On the informational basis of social choice

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<ul><li><p>Abstract. The notion of informational basis in social choice can be broad-ened so as to cover not only the standard notions related to interpersonalutility comparisons, but also information about utilities or preferences at(ir)relevant alternatives, non-utility features of alternatives, personalresponsibility, unconcerned subpopulations, and feasibility constraints. Thispaper proposes a unied conceptual framework for all these notions, andanalyzes the kind of information retained in each case. This new frameworkyields a deeper understanding of the diculties and possibilities of socialchoice. New welfarism theorems are also obtained.</p><p>1 Introduction</p><p>The notion of informational basis has been coined by Sen (1970) and dAs-premont and Gevers (1977), in order to describe the sets of data that are usedin the determination of social preferences over alternatives. Although, in hiswork as a whole, Sen has studied the issue of information in social choice frommany dierent angles, especially in the debate about welfarism, the notion ofinformational basis itself has often been conceived in a rather narrow way, thatis, in terms of interpersonal comparisons of utility (or any similar notion ofwell-being). Moreover, a large consensus still exists in social choice about thefact that interpersonally comparable indices of individual well-being are a</p><p>Soc Choice Welfare (2003) 21: 347384DOI: 10.1007/s00355-003-0263-5</p><p>I am indebted to W. Bossert, W. Gaertner, F. Gaspart, L. Gevers, N. Gravel, S. Kolm,M. Le Breton, Ph. Mongin, P. Hammond, K. Suzumura, K. Tadenuma, A. Trannoyand J. Weymark for many conversations on topics related to this paper. I havebeneted from reactions of the audience at the conference in honor of Louis Gevers,Namur 1999, and at a seminar at the LSE, and from comments by two referees and theeditor, F. Maniquet. None of them is responsible for the shortcomings of the paper.</p><p>On the informational basis of social choice</p><p>Marc Fleurbaey</p><p>Universite de Pau et des Pays de lAdour, UFR Droit, Economie, Gestion, BP 1633,64016, Pau cedex, France (e-mail: marc.eurbaey@univ-pau.fr) andCATT, THEMA, IDEP</p></li><li><p>necessary piece of information in order to obtain consistent social preferencesand to avoid the diculties so famously described in Arrows impossibilitytheorem (Arrow 1951).1 Such a consensus obviously reinforces the widespreadimpression that the question of the informational basis does indeed revolvearound this key issue of interpersonal comparisons.</p><p>In this paper, I propose broadening the concept of informational basis, soas to make it possible to rigorously discuss, in a unied framework, the use ofinformation not only about individual utilities but also about individualpreferences, individual talents and handicaps, and about other features of theeconomy such as the set of feasible allocations.</p><p>The purpose of the paper is not just to provide a more suitable toolbox forthe analysis of information in social choice. The broadened conceptual frameshould make it easier to think about how to introduce more information thanArrow allowed in his famous impossibility theorem, and should make it moretransparent that the consensus about the need for interpersonally comparableindices of well-being is questionable. In a variety of models, Samuelson (1977),Pazner (1979), Kaneko and Nakamura (1979), Dhillon and Mertens (1999),and Fleurbaey and Maniquet (1996, 2000, 2001) have articulated the dissidentview that ordinal non-comparable preferences may be sucient informationfor consistent and equitable social preferences. The concepts proposed in thispaper will clarify how this outlying thesis ts in a general coherent picture.</p><p>The broader conceptual framework proposed in this paper will not onlyexplain how social choice can be made possible in absence of interpersonallycomparable indices of well-being. It will also relate the diculties and possi-bilities of social choice to notions which are not usually thought to be centralto such issues. Social preferences may be more or less hard to constructdepending on the adoption of principles of responsibility, separability, andindependence of feasibility constraints. In particular, it will be shown belowthat Arrows theorem can be reinterpreted in terms of the following trilem-ma. Reasonable social preferences cannot at the same time hold individualsresponsible for their utility functions (as distinct from their preferences), dis-regard utilities at infeasible alternatives, and disregard feasibility constraints.</p><p>The paper is organized as follows. The next section presents the frameworkand the basic notions. The key concept is related to the information used in thesocial ranking of two alternatives. Then, Sect. 3 through 8 examine the use ofinformation about, successively, utilities (interpersonal comparisons), relevantor irrelevant alternatives, non-utility features of alternatives (Paretianism),personal responsibility, unconcerned subpopulations (separability), and feasi-bility constraints. Since the classical notion of informational basis is limited tothe rst item of this list, themain achievement in these sections is the integrationof all the other notions into a unied conceptual framework, and actually, oneand the same family of axioms serves to cover all these informational issues.Every section ismade upof a short formal analysis, followedby some comments</p><p>1 For a very elegant presentation of this consensual view, see Sen (1999).</p><p>348 M. Fleurbaey</p></li><li><p>on the ethical foundations of the informational approaches under consider-ation. Then, sect. 9 studies some interesting combinations of informationalrequirements introduced separately in the previous sections. Section 10 exam-ines the possibility of constructing consistent social preferences on the basis ofvarious kinds of information and interpersonal comparisons, and proposes adistinction between dierent kinds of welfarism and non-welfarism. Section 11summarizes the results and concludes.</p><p>2 Framework and basic notions</p><p>The standard social choice problem is the determination of a mapping whichdenes social preferences over a given set of alternatives as a function of theprole of preferences of a given population. This framework is too restrictiveif one wants to study how social preferences could depend on other charac-teristics of the individuals (such as utility functions, or talents), on the pop-ulation itself (demography), or on the set of alternatives.</p><p>A more general framework is the following. A social choice entry ise hN ;X ; where N is the set of individuals of the relevant population,hN hii2N is the prole of individual characteristics, and X the set of alter-natives. The social choice problem is to nd a social ordering function (SOF) Rsuch that, for every e 2 E; where E is a relevant domain of entries, Re is acomplete preorder on X . Social preferences dened by the preorder Rewill bedenoted xRey, with related strict preferences xP ey and indierence xIey:Let N [hN ;X 2EN denote the global population from which particular pop-ulations N are drawn.</p><p>The interest of this problem can be explained as follows. The set X mayconsist of all alternatives which are feasible in some general sense. But the actualdecisions the populationwill have tomakemay be limited to a strict subset ofX :For instance, X may be the set of all alternatives over which individual pref-erences are dened, and scarcity constraints may limit actual choices. Or X maybe the set of technically feasible alternatives, and incentive compatibility con-straints maymake it impossible to achieve all alternatives which are technicallyfeasible. Or the social decisionsmay always take the form of a piecemeal reformopposed to the status quo. There are a variety of possible constraints preventingsociety to choose directly from the whole set X . In all such contexts, a completepreorder over X is a very useful tool. Moreover, the theory would be quiteuseless if it were able to dene social preferences for a very small class of socialchoice entries. It must be able to solve the problem for a suciently wide class,covering all relevant cases onemay envision. This is why a function, not just onepreorder for one entry, is sought.2</p><p>2 As dened here, the social choice problem is still too narrow to study someimportant issues. In particular, it does not make it possible to study the issue ofoptimal population size, which requires a preorder over dierent alternatives fordierent populations. For a synthesis on this issue, see Blackorby et al. (1997).</p><p>On the informational basis of social choice 349</p></li><li><p>Obviously, the nature of the social choice problem depends a lot on thedomain of entries. Depending on E, the social choice problem may consist inan abstract collective problem of choice, or in a more concrete allocationproblem.3</p><p>It is assumed here that hi is a complete description of individual ischaracteristics. For any notion of utility, one can then dene a mapping Usuch that Uhi : Xi ! R is is utility function, dened over some relevantdomain Xi which contains X . We do not restrict the domain of denition ofutility functions to X , because in some problems, X describes a feasible setand individual utilities are dened over a much broader set. As it will appearin the sequel, it is not an innocuous restriction in such cases to disregardutilities outside X .</p><p>The central topic of this paper will be the information used by a SOF Rabout an entry e hN ;X when ranking two alternatives x; y 2 X . Typically,there is some information used by R; and the rest is disregarded. This can becaptured by introducing a function f , hereafter called a data lter, whichscreens out all irrelevant information and retains only what is consideredrelevant. In other words, the SOF R will satisfy the following axiom for somewell chosen data lter f :</p><p>Independence of Non-f Information INfI:</p><p>8e hN ;X ; e0 h0N 0 ;X 0 2 E; 8x; y 2 X 2; 8x0; y0 2 X 0 2;f e; x; y f e0; x0; y0 ) xRey , x0Re0y0 :</p><p>I claim that most if not all of the issues pertaining to the notion of infor-mational basis in social choice revolve around particular features and prop-erties of the data lter f . This claim will be illustrated, if not proved, in thefollowing sections.</p><p>3 Utility transformations</p><p>The notion of informational basis has been traditionally associated totransformations of proles of utility functions (Sen 1970; dAspremont andGevers 1977), and transformations of vectors of utility levels (Gevers 1979).The purpose of this section is to relate such notions to INf I, and in theprocess to clarify a few points.</p><p>3 The notion of social choice problem dened here is suciently general to cover thetheory of fair allocation, which has mostly been conned to the search of coarsepreorders distinguishing good allocations from bad ones. Such coarse preorders arenevertheless complete, and therefore the allocation rules of the theory of fairallocation can be described as SOFs for the social choice problem.</p><p>350 M. Fleurbaey</p></li><li><p>3.1 Formal analysis</p><p>Let U be a function deriving any individual is utility function from hercharacteristics hi. That is, if is utility function is ui, then ui Uhi. With anabuse of notation, let UhN denote the prole of utility functions uN suchthat ui Uhi for every i 2 N .</p><p>Let UONC denote the set of vectors of mappings u uii2N such that forevery i 2 N ;ui : R! R is increasing. The subscript ONC refers to theordinal non-comparable information setting that this set implies, as will beexplained below. With a slight abuse of notation, for any e hN ;X , letuuN denote the vector of utility functions u0N such that for every i 2 N , andevery x 2 Xi; u0ix uiuix. Similarly, let simultaneously denote theordinary composition of functions, and also the composition operation ap-plied to vectors of functions component-wise, so that u u0 ui u0ii2N .With this convention, UONC; is an algebraic group.4</p><p>In the literature, the sets of transformations have been used in the con-struction of invariance axioms such as the following one. Let U UONC .Invariance to U INV U:</p><p>8e hN ;X ; e0 h0N ;X 2 E; 8uN UhN ; u0N Uh0N ;9u 2 U; u0N uuN ) Re Re0:</p><p>As an illustration, the axiom INVUONC means that any transformation ofutility functions which does not alter individual ordinal preferences leavessocial preferences unchanged. In other words, under this axiom, the onlyinformation about individual utilities that is used is contained in individualnon-comparable preferences.</p><p>The question to be addressed here is how such an axiom can be translatedinto a requirement imposed on the data lter f of the INf I axiom. It isimmediate that R satises INVU whenever it satises INf I for a data lter fsuch that:</p><p>8e hN ;X ; e0 h0N ;X 2 E; 8uN UhN ; u0N Uh0N ;</p><p>9u 2 U; u0N uuN ) f e; : f e0; ::</p><p>But it is also useful to examine how a particular INf I axiom can serve toexpress the INVU condition, because this helps to understand the nature ofthe restrictions imposed by INVU. For any subset U UONC , let Ug; denote the subgroup of UONC ; generated by U (that is, the smallest supersetof U which is a group). This set is unique, as recalled in the following lemma.</p><p>4 The importance of algebraic groups in this area was emphasized by Roberts (1980).</p><p>On the informational basis of social choice 351</p></li><li><p>Lemma 1. Let U UONC, and U1 denote the set of inverses of elements of U.Then</p><p>Ug fu 2 UONC j9u1; . . . ;um 2 U [ U1; u u1 . . . umg:The following proposition deciphers the consequences of INVU over the datalter f involved in INf I, when the two axioms are made equivalent. First,notice that the set Ug generates an equivalence relation Ug on the domain E,which is dened by</p><p>hN ;X Ug h0N 0 ;X 0,N N 0 andX X 0 and 9u2Ug;Uh0N uUhN :For any e 2 E, let Ug e denote the equivalence class for Ug to which ebelongs.</p><p>Proposition 1. Consider any subset U UONC , and assume that E is rich en-ough so that, for any e hN ;X 2 E, and any u 2 Ug, there exists h0N ;X 2 Esuch that Uh0N uUhN . The axiom INV U is then equivalent to INfI for fdened by:</p><p>f e; x; y Ug e; x; y :Proof.5 INVU implies INf I. Let f e; x; y f e0; x0; y0. Then, by deni-tion of f , one has x x0; y y0;N N 0;X X 0; and</p><p>9u 2 Ug;Uh0N uUhN :By the above lemma, there exist u1; . . . ;um 2 U [ U1, such thatu u1 . . . um. One then has</p><p>Uh0N u1 . . . umUhN :By the richness assumption on E, and since um 2 Ug, there existsem hmN ;X 2 E such that UhmN umUhN , and thereforeUhN u1m UhmN . Since either um 2 U or u1m 2 U, then by INVU, onehas Re Rem. Similarly, let em1 hm1N ;X 2 E be such thatUhmN um1UhmN . By INVU, one has Rem Rem1. By iteration ofthis argument, one nally obtains Re Re0, and therefore Re and Re0coincide on fx; yg.</p><p>INf I implies INVU. Let e hN ;X ; e0 h0N ;X 2 E; uN UhN ;u0N Uh0N , be such that</p><p>9u 2 U; u0N uuN :Take any x; y 2 X 2. Since U Ug, one has f e; x; y f e0; x; y, so that,by INf I, Re and Re0 coincide on fx; yg. Since this applies to anyx; y 2 X 2;Re Re0. j</p><p>5 Prop. 1 just gives the simplest example of f such that INVU a...</p></li></ul>