On social welfare orders satisfying anonymity and asymptotic density-one Pareto
aa r X i v : . [ ec on . T H ] A ug On social welfare orders satisfying anonymity and asymptoticdensity-one Pareto
Ram Sewak Dubey ∗ Giorgio Laguzzi † Francesco Ruscitti ‡ August 14, 2020
Abstract
We study the nature (i.e., constructive as opposed to non-constructive) of social welfare orders oninfinite utility streams, and their representability by means of real-valued functions. We assume finiteanonymity and introduce a new efficiency concept we refer to as asymptotic density-one Pareto. Wecharacterize the existence of representable and constructive social welfare orders (satisfying the aboveproperties) in terms of easily verifiable conditions on the feasible set of one-period utilities.
Keywords:
Anonymity, Non-Ramsey set, Social Welfare Order, Asymptotic Density-OnePareto.
Journal of Economic Literature
Classification Numbers:
D60, D70, D90. ∗ Department of Economics, Feliciano School of Business, Montclair State University, Montclair, NJ 07043; E-mail:[email protected] † University of Freiburg in the Mathematical Logic Group at Eckerstr. 1, 79104 Freiburg im Breisgau, Germany; Email: [email protected] ‡ Department of Economics and Social Sciences, John Cabot University, Via della Lungara 233, 00165 Rome, Italy. Email:[email protected] Introduction
Imagine a social planner (e.g., a policy-maker) performing pairwise ranking of alternative policies that affectalso the welfare of generations in the far-off future. Examples of such policies are fiscal policies, issuesrelated to climate, environmental preservation, and sustainable development. Suppose the social plannerwants to do so relying on social preferences (i.e., a social welfare order) that satisfy simultaneously both“efficiency” and “equity” properties. The ideal situation would be a social welfare order (SWO, henceforth)that admits a real-valued numerical representation, i.e., a social welfare function (SWF, henceforth). If noSWF exists, it would still be possible to perform pairwise ranking of infinite utility streams, for the purposeof decision-making, as long as the SWO at hand can be described explicitly. The worst-case scenario,wherein the existence of a SWO satisfying a set of desiderata is of no practical use, is that a SWF doesnot exist, and the SWO under consideration cannot be operationalized. Which of the above three scenariosdoes the social planner find itself in? With this question in mind, and letting Y be the set of all possiblevalues that each generation’s utility can take, our main goal is to characterize the existence of representableand constructive SWO’s (that treat all generations “equally” and respect some form of the Pareto axiom) interms of conditions on Y that can be easily checked. The main results of the present paper, i.e., Theorems1 and 2, serve this purpose. What do we mean by treating all generations equally and the Pareto axiom?We consider the canonical (finite) anonymity axiom (AN, henceforth, which is a procedural equity conceptformalized in Diamond (1965)) and define a new efficiency concept, namely asymptotic density-one Pareto(ADP, henceforth). Interestingly, although a constructive SWO need not be representable, yet we find outthat there is a representable SWO satisfying AN and ADP if and only if there is a constructive SWO with thesame properties. In fact, it turns out that either way existence is equivalent to Y being finite (see Theorems1 and 2 below).Our focus on the AN and ADP concepts may be justified as follows. Given Basu and Mitra (2007b)and Demichelis et al. (2010), we know that AN is basically the weakest form of anonymity compatible withstrongly Paretian quasi-orderings on the set of infinity utility streams. Now, recall that in this work we dealwith SWO’s, and a SWO is indeed a quasi-ordering. Therefore, if we strengthened the inter-generationalequity concept to a stronger property than AN, we would risk ending up with a Paretian SWO which does notrespect “equity”. So, we use the AN axiom to formalize inter-generational equity, as is standard practice. Onthe other hand, we learn from Basu and Mitra (2003) that there does not exist any SWF satisfying AN andthe strong Pareto axiom when Y contains at least two distinct elements. Moreover, Basu and Mitra (2007a,Theorem 3) show that when Y is a subset of the set of non-negative integers, there is a SWF satisfying ANand weak Pareto. These results suggest that the choice of the efficiency axioms and the properties of Y areboth important factors in determining the existence of a numerical representation or, more in general, forestablishing a possibility result. Therefore, we are led to pursue the following research strategy: we do notchallenge the AN axiom, but we deem it worthwhile to address the following question: in moving fromstrong Pareto toward weak Pareto, how weak a notion of efficiency can one impose on anonymous SWO’sso as to obtain a possibility theorem that depends on easily verifiable properties of Y ? We would like toargue that the ADP axiom seems suitable to answer the previous question. To see this, observe, first of all,that ADP postulates that, given any two sequences x and y satisfying x > y , if x strictly dominates y ona subset of the natural numbers having asymptotic density equal to one, then x is socially preferred to y .Clearly, ADP is weaker than strong Pareto and stronger than weak Pareto. Furthermore, ADP is in a way“satisfactorily close” to weak Pareto. To see this, note that ADP is weaker than Weak Upper AsymptoticPareto which is an efficiency notion introduced in Dubey et al. (2020). In addition, recall that when theasymptotic density of a subset of natural numbers is well-defined, it coincides with the lower asymptoticdensity which, moreover, is bounded above by one. Hence, ADP is weaker than both Asymptotic Pareto2nd d -Asymptotic Pareto , and it is also weaker than all conceivable variants of LAP that could be definedby letting the threshold for the value of the lower asymptotic density vary between zero and one.Theorem 1 is a handy characterization result whose range of applicability may be appreciated by notingthat part 1 of Proposition 4.4 in Petri (2019) obtains as an immediate corollary of our Theorem 1. In terms ofthe analytical techniques employed to prove Theorem 1, we mention, in passing, that the proof of the latterhinges on Lemma 1 below, which asserts that there does not exist any social welfare function satisfying ADPand AN if Y is the set of natural numbers. While it suffices to establish an impossibility result over the set ofnatural numbers (which is the common practice, in the established literature, we stick to), we stress that theneed to exhibit a subset with asymptotic density equal to one prompts us to properly adjust the constructionused by Basu and Mitra (2003). More details can be found in section 4.1. As for Theorem 2, we deem it auseful characterization result. Let’s discuss why this is the case.Firstly, in view of part 1 of Theorem 4.1 in Petri (2019), there is no SWF satisfying LAP and AN if Y isinfinite, but from Svensson (1980) it follows that a SWO satisfying AN and any efficiency axiom weaker thanstrong Pareto, like LAP or DAP, is guaranteed to exist. Therefore, one naturally wonders whether a SWOthat respects LAP and AN on an infinite Y can be constructed. Incidentally, the constructive nature of socialwelfare orders, or the lack thereof, was first investigated by Fleurbaey and Michel (2003) who conjecturedthat “there exists no explicit (that is, avoiding the axiom of choice or similar contrivances) description ofan ordering which satisfies weak Pareto and indifference to finite permutations”. In a setting different fromthe present framework, Zame (2007) and Lauwers (2010) confirmed this conjecture showing the existenceof a non-measurable set and a non-Ramsey collection of sets, respectively. Going back to the above query,it turns out that the answer to it, which is in the negative, is a straightforward corollary of Theorem 2 (seeCorollary 1 below). We reach this conclusion by proving (see Lemma 2) the existence of a non-Ramseycollection of sets, therefore our approach follows in the footsteps of Lauwers (2010).Secondly, Dubey (2011) proved that a SWO satisfying AN and weak Pareto is constructive if and onlyif Y does not contain any subset order isomorphic to the set of integers. Arguably, it is not immediatelyobvious (without adequate inspection) that a given domain of one-period utilities does not contain a subsetorder isomorphic to the set of integers. It is easier to figure out whether it is finite or not. That’s where ourTheorem 2 comes into play. To be more concrete and fix ideas, while one can show, with some work, thatneither the set { /n : n ∈ N } nor the set { n/ ( n + ) : n ∈ N } contains a subset order isomorphic to the setof integers, it is evident that the previous sets both contain infinitely many elements. That said, consider aSWO satisfying ADP and AN. Remember that ADP is stronger than weak Pareto, so one could certainlyexploit Dubey (2011) to decide whether such a SWO is susceptible of explicit description if, say, Y is eitherof the above sets. As a matter of fact, by Dubey (2011) the SWO under consideration is a constructiveobject. Notice, however, that if one applies directly our Theorem 2 one can make the same determinationeffortlessly.The remainder of the paper is organized as follows. Section 2 gathers some preliminary concepts,notation, and presents a brief description of the notion of asymptotic density. In section 3, we define theconcept of SWO and various equity and efficiency axioms imposed on social welfare orders. In section 4,we state and prove our main results. Section 5 concludes and presents a table designed to help the readerposition our contribution in the established literature. Proofs are relegated to the Appendix. Let R , Q , and N be the set of real numbers, rational numbers, and natural numbers, respectively. For all y , z ∈ R N , we write y > z if y n > z n , for all n ∈ N ; y > z if y > z and y = z ; and y ≫ z if y n > z n The efficiency concepts of Asymptotic Pareto and d -Asymptotic Pareto (dAP, henceforth) were introduced in Petri (2019). Wewill refer to Asymptotic Pareto as lower asymptotic Pareto (LAP, henceforth). n ∈ N . Given any x ∈ R N and N ∈ N , we denote the vector consisting of the first N componentsof x by x ( N ) , and the upper tail sequence of x , from the element N + x [ N ] . Formally, x ( N ) = ( x , x , · · · , x N ) and x [ N ] = ( x N + , x N + , · · · ) .It is useful to recall the definition of asymptotic density of any S ⊂ N . As usual, let | · | denote thecardinality of a given finite set. The lower asymptotic density of S is defined as follows: d ( S ) = lim inf n → ∞ | S ∩ {
1, 2, · · · , n }| n .Similarly, the upper asymptotic density of S is defined as follows: d ( S ) = lim sup n → ∞ | S ∩ {
1, 2, · · · , n }| n .One says that S has asymptotic density d ( S ) if d ( S ) = d ( S ) , in which case d ( S ) is equal to this commonvalue. Formally, d ( S ) = lim n → ∞ | S ∩ {
1, 2, · · · , n }| n . Let Y ⊂ R be the set of all possible utilities that any generation can achieve. Then, X ≡ Y N is the set of allfeasible utility streams. We denote an element of X by x , or alternately by h x n i , depending on the context.If h x n i ∈ X , then h x n i = ( x , x , · · · ) . For all n ∈ N , x n ∈ Y represents the amount of utility that period- n generation earns.A binary relation on X is denoted by % . Its symmetric and asymmetric parts, denoted by ∼ and ≻ ,respectively, are defined in the usual manner. A social welfare order (SWO) is, by definition, a completeand transitive binary relation on X . Given a SWO % , one says that % can be represented by a real-valuedfunction, called a social welfare function (SWF), if there is a mapping W : X → R such that for all x , y ∈ X , x % y if and only if W ( x ) > W ( y ) .We will be dealing with the following equity and efficiency axioms we may want the SWO to satisfy. Definition 1.
Anonymity (AN henceforth): If x , y ∈ X , and there exist i , j ∈ N such that y j = x i and x j = y i ,while y k = x k for all k ∈ N \ { i , j } , then x ∼ y . Definition 2.
Lower Asymptotic Pareto (LAP henceforth): Given x , y ∈ X , if x > y and x i > y i for all i ∈ S ⊂ N with d ( S ) >
0, then x ≻ y . Definition 3.
Asymptotic Density-one Pareto (ADP henceforth): Given x , y ∈ X , if x > y and x i > y i forall i ∈ S ⊂ N with d ( S ) =
1, then x ≻ y .It’s easy to see that ADP is strictly weaker than LAP (i.e., LAP implies ADP but the converse is nottrue). In this section we state and prove the main results of this paper. The following example exhibits a SWFsatisfying ADP and AN if Y is finite. This function, along with the remarks below, will be instrumental forthe proof of the two main results of this paper, namely Theorems 1 and 2.4 xample 1. Let X = Y N , Y being finite, and let W : X → R be defined as follows: W ( x ) := lim inf n → ∞ P nk = x k n . (1)Next, define the following binary relation % on X : x % y if and only if W ( x ) > W ( y ) . (2)Clearly, (1) is a SWF that represents the SWO given by (2). Petri (2019, p. 860) has shown that thelatter satisfies AN and LAP. Since ADP is weaker than LAP, the SWO defined by (2) also satisfies ADP andAN (when Y is finite). Of course, since (1) is an explicit formula for the SWF, the underlying SWO (2) canbe explicitly described (it is a constructive object). In this section we prove that a necessary and sufficient condition for the existence of a SWF satisfying ADPand AN on X = Y N is that Y be finite. The bulk of Theorem 1 involves showing that there is no SWFsatisfying ADP and AN if Y is infinite. To this end, it will be helpful to prove first (in Lemma 1 below) thatno SWF satisfying the foregoing properties exists if Y = N . Indeed, as will become apparent in the proofof Theorem 1, Lemma 1 yields almost automatically the desired result. This is because any infinite subsetof the real numbers contains a subset which is order isomorphic to N or to the set of negative integers: (seePetri (2019, Lemma A.3 p. 869) for a proof).Note that in the proof of Lemma 1 we take advantage of a technique used in Basu and Mitra (2003).More specifically, we begin by noting that in order to invoke the ADP axiom we need to somehow exhibit asubset of the natural numbers (on which a strict Pareto-improvement takes place) that has asymptotic densityequal to one. We construct such a set so that it is the complement of a set with zero asymptotic density. Toget zero density, in turn, we reproduce the same construction as in Basu and Mitra (2003) except that wetransform the natural numbers considered therein into factorials. It turns out that this tweak suffices for ourpurposes. This is because, loosely put, factorial numbers are far enough apart to yield a “sparse” subset ofthe natural numbers. Lemma 1.
There does not exist any social welfare function satisfying ADP and AN on X = Y N , with Y = N . Combining Example 1 (and the related remarks) with Lemma 1, yields the following theorem:
Theorem 1.
There exists a social welfare function satisfying ADP and AN on X = Y N if and only if Y isfinite. Let T be an infinite subset of N . We denote by Ω ( T ) the collection of all infinite subsets of T , and we let Ω denote the collection of all infinite subsets of N . Definition 4.
A collection of sets Γ ⊂ Ω is said to be non-Ramsey if for every T ∈ Ω , the collection Ω ( T ) intersects both Γ and its complement Ω (cid:31) Γ .The bulk of Theorem 2 below involves showing that there is no constructive SWO satisfying ADP andAN on X = Y N if Y is infinite. We achieve this goal by proving that the existence of a SWO (defined on X = Y N , where Y is an infinite set) satisfying AN and ADP implies the existence of a non-Ramsey collection5f set. Observe, again, that any infinite set of real numbers contains a subset order isomorphic to the set ofnatural numbers (or to the set of negative integers). Therefore, it will suffice to prove (in Lemma 2 below)that when Y = N , the existence of a SWO satisfying AN and ADP entails the existence of a non-Ramseycollection of set.The proof of Lemma 2 below is inspired by Lauwers (2010). We start from the same sequence of naturalnumbers as Lauwers’s, but we convert its elements into factorial numbers. Then, following Lauwers’sapproach, we use that sequence to construct a partition of the set of natural numbers. However, we make thefollowing change to Lauwers’s partition: loosely put, we expand and shrink any two consecutive subsets,respectively, that belong to the aforementioned partition. We do that without altering the union of suchsets. We resort to this particular partition as it allows us to prove that the asymptotic density of the key setsinvolved in the proof of Lemma 2 (where we appeal to the ADP axiom) is equal to one. Lemma 2.
Let Y = N , and assume that there is a social welfare order on X = Y N satisfying AN and ADP.Then, there exists a non-Ramsey set. Combining Example 1 with Lemma 2, we get the following theorem.
Theorem 2.
There exists a constructive social welfare order satisfying ADP and AN on X = Y N if and onlyif Y is finite. The following corollary is an easy consequence of Theorem 2 and Example 1 (see the comments below ( ) ): Corollary 1.
There exists a constructive social welfare order satisfying LAP and AN on X = Y N if and onlyif Y is finite. In this paper we have focused on SWOs, on infinite utility streams, satisfying the ADP and AN axioms.Apart from the considerations already articulated in the introduction, it is also worth considering the fol-lowing table in order to highlight the scope of our results in relation to the existing literature. The tablesummarizes some findings (concerning the representability and nature of social welfare orders) obtainedwhen the anonymity axiom is combined with various forms of the Pareto axiom.Table
Pareto axiom | Y | Representation Constructive natureStrong > > > eferences K. Basu and T. Mitra. Aggregating infinite utility streams with intergenerational equity: The impossibilityof being Paretian.
Econometrica , 71(5):1557–1563, 2003.K. Basu and T. Mitra. Possibility theorems for aggregating infinite utility streams equitably. In J. Roemer andK. Suzumura, editors,
Intergenerational Equity and Sustainability (Palgrave) , pages 69–74. (Palgrave)Macmillan, 2007a.K. Basu and T. Mitra. On the existence of Paretian social welfare quasi orderings for infinite utility streamswith extended anonymity. In J. Roemer and K. Suzumura, editors,
Intergenerational Equity and Sustain-ability (Palgrave) , pages 85–99. (Palgrave) Macmillan, 2007b.J. A. Crespo, C. Núñez, and J. P. Rincón-Zapatero. On the impossibility of representing infinite utilitystreams.
Economic Theory , 40(1):47–56, 2009.S. Demichelis, T. Mitra, and G. Sorger. Intergenerational equity and stationarity.
Vienna Economics Papers1003 , 2010.P. A. Diamond. The evaluation of infinite utility streams.
Econometrica , 33(1):170–177, 1965.R. S. Dubey. Fleurbaey-Michel conjecture on equitable weak Paretian social welfare order.
Journal ofMathematical Economics , 47(4-5):434–439, 2011.R. S. Dubey, G. Laguzzi, and F. Ruscitti. On the representation and construction of equitable social welfareorders.
Mathematical Social Sciences , forthcoming, 2020.M. Fleurbaey and P. Michel. Intertemporal equity and the extension of the Ramsey criterion.
Journal ofMathematical Economics , 39(7):777–802, 2003.L. Lauwers. Ordering infinite utility streams comes at the cost of a non-Ramsey set.
Journal of MathematicalEconomics , 46(1):32–37, 2010.H. Petri. Asymptotic properties of welfare relations.
Economic Theory , 67:853–874, 2019.L. G. Svensson. Equity among generations.
Econometrica , 48(5):1251–1256, 1980.W. R. Zame. Can utilitarianism be operationalized?
Theoretical Economics , 2:187–202, 2007.
We begin by stating two claims which are useful for the proof of Lemma 1. The easy proof of both of themis left to the reader.
Claim 1.
Given the set A := { n ! : n ∈ N } , we have that d ( A ) =
0. Therefore, d ( B ) =
1, where B := N \ A . Claim 2.
Let A , B ⊆ N be such that A △ B is finite. Then d ( A ) = d ( B ) . Proof of Lemma We establish the claim by contradiction. Let W : X → R be a SWF satisfying ADP andAN. Let q , q , · · · be any arbitrary enumeration of rational numbers in (
0, 1 ) . We keep this enumerationfixed throughout the proof. Pick r ∈ (
0, 1 ) and let u ( r ) := min { n ∈ N : q n ∈ [ r , 1 ) } . Having defined u ( r ) ,for every k > u k + ( r ) := min { n ∈ N \ { u ( r ) , u ( r ) , · · · , u k ( r ) } : q n ∈ [ r , 1 ) } .Note that u ( r ) < u ( r ) < · · · < u k ( r ) < · · · . Thus, we can define u ( r ) as follows: u ( r ) = { u ( r ) , u ( r ) , · · · , u k ( r ) , · · · } .Let U ( r ) := { u ( r ) !, u ( r ) !, · · · , u k ( r ) !, · · · } . Define L ( r ) := N \ U ( r ) = { l ( r ) , l ( r ) , · · · , l k ( r ) , · · · } , We denote the symmetric set difference for sets A and B as A △ B := ( A \ B ) ∪ ( B \ A ) . l ( r ) < l ( r ) < · · · < l k ( r ) < l k + ( r ) < · · · .We define the utility stream h x ( r ) i as follows: x t ( r ) = (cid:12) t ∈ U ( r ) , m + t = l m ( r ) , m ∈ N . (3)Observe that the sequence h x ( r ) i takes values 2, 3, · · · in increasing order at coordinates in the set L ( r ) andtakes constant value of one for all coordinates in the set U ( r ) . Thus, x l = x l = x u k ( r ) ! = k ∈ N . Next, we define utility stream h z ( r ) i in an identical fashion by taking U ( r ) = U ( r ) \ { u ( r ) ! } , and so L ( r ) = L ( r ) ∪ { u ( r ) ! } : z t ( r ) = (cid:12) t ∈ U ( r ) , m + t = l m ( r ) , m ∈ N , (4)where we consider as above the usual increasing enumeration U ( r ) := { u m ( r ) : m ∈ N } and L ( r ) := { l m ( r ) : m ∈ N } . Thus, l ( r ) = l ( r ) , l ( r ) = l ( r ) , · · · , l u ( r ) ! − ( r ) = u ( r ) ! − = l u ( r ) ! − ( r ) , and, l u ( r ) ! ( r ) = u ( r ) ! = l u ( r ) ! ( r ) − · · · , l k ( r ) = l k ( r ) −
1, for all k > u ( r ) !.Hence, • x u ( r ) ! ( r ) = < z u ( r ) ! ( r ) • ∀ k < u ( r ) ! ( x k ( r ) = z k ( r )) • ∀ k ∈ U ( r )( k > u ( r ) ! ⇒ x k ( r ) = = z k ( r )) • ∀ k ∈ L ( r )( k > u ( r ) ! ⇒ x k ( r ) < z k ( r )) .Let S := [ u ( r ) !, u ( r ) ! ) ∩ N andfor all k ∈ N , k >
1, let S k := ( u k ( r ) !, u k + ( r ) ! ) ∩ N and S := ∞ [ k = S k It follows from above that for all t ∈ S , we have z t ( r ) > x t ( r ) . Applying Claim 1 and 2 above, one caneasily show that d ( S ) =
1. Hence, by ADP we get x ( r ) ≺ z ( r ) , therefore W ( x ( r )) < W ( z ( r )) . (5)Next, we pick s ∈ ( r , 1 ) for which h x ( s ) i and h z ( s ) i are defined using the same construction as above.Observe that U ( s ) ⊂ U ( r ) and let U ( rs ) := U ( r ) \ U ( s ) . Note that there are infinitely many natural numbersin U ( rs ) and U ( s ) since there are infinitely many rational numbers q n ∈ [ r , s ) and q n ∈ [ s , 1 ) respectively.Let u ( rs ) ! := min { n ! : n ! ∈ U ( rs ) } and u ( rs ) ! := min { n ! : n ! ∈ U ( rs ) \ { u ( rs ) ! }} .More generally, it is clear that we can consider the usual enumerations U ( s ) = { u k ( s ) ! : k ∈ N } and U ( rs ) = { u k ( rs ) ! : k ∈ N } . Also notice that U ( s ) and U ( rs ) form a partition of U ( r ) , which means U ( s ) ∩ U ( rs ) = ∅ and U ( s ) ∪ U ( rs ) = U ( r ) . As a consequence, for every k ∈ N , any t ∈ U ( r ) such that u k ( rs ) ! < t = z u N ( r ) ! ( r ) • ∀ k < u N ( r ) ! ( x k ( s ) = z k ( r )) • ∀ k ∈ U ( s )( k > u N ( r ) ! ⇒ z k ( r ) = = x k ( s )) • ∀ k ∈ L ( s )( k > u N ( r ) ! ⇒ z k ( r ) < x k ( s )) .By construction, U ( s ) ⊂ U ( r ) which gives L ( s ) ⊃ L ( r ) ; then 1 = d ( L ( r )) d ( L ( s )) and so by Claim 2also d ( S ) =
1, where S := { n ∈ L ( s ) : n > u N ( r ) ! } . By ADP we then get the desired z ( r ) ≺ x ( s ) .Case (b): u ( s ) ! = u ( r ) !, i.e., q u ( r ) ∈ [ s , 1 ) . In this case U ( s ) \ { u ( r ) ! } ( U ( r ) , which means we canstill rely on the above argument, but in this case we have to permute finitely many elements in order toswitch the position k = u ( r ) ! and for other finitely many k ’s, where x k ( s ) < z k ( r ) . More precisely,comparing the streams x ( s ) and z ( r ) , the following hold: • ∀ t < u ( r ) !, z t ( r ) = x t ( s ) and z u ( r ) ! ( r ) > = x u ( r ) ! ( s ) , • ∀ t ∈ ( u ( r ) !, u ( rs ) ! ) ∩ U ( r ) , z t ( r ) = x t ( s ) =
1, and z u ( rs ) ! ( r ) = < x u ( rs ) ! ( s ) , • ∀ t ∈ ( u ( r ) !, u ( rs ) ! ) ∩ L ( r ) , z t ( r ) = x t ( s ) + > x t ( s ) > • ∀ t ∈ ( u ( rs ) !, u ( rs ) ! ) ∩ U ( r ) , z t ( r ) = x t ( s ) =
1, and z u ( rs ) ! ( r ) = < x u ( rs ) ! ( s ) , • ∀ t ∈ ( u ( rs ) !, u ( rs ) ! ) ∩ L ( r ) , z t ( r ) = x t ( s ) > • ∀ t > u ( rs ) !, t ∈ L ( s ) , x t ( s ) > z t ( r ) + > z t ( r ) , and • ∀ t > u ( rs ) !, t ∈ U ( s ) , x t ( s ) = z t ( r ) = π of coordinates t ∈ [ u ( r ) !, u ( rs ) ! ] ∩ L ( r ) in sequence h z ( r ) i to obtain h z ′ i := h z π ( t ) : t ∈ N i . More precisely, let { a , a , . . . a K } be an increasing enumeration of [ u ( r ) !, u ( rs ) ! ] ∩ L ( r ) , then we can define the finite permuntation π as follows: π ( t ) := t if t / ∈ [ u ( r ) !, u ( rs ) ! ] u ( rs ) ! if t = u ( r ) ! a j − if t = a j (for j > z ′ satisfies the following properties: • z ′ u ( r ) ! = z u ( rs ) ! ( r ) = • ∀ t ∈ ( u ( r ) !, u ( rs ) ! ] ∩ U ( r ) , z ′ t = z t ( r ) , i.e., no change, • since | ( u ( r ) !, u ( rs ) ! ] ∩ L ( r ) | = | [ u ( r ) !, u ( rs ) ! ) ∩ L ( r ) | , z ′ t = z t ′ ( r ) , where t ′ and t occupy thesame position in the increasing enumerations of the sets [ u ( r ) !, u ( rs ) ! ) ∩ L ( r ) and ( u ( r ) !, u ( rs ) ! ] ∩ L ( r ) respectively, • z ′ t = z t ( r ) for remaining t ∈ N .Hence, by AN, we have z ′ ∼ z ( r ) , and W ( z ′ ) = W ( z ( r )) . Also x t ( s ) > z ′ t for all t ∈ N and furthermore • ∀ t < u ( rs ) !, z ′ t x t ( s ) , • z ′ u ( rs ) ! = < x u ( rs ) ! ( s ) . 9 ∀ t > u ( rs ) !, t ∈ L ( s ) , x t ( s ) > z ′ t + > z ′ t , and • ∀ t > u ( rs ) !, t ∈ U ( s ) , x t ( s ) = z ′ t = S := { n ∈ L ( s ) : n > u ( rs ) ! } we havethat for all t ∈ S , x t ( s ) > z ′ t and d ( S ) =
1. Therefore, by ADP z ′ ≺ x ( s ) , hence W ( z ′ ) < W ( x ( s )) . (7)Combining z ( r ) ∼ z ′ and z ′ ≺ x ( s ) , we get z ( r ) ≺ x ( s ) , hence W ( z ( r )) < W ( x ( s )) . (8)Consequently, in both cases, we obtain W ( z ( r )) < W ( x ( s )) . (9)Therefore, (5) and (9) imply that ( W ( x ( r )) , W ( z ( r )) and ( W ( x ( s )) , W ( z ( s )) are non-empty and disjointopen intervals. Hence, because r and s , with r < s , were arbitrary, by density of Q in R we conclude that wehave found a one-to-one mapping from (
0, 1 ) to Q , which is impossible as the latter set is countable. Proof of Theorem Suppose that there is a social welfare function W satisfying ADP and AN on X = Y N .We must show that Y is finite. To see this, suppose, by way of contradiction, that Y is infinite, and let A bea subset of Y which is order isomorphic to N . Then, the restriction of W to A N , say W , is still a SWF thatsatisfies ADP and AN. Next, observe that there is an order-preserving bijection between A and N , let’s say f : N → A . Therefore, it’s routine matter to verify that a suitable composition of f with W maps N N into R and satisfies ADP and AN, which contradicts Lemma 1.Conversely, assume that Y is finite. Then, it readily follows from Example 1 and the relative remarksthat the mapping defined by 1) is a SWF that satisfies ADP and AN.The following results (regarding the lower asymptotic density of subsets of N ) will be used in the proofof Lemma 2. Let N be a sequence of natural numbers such that n < n < n < · · · denoted as N := { n k : k ∈ N } . We partition the set of natural numbers, N , in two infinite subsets based on N . U k ( N ) := (cid:18) n k − !, n k + ! − n k + ! n k ! (cid:21) ∩ N , k ∈ N , U ( N ) := ∞ [ k = U k ( N ) , L ( N ) := [ n ! ] ∩ N , L k + ( N ) := (cid:18) n k + ! − n k + ! n k ! , n k + ! (cid:21) ∩ N , k ∈ N , and L ( N ) := ∞ [ k = L k ( N ) .Note that for all k ∈ N , we have, n k + ! − n k + ! n k ! > n k !.To see this, observe that the inequality above is equivalent to the following inequality,1 − n k ! > n k ! n k + ! .10ince, n k > n k + > n k + n k + >
3, we obtain1 − n k ! > − = > > n k + > n k ! n k + ! .Furthermore, | U ( k ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) n k − !, n k + ! − n k + ! n k ! (cid:21) ∩ N (cid:12)(cid:12)(cid:12)(cid:12) = n k + ! − n k + ! n k ! − n k − !.Hence, | U ( k ) | n k + ! = n k + ! − n k + ! n k ! − n k − ! n k + ! = − n k ! − n k − ! n k + ! .Observe that n k + > n k + > n k − +
2. Thus, n k − ! n k + ! n k − ! ( n k − + ) ( n k − + ) ( n k − ! ) = ( n k − + ) ( n k − + ) →
0, when k → ∞ .Therefore, lim k → ∞ | U ( k ) | n k + ! = l < k , | U ( l ) | n k + ! = n l + ! − n l + ! n l ! − n l − ! n k + ! →
0, when k → ∞ .Therefore, d ( U ( N )) = lim inf m → ∞ (cid:12)(cid:12)(cid:12)(cid:12) m S k = U k ( N ) (cid:12)(cid:12)(cid:12)(cid:12) n m + ! = lim m → ∞ (cid:18) | U m ( N ) | n m + ! (cid:19) = d ( U ( N )) =
1. (10)We also consider sequence N \ { n } . Corresponding partitions the set of natural numbers N are b U k ( N ) := (cid:18) n k !, n k + ! − n k + ! n k + ! (cid:21) ∩ N , k ∈ N , b U ( N ) := ∞ [ k = b U k ( N ) , b L ( N ) := [ n ! ] ∩ N , b L k + ( N ) := (cid:18) n k + ! − n k + ! n k + ! , n k + ! (cid:21) ∩ N , k ∈ N , and b L ( N ) := ∞ [ k = b L k + ( N ) .By using a similar argument as for U ( N ) above, one can obtain d (cid:16) b U ( N ) (cid:17) =
1. (11)11 roof of Lemma Given sequence N defined above, we describe the set U ( N ) as U ( N ) = { u k : u k < u k + , for all k ∈ N } .We define the utility stream x ( N ) whose components are, x t ( N ) = t ∈ L ( N ) ,2 if t = u x u l + t = u l + , for l ∈ N . (12)We also construct the sequence y ( N ) using the subset N \ { n } , with its components being defined as follows y t ( N ) = t ∈ b L ( N ) ,2 if t = b u y b u l + t = b u l + , for l ∈ N . (13)Let % be a social welfare order satisfying AN and ADP. We claim that the collection of sets Γ ≡ { N ∈ Ω : x ( N ) ≺ y ( N ) } is non-Ramsey. We must show that for each T ∈ Ω , the collection Ω ( T ) intersects both Γ and Ω (cid:31) Γ . To this end, it is sufficient to show that for every T ∈ Ω (1) if T ∈ Γ , then there exists S ∈ Ω ( T ) such that S / ∈ Γ , and (2) if T / ∈ Γ , then there exists S ∈ Ω ( T ) such that S ∈ Γ . As the binary relation isassumed to be complete, one of the following cases must be true: (a) x ( T ) ≺ y ( T ) ; (b) y ( T ) ≺ x ( T ) ; and (c) x ( T ) ∼ y ( T ) . We now consider each of these three cases.(a) Let x ( T ) ≺ y ( T ) or T ∈ Γ . Take S = T − { t } = { t , t , t , · · · } . Note that S ∈ Ω ( T ) .(i) Since y t ( T ) = x t ( T − { t } ) = x t ( S ) ∀ t ∈ N , we get y ( T ) ∼ x ( S ) . (14).(ii) y t ( S ) = x t ( T − { t , t } ) yields • x t ( T ) > y t ( S ) for all t ∈ N , • y t ( S ) = x t ( T ) = t ∈ L ( T ) , • x t ( T ) > = y t ( S ) for all t ∈ U ( T ) which implies x t ( T ) − y t ( S ) >
1, and • x t ( T ) > y t ( S ) >
1, for all t ∈ U ( T ) .Applying (10) we know that d ( U ( T )) =
1. Using ADP, we obtain y ( S ) ≺ x ( T ) (15)Therefore, by (14) and (15) we get y ( S ) ≺ x ( T ) ≺ y ( T ) ∼ x ( S ) . By transitivity, y ( S ) ≺ x ( S ) , whichestablishes that S / ∈ Γ , as was to be proven.(b) Let y ( T ) ≺ x ( T ) or T / ∈ Γ . We drop t and t , t , · · · , t m , t m + (with m >
2) to obtain the set S = { t , t , t m + , t m + , · · · } such that | U ( T ) | < | L ( T ) | + | L ( T ) | + · · · + | L m + ( T ) | .Note that S ∈ Ω ( T ) . We claim (cid:12)(cid:12)(cid:12)b L ( T ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)b L ( T ) (cid:12)(cid:12)(cid:12) + · · · + (cid:12)(cid:12)(cid:12)b L m + ( T ) (cid:12)(cid:12)(cid:12) < | L ( S ) | . (16)12o see this, observe that (cid:12)(cid:12)(cid:12)b L ( T ) (cid:12)(cid:12)(cid:12) = t ! t ! , (cid:12)(cid:12)(cid:12)b L ( T ) (cid:12)(cid:12)(cid:12) = t ! t ! , · · · , (cid:12)(cid:12)(cid:12)b L m + ( T ) (cid:12)(cid:12)(cid:12) = t m + ! t m + ! , | L ( S ) | = t m + ! t ! .Also the desired inequality is equivalent to the following condition t m + ! t ! > t m + ! t m + ! + · · · + t ! t ! . (17)The right hand side member of the above inequality (17) contains m terms. We rearrange (17) as thesum of m terms (cid:18) t m + ! t ! − m t m + ! t m + ! (cid:19) + (cid:18) t m + ! t ! − m t m ! t m − ! (cid:19) + · · · + (cid:18) t m + ! t ! − m t ! t ! (cid:19) >
0. (18)To prove the above inequality, it suffices to show that each of the m expressions inside the parenthesesare positive. The expression in the first parenthesis is (cid:18) t m + ! t ! − m t m + ! t m + ! (cid:19) = (cid:18) t m + ! t m + ! (cid:19) (cid:18) t m + ! t ! − m (cid:19) . (19)Since t m + > t + ( m − ) , and t > t m + ! t ! > ( t + m − ) · · · ( t + ) t ! t ! > ( + m − )( + m − ) · · · ( + ) = ( m + )( m ) · · · ( ) > m .Thus, each of the two terms on the right hand side member of (19) is positive, hence the expressionin the first parenthesis of (18) is positive. The argument to show that the expressions inside the otherparentheses goes as follows (we focus on the the second parenthesis). (cid:18) t m + ! t ! − m t m ! t m − ! (cid:19) = (cid:18) t m ! t m − ! (cid:19) (cid:18) t m + ! t m ! t m − ! t ! − m (cid:19) . (20)Since t m + > t m + t m − > t , and t m > m , we get t m + ! t m ! t m − ! t ! > ( t m + ) ( t m + ) ( ) > ( m + )( m + ) > m .(i) For y ( T ) and x ( S ) , we get • [ t m + ! )] = b L ( T ) ∪ b U ( T ) ∪ · · · ∪ b L m ( T ) ∪ b U m ( T ) ∪ b L m + ( T ) = L ( S ) ∪ U ( S ) ∪ L ( S ) , b L ( T ) = L ( S ) and therefore, b U ( T ) ∪ · · · ∪ b L m ( T ) ∪ b U m ( T ) ∪ b L m + ( T ) = U ( S ) ∪ L ( S ) , • y t ( T ) = x t ( S ) = t ∈ b L ( T ) and y t ( T ) = x t ( S ) > t ∈ b U ( T ) , • | b L ( T ) ∪ b L ( T ) ∪ b L m + ( T ) | < | L ( S ) | and therefore, | b U ( T ) ∪ b U ( T ) ∪ b U m ( T ) | > | U ( S ) | . Inother words, there are fewer coordinates in [ t !, t m + ! ] ∩ N , with y t ( T ) = < x t ( S ) and astrictly larger number of coordinates with y t ( T ) > x t ( S ) > • y t ( T ) − x t ( S ) > ∞ S k = m + b U k ( T ) , and y t ( T ) = x t ( S ) = ∞ S k = m + b L k ( T ) .We switch those coordinates of x ( S ) in [ t !, t m + ! ] having y t ( T ) = < x t ( S ) with an equalnumber of elements from the remaining coordinates satisfying x t ( S ) = < y t ( T ) to obtain a13equence x ′ such that x ′ t y t ( T ) for all t ∈ N . Since only finitely many terms of x ( S ) have beenpermuted to obtain x ′ , AN implies x ′ ∼ x ( S ) . (21)Furthermore, for all coordinates in ∞ S k = b U k ( T ) , y t ( T ) − x ′ t >
1. Since by (11), d (cid:16) b U ( T ) (cid:17) =
1, andby Claim 2, d (cid:18) ∞ S k = b U k ( T ) (cid:19) =
1, using ADP, we have obtained x ′ ≺ y ( T ) . (22)Thus, it follows from (21), (22), and transitivity that x ( S ) ≺ y ( T ) . (23)(ii) For x ( T ) and y ( S ) , we note that • [ t m + ! ] = L m + ( T ) ∪ m + S k = L k ( T ) ∪ U k ( T ) = b L ( S ) ∪ b U ( S ) ∪ b L ( S ) , b L ( S ) = L ( T ) ∪ U ( T ) ∪ L ( T ) and b L ( S ) = L m + ( T ) and therefore, b U ( S ) = U m + ( T ) ∪ m + [ k = L k ( T ) ∪ U k − ( T ) . • x t ( T ) = y t ( S ) = t ∈ L ( T ) ∪ L ( T ) , for all t ∈ U ( T ) , y t ( S ) = < x t ( T ) , and forall t ∈ L ( T ) ∪ · · · ∪ L m + ( T ) , y t ( S ) > = x t ( T ) . • since | U ( T ) | < | L ( T ) | + | L ( T ) | + · · · + | L m + ( T ) | ,we get | b U ( S ) | > | U ( T ) ∪ U ( T ) ∪ · · · ∪ U m + ( T ) | .In other words, there are fewer coordinates in [ t !, t m + ! ] ∩ N , with y t ( S ) = < x t ( T ) anda strictly larger number of coordinates with y t ( S ) > x t ( T ) > • therefore, y t ( S ) − x t ( T ) > t ∈ ∞ S k = m + U k ( T ) , and y t ( S ) = x t ( T ) = t ∈ ∞ S k = m + L k ( T ) .As in (i) above, we can implement a finite permutation of y ( S ) (among coordinates in b L ( S ) ∪ b U ( S ) ) and invoke AN and ADP to obtain x ( T ) ≺ y ( S ) . (24)Therefore, by (23) and (24) we get x ( S ) ≺ y ( T ) ≺ x ( T ) ≺ y ( S ) . By transitivity, x ( S ) ≺ y ( S ) , whichyields S ∈ Γ , as was to be proven.(c) Let x ( T ) ∼ y ( T ) , i.e., T / ∈ Γ . We drop t , t , t , t , t , · · · , t m and t m + to obtain the set S = { t , t , t m + , t m + , · · · } . Note that S ∈ Ω ( T ) and | L ( T ) | + · · · + | L m + ( T ) | > | U ( T ) | + | U ( T ) | . (25)Using the technique relied upon in case (b) above, we can show (cid:12)(cid:12)(cid:12)b L ( T ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)b L ( T ) (cid:12)(cid:12)(cid:12) + · · · + (cid:12)(cid:12)(cid:12)b L m + ( T ) (cid:12)(cid:12)(cid:12) < | L ( S ) | . (26)14i) For x ( S ) and y ( T ) , we get (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m + [ k = b L k ( T ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m [ k = b U k ( T ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = | L ( S ) ∪ U ( S ) ∪ L ( S ) | . • y t ( T ) = x t ( S ) = b L ( T ) ∪ b L ( T ) , • y t ( T ) > x t ( S ) = t ∈ b U ( T ) , • in view of (26) there is a strictly larger number of coordinates in m S k = b U k ( T ) , such that y t ( T ) > U ( S ) for which x t ( S ) >
1, and • for all elements in t ∈ ∞ S k = m + b U k ( T ) , y t ( T ) > x t ( S ) .As in part (i) of case (b) above, we can perform a finite permutation of x ( S ) and use AN and ADPto obtain x ( S ) ≺ y ( T ) . (27)(ii) For x ( T ) and y ( S ) , we note that • for all t ∈ L ( T ) ∪ L ( T ) ∪ L ( T ) , y t ( S ) = = x t ( T ) , • for all t ∈ U ( T ) ∪ U ( T ) , y t ( S ) = < x t ( T ) , and • for all t ∈ U ( T ) , 1 < y t ( S ) < x t ( T ) , • furthermore, for all t ∈ L ( T ) ∪ L ( T ) ∪ · · · ∪ L m + ( T ) , 1 = x t ( T ) < y t ( S ) .As in case (b)-(i) above, we can apply a finite permutation of y ( S ) (among coordinates in U ( T ) ∪· · · U m ( T ) ∪ L ( T ) ∪ · · · L m + ( T ) ) and invoke AN and ADP to obtain x ( T ) ≺ y ( S ) . (28)Therefore, by (27) and (28) we get x ( S ) ≺ y ( T ) ∼ x ( T ) ≺ y ( S ) . By transitivity, x ( S ) ≺ y ( S ) , whichyields S ∈ Γ , as was to be proven. Proof of Theorem Assume that Y is infinite. We must show that there does not exist any constructiveSWO satisfying ADP and AN on Y N . To this end, suppose, by way of obtaining a contradiction, that there isa constructive SWO satisfying ADP and AN on Y N . Let A ⊂ Y be order isomorphic to N , thus, let f : N → A be an order-preserving bijection. Consider the restriction of the foregoing SWO to A N , and call it (cid:23) . Next,we define an induced binary relation on N N , denoted by (cid:23) f , as follows: for all h n s i and h ¯ n s i in N N , ( n , n , n , · · · ) (cid:23) f ( ¯ n , ¯ n , ¯ n , · · · ) if and only if ( f ( n ) , f ( n ) , f ( n ) , · · · ) (cid:23) ( f ( ¯ n ) , f ( ¯ n ) , f ( ¯ n ) , · · · ) .Since f is increasing, it’s easy to show that also (cid:23) f is a constructive SWO that satisfies ADP and AN, whichcontradicts Lemma 2.For the converse, suppose that Y is finite. Then, it follows from Example 1 that the SWO defined by ( ))