^y is equivalent to

(*a(l)>£2X«(2)> ··· ,δπΧβ(π) <l(.ye(l),ß2.y«(2)>··· >ê(n)).

(ii) ̂ satisfies (II) and (P) if and only if there exist n<m and a permutation α of { 1 , . . . , n) such that for all xje Rm x^y is equivalent to

(X«(l), ... ,Χβ(η)) ^l(>Ot(l)> ·■· >ya(n))-

(iii) =< satisfies (II) and (SP) if and only if there exists a permutation ao/{l,...,m} such that for all x9ye Rm x^y is equivalent to

(Xa(l), ... ,X«(m)) î(.Va(l)> ··· »^«(m))·




This Corollary shows how the processes of reversing order and of cutting coordinates, admitted in Theorem 2, disappear by requiring a Pareto principle in increasing strength. Statement (iii) of the Corollary is a well-known result about lexical individual dictatorship obtained by Gevers [7, Theorem 3] and d'Aspremont [5, Theorem 3.3.6]. (Condition (II) corresponds to the mutually equivalent assumptions ON*, CN* and COC* in [5, 7].) Thus, Theorem 2 is a generalization of this result of d'Aspremont and Gevers. Statement (i) of the Corollary sharpens a result of d'Aspremont [5, Theorem 3.1]. The proof of The- orem 2, which uses a specialization of the general Theorem 1, is different from those given in the literature. The proofs given in [5, 7] rely substantially on the strong Pareto principle. (The proof as given in [7, p. 87] is somewhat sketchy.) The history of the problem is traced back in [5, 7] to Luce and Raiffa [12] and their treatment of Arrow's impossibility theorem [1] within the context of the theory of decision making under uncertainty [12, p. 342]. Luce and Raiffa suggest that under Arrow-like conditions it follows from a simple modification of a theorem due to Blackwell and Girshick that only a lexicographical ordering is possible [12, p. 344]. The theorem of Black well and Girshick [4, p. 118] also uses substantially a Pareto principle (in a weaker form) and relies on the vector space structure by applying the Hahn-Banach theorem. d'Aspremont as well as Gevers relate their above mentioned results to a weak version of Arrow's [1] classical theorem. "Weak", because the strong form of the Pareto principle is assumed. In the same spirt, Theorem 2 may be considered to be a strong version of Arrow's theorem in that no Pareto principle is assumed at all, while a neutrality axiom is implicit.

4. Conclusion

A new approach to lexicography has been developed by employing ordered blocks having a monotonie basis. For those blocks it is shown that a nontrivial ordering is translation-invariant precisely if it is essentially lexicographic of some degree. By specializing to ordered blocks given by a nontrivial social welfare ordering on Euclidean space it is shown that such an ordering possesses a certain invariance property if and only if the ordering can be obtained from ordinary lexicography by permutation, cutting-off and order reversal with respect to components. This generalizes the characterization of lexical individual dictatorship obtained by Gevers and d'Aspremont and provides a strong version of Arrow's impossibility theorem by not invoking any Pareto principle at all.

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Essentially lexicographic aggregation