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Journal of Economic Theory 106, 126–150 (2002)doi:10.1006/jeth.2001.2859
Min, Max, and Sum1
1 Uzi Segal thanks SSHRCC and Joel Sobel thanks NSF and CASBS for financial support.
Uzi Segal
Department of Economics, Boston College,Chestnut Hill, Massachusetts 02467
and
Joel Sobel
Department of Economics, University of California,San Diego, La Jolla, California 92093
Received November 22, 2000; final version received May 28, 2001;published online February 14, 2002
This paper provides characterization theorems for preferences that can be repre-sented by U(x1, ..., xn)=min{xk}, U(x1, ..., xn)=max{xk}, U(x1, ..., xn)=; u(xk),or combinations of these functionals. The main assumption is partial separability,where changing a common component of two vectors does not reverse strict pref-erences, but may turn strict preferences into indifference. We discuss applications ofour results to social choice. Journal of Economic Literature Classification Numbers:C0, D1, D6. © 2002 Elsevier Science (USA)
1. INTRODUCTION
This paper provides characterization theorems for preferences over Rn
that can be represented byU(x1, ..., xn)=min{xk},U(x1, ..., xn)=max{xk},U(x1, ..., xn)=; u(xk), or combinations of these functional forms.
The problem of finding characterization theorems for the additive andminimum functional forms has been studied from the perspective ofsocial-choice theory, decision making under ignorance, and consumertheory. In social choice theory, the arguments of the utility function U are
(von Neumann and Morgenstern) utilities of the members of the society.In decision making under ignorance, the arguments are state-contingentutilities, while in consumer theory, the arguments are commodities.
The standard assumptions on preferences made in this literature aresome form of continuity, monotonicity, symmetry, linearity, and separa-bility. The interpretation and importance of continuity and monotonicity2
2 The social choice literature often refers to monotonicity assumptions as the Paretoprinciple.
are familiar and well understood. The linearity assumption3 requires that
3 In the decision theoretic literature, this assumption is known as constant (absolute andrelative) risk aversion.
rankings do not change when one applies a common affine transformationto all components. In the social-choice context it defines the extent towhich individual utilities are comparable. The assumption of completeseparability states that changing a common component of two vectors doesnot reverse a weak preference. The main technical contribution of thispaper is to study the implications of a weaker version of the separabilityassumption. Under this weaker assumption, called partial separability,changing a common component of two vectors does not reverse strictpreferences, but may turn strict preferences into indifference (see Blackorbyet al. [4] and Mak [16]). Complete separability suggests that changes inone component should have no affect over the desirability of the rest of thecomponents. Partial separability replaces this with a limited influence,namely that a change in one component may not reverse preferences, butmay make the rest of the components immaterial.
This paper provides novel characterizations of the min and max criteriaand conditions under which preferences are represented by the sum of xiover a subset of the domain and either the max or min of xi over the rest ofthe domain. To put these results in perspective, we briefly review relatedliterature. Luce and Raiffa [15] survey the research of several authors onthe problem of decision making under ignorance done in the 1950s.Perhaps the most elegant contribution in this group is due to Milnor [19].He presents axiomatic treatments of the maxmin and sum criteria. Hischaracterization of the Hurwicz (see [15]) a criterion (weighted average ofmaxmax and maxmin) is based on a ‘‘column duplication’’ assumption thatstipulates that the ranking between a pair of vectors does not change if acomponent is added to each vector that duplicates an existing component.This assumption is plausible for problems of choice under ignorance (wherethe duplicate component could be the result of an arbitrary redefinition ofthe states of the world), but is harder to justify in other applications,especially to the theory of social choice. His characterization of the sumcriterion (in the context of decision making under ignorance, this is the
MIN, MAX, AND SUM 127
principal of insufficient reason) depends on monotonicity and completeseparability assumptions.
Maskin [18] provides a characterization of the sum and maxmin criteriasimilar to Milnor’s. He relates his results to the social choice literature.Maskin relaxes the continuity assumption to provide characterizations oflexicographic maxmin and lexicographic maxmax criteria as well.
We assume throughout that preferences are symmetric, continuous,weakly increasing, and satisfy partial separability. We supplement theseassumptions with different conditions to obtain the representation func-tions described above. We introduce the assumptions formally in Section 2.By focusing on partial separability rather than the stronger completeseparability axiom, this paper provides axioms that imply not only the sumrepresentation, but also max, min, and a combination of these two andsum.
If preferences are strictly monotonic, then partial separability impliescomplete separability (see Färe and Primont [9]). In order to get morethan just additively separable representations we must permit some flat-ness. In Section 3 we analyze the implications of local flatness with respectto one variable at a point along the main diagonal. We show that if sym-metry is assumed, then the indifference curve through this point is eithermin or max. Section 4 shows how an additional indifference monotonicityaxiom implies that if one indifference curve is described by the min func-tion, then all lower indifference curves are min as well and that higherindifference curves are either min or additively separable. Similarly, if oneindifference curve is described by the max function, then all higher indif-ference curves are max as well, and all lower indifference curves are eithermax or additively separable.
The third result, presented in Section 5, shows that a linearity axiomcombines with the partial separability axiom to guarantee that preferencesmust be represented throughout their domain by min, max, or ; xi. In thissection we also point out that the linearity axiom is stronger than necessaryto obtain this trichotomy. Under a weaker condition, preferences must berepresented throughout their domain by min, max, or ; u(xi).
Section 6 discusses the connection between our results and a socialchoice problem, where the question is how should society allocate indivi-sible goods when it decides to use a lottery.
2. AXIOMS AND PRELIMINARY RESULTS
Consider a preference relation R on Rn. We denote by P and ’ thestrict and indifference relations, respectively. The preferences R are
128 SEGAL AND SOBEL
assumed to be complete, transitive, and continuous. Denote e=(1, ..., 1)¥Rn and let e i=(0, ..., 1, ..., 0). Assume:
(M) Monotonicity: x± y implies xP y.4
4 For x, y ¥Rn, (x1, ..., xn)± (y1, ..., yn) iff, for all i, xi > yi.
(S) Symmetry: For every permutation p of {1, ..., n} and for every x,
(x1, ..., xn) ’ (xp(1), ..., xp(n)).
Define x−k to be the vector in Rn−1 that is obtained from x by eliminat-ing component k, and let (x−k, yk) be the vector obtained from x byreplacing xk with yk.
All of the results of the paper depend on the following axiom.
(PS) Partial separability: For all x, y ¥Rn and every k, if x=(x−k, xk)P (y−k, xk) then (x−k, yk)R (y−k, yk)=y.
The partial separability axiom is weaker than the common separabilityaxiom:
(CS) Complete separability: For all x and y and for every k,(x−k, xk)R (y−k, xk) iff (x−k, yk)R (y−k, yk).
It follows immediately that complete separability implies partial separa-bility. The converse is of course not true (see for example Theorem 1).However, if we make a stronger monotonicity assumption, such an equiva-lence will follow.
(SM) Strict monotonicity: x j y implies xP y.
Färe and Primont [9] show that if the preferences R satisfy strictmonotonicity and partial separability, then they satisfy complete separa-bility.
It is useful to note equivalent forms of the partial separability assump-tion. For x ¥Rn, let ck(x)={x̄−k: (x̄−k, xk)R (x−k, xk)=x}. That is, ck(x)is the intersection of the upper set of x with the hyperplane where the kthcomponent equals xk. Let N denote the set {1, ..., n} and N−k denoteN0{k}.
Proposition 1. Let U be a continuous utility function that represents R.The following three conditions are equivalent.
1. For all x, y ¥Rn, if x=(x−k, xk)P (y−k, xk), then (x−k, yk)R(y−k, yk)=y.
2. There exist continuous functions f: Rn−1QR and g: R2QR suchthat U(x−k, xk)=g(f(x−k), xk), where g is nondecreasing in its first argu-ment.
3. For each x, xŒ ¥Rn, either ck(xŒ) ı ck(x) or ck(x) ı ck(xŒ).
MIN, MAX, AND SUM 129
Proof. Bliss [5] and Blackorby et al. [4, Theorem 3.2b, p. 57 andTheorem 3.3b, p. 65] prove that the last two conditions are equivalent.Mak [16, Proposition (2.11)] proves the equivalence of the first and thethird conditions. L
If xk=x−
k, then ck(x) and ck(xŒ) are upper sets of the same inducedpreferences on Rn−1 and are of course nested. Condition 3 of Proposition 1is therefore restrictive only when xk ] x
−
k. This condition does not requirethat indifference curves of the induced preferences on Rn−1 at the levelwhere the kth component is xk or x −k will be the same. Blackorby et al. [4,pp. 43–46] and Mak [16] define the set of variables N−k to be separablefrom {k} if and only if the third condition in Proposition 1 holds.5
5 It is possible to amend the conditions of Proposition 1 and the partial separability axiomto obtain the stronger notion of partial set separability, where a set of variables I is partiallyseparable from the set N0I. We do not make this assumption, but discuss it briefly inSection 6.
Some of our arguments derive local implications of our assumptions. Inorder to extend these properties, we make other assumptions. The follow-ing one, which we do not use until Section 5, is the strongest.
(L) Linearity: For all x, y ¥Rn,1. For all a, xR y iff x+ae R y+ae.2. For all b > 0, xR y iff bxR by.
In the decision theoretic literature this axiom is called constant riskaversion where together with the independence axiom it is known to implyexpected value maximization.6 It is also widely used in the literature con-
6 For an analysis of constant risk aversion without the independence axiom, see Safra andSegal [23].
cerning income distribution. In the social choice literature, where prefer-ences are defined over individual utilities, the axiom states that the orderingis invariant with respect to a common positive affine transformation ofutilities (see Maskin [17]). In Section 6 we provide an example where thisaxiom may seem acceptable.
To investigate the implications of the partial separability axiom, we needthe following definition.
Definition 1. The two vectors x, y ¥Rn are comonotonic if, for all kand kŒ, xk \ xkŒ iff yk \ ykŒ. For x ¥Rn, the comonotonic sectorM(x) is theset of all points y such that x and y are comonotonic.
Two vectors are comonotonic if they have the same ranking of theircomponents. Observe that for a and b as in the definition of axiom L, thethree vectors x, x+ae, and bx are comonotonic.
130 SEGAL AND SOBEL
Definition 2. Preferences R satisfy strict monotonicity at x if andonly if for all xŒ, xœ ¥Rn, xŒ j x j xœ implies xŒP xP xœ.
If the preferences R satisfy strict monotonicity at x, then the linearityaxiom implies that R satisfy strict monotonicity for all xŒ ¥M(x) andhence symmetry implies strict monotonicity. In combination with the otheraxioms, therefore, the linearity axiom implies that preferences will violatestrict monotonicity everywhere or satisfy the complete separability axiom.Weaker assumptions suffice for most of our analysis.
We say that the preferences R are strictly monotonic with respect to thekth component at the point x if for all x −k > xk > x
'
k , (x−k, x−
k)P xP(x−k, x
'
k ).
(CF) Comonotonic flatness: For every x, if the preferences are strictlymonotonic with respect to the kth component at the point x, then they arestrictly monotonic with respect to this component for all y ¥M(x).
Axioms relating to comonotonic vectors are popular in the literatureconcerning income distribution and decision making under uncertainty.The most popular alternative to expected utility theory, called rankdependent (Quiggin [20]), assumes that the utility from an outcome xi,which is to be obtained with probability pi, is multiplied by a function ofpi, and the adjusted value of pi depends on xi’s rank. Formally, denotep0=0 and let x1 [ · · · [ xn. The value of the lottery (x1, p1; ...; xn, pn) isgiven by
Cn
i=1u(xi) 5f 1 C
i
j=0pj 2−f 1 C
i−1
j=0pj 26 ,
where f(0)=0, f(1)=1, and f is increasing and continuous. Considernow the set of lotteries (x1,
1n ; ...; xn,
1n), and suppose that at a point where
all outcomes are distinct, (...; xi+e,1n ; ...) ’ (...; xi,
1n ; ...). In expected
utility theory this indifference implies u(xi+e)=u(xi); therefore, the indif-ference holds regardless of the rest of the outcomes. In the rank dependentmodel, on the other hand, this indifference may hold because f( in)=f(
i−1n )
(assume that the outcomes are ordered from lowest to highest). In thatcase, indifference need not hold at xi if the outcome xj changes for j ] i.On the other hand, it does hold for all outcomes that are ranked ith frombelow. The comonotonic indifference axiom complies with this model.
The linearity axiom implies comonotonic flatness, but of course notconversely. In combination with the symmetry axiom, however, comono-tonic flatness implies that either preferences satisfy strict monotonicityglobally or they violate this axiom globally. For Theorem 1 we need aneven weaker assumption.
MIN, MAX, AND SUM 131
(IM) Indifference monotonicity: Let x and y be comonotonic, andsuppose that x ’ y. The preferences R satisfy strict monotonicity at x if,and only if, they satisfy strict monotonicity at y.
This axiom is weaker than comonotonic flatness in two respects. Let xand y be comonotonic vectors. If at x the preferences are strictly mono-tonic with respect to the kth component and flat with respect to the athcomponent, then axiom CF requires the same at y, while indifferencemonotonicity is satisfied even if preferences at y are strictly monotonic withrespect to the ath component and flat with respect to the kth one. Second,unlike the comonotonic flatness axiom, where uniform behavior is requiredover the whole comonotonic sectorM(x), axiom IM only restricts behavioralong the comonotonic part of the indifference curve through x. Togetherwith symmetry, indifference monotonicity implies that if x ’ y, then thepreferences R satisfy strict monotonicity at x if, and only if, they satisfystrict monotonicity at y. In the remainder of the paper, we use this versionof axiom IM.
Under the partial separability axiom, weak monotonicity of preferences(that is, x± y implies xR y) guarantees that the functions f and g in therepresentation in Proposition 1 can be taken to be weakly monotonic.Under the stronger monotonicity condition M and under symmetry, we canguarantee that both of the functions in the representation of Proposition 1are increasing.7 Formally:
7 The function h: RmQR is weakly monotonic if x F y implies that h(x) \ h(y). It isincreasing if x± y implies that h(x) > h(y).
Proposition 2. Assume that the preferences R satisfy monotonicity,symmetry, and partial separability, and let U be a continuous utility functionthat represents R. Then there exist increasing and continuous functionsf: Rn−1QR and g: R2QR such that U(x−k, xk)=g(f(x−k), xk).
Proof. Take x−k ± y−k and assume f(x−k) [ f(y−k). To simplifynotation, assume k > 1 and a=y1=minj ] k yj. Let z be obtained by per-muting the first and kth components of (x−k, a). It follows that x−k \z−k \ y−k. Weak monotonicity and Proposition 1(2) imply that f is non-decreasing, hence
f(x−x)=f(z−k)=f(y−k) (1)
and therefore
U(y−k, a)=g(f(y−k), a)=g(f(x−k), a)=U(x−k, a)=U(z)=U(z−k, x1)
=g(f(z−k), x1)=g(f(x−k), x1)=U(x−k, x1), (2)
132 SEGAL AND SOBEL
where the second and seventh equations follow from Eq. (1), the first,third, sixth, and eighth equations follow from the definition of g and f, thefourth equation follows from the symmetry axiom, and the fifth equationbecause the kth component of z is x1. It follows from Eq. (2) thatU(y−k, a)=U(x−k, x1), which contradicts M since a < x1. This establishesthat f is increasing. That g is increasing (when its first argument is in therange of f) now follows directly from axiom M. L
3. MONOTONICITY ALONG THE MAIN DIAGONAL
Although partial separability restricts upper sets, and its verificationtherefore needs the analysis of the preferences R at many points, it turnsout that when monotonicity and symmetry are also assumed, much infor-mation is contained in the preferences’ behavior along the main diagonal{le: l ¥R}. The present section is devoted to this analysis.
Fix a point l0e. The next proposition, which does not require linearity,comonotonic flatness, or indifference monotonicity, shows that preferencesare either strictly monotonic at this point, or the indifference curve throughthe point is either min or max. As before, e i=(0, ..., 1, ..., 0).
Proposition 3. Let n \ 3. Assume monotonicity, symmetry, and partialseparability, and suppose that there exist l0, e > 0, and m such that l0e+eem ’ l0e. Then x ’ l0e if, and only if, min{xi}=l0.
Axiom PS places no restriction on monotonic preferences when n=2,and simple examples show that Proposition 3 does not hold when n=2(see, e.g., the preferences that are represented by Eq. (20)). We will proveProposition 3 using Lemmas 1 and 2. A symmetric argument establishesthe next result.
Proposition 4. Let n \ 3. Assume monotonicity, symmetry, and partialseparability, and suppose that there exist l0, e > 0, and m such thatl0e−eem ’ l0e. Then x ’ l0e if, and only if, max{xi}=l0.
Lemma 1. Assume monotonicity, symmetry, and partial separability, andsuppose that there exist l0, a > 0, and 1 [ m < n−1 such that l0e+;m
i=1 ae i ’l0e. Then there exists h > 0 such that l0e+;m−1
i=1 ae i+;n−1i=m he i ’ l0e.
This lemma implies, in particular, that if it is possible to increase thevalue of one component of a point l0e on the main diagonal withoutmoving to a strictly better point, then in an open neighborhood of l0e, theindifference curve through l0e is derived from the min representationfunction.
MIN, MAX, AND SUM 133
Proof. Suppose that there exist b > 0 and k such that
l0e+Cm−1
i=1ae i+C
k−1
i=mbe i ’ l0e (3)
but, for all e > 0,
l0e+Cm−1
i=1ae i+C
k−1
i=mbe i+eekP l0e. (4)
We want to show that k=n. Assume that k < n. Let x−k=(x1, ..., xk−1,xk+1, ..., xn) where xi=l0+a for i < m, xi=l0+b for m [ i < k, andxi=l0 for i > k. It follows from (3) that
(x−k, l0) ’ l0e (5)
and for all 0 < e < a,
(x−k, l0+e)P l0e ’ l0e+Cm−1
i=1ae i+eek, (6)
where the strict preference follows from (4) and the indifference followsfrom the assumption of the lemma and axiom S. By continuity, (6) impliesthat there exists d > 0 such that if y=(l0+d) e+;m−1
i=1 ae i, then
(x−k, l0+e)P (y−k, l0+e). (7)
Furthermore, unless the statement of the lemma holds true, it must be thecase that (y−k, l0)P l0e. It follows from (5) that
(y−k, l0)P (x−k, l0). (8)
Equations (7) and (8) violate axiom PS. Hence we have a contradiction,proving that k=n. L
Lemma 2. Assume monotonicity, symmetry, and partial separability, andsuppose that there exist l0, a > 0, and 1 [ m < n−1 such that l0e+;m
i=1 ae i ’l0e. Then l0e+;n−1
i=1 ae i ’ l0e.
In other words, the value of h in Lemma 1 is at least a.
Proof. Suppose that for some k \ 2,
l0e+Ck−1
i=1ae i ’ l0e (9)
134 SEGAL AND SOBEL
but l0e+;ki=1 ae iP l0e. We want to show that k=n. Assume that k < n
and argue to a contradiction. First, we claim that there exists b > 0 suchthat
l0e+Ck−1
i=1ae i+bek ’ l0e. (10)
If (10) did not hold, then it follows from (9) and continuity that for alle > 0, there exists d > 0 such that
l0e+Ck−1
i=1ae i+eekP l0e+C
k−1
i=1ae i+C
n
i=kde i. (11)
Applying axioms PS and M, we can conclude from (11) that
l0e+Ck−2
i=1ae i+dek−1+eekR l0e+C
k−2
i=1ae i+ C
n
i=k−1de iP l0e. (12)
But if k < n and d and e are sufficiently small, then by Lemma 1,
l0e+Ck−2
i=1ae i+dek−1+eek ’ l0e. (13)
Since (13) contradicts (12), (10) must hold.Now let b̄ — sup{b: l0e+;k−1
i=1 ae i+bek ’ l0e}. We know that b̄ > 0 andwe want to show that b̄ \ a. Assume b̄ < a. Again by continuity it followsthat for all e > 0, there exists d ¥ (0, b̄) such that
l0e+Ck−1
i=1ae i+(b̄+e) ekP l0e+C
k−1
i=1ae i+C
n
i=kde i.
By PS
l0e+Ck−2
i=1ae i+dek−1+(b̄+e) ekR l0e+C
k−2
i=1ae i+ C
n
i=k−1de i. (14)
However, by axiom S,
l0e+Ck−2
i=1ae i+dek−1+(b̄+e) ek ’ l0e+C
k−2
i=1ae i+(b̄+e) ek−1+dek (15)
MIN, MAX, AND SUM 135
and, provided that b̄+e [ a, axiom M implies that
l0e+Ck−2
i=1ae i+(b̄+e) ek−1+dekQ l0e+C
k−1
i=1ae i+dekQ l0e+C
k−1
i=1ae i+b̄ek.
(16)
Since
l0e+Ck−1
i=1ae i+b̄ek ’ l0e
by the definition of b̄, (14), (15), and (16) combine to imply that
l0e R l0e+Ck−2
i=1ae i+ C
n
i=k−1de i,
a violation of axiom M. This contradiction establishes the lemma. L
Proof of Proposition 3. From Lemma 2 it is sufficient to show thatsup{x1: l0e+x1e1 ’ l0e}=.. Denote the supremum by a and assume thata is finite. By assumption, a > l0. By Lemma 2 and n > 2, there exists d > 0such that l0e+ae1+den ’ l0e. Therefore, by continuity and monotonicity,there exists eŒ > 0 such that
(l0+d) e P l0e+(a+eŒ) e1+den. (17)
On the other hand, for sufficiently small d,
l0e+(a+eŒ) e1P l0e ’ l0e+d Cn−1
i=1e i, (18)
where the indifference follows from Lemma 1 and the strict preferencefollows from the definition of a. Equations (17) and (18) violate axiom PS.This contradiction establishes the proposition. L
4. INDIFFERENCE MONOTONICITY
Consider the function U: RnQR, given by
U(x)=˛D xi x ¥Rn++
min{xi} otherwise.(19)
136 SEGAL AND SOBEL
FIG. 1. The function U for n=2.
(See Fig. 1 for the case n=2.) This function satisfies monotonicity, sym-metry, partial separability, and indifference monotonicity. As we show inthis section, to a certain extent, it is typical of the functions satisfying theseaxioms.
Theorem 1. Let n \ 3. The following two conditions on R are equivalent.
1. R satisfy monotonicity, symmetry, partial separability, and indif-ference monotonicity.
2. R satisfy one of the following conditions.
(a) There exists lg ¥ [−.,.] and a function u: (lg,.)QR withlimx a l* u(x)=−., such that for l [ lg, x ’ le iff min{xi}=l, and forl > lg, x ’ le iff ; u(xi)=nu(l).
(b) There exists lg ¥ [−.,.] and a function u: (−., lg)QR withlimx ‘ l* u(x)=., such that for l \ lg, x ’ le iff max{xi}=l, and forl < lg, x ’ le iff ; u(xi)=nu(l).
Adding a quasi-concavity assumption in Condition 1 eliminates possi-bility 2(b). Likewise, assuming quasi-convexity eliminates 2(a). As notedabove, axiom PS is not restrictive when n=2. Consequently the theoremrequires that n > 2. Even if we invoke the stronger linearity axiom insteadof indifference monotonicity, when n=2, indifference curves in a comono-tonic sector must be parallel straight lines, but they are otherwise notrestricted (see Roberts [22, p. 430]). For example, the preferences over R2
that are represented by the utility function
U(x1, x2)=˛x1+2x2 x1 [ x22x1+x2 x1 \ x2
(20)
MIN, MAX, AND SUM 137
satisfy axioms M, S, L, and PS but cannot be represented by any of theutility functions in the theorem.
By Propositions 3 and 4 we know that lack of strict monotonicity at apoint along the main diagonal implies that the indifference curve throughthis point is either max or min. The next lemma utilizes the indifferencemonotonicity axiom to obtain restrictions on upper and lower sets of suchindifference curves.
Lemma 3. Assume that the preferences R satisfy monotonicity, symmetry,partial separability, and indifference monotonicity. If for some point l0e and forsome e > 0, l0e+ee i ’ l0e, then for all l [ l0, x ’ le if, and only if,min xi=l.
Proof. Let V be a continuous representation of R. Suppose that forsome l0 and e > 0, l0e+ee i ’ l0e. By Proposition 3, x ’ l0e iff min xi=l0.Suppose that for some l < l0, j, and eŒ > 0, le+eŒe jP le. By symmetry,these preferences hold for every j. We show that such preferences contra-dict our assumptions. Suppose first that for some eœ > 0 and j, and for thesame l, le− eœe j ’ le. Then by Proposition 4, x ’ le iff max xi=l. Letl̃=sup{l [ l0 : x ’ le iff max xi=l}. The sup is attained by continuity. Ifl̃=l0, then l0e+;n−1
i=1 e i ’ l0e− en, a violation of monotonicity. Likewise,we assume, wlg, that l0=min{l \ l̃ : x ’ le iff min xi=l}. It follows thatfor every l ¥ (l̃, l0), the preferences R are strictly monotonic at le, and byaxiom IM, the preferences are strictly monotonic along the indifferencecurve through le.
By axiom IM and continuity it is possible to find an open box B=(l0−d, l0−d)n around l0e such that the preferences R are strictly mono-tonic on the set C=B 5 {x: min xi < l0}. Since, by monotonicity, indif-ference curves are connected on C, and since strict monotonicity and axiomPS imply axiom CS, it follows from Segal [24] that R on C can be repre-sented by a transformation of an additively separable function. Usingsymmetry we obtain that on C, V(x)=h(;n
i=1 v(xi)) for continuous,strictly increasing functions h and v. Choose 0 < aŒ < d, and for a ¥ (0, aŒ]construct a sequence xm(a) ¥ C of the form xmj (a)=l
0+a for j > 1 andxm1 (a) ‘ l
0. Let x(a)=limmQ. xm(a)=l0e+a;ni=2 e i. By assumption, V is
continuous at x(aŒ), and since l0+a ¥ (l0−d, l0+d), it follows that v iscontinuous at l0+a for all a ¥ (0, aŒ]. Moreover, for all a ¥ (0, aŒ], h iscontinuous at v(l0)+(n−1) v(l0+a). To see this, note that by the strictmonotonicity of v, v(a) < v(aŒ) for all a ¥ (0, aŒ). Hence v(l0)+(n−1)×v(l0+a)=v(x1)+(n−1) v(l0+aœ) for some x1 < l0 sufficiently close to l0
and aœ ¥ (a, aŒ). Since x1e1+aœ;ni=2 e i ¥ C, h is continuous at v(l0)+
(n−1) v(l0+a). Since V(xm(a))=h(v(xm1 (a))+(n−1) v(l0+a)), we have,
by the continuity of V, h, and v,
V(x(a))=h((n−1) v(l0+a)+v(l0)).
138 SEGAL AND SOBEL
Since V(x(a)) is constant for a > 0, this equation contradicts the strictmonotonicity of h and v. L
Proof of Theorem 1. (2) S (1): Monotonicity, symmetry, and indif-ference monotonicity are obviously satisfied. We will use the second part ofProposition 1 to obtain the PS axiom. For part 2(a), let
f(x1, ..., xn−1)=˛min{xk} min{xk} [ lg
exp 1 Cn−1
k=1u(xk)2+lg min{xk} > lg
and
g(y1, y2)=˛min{y1, y2} min{y1, y2} [ lg
exp(ln(y1−lg)+u(y2))+lg min{y1, y2} > lg.
For part 2(b) of the theorem, let
f(x1, ..., xn−1)=˛max{xk} max{xk} > lg
− exp 1− Cn−1
k=1u(xk)2+lg max{xk} [ lg
and
g(y1, y2)=˛max{y1, y2} max{y1, y2} > lg
− exp(ln(−y1+lg)−u(y2))+lg max{y1, y2} [ lg.
(1) S (2): If, for all l, R are strictly monotonic at le, then by IM thepreferences R satisfy SM. By symmetry it follows that such preferences canbe represented by ; u(xi) for a strictly increasing, continuous function u(see Debreu [6] and Gorman [10]).
If there exists l such that the preferences R are not strictly monotonic atle, then, by symmetry, there are three possibilities.
1. There exists e > 0 such that for all eŒ < e, le+eŒe1 ’ le P le− eŒe1.2. There exists e > 0 such that for all eŒ < e, le+eŒe1P le ’ le− eŒe1.3. There exists e > 0 such that for all eŒ < e, le+eŒe1 ’ le ’ le− eŒe1.
As in the proof of Lemma 3, it follows by Propositions 3 and 4 that thethird case violates monotonicity. We show here that the first case impliesthe representation of part 2(a) of the theorem. The proof that the secondcase implies 2(b) is similar.
MIN, MAX, AND SUM 139
Suppose the first case holds, and let lg=sup{l : ,e > 0 such that le+ee1 ’ le}. If lg=., we are through. Otherwise, it follows by Lemma 3that for all l [ lg, x ’ le iff min{xi}=l.
By definition, for all l > lg the preferences R are strictly monotonic atle. By IM, they are strictly monotonic on (l,.)n. Again by Debreu [6]and Gorman [10], there exists a function u: (lg,.)QR such that thesepreferences satisfy x ’ le iff ; u(xi)=nu(l). Finally, if limx a l u(x) > −.,then there are two points y, yŒ such that min{yi}=min{y −i}=l, butlimx a y ; u(xi) ] limx a yŒ ; u(xi), while y ’ yŒ, a contradiction. L
Note the role that axiom IM plays in the proof.
Example 1. Define U: RnQR by
U(x)=˛min{xi} min{xi} > 1
Dn
i=1min{xi, 1} min{xi} ¥ [0, 1]
min{xi} min{xi} < 0.
The preferences that are represented by this function are monotonic,symmetric, quasi-concave, and satisfy axiom PS. They fail to satisfy theconclusion of the theorem, as an interval over which preferences are strictlymonotonic along the diagonal is sandwiched between two nonempty sets inwhich preferences are min xi. The example fails to satisfy the assumptionsof the theorem. Although preferences are strictly monotonic at le forl ¥ (0, 1), preferences are not strictly monotonic for all points on such anindifference curve.
5. LINEARITY AND COMONOTONIC FLATNESS
In this section we discuss the implications of axiom L on our results.
Theorem 2. Let n \ 3. The following two conditions on the preferencesR over Rn are equivalent.
1. R satisfy monotonicity, symmetry, linearity, and partial separa-bility.
2. R can be represented by one of the following functions.
(a) U(x1, ..., xn)=max{xk}.(b) U(x1, ..., xn)=min{xk}.(c) U(x1, ..., xn)=; xk.
140 SEGAL AND SOBEL
When partial separability is replaced with complete separability, Maskin[17] proves the equivalence of the first condition and the third possiblerepresentation in a social choice framework.
Proof. (2) S (1): Since on comonotonic sectors, the three functionssuggested by the theorem are linear, and since the changes that are per-mitted by axiom L do not take a point to a new comonotonic sector, itfollows that all three functions satisfy axiom L. Monotonicity and symme-try are obviously satisfied. Using Proposition 1, the PS axiom followseasily. For case (a), let f(x1, ..., xn−1)=max{xk} and g(x1, x2)=max{x1, x2}; for case (b), let f(x1, ..., xn−1)=min{xk} and g(x1, x2)=min{x1, x2}; and for case (c), let f(x1, ..., xn−1)=; xk and g(x1, x2)=x1+x2.
(1) S (2): Since axiom L is a property of preferences, rather than ofrepresentation functions, we can choose U such that U(le)=l. Then forevery x,
U(bx)=bU(x) and U(x+ae)=U(x)+a. (21)
We first analyze preferences that satisfy the complete separability axiom.
Lemma 4. Let M be a comonotonic section of Rn. If the preferences RonM satisfy monotonicity, linearity, and complete separability, then they canbe represented by a function of the form ; akxk.
Maskin [17] proved this lemma (with a1=·· ·=an=1) under the addi-tional assumption of symmetry.
Proof. By an extension of theorems of Debreu [6] and Gorman [10](see Wakker [26]), we know that R onM can be represented by ; uk(xk),so on M, U(x)=h(; uk(xk)) for some increasingly monotonic function h.By Eq. (21) we obtain for all x ¥M and sufficiently small a
h 1C uk(xk+a)2=h 1C uk(xk)2+a (22)
and for b sufficiently close to 1
h 1C uk(bxk)2=bh 1C uk(xk)2 . (23)
The functions h and uk, k=1, ..., n, are monotonic; therefore almosteverywhere differentiable. The rhs of Eq. (22) is differentiable with respect
MIN, MAX, AND SUM 141
to a; therefore h and uk are differentiable functions. Differentiate thisequation with respect to a to obtain
hŒ 1C uk(xk+a)2 ·C u −k(xk+a)=1. (24)
In particular, for a=0 we obtain
hŒ 1C uk(xk)2 ·C u −k(xk)=1. (25)
Differentiate Eq. (22) with respect to xa and obtain
hŒ 1C uk(xk+a)2 u −a(xa+a)=hŒ 1C uk(xk)2 u −a(xa). (26)
From Eqs. (24) and (25) it follows that hŒ(; uk(xk+a)) and hŒ(; uk(xk))are not zero. Therefore, if u −a(xa)=0, then by Eq. (26), for all a,u −a(xa+a)=0 and ua(xa) — aa. If, for all a, u −a(xa)=0, then by the aboveargument the claim is satisfied with a1=·· ·=an=0. Otherwise, supposewlg that u −1(x1) ] 0. If, for all other a, u −a(xa)=0, then R are representedby v(x1) :=h(u1(x1)+;n
k=2 ak), where for every a,
v(x1+a)=h 1u1(x1+a)+Cn
k=2ak 2
=h 1u1(x1)+Cn
k=2ak 2+a=v(x1)+a.
(The second equation follows by Eq. (22)). Similarly, by Eq. (23),v(bx1)=bv(x1). Hence v is linear. So suppose u −1(x1) ] 0 and u −2(x2) ] 0.From Eq. (26) it follows that
u −1(x1+a)u −1(x1)
=hŒ(; uk(xk))hŒ(; uk(xk+a))
=u −2(x2+a)u −2(x2)
.
Fix x2 and consider x1 and a as variables to obtain that
u −1(x1+a)u −1(x1)
=g(a)S u −1(x1+a)=g(a) u−
1(x1). (27)
The solution of this functional equation is u −1(x1)=gezx1 and g(a)=eba (see
Aczél [1, p. 143, Theorem 2]). Hence u1(x1)=a1eb1x1+c1 if z ] 0, andu1(x1)=a1x1+c1 if z=0. Similarly, for every k, if u −k(xk) ] 0, then eitheruk(xk)=akebkxk+ck, bk ] 0, or uk(xk)=akxk+ck. Since we can definehg(z)=h(z+; ck), we can assume wlg that ck=0 for all k.
142 SEGAL AND SOBEL
If for some k, uk(xk)=akxk, then from Eq. (27) it follows that g(a)=1.On the other hand, if uk(xk)=akebkxk, then g(a)=ebka. In other words,either for every k such that u −k ] 0 we have uk(xk)=akebkxk or for all such k,uk(xk)=akxk.
Suppose the first case. By Eq. (25), if ; uk(xk) does not change, thenneither should ; u −k(xk). Moreover, since by monotonicity hŒ is almosteverywhere nonzero, ; uk(xk) and ; u −k(xk) have the same indifferencecurves. In particular, at each point ; uk and ; uŒ should have the sameMRS (marginal rate of substitution). The MRS between k and a for ; uk isgiven by akbkebkxk/aabaebaxa, while the corresponding MRS for ; u −k is givenby akb
2kebkxk/aab
2aebaxa; hence bk=ba=b. Differentiate Eq. (23) with respect
to b, set b=1, and obtain
hŒ 1C uk(xk)2 ·C xku −k(xk)=h 1C uk(xk)2 .
Therefore, the MRS between k and a for the function ; xku −k(xk) must bethe same as the corresponding MRS for ; uk(xk). It follows that
akbebxk+akb2xkebxk
aabebxa+aab2xaebxa=akbebxk
aabebxaS b=0
and hence the claim of the lemma. L
Conclusion 1. If we add symmetry to the assumptions of Lemma 4 weobtain that R can be represented by ; xk (see Maskin [17]).
Suppose now that there exists a point x where the preferences R satisfystrict monotonicity. As noted above (see discussion after Definition 2), thelinearity axiom then implies that the preferences R are strictly monotonic.Lemma 4 then implies the third possible representation of the theorem. Onthe other hand, if for some x, i, and e > 0, x+ee i ’ x, then linearity impliesthat for all l, le+ee i ’ le. Proposition 3 then implies the second possiblerepresentation of the theorem. Finally, if for some x, i, and e > 0,x− ee i ’ x, then the first possible representation of the theorem likewisefollows by Proposition 4. L
The linearity axiom plays two roles in the proof of Theorem 2. First, itenables us to show that when preferences are completely separable, thenthey can be represented by a linear function. Second, it guarantees that afailure of monotonicity will hold throughout a comonotonic sector.Theorem 1 provides one characterization of preferences when the linearityaxiom does not hold, while Example 1 demonstrates that some assumptionis needed to have control over the way in which local violations of mono-tonicity influence the global behavior of preferences. To isolate the second
MIN, MAX, AND SUM 143
FIG. 2. Theorems 1–3.
role of the linearity axiom, we discuss the implications of replacing linearitywith the weaker condition of comonotonic flatness.
Theorem 3. Let n \ 3. The following two conditions on the preferencesR are equivalent.
1. R satisfy monotonicity, symmetry, comonotonic fatness, and partialseparability.
2. R can be represented by one of the following functions.
(a) U(x1, ..., xn)=max{xk}.(b) U(x1, ..., xn)=min{xk}.(c) U(x1, ..., xn)=; u(xk) for some strictly increasing u.
Proof. Axiom CF implies IM; therefore Propositions 3 and 4 guaranteethat when preferences fail to be strictly monotonic, they can be representedby either the min or the max function. By Proposition 2, if preferencessatisfy axiom SM, then they satisfy axiom CS. By Debreu [6] and Gorman[10] the theorem follows. L
Figure 2 summarizes Theorems 1–3. In all three cases we assume that thepreferences satisfy monotonicity, symmetry, and partial separability.
Remark. All the results of the paper can be obtained for a symmetricbox (a, b)n …Rn. The only place where a more detailed (but trivial)
144 SEGAL AND SOBEL
argument is needed is when the linearity axiom is invoked.8 We omit this
8 In the context of social choice, Blackorby and Donaldson [3] assume complete separa-bility and split linearity into ratio-scale comparability and translation-scale comparability. Ifthey assume only the multiplicative part of linearity (ratio-scale comparability) their assump-tions characterize a form of generalized mean. Their arguments are complicated by the sign ofxi, hence domain restrictions could be important when one assumes only the second conditionin the linearity assumption.
discussion. Note, however, that the box needs to be symmetric, as ourresults strongly depend on the symmetry axiom. This is in contrast withpapers dealing with complete separability, where the main diagonal usuallyplays no important role.9
9 An exception is Wakker [26], where preferences are defined over a comonotonic sector.
6. HARSANYI AND RAWLS
Social welfare functions are usually functions of individual (vonNeumann and Morgenstern, vN&N) utilities. This approach and its axio-matic foundations have a serious flaw. vN&M utility functions are uniqueup to increasing affine transformations, but only one version of each utilityfunction appears in the social welfare function. It is not clear which repre-sentation of individual preferences should enter the social welfare function.(For a rigorous discussion of this problem, see Weymark [27].) Recently,three similar (axiomatic) choices of affine transformations were offered byDhillon and Mertens [7], Karni [14], and Segal [25]. Here we suggestanother possible choice of the vN&M utility functions and illustrate someof the paper’s results. The trick is to have social preferences defined directlyover allocations, without making references to individual preferences. This ispossible because of the specific setup of the social choice problem we discuss.
An n-person society has to allocate, with probability p, m < n units of anindivisible good. For example, there is a p-probability that the country willhave to go to war, in which case the army will need to draft m extra sol-diers. If each person can receive at most one unit of this good,10 then there
10 If individuals could receive more than one unit (while others received none), then theassumption that society is indifferent regarding the actual receiver of each unit (see below) ismuch less appealing.
are (mn ) pure social policies, none of which are egalitarian. Indeed, by defi-nition, it is impossible to obtain an egalitarian ex post distribution of the munits. But society is not bound, ex ante, to choose such a pure policy. Asargued by Harsanyi [11–13], society can randomize and may have pref-erences over such randomizations. In our case, society may assign eachmember i a probability pi denoting the probability this person will be drafted.
MIN, MAX, AND SUM 145
The constraints are
1. ; pi=pm ; and2. pi [ p, i=1, ..., n.
Society can randomize over pure social policies (that is, over m personselections) such that exactly m people will be drafted, and the probabilityperson i will be selected is pi (see the Appendix for such a procedure). Weassume that society has preferences R over distributions of the form(p1, ..., pn).11 Assuming that society’s preferences depend only on
11 Observe that although these are probabilities, the vector (p1, ..., pn) is not a probabilitydistribution, as ; pi=pm. Further, since precisely m people are drafted, the probabilities arenot stochastically independent.
(p1, ..., pn) is a strong assumption. It requires that society is indifferentbetween two random social policies that lead to the same distribution ofprizes. Such preferences may, but need not, favor egalitarian distributionsover uneven ones. These preferences are ex ante, and they compare proba-bility allocations that cannot be compared ex post. Specifically, oncesociety knows whether it is in war, the probability p is no longer relevant,and the new social choice problem becomes to draft m people into thearmy. One may indeed suggest deferring such decisions till later, moreover,since m too may change, to wait until we know exactly how many soldiersneed to be drafted.
The separation of the actual social choice problem from other situationsthat could have, but did not, happen is far from being obvious. Harsanyi[12] is usually understood to assume such separation. Yaari [28] pointsout that such a separation may make (weighted) utilitarian and Rawlsianallocations identical. The issue is formally explored in Segal [25], where itis assumed that a solution must apply simultaneously to all possible sets ofresources. We offer a similar approach and would like social preferences tohold at the point where the eventuality of war is still uncertain and thenumber of recruits m may change. Next we try to justify the assumptions ofTheorem 1 in the context of this social choice problem.
Monotonicity and partial separability need some explanation. If p and mare fixed, then the first constraint makes the ‘‘if’’ part of the monotonicityaxiom empty. We therefore assume that p and m can vary and that socialpreferences are over the set (0, 1)n. The monotonicity axiom asserts that ifp or m are increasing, and each member of society receives at least some ofthe added probability, then society is better off.12
12 Of course, if the items to be allocated are considered ‘‘bads’’ ( for example, draft service),then either the monotonicity axiom should be reversed or one should redefine the commodityto be allocated. In the draft example, the good should be ‘‘not serving in the army,’’ andsociety will have n−m units of this good.
146 SEGAL AND SOBEL
TABLE I
i=1, ..., 14 i=15 i=16, ..., 100
A pi=12 pi=
12 pi=
12
B pi=12 pi=1 pi=
12−
1170
C pi=e pi=12 pi=
12
D pi=e pi=1 pi=12−
1170
Partial separability too is reasonable in the present context. Changingthe probability for one person should not strictly reverse the induced orderon the rest of the probability distribution. This assumption makes senseeven if there are special links between individuals. Suppose, for example,that 15% or the population are of race r1, while the remaining 85% are ofrace r2. For simplicity, let n=100. Let e be close to zero, and consider thefour distributions in Table I.
If the allocation is not racially biased (as is the case with allocations Aand B), then society will probably (strictly) prefer the more egalitarian dis-tribution A to B. However, if the allocation favors group r2, society mayprefer to compensate at least some of the members of group r1; henceDP C. Partial separability alone does not rule out such preferences, sincechanging person 1’s outcome from 1
2 to e may make society indifferent andthen changing person 2’s outcome from 1
2 to e may reverse these prefer-ences. Note, however, that such a reversal of preferences is ruled out bypartial set separability (see footnote 5).13
13 A strict reversal of preferences is not consistent with the assumptions used in any of ourtheorems. For all of the functional forms identified in these theorems, it is impossible forsociety to strictly prefer A to B and to strictly prefer D to C.
The linearity axiom has two parts, homogeneity and additivity. In thepresent context, homogeneity suggests that social preferences for probabil-ity distributions conditional on p are always the same. In other words,society has preferences for distributions of the probabilities needed to selectm individuals out of n. These preferences do not depend on the probabilitythat society will actually need to select these people. The additivity partsuggests that if m increases, and the added probability is equally distrib-uted, the preferences between two distributions do not change.
The most controversial of our assumptions is symmetry. This axiom isoften used in the social choice literature in reference to utilities.14 Our
14 See, for example, Diamond [8], or Ben-Porath et al. [2].
model so far has no utilities (in fact, we did not even introduce individualpreferences), so the symmetry axiom needs a fresh defense. In the absence
MIN, MAX, AND SUM 147
of information about individual well-being, it is plausible for the socialplanner to treat individual members of society as having equal rights toallocations. That is, the preferences over allocations should be independentof the identities of the agents. The symmetry axiom requires precisely thislevel of anonymity, namely that probability distributions are ranked withno reference to the identity of the individuals receiving these probabilities.15
15 For a similar intuition, but with a different notion of symmetry, see Segal [25].
Given our assumptions, Theorem 2 implies16 that society ranks probabil-
16 See the remark after Theorem 3.
ity distributions of the form (p1, ..., pn) by one of the following three socialwelfare functions: (1) ; pi, (2) min{pi}, and (3) max{pi}. Note that giventhe constraint ; pi=pm, rule 1 effectively says that society is indifferentover all feasible probability distributions.
Suppose that all members of society are expected utility maximizers.Choose a normalization of the utilities from the indivisible good such thatu1(0)=· · ·=un(0)=0 and u1(1)=· · ·=un(1)=1. Then pi stands notonly for the probability that person i will receive a unit of the good, butalso for his or her expected utility from the lottery this probability gener-ates. Rule 1 above is therefore the same as Harsanyi’s [11, 12] socialwelfare function, while rule 2 is a Rawlsian-like [21] function. Note thatwe do not claim that rule 1 yields a utilitarian social ranking, becausenothing in our model enables us to compare individual utilities. Indeed, asargued by Weymark [27], utilitarianism is inconsistent with the abovenormalization unless initially u1(1)=· · ·=un(1).
If instead of linearity we assume indifference monotonicity we may get acombination of Harsanyi and Rawls. For example, define
Ua, y(p1, ..., pn)=˛min pi min pi [ a
exp 1C uy, a(pi)2+a min pi > a,
where
ua, y(p)=˛log(p−a) a < p [ ylog(y−a)y
p p > y.
As yQ a, the area where indifference curves of Ua, y are linear becomesalmost the whole upper set of the indifference curve min{pi}=a. In otherwords, these social preferences are Rawlsian up to a certain threshold,beyond which they are Harsanyian.
148 SEGAL AND SOBEL
APPENDIX: CHOOSING m OUT OF n PEOPLE
To simplify the analysis, we assume throughout that p=1; that is,society knows for sure that it will need to draft m people.
For a set An={1, ..., n}, let Sm, n be the set of all m-elements subsets ofAn. Let p=(p1, ..., pn) ¥Rn
+ such that ; pi=m. We want to show thatthere are qS, S ¥Sm, n such that
1. For every S, qS ¥ [0, 1].2. ;S qS=1.3. For every i, ;S: i ¥ S qS=pi.
The task is obvious either when m=1 (q{i}=pi) or when m=n (forevery i, pi must be 1, and qAn=1). Let m < n, and suppose we foundappropriate q-vectors for all (mŒ, nŒ) ] (m, n) such that mŒ [ m and nŒ [ n.We now show how to construct the vector q for the pair m, n.
If for some i, pi=1, then assume, wlg, that pn=1, get the vector qŒ forthe pair m, n−1, and define qS=0 if n ¨ S and qS=q
−
S0{n} if n ¥ S. If forsome i, pi=0, then assume, wlg, that pn=0, get the vector qŒ for the pairm, n−1, and define qS=0 if n ¥ S and qS=q
−
S0{n} if n ¨ S. Otherwise,assume, wlg, that for all i, 0 < p1 [ p2 [ · · · [ pn < 1. Define qg=q{1, ..., m}=min{p1, 1−pn}, and set ri=(pi−qg)/(1−qg) for i=1, ..., m and ri=pi/(1−qg) for i=m+1, ..., n. Clearly, for all i, ri ¥ [0, 1], ; ri=m, andeither r1=0 or rn=1. In the first case, define for i=2, ..., n, r −i=ri+1, finda vector qŒ that solves the problem for the pair m, n−1 with the vector rŒ,define qS=p1 for S={1, ..., m}, qS=0 for all other sets S containingperson 1, and qS=(1−p1) q
−
SŒ otherwise, where j ¥ SŒ iff j+1 ¥ S. Since;S … {2, ..., n} q
−
S=1, it follows that ;S qS=1. By definition, ;S: 1 ¥ S qS=p1. For all other i=2, ..., m, ;S: i ¥ S qS=p1+(1−p1);S … {2, ..., n}: i ¥ S q
−
SŒ=p1+(1−p1) ri=pi. Finally, for i=m+1, ..., n, ;S: i ¥ S qS=(1−p1)×;S … {2, ..., n}: i ¥ S q
−
SŒ=(1−p1) ri=pi.If rn=1, then solve the problem for m−1, n−1 for r1, ..., rn−1, and get
the vector qŒ. Define qS=pnq−
S0{n} if n ¥ S, and qS=0 otherwise.
REFERENCES
1. J. Aczél, ‘‘Lectures on Functional Equations and Their Applications,’’ Academic Press,New York, 1966.
2. E. Ben-Porath, I. Gilboa, and D. Schmeidler, On the measurement of inequality underuncertainty, J. Econ. Theory 75 (1997), 194–204.
3. C. Blackorby and D. Donaldson, Ratio-scale and translation-scale full interpersonalcomparability without domain restrictions: Admissible social evaluation functions, Int.Econ. Rev. 23 (1982), 249–268.
MIN, MAX, AND SUM 149
4. C. Blackorby, D. Primont, and R. Russell, ‘‘Duality, Separability, and FunctionalStructure: Theory and Economic Applications,’’ North-Holland, New York, 1978.
5. C. Bliss, ‘‘Capital Theory and the Distribution of Income,’’ North-Holland–AmericanElsevier, New York, 1975.
6. G. Debreu, Topological methods in cardinal utility theory, in ‘‘Mathematical Methods inthe Social Sciences’’ (K. Arrow, S. Karlin, and P. Suppes, Eds.), Stanford UniversityPress, Stanford, 1960.
7. A. Dhillon and J.-F. Mertens, Relative utilitarianism, Econometrica 67 (1999), 471–498.8. P. A. Diamond, Cardinal welfare, individualistic ethics, and interpersonal comparison of
utility, J. Polit. Econ. 75 (1967), 765–766.9. R. Färe and D. Primont, Separability vs. strict separability: A further result, J. Econ.Theory 25 (1981), 455–460.
10. W. M. Gorman, The structure of utility functions, Rev. Econ. Stud. 35 (1968), 367–390.11. J. C. Harsanyi, Cardinal utility in welfare economics and in the theory of risk-taking,J. Polit. Econ. 61 (1953), 434–435.
12. J. C. Harsanyi, Cardinal welfare, individualistic ethics, and interpersonal comparisons ofutility, J. Polit. Econ 63 (1955), 309–321.
13. J. C. Harsanyi, ‘‘Rational Behavior and Bargaining Equilibrium in Games and SocialSituations,’’ Cambridge Univ. Press, Cambridge, UK, 1977.
14. E. Karni, Impartiality: Definition and representation, Econometrica 66 (1998), 1405–1415.15. R. D. Luce and H. Raiffa, ‘‘Games and Decisions,’’ Wiley, New York, 1957.16. K.-T. Mak, Notes on separable preferences, J. Econ. Theory 33 (1984), 309–321.17. E. Maskin, A theorem on utilitarianism, Rev. Econ. Stud. 45 (1978), 93–96.18. E. Maskin, Decision-making under ignorance with implications for social choice, Theoryand Decision 11 (1979), 319–337.
19. J. Milnor, Games against nature, in ‘‘Decision Processes’’ (R. M. Thrall, C. H. Coombs,and R. L. Davis, Eds.), Wiley, New York, 1954.
20. J. Quiggin, A theory of anticipated utility, J. Econ. Behavior and Organization 3 (1982),323–343.
21. J. Rawls, ‘‘A Theory of Justice,’’ Harvard University Press, Cambridge, MA, 1971.22. K. Roberts, Interpersonal comparability and social choice theory, Rev. Econ. Stud. 47
(1980), 421–439.23. Z. Safra and U. Segal, Constant risk aversion, J. Econ. Theory 83 (1998 ), 19–42.24. U. Segal, Additively separable representations on non-convex sets, J. Econ. Theory 56
(1992), 89–99.25. U. Segal, Let’s agree that all dictatorships are equally bad, J. Polit. Econ. 108 (2000),
569–589.26. P. P. Wakker, Continuous subjective expected utility with nonadditive probabilities,J. Math. Econ. 18 (1989), 1–27.
27. J. A. Weymark, A reconsideration of the Harsanyi–Sen debate on utilitarianism, in‘‘Interpersonal Comparisons of Well-Being’’ (J. Elster and J. Roemer, Eds.), Chap. 8,Cambridge Univ. Press, Cambridge, UK, 1991.
28. M. E. Yaari, Rawls, Edgeworth, Shapley Nash: Theories of distributive justicere-examined, J. Econ. Theory 24 (1981), 1–39.
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