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.
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
Soc Choice Welfare (2003) 21: 347384DOI: 10.1007/s00355-003-0263-5
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.
On the informational basis of social choice
Universite de Pau et des Pays de lAdour, UFR Droit, Economie, Gestion, BP 1633,64016, Pau cedex, France (e-mail: email@example.com) andCATT, THEMA, IDEP
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.
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.
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.
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.
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
1 For a very elegant presentation of this consensual view, see Sen (1999).
348 M. Fleurbaey
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.
2 Framework and basic notions
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.
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.
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
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).
On the informational basis of social choice 349
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
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 .
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 :
Independence of Non-f Information INfI:
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 :
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.
3 Utility transformations
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.
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.
350 M. Fleurbaey
3.1 Formal analysis
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 .
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
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:
8e hN ;X ; e0 h0N ;X 2 E; 8uN UhN ; u0N Uh0N ;9u 2 U; u0N uuN ) Re Re0:
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.
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:
8e hN ;X ; e0 h0N ;X 2 E; 8uN UhN ; u0N Uh0N ;
9u 2 U; u0N uuN ) f e; : f e0; ::
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.
4 The importance of algebraic groups in this area was emphasized by Roberts (1980).
On the informational basis of social choice 351
Lemma 1. Let U UONC, and U1 denote the set of inverses of elements of U.Then
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
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.
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:
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
9u 2 Ug;Uh0N uUhN :By the above lemma, there exist u1; . . . ;um 2 U [ U1, such thatu u1 . . . um. One then has
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.
INf I implies INVU. Let e hN ;X ; e0 h0N ;X 2 E; uN UhN ;u0N Uh0N , be such that
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
5 Prop. 1 just gives the simplest example of f such that INVU and INf I are equivalent.There is a more general theorem saying that this equivalence is obtained for all fwhich are isomorphic to the one given here. This is true for all results below.
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The formula f e; x; y Ug e; x; y captures the substance ofINVU. The only information that is retained about e is the equivalence classto which it belongs, and this implies lumping together proles of utilityfunctions which are related by transformations of Ug (not just U).
A corollary of this result, then, is that INVU is equivalent to INVUg,which means that there is no obligation to restrict attention to sets oftransformations U which are algebraic groups, and also no limitation in doingso. This corollary was proved in a particular case by dAspremont and Gevers(1977), when they showed (their Theorem 1) the equivalence of referring tothe set of cardinal non-comparability
UCNC u 2 UONCj9a 2 RN ; b 2 RN; 8i 2 N ; 8u 2 R;uiu ai biun o
and to the set
UAG u 2 UONCj9a 2 R; b 2 RN; 8i 2 N ; 8u 2 R;uiu a biun o
The equivalence between INVUAG and INVUCNC is simply the consequence ofUAGg UCNC:
For the rest of the discussion, it is useful to introduce the set of cardinalunit comparability
UCUC u 2 UONC j9a 2 RN ; b 2 R; 8i 2 N ; 8u 2 R;uiu ai bun o
The axiom INVUCUC , in particular, deletes any information about interper-sonal comparisons of utility levels (because the ai may change the zeros ofindividual utility functions independently), but preserves information aboutcomparisons of utility dierences. It is satised by the utilitarian SOF, whichis dened here by: 8e hN ;X 2 E; 8uN UhN ; 8x; y 2 X ;
And the set of ordinal measurability and full comparability
UOFC u 2 UONC j8i; j 2 N ;ui uj
yields an axiom INVUOFC which preserves information about interpersonalcomparisons of utility levels, but not more, and is satised in particular by themaximin SOF, dened by: 8e hN ;X 2 E;8uN UhN ; 8x; y 2 X ;
xRey , mini2N
3.2 Ethical comments
An axiom like INVU restricts the kind of information that can be used by theSOF about the prole of individual utility functions. The literature6 studying
6 For two excellent surveys, see Bossert and Weymark (1998) and dAspremont andGevers (2002).
On the informational basis of social choice 353
such axioms has usefully claried the informational content of various socialwelfare functions, most notably utilitarianism and themaximin criterion. But itusually left unclear why an axiom of the INVU sort should be viewed asappealing, in the normative perspective of the construction of good socialpreferences. A supercial reading might give the impression that it all has to dowith the fact that some information about individual utilities may just happennot to be available, so that an axiom like INVU may help analyzing the con-sequences of this information shortage on the social objective. This idea is quitequestionable. The construction of good social preferences should involve allethically relevant information, independently of what is available or possible.7
As an example, suppose a utilitarian social planner is told that the only infor-mation available is about individual levels of utility, without any clue aboututility dierences and intensities. Should thismake the social planner accept theaxiom INVUOFC? This axiom is not satised by the utilitarian SOF. Should theplanner abandon her utilitarian preferences and become an egalitarian, sincethe maximin SOF satises INVUOFC? A more consistent attitude would be forher to keep her utilitarian social preferences and to conclude that the imple-mentation of these preferences will be dicult. Trying to collect the relevantdata might be the rst way out to probe.
In conclusion, one should reserve the use of INVU for discarding irrelevantinformation, not for addressing information shortages. The various axiomsINV U, for dierent sets U, need not appear equally justiable, in this respect.For instance, it is not very hard to justify INVUONC , which excludes anyinformation about utilities except that contained in ordinal non-comparablepreferences, by invoking the individuals responsibility for their subjectivesatisfaction. One may want to respect individual preferences and therefore takethem into account, but disregard utilities as a purely private matter. This is justwhat INVUONC stipulates.8 It seems harder to defend INVUCUC . Truly enough,this axiom is logically weaker than INVUONC, so that, rigorously, it should beeasier to justify. But it is hard to defend it without justifying INVUONC in theprocess. How could one argue that utility levels do not matter whereas utilitydierences may matter?9 It is also quite hard to defend INVUOFC , exceptthrough a direct defense of the absolute priority of theworst-o, as embodied inthe maximin and leximin criteria, or in an equity axiom such as Hammonds(1976). But deriving a justication of INVUOFC from the maximin criterion is
7 This viewpoint is defended e.g., in Kolm (1996).8 This line of argument can be found in Rawls (1982), Dworkin (2000).9 One nds awkward arguments to this eect in Harsanyi (1976, p. 72 and pp. 7576).One is that medical treatment should go to whom it benets more, and not in priorityto a poor (as opposed to a millionaire). A second example is about giving a smallpresent to one of two boys, a happy one who derives great joy from presents, and anunhappy one who does not derive much pleasure from small presents. Harsanyi arguesin favor of giving the present to the rst one, on the ground that the giver is notresponsible for the initial inequality. In both examples, the intuition seems to be drivenby a matter of dierent spheres of justice.
354 M. Fleurbaey
unwelcome for a literaturewhich goes the otherway, deriving themaximin fromINVUOFC. In conclusion, the ethical foundations of the INV/ approach arerather fragile, with at least one exception: INVUONC. Thewidespread belief that,in view of Arrows impossibility, the INVUONC axiom leads social choice to adead end, has restricted its use. In view of the new possibilities discussed below,it may be rehabilitated.
This criticism of the INVU approach should not be understood as meaningthat introducing interpersonally comparable indices of well-being is not sound.Quite to the contrary, the idea of constructing such indices, as defended forinstance in Sens broad theory of equality, is very respectable. The point here isthat particular axioms such as INVUCUC or INVUOFC maynot be very helpful. Itis not on the basis of INVUOFC that Sendefends an egalitarian view, for instance,but because of the direct ethical value of equality.
4 Irrelevant alternatives
The INV U axioms (for various sets U) are the core of the traditional notion ofinformational basis. But they are just the starting point of our analysis. A rstbroadening of the notion of informational basis is now proposed, byencompassing the condition of independence of irrelevant alternatives, acondition which has usually been used jointly with axioms bearing on trans-formations of utility functions.
4.1 Formal analysis
The following axiom says that a change in the population prole which doesnot alter the levels of utility at two alternatives should not modify socialpreferences on these two alternatives.
Independence of Other Alternatives (IOA):
8e hN ;X ; e0 h0N ;X 2 E; 8uN UhN ; u0N Uh0N ; 8x; y 2 X ;uN fx;yg u0N
fx;yg) Re fx;yg Re0
This axiom is usually called independence of irrelevant alternatives. Thisname is, however, the source of two confusions. First, it gives the wrongimpression that it is just an innocuous adaptation of Arrows axiom of inde-pendence, which requires social preferences over two alternatives not to changewhen individualpreferenceson these twoalternativesdonot change.This axiomis actually a very substantial weakening of Arrows axiom, as shown below.10
10 A formal denition of Arrows axiom is given in Subsect. 9.2. The idea that IOA isa faithful translation of Arrows axiom is tempting when one presents the introductionof utilities as a change of framework, rather than just a change of axioms in a moregeneral framework. See e.g., Sen (1986, p. 1114): For a SWFL the Arrow conditionsare readily redened. Hammond (1987) proposed the alternative name Independenceof Irrelevant Utilities.
On the informational basis of social choice 355
Second, it makes a confusion between the uncontroversial requirement thatirrelevant information should be disregarded, and the controversial denitionof the irrelevant information. From the standpoint of deontology, it isprobably preferable to use names expressing the content of an axiom in aneutral and transparent manner.
Now we can begin to see the power of the INf I axiom, since it can alsoexpress this new condition. What requirements on the data lter f does IOAimply? The SOFR satises IOAwhenever it satises INf I for a data lter f suchthat: 8e hN ;X ; e0 h0N ;X 2 E; 8uN UhN ; u0N Uh0N ; 8x; y 2 X ;
uN fx;yg u0N
fx;yg) f e; x; y f e0; x; y:Again, it is useful to look for conditions under which the axioms IOA andINf I are actually equivalent.
Proposition 2. A SOF R satises IOA if and only if it satises INfI for f denedby:
f e; x; y uN x; uN y;X ; x; y;where e hN ;X ; uN UhN .Proof. IOA implies INf I. Let f e; x; y f e0; x0; y0. This impliesx x0; y y0;N N 0;X X 0; uN x u0N x; uN y u0N y, and therefore,by IOA, Re fx;yg Re0
INf I implies IOA. Let uN fx;yg u0N
fx;yg. Then f e; x; y f e0; x; y,so that by INf I, Re fx;yg Re0
As this proposition shows, IOA retains very little information aboutindividual situations at x and y. And one may want to introduce someadditional information, in particular about the prole of utility functions,which describes the kind of population concerned with the construction ofsocial preferences. This can be done by taking account of utilities at otheralternatives. There are many possible subsets of other alternatives which maybe introduced. We limit our attention here to two typical examples.11 The rstone consists in taking account of utilities over the whole set X , and disre-garding utilities outside X .
Independence of Non-Feasible Alternatives (INFA):
8e hN ;X ; e0 h0N ;X 2 E; 8uN UhN ; u0N Uh0N ,uN X u0N
X) Re Re0:This axiom is equivalent to INf I for
f e; x; y uN jX ;X ; x; y:
11 For a more extensive study along these lines, see Fleurbaey et al. (2002).
356 M. Fleurbaey
The second example introduces a condition stipulating that the upper andlower contour sets remain the same.12 Dene
UCuiu fz 2 Xijuiz ug LCuiu fz 2 Xijuiz ug:The following axiom allows social preferences to disregard the precise levelsof utilities at all alternatives which are not indierent to x and y, except forchecking that they remain better, or worse, than x or y.
Independence of Non-Indierent Alternatives (INIA):
8e hN ;X ; e0 h0N ;X 2 E; 8uN UhN ; u0N Uh0N ; 8x; y 2 X ,
8i 2 N ;
UCuiuix UCu0iuixUCuiuiy UCu0iuiyLCuiuix LCu0iuixLCuiuiy LCu0iuiy
) Re fx;yg Re0
This axiom is equivalent to INf I for13
f e; x; y uN x; uN y; UCuiuixi2N ; UCuiuiyi2N ;LCuiuixi2N ; LCuiuiyi2N ;X ; x; y
This axiom allows the social comparison of x and y to depend not only ontheir utility levels but also on how they are ranked by individuals with respectto other alternatives.
It may be useful at this point to provide examples of reasonable SOFssatisfying INFA or INIA but not IOA. In an abstract voting problem, theBorda rule, which applies the utilitarian criterion to individual Borda scores14
vix #uiLCuiuix \ X ;(by standard convention, uiA denotes the range of ui on any subset A) doessatisfy INFA and INIA, but not IOA.
Less classical examples may be provided for the economic problem ofdividing a total bundle X 2 R of goods among n individuals whoseconsumption set is R, and whose preferences bear only on their personalconsumption bundle xi (for i 1; . . . ; n). Pazner (1979) proposed to apply themaximin criterion to the vector of individual vi dened by
vixi minfk 0juikX uixig:
12 The idea of such a condition can be traced back to Hansson (1973).13 Notice that in INIA the upper and lower contour sets are dened for utility levels,so that for instance, UCuiuix UCu0iuix and LCuiuix LCu0iuix implyuix u0ix.14 There are several possible denitions of Borda scores when individuals may beindierent between alternatives. The denition proposed here counts the number ofindierence classes, in X , at and below x. For instance, if X fx; y; z; tg, and if i haspreferences dened by uix > uiy uiz > uit, then vix 3 > viy viz 2 > vit 1.
On the informational basis of social choice 357
This gives a SOF which does not satisfy IOA but satises INIA. It alsosatises INFA when the set X is dened as
X fx x1; . . . ; xn 2 Rn jx1 . . . xn Xg:Fleurbaey and Maniquet (1996) proposed another example of SOF, whichcomputes the social value of an allocation as the smallest fraction of X whichbelongs to the convex hull of the union of individual upper contour sets in theconsumption space:
min k 0jkX 2 co[
i2Nfz 2 R j uiz uixig
This SOF is axiomatically justied in Fleurbaey and Maniquet (2001), and itsmain attractive feature is that, in economies with convex preferences, its rstbest subset of allocations, in X , is always exactly the subset of egalitarianWalrasian allocations (i.e. competitive equilibria with equal budgets for allindividuals). This SOF satises INIA, but does not satisfy INFA nor IOA.
4.2 Ethical comments
f e; x; y uN x; uN y;X ; x; y exactly describes what information is retained under the operation of IOA.First, the content of the two alternatives x; y under consideration (and the setX ) is fully registered. No relation is established by this axiom between dif-ferent pairs of alternatives. For instance, the social ordering may be utili-tarian for a particular pair x; y and egalitarian for another pair x0; y0.Second, for the contemplated pair x; y, the only information that is retainedabout their consequences over individuals is contained in the levels of utilityat these alternatives. The content of this information is not supplemented byany other individual characteristics, not even by a fuller description of theindividual utility functions. This axiom therefore embodies a very restrictivekind of welfarism, which is utterly implausible a priori.
But two remarks may alleviate this negative impression. First, the map-ping U which derives utility functions ui from individual characteristics himay be anything, and may take into account any feature of individual sub-jective or objective well-being. In other words, the kind of welfarismembodied in IOA is purely formal, and is compatible with any empiricalcontent given to the notion of utilities. Second, if one assumes U is wellchosen so that any relevant information about individual situations at x and yis correctly summarized into the gures uN x; uN y, then it is much less clearthat IOA is restrictive. By tautology, if anything that counts is in thosegures, the rest does not count, and can safely be disregarded.
If one follows this line of reasoning, the appeal of IOA is conditionalon the availability of a good function U capturing all that counts about
358 M. Fleurbaey
individual situations. This suggests two critical points. By assuming that,somewhat miraculously, we have a function U which gives the perfect mea-surement of individual well-being, the theory of social choice loses any gripon the substantial debate of how U should be dened. Since IOA somehowasserts the perfection of U , this leaves no space within the theory of socialchoice to examine alternative denitions of U . As soon as one adopts IOA fora particular U , all other measures of well-being are excluded. This drasticreduction of possible choices about U can be avoided only by treating IOAnot as an (initial) axiom, but as a result of some anterior analysis.
A second critical point is that the construction of uix Uhix meansthat individual well-being is dened by relying only on individual charac-teristics. This is rather appealing, but, nevertheless, a little restrictive. Apriori, one may think of dening individual well-being as a function of thewhole population prole, or of the situation of all individuals in x. In otherwords, a more general approach would authorize uix Ui; hN x. IOAcould be applied to this broader notion of well-being.
Let us now come back to the usual economists world, in which ui is anordinary measure of subjective utility. In this context, IOA is quite unac-ceptable. Consider the following example. Two goods, 1 and 2, have to bedistributed to two individuals, Ann and Brian. Allocation x gives Ann thebundle (4,6) and Brian the bundle (7,5). Allocation y gives them the bundles(5,7) and (6,4), respectively. We are told that Anns utility is 20 in x, and 24 iny, while Brians utility is 24 in x and 20 in y. According to IOA, this should beenough information to compare x and y. In summary, under IOA, theinformation which can be used is (ignoring the feasible set X ):
In view of the perfect symmetry of this table, social indierence is theunavoidable conclusion. But this is very unsatisfactory. Compare the fol-lowing two utility proles, which both yield the above utility gures at thetwo allocations:
hN :uA xA1 3xA2 2uB xB1 3xB2 2
u0A 3xA1 xA2 2u0B 3xB1 xB2 2:
In prole hN , allocation x gives individuals bundles they both deem equiva-lent, since
uA4; 6 uA7; 5 20 uB4; 6 uB7; 5 24;and allocation x can actually be obtained as a Walrasian equilibrium in whichthe two agents have the same budget set. In contrast, allocation y gives Briana bundle that both deem inferior to Anns bundle, so that, in particular, Brian
Allocation x Allocation y
Anns bundle xA1; xA2 (4,6) (5,7)Brians bundle xB1; xB2 (7,5) (6,4)Anns utility 20 24Brians utility 24 20
On the informational basis of social choice 359
envies Ann (in the sense that he would rather have her bundle). This seems togive x a serious ethical advantage over y. Now, with prole h0N , the utilitygures are the same at x and y, but the roles of x and y are inverted, since y isnow an egalitarian Walrasian allocation, while in x Ann envies Brian.
If one believes that these considerations are relevant, then IOA is toorestrictive and unduly eliminates relevant information. The additionalinformation that has been used in this example would have been availableunder the weaker axioms INFA or INIA. Notice that we referred only toindierence curves at x and y, so that INIA seems to focus on the appropriatekind of information for many equity considerations. INFA retains informa-tion about all utilities over X , which is too much when other indierencecurves are irrelevant, and is also too little for some equity notions. In par-ticular, when X is an Edgeworth box, knowing whether an allocation isWalrasian or not, when it is not interior (more precisely, when one agentconsumes all of one good), is not generally possible by looking only atindierence curves within the Edgeworth box. This explains why the Fleur-baey-Maniquet example of SOF dened above does not satisfy INFA.
The Pareto principle is usually presented in relation to democratic principles,the respect of unanimous preferences, but it is also well known that it limitsthe possibility to rely on non-utility information, and therefore pushes socialchoice in the direction of welfarism.
5.1 Formal analysis
The most relevant Pareto condition for the discussion of this informationalfeature is Pareto-Indierence:
8e hN ;X 2 E; 8uN UhN ; 8x; y 2 X ,uN x uN y ) xIey:
This axiom is implied by INf I whenever f satises the following neutralityproperty: 8e hN ;X 2 E; 8uN UhN ; 8x; y; x0; y0 2 X ,
uN x uN x0uN y uN y0
) f e; x; y f e; x0; y0:
Equivalence between PI and INf I is obtained as follows.
Proposition 3. A SOF R satises PI if and only if it satises INfI for f denedby:
f e; x; y e; uN x; uN y:
360 M. Fleurbaey
Proof. PI implies INf I. Assume f e; x; y f e0; x0; y0. This impliese e0, and uN x; uN y uN x0; uN y0. By PI, this implies xIex0 andyIey0. And therefore xRey , x0Rey0.
INf I implies PI. Assume uN x uN y. Let e0 e and x0 y; y0 x. Thenf e; x; y f e0; x0; y0. By INf I, one has xRey , yRex. This leavesxIey as the only logical possibility. j
5.2 Ethical comments
From the formula
f e; x; y e; uN x; uN y;it is clear that PI entails the impossibility for social preferences to take ac-count of any non-utility or non-preference feature of individual situations atthe alternatives considered. In this way, PI implies a good deal of welfarism.On the other hand, the fact that all information about e remains available foruse in social preferences means that a lot can be done in relating uN x anduN y to corresponding features of individual situations. For instance,knowing that i is at uix in x entails a full knowledge of the content of isupper and lower contour sets. More generally, all characteristics of thepopulation (and also of the set of alternatives X ) can be used in order to takeaccount of the nature of the problem and of the types of individuals involved.
It is instructive to compare the above formula with the similar formula forIOA:
f e; x; y uN x; uN y;X ; x; y:In the latter one sees that all information is lost about individual character-istics, so that it is impossible to put the levels of utility in perspective. On theother hand, knowledge of x and y makes it possible to have social preferencesdepend directly on features of x and y, which is not allowed by PI.
Individual responsibility has already been mentioned, when discussingINVUONC and responsibility for subjective satisfaction. When individuals aredeemed responsible for some part of their situation in an alternative, thispresumably means that social preferences may legitimately disregard thataspect of individual situations. Individual responsibility over something meansthat it belongs to the private sphere, and that social preferences need notbother about it. For instance, social preferences may focus on opportunity setsoered to individuals, and disregard the particular choices made in these setsby individuals, or they may focus on initial resources granted to individuals,and disregard what individuals make of these resources. Again, this is directlyrelated to the distinction between relevant and irrelevant information.
On the informational basis of social choice 361
6.1 Formal analysis
Assume that all features of individual is situation at alternative x, for whichindividual i is not responsible, are summarized in some (possibly multi-dimensional) measure cix, and that the function ci is itself obtained fromindividual characteristics hN through a mapping ci Ci; hN . The fact that Cmay depend on hN and not just on hi reects the possibility that non-responsibility features of an individual may be jointly determined by char-acteristics of the whole population.
It is possible to express the fact that individual i is responsible for anythingelse by letting social preferences disregard all that is not recorded in ci. Again,with an abuse of notation, ChN denotes Ci; hN i2N .Independence of Responsibility Features (IRF):
8e hN ;X ; e0 h0N ;X 2 E; 8cN ChN ; c0N Ch0N ,cN c0N ) Re Re0:
The formal similarity between this axiom and INFA enables us to seeimmediately that the relevant formula for the data lter f , when IRF isequivalent to INf I, is:
f e; x; y cN ;X ; x; y:When alternatives x; y are so precisely described that they contain adescription of features for which individuals are responsible (for instance,particular consumption bundles, or utility levels), cN x; cN y can erase theirrelevant data and focus on the relevant parameters of individual situationsat x and y (e.g. budget sets).
An axiom like INVUONC, which conveys the idea that individuals areresponsible for their utility functions (as distinct from their preferences), canalso be reformulated as an IRF condition, simply by letting Ci; hN record allindividual characteristics except the utility function.
6.2 Ethical comments
Depending on the denition of C, IRF may be more or less restrictive. WhenCi; hN retains all information and, for instance, equals the constant functionci hi, IRF does not impose any restriction on social preferences.
One encounters axioms of the IRF sort in all branches of the literature onresponsibility-sensitive egalitarianism.15 For instance, there is a literature onfair allocation, dealing with the allocation of money to individuals whoseutility function depends on money and a personal talent parameter. In thisapproach individuals are supposed to be responsible for their utility function,and INVUONC is retained as an expression of this responsibility. A stronger
15 See the surveys by Fleurbaey (1998) and Fleurbaey and Maniquet (1999).
362 M. Fleurbaey
axiom requiring the allocation of money to be independent of individualpreferences is even sometimes required. Another part of the literature studiessocial welfare functions which evaluate individual opportunity sets,16 and anIRF axiom is satised in this approach since the social evaluation dependsonly on opportunity sets and disregards what exact option is chosen byindividuals in their opportunity sets.
The literature does contain other conditions related to the idea ofresponsibility. For instance, some conditions request some preference forequality of budget sets between individuals who dier only in their respon-sibility characteristics. Such conditions express a neutrality requirement overresponsibility characteristics, and prevent social preferences from expressing abias in favor or against some particular exercise of responsibility by indi-viduals. This idea of neutrality17 bears on the desirable kind or degree ofredistribution, and goes farther than the mere idea of responsibility. In adierent vein, the literature dealing with social welfare functions computed onopportunity sets interprets the idea of responsibility as implying that socialpreferences should display no inequality aversion over individuals who havethe same opportunities, and should therefore simply seek to maximize the sumof outcomes reached by such subpopulation. Again, this approach goes far-ther than the mere idea of responsibility, and indirectly advocates a particular(non-neutral) distribution of rewards.
Social preferences are separable when, in the comparison of two alternatives xand y, they disregard the fate of individuals who are not aected by thechange from x to y or vice versa. The situation of subpopulations can then bestudied separately.
7.1 Formal analysis
There aremanyways to dene the fact that an individual is not concerned by thechoice between x and y. It all depends on how one should evaluate the situationof an individual. The traditional way refers to utility, and, as emphasized above,
16 For instance, in Roemer (1998), an individual opportunity set is measured by thestatistical distribution of outcomes in the subpopulation with identical non-respon-sible characteristics. This distribution depends on government policy and on thebehavior of the population.17 Fleurbaey (1995, 1998) calls this kind of neutrality the principle of natural reward(dont over-punish or over-reward those who exercise their responsibility in aparticular way). When IRF is applied to functions ci which are constant in x, and xdescribes the allocation of resources, then natural reward is entailed by IRF (and ananonymity requirement) since the optimal allocation of resources becomes indepen-dent of changes in individual responsibility characteristics. Individuals who dier onlyin their responsibility characteristics will then obtain similar resources.
On the informational basis of social choice 363
this may cover any relevant notion of well-being. Now, when the fate ofunconcerned individuals is measured by their utility, it seems consistent to usethe same measure for the rest of the population. This justies the followingaxiom, which says that the individuals who are indierent between x and y canbe disregarded (except for their mere existence), and that social preferences canthen only look at the utility functions of the rest of the population.
Independence of Indierent Individuals (III):
8e hN ;X ; e0 h0N ;X 2 E; 8uN UhN ; u0N Uh0N ; 8M N ; 8x; y 2 X ,uM x uM yu0M x u0M yuNnM u0NnM
;) Re fx;yg Re0
Proposition 4. A SOF R satises III if and only if it satises INfI for f denedby:
f e; x; y uNnMe;x;y;N ;X ; x; y;it where Me; x; y fi 2 N juix uiyg.Proof. III implies INf I. Let f e; x; y f e0; x0; y0. Then N N 0;X X 0; x x0 and y y0. In addition, for all i 2 M Me; x; y Me0; x; y; uix uiy; u0ix u0iy, whereas uNnM u0NnM . Therefore, byIII, Re fx;yg Re0
INf I implies III. Let uM x uMy; u0M x u0M y; uNnM u0NnM . Onetherefore has M Me; x; y \Me0; x; y, so that NnMe; x; y NnM andNnMe0; x; y NnM . As a consequence, uNnMe;x;y u0NnMe0;x;y. Thenf e; x; y f e0; x0; y0, and by INf I one gets Re fx;yg Re0
There are several variants of this axiom in the literature.18 When dealingwith an economic model, it is usually possible to describe an alternative x byindividual bundles xi for i 2 N . Let xN xii2N then denote the relatedalternative. One can then formulate an axiom saying that individuals whosebundle does not change can be disregarded and even removed from thepopulation.19
Independence of Unconcerned Individuals (IUI):
8e hN ;X 2 E; 8M N ; 8xN ; yN 2 X ,xM yM ) xNReyN , xNnMReryNnM
where er hNnM ;X xM and X xM fzNnM jxM ; zNnM 2 Xg.
18 For instance, dAspremont and Gevers (1977) have an axiom which only disregardsthe utility of individuals who are totally indierent over all alternatives of X .19 This axiom is closely linked to the consistency condition of the theory of fairallocation. On this condition, see e.g., Thomson (1996).
364 M. Fleurbaey
This axiom can be shown to be equivalent20 to INf I for
f e; xN ; yN hNnMxN ;yN ;X xMxN ;yN ; xNnMxN ;yN ; yNnMxN ;yN
where MxN ; yN fi 2 N jxi yig. Notice that, compared to the previousformula, N has disappeared, reecting the fact that unconcerned individualsare dealt with as if they simply did not exist. More interestingly, this axiom ismuch less welfarist than III, since it minimally denes the fact of beingunconcerned in terms of bundles, which leaves open many possibilities for theevaluation of individual well-being. This is why hNnMxN ;yN appears in the datalter f .
7.2 Ethical comments
Separability, and axioms like III and IUI, can be motivated in several ways.There is rst an issue of informational parsimony and simplicity. Separablesocial preferences allow simple computations over subpopulations, andguarantee the consistency between separate studies at local levels and globalstudies for the whole population. A second idea is related to how democracyworks. In any contest between two alternatives, the unconcerned individualsare likely not to express any preference in favor of any of them, and thereforetheir vote will not inuence the nal decision. A third, more normativeviewpoint, is that the principle of subsidiarity requires decisions to be underthe control of concerned individuals only. Unconcerned individuals may giveadvice and recommendations, but the ultimate decision power should remainentirely in the hands of concerned individuals.
Against these arguments, it is sometimes said that the evaluation of whatis going on in a subpopulation may depend on how it fares with respect to therest of the population. For instance, social preferences may be more egali-tarian if the subpopulation under consideration is poor compared to the rest,and be less egalitarian if it is much richer than the rest.
Many axioms, in the literature of social choice and fair allocation, havesome avor of separability. For instance, the Strong Pareto principle adds toPI the statement that x is strictly better than y whenever part of the populationis indierent while the rest strictly prefers x. This can be derived from a versionof III in which indierent individuals can be removed from the population (asin IUI), combined with the Weak Pareto principle according to which unan-imous strict preferences for x over y, in the population, entails the same forsocial preferences. Other examples are given by the Pigou-Dalton principle oftransfer, and Hammonds (1976) equity axiom, which focus on two-individualsubpopulations.
Separability axioms will generally be incompatible with non-individual-istic denitions of utility ui Ui; hN or non-responsibility features
20 The proof is quite dierent from the previous one, and is given in the Appendix.
On the informational basis of social choice 365
ci Ci; hN . This potential conict will be illustrated below, concerningsocial preferences favoring Walrasian allocations.
As explained in Sect. 2, the set X may be the set of feasible alternatives, andfeasibility may be conceived of in various ways. It may simply be the set ofalternatives over which individual utilities are dened (e.g., individual bundlesmust belong to individual consumption sets), or the set of technically feasiblealternatives, or the set of incentive-compatible alternatives, etc. Should socialpreferences over two alternatives depend on the general shape of the set X towhich they belong? This question is again related to informational issues.
8.1 Formal analysis
When social preferences do not depend on the particular shape of X , theymay satisfy the following axiom.
Independence of Feasible Set (IFS):
8e hN ;X ; e0 hN ;X 0 2 E; 8x; y 2 X \ X 0,Re fx;yg Re0
There are variants of this condition for particular contexts. For instance,when individual characteristics hi comprise productive talents, the domain Emay be such that there is only one set X for any given prole hN . In this caseIFS is vacuously satised, but one may then want to modify IFS in order tosay that social preferences should not depend on the prole of talents, butonly on the prole of preferences, for instance.
Translating this axiom into the INf I language is done as follows:
Proposition 5. A SOF R satises IFS if and only if it satises INfI for f denedby:
f e; x; y hN ; x; y:
Proof. IFS implies INf I. Let f e; x; y f e0; x0; y0. This impliesx x0; y y0, and therefore x; y 2 X \ X 0. In addition, hN h0N . By IFS, onethen has Rejfx;yg Re0jfx;yg.
INf I implies IFS. Let hN h0N . Then f e; x; y f e0; x; y. By INf I,one has Rejfx;yg Re0jfx;yg. j
8.2 Ethical comments
The appeal of IFS depends on the context and the notion of feasibility whichdetermines the set X . The initial aim of the theory of social choice, as posited
366 M. Fleurbaey
by Arrow (1951), was probably to construct social preferences over the wholeset of alternatives X for which individual preferences are well-dened. Thedomain of social choice entries E then had a xed set X X , and changes ofindividual preferences were the only source of variability in the domain. Inthis way IFS was vacuously satised, but, more substantially, it was indeedthe case that social preferences over two alternatives did not depend at all onfeasibility constraints. In other words, from Arrovian social preferences overthe global set of alternatives X , one can derive a SOF R on any domain ofentries e hN ;X with X X , by letting Re coincide with the Arroviansocial preferences on X , and this SOF does indeed satisfy IFS in a non-trivialway.
The availability of general social preferences on a large set X is indeedan alluring perspective, but the constraints IFS imposes must not be ne-glected. Consider the problem of distributing bread and water to a givenpopulation. When there is no water, a particular ranking of allocations ofbread will be formed. According to IFS, this ranking should be retainedeven if water became available. This is questionable, for the followingreason. In absence of water, presumably some simple egalitarian rankingwould seem reasonable for the allocation of bread. But when water isavailable, the allocation of bread could legitimately take account of howmuch individuals are willing to substitute water for bread.
The problem becomes acute under Pareto-Indierence. For simplicity,consider a population with two individuals, Ann and Brian. Assume for in-stance that, for the allocations of bread only, in absence of water, giving 10 toAnn and 8 to Brian is better than 12 and 6, respectively. Now suppose thatAnn and Brian are indierent between one-good bundles as described in thetable:
By IFS, the above ranking of allocations of bread should be retained evenwhen water is available. In an economy where both goods are available, PIentails that, if giving 10 of bread to Ann and 8 to Brian is better than 12 and 6,then giving 6 of water to Ann and 12 to Brian is better than 8 and 10. By IFS,this ranking of allocations of water should be retained even in the case whenthere is no bread.
This shows that IFS is very restrictive and questionable in such a context.It prevents social preferences from taking account of the relative scarcity ofgoods, and from focusing on the appropriate parts of individual preferences.For instance, if one thinks that an egalitarian Walrasian allocation is a goodsocial objective, it makes little sense to look for social preferences that areindependent of the relative scarcity of goods, since individual situations have
Ann is indierent between: 12 810 6
Brian is indierent between: 8 126 10
On the informational basis of social choice 367
to be evaluated in terms of budgets, and the relative prices of goods willdepend on total supply.21
In this section, we study how the combination of various informationalaxioms shapes the available information. It is impossible to examine allcombinations here. But the translation of informational axioms into theINf I format makes it often quite easy to see the consequences of combiningseveral such axioms. When a SOF satises INf1I and INf2I, for two ltersf1 and f2, one may simply look at the information which is retained by bothlters. For instance, combining PI and IFS means using the two lters
e; uN x; uN y and hN ; x; y;which implies retaining only
hN ; uN x; uN y:
9.1 IOA and PI
If one combines the lters for IOA and PI,
uN x; uN y;X ; x; y and e; uN x; uN y;
in order to extract the common information, one gets
uN x; uN y;X ;
which expresses the welfarist approach according to which, in a given set X ,only vectors of utilities are taken into account, irrespectively of any otherinformation.
However, INf I for this third data lter is not in general equivalent to thecombination of IOA and PI. This equivalence is obtained only on sucientlyrich domains. The classical welfarism lemma was obtained by dAspremontand Gevers (1977) under the assumption that all utility functions on X areadmissible. This assumption of universal domain makes it impossible to applythe result to economic domains and raises the question of how general it is.Fortunately, a weaker richness assumption, which is satised on many eco-nomic domains, is sucient.
21 Except in the subdomain of homothetic preferences. See Eisenberg (1961),Fleurbaey and Maniquet (1996).
368 M. Fleurbaey
Let us assume that for every e hN ;X 2 E, with uN UhN , and anyx; y; x0; y0 2 X , one can nd e1 h1N ;X ; e2 h2N ;X 2 E; u1N Uh1N ;u2N Uh2N , and x00; y00 2 X such that
u1N x u1N x00 u2N x00 u2N x0 uN x;u1N y u1N y00 u2N y00 u2N y0 uN y:
This assumption means that any pair of alternatives x; x0 can be connected byindierence to a third alternative x00 for two proles of preferences, and thatthis connection can be done for two pairs at the same time.22
Proposition 6. Under the above richness assumption, a SOF R satises IOA andPI if and only if it satises INfI for f dened by:
f e; x; y uN x; uN y;X :
Proof. IOA and PI jointly imply INf I. Let f e; x; y f e0; x0; y0. Thismeans that X X 0; uN x u0N x0; uN y u0N y0. By the richness assump-tion, one can nd e1 h1N ;X ; e2 h2N ;X 2 E; u1N Uh1N ; u2N Uh2N ,and x00; y00 2 X such that
u1N x u1N x00 u2N x00 u2N x0 uN x;u1N y u1N y00 u2N y00 u2N y0 uN y:
By PI, xIe1x00; yIe1y00; x0Ie2x00; y0Ie2y00. By IOA,Re fx;yg
Therefore one has
xRey , xRe1y by IOA, x00Re1y00 by PI, x00Re2y00 by IOA, x0Re2y0 by PI, x0Re0y0 by IOA:
22 It is not satised in domains where an alternative is strictly worse than all others forall admissible preferences, such as domains with strictly monotonic preferences andallocations containing a zero bundle for some agents. One can check that on suchdomains the welfarism result does not hold. For instance, the SOF which simply ranksall allocations in which no individual has a zero bundle over all other allocations doessatisfy IOA and PI, but is not welfarist.
On the informational basis of social choice 369
INf I implies PI. Let uN x uN y. Then f e; x; y f e; y; x,implying xRey , yRex, and therefore xIey.
INf I implies IOA. Let uN x u0N x; uN y u0N y. Thenf e; x; y f e0; x; y and f e; y; x f e0; y; x so that by INf I,Rejfx;yg Re0jfx;yg. . j
9.2 INVU and IOA
In the literature about social choice with interpersonal comparisons of utility,the axioms INVU and IOA are seldom used separately, and it is interesting toanalyze the consequences of this association. This is rarely done because theusual axiomatic analysis rst exploits the abovewelfarist result (combining IOAand PI) in order to obtain a preorder over utility vectors inRN , and then studiesthe consequences of INVU over this preorder. As it turns out, combining INVUand IOA has very clear consequences, independently of Pareto conditions.
First, for any U UONC , any u 2 U, any N and any vector aN 2 RN , letuaN denote the vector uiaii2N . The set Ug generates an equivalencerelation Ug on [NNRN RN , which is dened by
aN ; bN Ug a0N 0 ; b0N 0 , N N 0 and 9u 2 Ug; a0N uaN ; b0N ubN :
For any aN ; bN , let Ug aN ; bN denote the equivalence class for Ug towhich aN ; bN belongs. Interestingly, dierent groups Ug may yield the sameequivalence relation. A famous example, due to Sen (1970), is given by UONCand UCNC.
Proposition 7. 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 existsh0N ;X 2 Esuch that Uh0N uUhN . A SOF R satises INV U and IOA ifand only if it satises INfI for f dened by:
f e; x; y Ug uN x; uN y;X ; x; y;where e hN ;X ; uN UhN .Proof. INVU and IOA jointly imply INf I. Let f e; x; y f e0; x0; y0.This implies x x0; y y0;N N 0;X X 0 and
9u 2 Ug; Uh0N x uUhN x; Uh0N y uUhN y:By the richness assumption, there exists e00 h00N ;X 2 E such that Uh00N uUhN . One therefore has
Uh0N x Uh00N x; Uh0N y Uh00N y:By IOA, this implies Re00jfx;yg Re0jfx;yg. In addition, INVU, which is, asnoted above, equivalent to INVUg, entails that Re00 Re. ThereforeRe0jfx;yg Rejfx;yg.
370 M. Fleurbaey
INf I implies INVU. This immediately follows from the fact that:8e hN ;X ; e0 h0N ;X 2 E; 8uN UhN ; u0N Uh0N ,
9u 2 U; u0N uuN ) f e; : f e0; ::INf I implies IOA. This follows from the fact that: 8e hN ;X ;
e0 h0N ;X 2 E; 8uN UhN ; u0N Uh0N ; 8x; y 2 X ;uN jfx;yg u0N jfx;yg ) f e; x; y f e0; x; y: j
The formula f e; x; y Ug uN x; uN y; x; y shows how thecombination of INVU and IOA limits the information about the utility bi-vector uN x; uN y: It is also worth noting that, under the richnessassumption of this proposition, INVU and IOA are, jointly, equivalent 23 tothe following simple axiom:Binary invariance to U BINV U):8e hN ;X ; e0 h0N ;X 2 E; 8uN UhN ; u0N Uh0N ; 8x; y 2 X ;
9u 2 U; u0N x uuN xu0N y uuN y
) Rejfx;yg Re0jfx;yg:
In view of the above proposition, BINVU is equivalent to BINVU0 wheneverUgU0g . As an example, BINVUCNC is equivalent to BINVUONC .24
The equivalence relation UONC , restricted to a given population N forbrevity of notations, may be described in the following, equivalent ways:
aN ; bN UONC a0N ; b0N , 9u 2 UONC; 8i 2 N ;a0i uiaib0i uibi
, 8i 2 N ; a0i b0i
ai bi > 0 or a0i b0i ai bi 0:, a0N b0N is in the same orthant(s) as aN bN :
Recalling that the application of this equivalence relation is made on
aN ; bN uN x; uN ya0N ; b0N u0N x; u0N y;
one sees that the axiom BINVUONC is actually Arrows axiom of Indepen-dence of Irrelevant Alternatives. Let Rhi denote individual is preferenceson Xi, and let RhN denote Rhii2N .Arrow Independence of Irrelevant Alternatives (Arrow IIA):
8e hN ;X ; e0 h0N ;X 2 E; 8RN RhN ;R0N Rh0N ; 8x; y 2 X ;RN jfx;yg R0N jfx;yg ) Rejfx;yg Re0jfx;yg:
23 The proof of this fact is given in the appendix.24 Bossert (1999) studies the related phenomenon that combining invariance condi-tions about utility proles with Pareto indierence and IOA may entail much largerinvariance conditions.
On the informational basis of social choice 371
As another illustration, consider the case of cardinal unit comparability,related to the set of transformations UCUC (dened above). UCUC ; is analgebraic group, and the equivalence relation UCUC , restricted to a givenpopulation N , may be described in the following, equivalent ways:
aN ; bN UCUC a0N ; b0N , 9a 2 RN ; b 2 R; 8i 2 N ;a0i ai baib0i ai bbi
, 8i; j 2 N ; a0i b0iai bi
a0j b0jaj bj or
a0i b0i ai bi 0 or a0j b0j aj bj 0, 8i; j 2 N ; a
a0j b0j ai bi
aj bj or a0j b0j aj bj 0
, a0N b0N is proportional to aN bN :As a consequence, BINVUCUC involves much more precise information thanthe mere comparisons of utility dierences with which utilitarianism iscommonly associated.25 The ratios of dierences have to remain unchangedunder UCUC; so that the direction of the vector of utility dierencesuN x uN y is unaltered.26
9.3 INFA, PI and IFS
There is an obvious tension between INFA and IFS. The former excludesinformation about individual utilities outside X , while the latter excludes anyinformation about X . The combination of the two excludes a lot of infor-mation. This is illustrated by the following proposition.
1. The domain E is a Cartesian product H N, where H is the set of proleshN and N is the set of feasible sets X ;
2. The set N is such that for any X ;X 0 2 N, there is X 00 2 N such thatX [ X 0 X 00;
3. There exist N1;N2 N such that:(a) for any X 1 2 N1;X 2 2 N2;X 1 \ X 2 ;;(b) for any e hN ;X 2 E; uN UhN , any x 2 X , there exist X 1 2 N1;
x1 2 X 1;X 2 2 N2; x2 2 X 2; uN x uN x1 uN x2;
25 This point is emphasized in Bossert (1991) and Bossert and Weymark (1998). Forinstance, with three individuals, knowing that
u1x u1y > u2y u2x > u3y u3x > 0tells everything about individual preferences and comparisons of dierences, but isinsucient for the utilitarian SOF to rank x and y:26 Although the association of INVU and IOA into BINVU is worth analyzing,without Pareto conditions, it does not yield a direct characterization of the traditionalSOFs. BINVU alone does not prevent social preferences from being imposed or biasedin favor of particular individuals.
372 M. Fleurbaey
(c) for any X 1 2 N1;X 2 2 N2, any hN ; h0N 2 H, there exists h00N 2 H such thatUh00N jX 1 UhN jX 1 and Uh00N jX 2 Uh0N jX 2 :
In the third part27 of this assumption, the idea is that the feasible setsX 1;X 2 are in the outskirts of the global domain, with an empty intersectionand no constraint about connecting utility proles. For instance, in a problemof division of unproduced commodities, X 1 may contain only allocations ofgood 1, and X 2 allocations of good 2.28 The example with bread and water inSubsect. 8.2 may help in getting the intuition of the next result.
One indeed obtains the following new welfarism proposition.
Proposition 8. Under the above assumption, a SOF satises INFA, PI and IFSif and only if it satises INfI for f dened by:
f e; x; y uN x; uN y:
Proof. INFA, PI and IFS jointly imply INf I. Let f e; x; y f e0; x0; y0,that is, uN x; uN y u0N x0; u0N y0. Let e1 hN ;X 1 2H N1; x1; y1 2 X 1, and e2 h0N ;X 2 2 H N2; x2; y2 2 X 2, be such that
uN x1; uN y1
uN x; uN y u0N x2; u0N y2
u0N x0; u0N y0
Let e hN ;X ; e0 h0N ;X 2 E be such thatX [ X 0 [ X 1 [ X 2 X :
By PI, xIex1; yIey1; x0Ie0x2; y0Ie0y2. ThereforexRey , x1Rey1; x0Re0y0 , x2Re0y2:
By IFS, one actually obtains
xRey , x1Re1y1; x0Re0y0 , x2Re2y2:Let e hN ;X 2 E; uN UhN , be such that uN jX 1 uN jX 1 ; uN jX 2 u0N jX 2 . One then has
uN x1 uN x1 uN x u0N x0 u0N x2 uN x2;uN y1 uN y1 uN y u0N y0 u0N y2 uN y2:
As a consequence, by PI x1Iex2 and y1Iey2. Besides, by INFA,x1Re1y1 , x1Rey1; x2Re2y2 , x2Rey2:
27 Notice that the third part implies the rst, which is written down only for clarityssake. Indeed, Uh00N jX 1 UhN jX 1 and Uh00N jX 2 Uh0N jX 2 entail Uh00N jX 1\X 2 UhN jX 1\X 2 Uh0N jX 1\X 2 , which cannot hold for any hN ; h0N if X 1 \ X 2 6 ;.28 Part 3.b of the assumption is not satised in an abstract domain containingindividual preferences without any indierence. One can obtain similar results on suchdomains by relying on dierent Pareto conditions.
On the informational basis of social choice 373
x1Re1y1 , x2Re2y2:Finally,
xRey , x0Re0y0:INf I implies INFA. When uN jX u0N jX ; uN x; uN y u0N x; u0N y
for any x; y 2 X , so that f e; x; y f e0; x; y for any x; y 2 X , andtherefore Re Re0.
INf I implies PI. Same argument as in Proposition 6.INf I implies IFS. This is obvious from the denition of f . j
Combining this result with the welfarism result of Subsect. 9.1, one candeduce that, under PI and IFS, the axioms INFA and IOA are equivalent.29
10 Information and the possibility of social choice
10.1 Two routes
It is now well understood that Arrows impossibility theorem comes from thefact that the axioms of the theorem delete too much information compared towhat is needed for the construction of consistent social preferences. Moreprecisely, Arrow IIA is the culprit. As explained above, Arrow IIA isequivalent to the combination of INVUONC and IOA. This decompositionindicates two routes for retrieving possibility results.
The most trodden route has consisted in weakening INVUONC andretaining information about interpersonal comparisons of utility (or anynotion of well-being represented by U ). It leads to traditional social welfarefunctions such as utilitarianism and the maximin criterion. The other route,suggested somewhat obscurely by Samuelson (1977)30 and very clearly byPazner (1979), consists in weakening IOA. Pazner actually proposed toweaken Arrow IIA into an axiom which is exactly the combination ofINVUONC and INIA, and, in an economic domain (with monotonic prefer-ences), requires the social ranking of two allocations to depend only onindierence curves at bundles obtained by individuals at the two allocations.While these authors gave examples of social preferences satisfying INVUONC,INIA, and PI, such as the Pazner example presented in Subsect. 4.1, moreprecise axiomatic derivations of social preferences from these axioms and
29 Along these lines a strengthening of Arrows impossibility theorem can be obtainedSee Fleurbaey et al. (2002).30 Samuelson (1977) focused on Kemp and Ngs (1976) single-prole independenceaxiom, and did not criticize Arrow IIA directly, under the dubious argument thatmulti-prole issues were largely irrelevant in welfare economics. A more explicitcriticism of Arrow IIA appeared however in Samuelson (1987).
374 M. Fleurbaey
other equity principles were obtained much more recently, by Fleurbaey andManiquet (2000, 2001).31
In an economic model (with monotonic preferences), where individual isindierence curve at allocation x may be denoted Iix, and IN x may denoteIixi2N , the combination of INVUONC , INIA, and PI is equivalent to INf Ifor
f e; x; y IN x; IN y;X :This can be compared to the combination of IOA and PI, which, in the samecontext, yields
f e; x; y uN x; uN y;X :In other words, while traditional social welfare functions process and com-pare utility vectors, the new kind of social welfare functions obtained in thealternative approach would have vectors of indierence curves as their argu-ment.
10.2 Welfarism and quasi-welfarism
The relationship of these approaches to welfarism deserves some scrutiny. Asexemplied by the Borda rule or the Pazner example of SOF, some of the newkind of social welfare functions evaluate any given vector of indierencecurves by rst putting a real number on each of them, and then applying atraditional social welfare function to this vector of real numbers. Is thatwelfarist or not? Does it involve interpersonal utility comparisons just as thetraditional approach? The confusion is reinforced by the fact that an ap-proach may be formally similar to welfarism, but philosophically quite faraway from it. I suggest distinguishing the following approaches, starting fromthe strongest form of welfarism:
Real welfarism. The real thing, in matters of welfarism, consists in lettingU measure subjective utility or satisfaction (there are several possibilities,and therefore several variants of real welfarism), and requiring the SOF tosatisfy IOA and PI.A typical example is classical utilitarianism.Real welfarism involves interpersonal comparisons of subjective utilities.
Formal welfarism. It adopts an exogenous denition of U , and requires theSOF to satisfy IOA and PI. But U may measure any objective or subjectivenotion of well-being, and it is provided by moral philosophy or the socialdecision-maker.
31 Results based on dierent weakenings of IOA were obtained by Kaneko andNakamura (1979) and Dhillon and Mertens (1999) in an abstract model withuncertainty.
On the informational basis of social choice 375
An example, where moral philosophy provides U , is Sens (1985, 1987)approach in terms of capabilities and functionings. Another example,involving the social decision-maker as the purveyor of U , is given by manypublications in public economics.32
Formal welfarism involves interpersonal comparisons of whatever Umeasures.
Individualistic quasi-welfarism. It lets U be any representation of individualpreferences, and requires the SOF to satisfy INVUONC and PI. But the SOFwhich is nally obtained satises INf I for
f e; x; y vN x; vN y;X ;
where vi is a real-valued function derived from individual characteristics:vi V hi. The satisfaction of PI forces V hi to be ordinally equivalent toUhi. But the SOF does not satisfy IOA. It satises INIA when the valueof vix depends only on the upper and lower contour sets at x.The Borda rule is an example, where vi is the individual Borda score.Bergson and Samuelsons concept of social welfare also belonged to thisapproach, although these authors did not venture to defend one particularfunction V . A more precise example is Pazners (1979), which, as denedabove, relies on
vixi minfk 0juikX uixig:
One sees that vi is ordinally equivalent to ui, but this SOF satisesINVUONC because a change of ui which does not alter ordinal preferenceswill leave the vi unchanged.This approach involves interpersonal comparisons of vi. But one shouldresist the temptation to view vi as a measure of subjective utility. The onlytint of welfarism here comes from PI, and the best interpretation of the vi,in the Pazner example for instance, is not in terms of satisfaction, but interms of resources. The quantity vixi does not measure is satisfaction,but measures the value of bundle xi, in terms of kX, according to isopinion. Similarly, in the Borda rule, the Borda score is less a measure ofsubjective satisfaction than a measure of the value of an alternative, esti-mated, roughly, by the number of alternatives which are worse. This jus-ties replacing the word welfarism by quasi-welfarism. For the Paznerexample, one could even propose the word wealthfarism.
Non-individualistic quasi-welfarism. It lets U be any representation ofindividual preferences, and requires the SOF to satisfy INVUONC and PI.But the SOF which is nally obtained satises INf I for
32 See e.g., Atkinson (1995), where U may in particular embody the social plannersaversion to inequality. A general discussion of formal welfarism in relation to realwelfarism is made in Mongin and dAspremont (1998).
376 M. Fleurbaey
f e; x; y vN hN ; x; vN hN ; y;X ;
where vi is a real-valued function depending on the whole prole and thewhole allocation.An example is the Fleurbaey-Maniquet example of SOF, dened above,and which can be alternatively formulated with the maximin criterionapplied to
vihN ; x vi xi; phN ; x;
vi xi; p minfk 0j9z; pz kpX; uiz uixig;phN ; x argmax
vi xi; p:
That is, vihN ; x is the money-metric utility function of i measured interms of the initial endowment kX which would give i the same satis-faction as xi, at prices p computed so as to give allocation x the bestmaximin value.This approach is quite far away from welfarism. First, like individualisticquasi-welfarism, it measures the value of resources rather than subjectivesatisfaction as such. Second, unlike individualistic quasi-welfarism, it doesnot evaluate the situation of an individual in isolation from the rest of thepopulation.
One could moreover distinguish real and formal variants of quasi-welfarism, since the indierence curves which serve as the main input in theevaluation of an alternative may belong to the actual subjective indierencemaps of the individuals, or instead represent any ordinal representation ofindividuals objective interests.
10.3 Separability and Walrasian allocations
Interestingly, one may conjecture that a SOF from the last non-individualisticapproach cannot in general satisfy the separability axiom III. This is easilychecked for the Fleurbaey-Maniquet example. This is actually true for anySOF relying on the maximin criterion. But in general, a leximin renement ofthe maximin may satisfy III, whereas this cannot be achieved with this par-ticular example. As can be checked easily, no SOF for which the maximumvalue over the feasible set is obtained only by egalitarian Walrasian alloca-
On the informational basis of social choice 377
tions can satisfy III or IUI.33 There is a basic dilemma between separability(III or IUI) and orienting social preferences toward Walrasian allocations.This dilemma has been ignored in the theory of fairness because the allocationrule which selects the subset of egalitarian Walrasian allocations does satisfythe consistency requirement (which is logically weaker than IUI) thatremoving some individuals and their bundles still leaves the allocation (for therest of the population) optimal.
10.4 A trilemma
The results of Subsect. 9.3 suggest a slightly more complex picture than thebinary analysis of the possibility of social choice which has just been pro-posed. From these results, one can deduce that no ordinal Paretian SOF cansatisfy INFA and IFS without satisfying Arrow IIA, and therefore running intothe troubles of Arrows theorem.
For instance, the Borda rule dened as above with individual Borda scores
vix #uiLCuiuix \ X is ordinal (it satises INVUONC), and satises INFA, but not IFS. A variant ofthis rule would dene Borda scores by
vix #uiLCuiuix;and obtain a modied Borda rule which satises IFS but not INFA. Theresults obtained in 9.3 show that there is no hope of nding another variantsatisfying simultaneously INFA and IFS in general.
The Pazner example of SOF satises INFA but not IFS, when the feasibleset X is dened as
X fx x1; . . . ; xn 2 Rn j x1 xn Xg:The Fleurbaey-Maniquet example of SOF satises neither INFA nor IFS. Anexample of SOF satisfying IFS but not INFA is obtained by modifyingPazners example, letting vi be dened with respect to a xed X0, independentof available resources.
This suggests that Arrows impossibility can be understood as reectingnot only a dilemma between ordinalism (INVUONC) and the restriction tolocal data (IOA), but also, and more deeply, a trilemma between ordinalism
33 The core argument is given in Fleurbaey and Maniquet (2001). In a nutshell,suppose x1; x2; x3 is an egalitarian Walrasian allocation and x01; x02; x3 is not, withx3 X=3, and that both x1; x2 and x01; x02 are egalitarian Walrasian allocations whentotal resources are 2=3X. This violates IUI, since in the reduced economy without thethird individual and with resources X x3 2=3X, social preferences are indierentbetween x1; x2 and x01; x02, whereas x1; x2; x3 is strictly preferred to x01; x02; x3 in thelarge economy. Besides, just by changing individual 3s preferences without removinghim, one may reverse the situation so that x01; x02; x3 is an egalitarian Walrasianallocation and x01; x02; x3 is not. Which violates III.
378 M. Fleurbaey
(INVUONC), the restriction to local data (INFA) and disregarding feasibilityconstraints (IFS). Since IFS was somehow built in Arrows framework, whileIOA was an extreme kind of restriction to local data, this could not beapparent in the classical approach to social choice.
In more general terms, we can then say that, in a Paretian approach, thereis a tension between three broad ethical principles:
Individual well-being should be measured in an ordinal non-comparableway (e.g. individuals should be held responsible for their numerical level ofwell-being).
Individual well-being at distant or infeasible alternatives is irrelevant. Social preferences should not be inuenced by feasibility constraints or be
limited to local alternatives.
As a consequence, there are not two but three dierent ways of obtainingpossibility results in social choice: 1) introducing interpersonally compa-rable utilities; 2) introducing non-local information about individual pref-erences; 3) restricting social preferences to local or feasible alternatives.These three ways are not exclusive of each other, and can be combined invarious degrees. For instance, the Pazner example of SOF satises INFAbut not IOA, which means that it introduces some non-strictly localinformation, but without going beyond the feasible set, while it violatesIFS.
In conclusion, the INf I axiom is a convenient instrument for the analysis ofthe informational content of various axioms. It disentangles the notion ofinformational basis from its historical origin tied to the perspective ofinterpersonal comparisons of utilities, and enabled us here to obtain someinteresting insights in the general outlook and understanding of the problemsand perspectives of social choice.
The following table summarizes the main informational dimensions andthe related data lters (see the relevant sections for the notations).
Information Axiom Data lter f e; x; yUtility INVU Ug e; x; y
Local alternatives IOA uN x; uN y;X ; x; y INFA uN jX ;X ; x; y
Pareto PI e; uN x; uN y Responsibility IRF cN ;X ; x; y Separability III uNnMe;x;y;N ;X ; x; y
Feasibility IFS hN ; x; y
On the informational basis of social choice 379
In this table, all axioms except PI restrict the information retained about ein f e; x; y. The next table summarizes the compatibility patterns obtainedwhen dierent informational restrictions are jointly imposed.
As obvious from this table, the combination of informational restrictionsabout utilities, non-local alternatives, non-utility features of alternatives(Pareto), and feasibility is fateful for the possibility of social choice.Retaining information about utilities has been the most favored escape fromthe impossibility problem, including when utility in reinterpreted in a non-welfarist way, as in Sens theory of capabilities. Another route consists inretaining information about individual characteristics at non-local alterna-tives, as proposed by Samuelson, Pazner and others. Along this route,retaining information about feasibility constraints is also often helpful, asexemplied by several SOFs presented above, which violate IFS by takingaccount of the relative scarcity of goods.
It is therefore important to broaden the concept of informational basisnot only for a better conceptual foundation of the theory of social choice,but also to get a better picture of the dilemmas and possibilities of socialchoice in relation to informational restrictions. As emphasized in this pa-per, the selection of data to be retained must primarily be a question ofethical relevance. In the comparison of the two (or three) main possibilityroutes for social choice which have just been outlined, the relative relevanceof utility information, non-local information and feasibility constraints hasto be assessed in order to decide which route is the most fruitful. Thispaper has examined and provided a few arguments touching on thesequestions, and distinguished in particular various kinds of welfarism andquasi-welfarism, but its purpose is to point out the issue, not to settle it.
Proposition 9. A SOF R satises IUI if and only if it satises INfI for f denedby:
f e; xN ; yN hNnMxN ;yN ;X xMxN ;yN ; xNnMxN ;yN ; yNnMxN ;yN
where MxN ; yN fi 2 N jxi yig.
Information Axioms Data lter f e; x; yLoc. alt. & Pareto IOA & PI uN x; uN y;X Loc. alt. & Utility IOA & INVU Ug uN x; uN y ;X ; x; y
e.g.: Arrow IIA IOA & INVUONCorthant uN x uN y ;
X ; x; y
Loc. alt., Feas. & Par. INFA, IFS & PI uN x; uN y
380 M. Fleurbaey
Proof. IUI implies INf I. Let f e; xN ; yN f e0; x0N ; y0N . By IUI,xNReyN , xNnMxN ;yN ReryNnMxN ;yN
with er hNnMxN ;yN ;X xMxN ;yN andX xMxN ;yN fzNnMxN ;yN jxMxN ;yN ; zNnMxN ;yN 2 Xg:
x0NRe0y0N , x0N 0nMx0N ;y0N Re0ry0N 0nMx0N ;y0N
with e0r h0N 0nMx0N ;y0N ;X0 x0Mx0N ;y0N and
X x0Mx0N ;y0N fzNnMxN ;yN jx0MxN ;yN ; zNnMxN ;yN 2 Xg:
Now, since er e0r and xNnMxN ;yN x0N 0nMx0N ;y0N ; yNnMxN ;yN y0N 0nMx0N ;y0N , one
xNnMxN ;yN ReryNnMxN ;yN , x0N 0nMx0N ;y0N Re0ry0N 0nMx0N ;y0N ;
and as a result
xNReyN , x0NRe0y0N :INf I implies IUI. Let xM yM . One has
f e; xN ; yN hNnMxN ;yN ;X xMxN ;yN ; xNnMxN ;yN ; yNnMxN ;yN
f er; xNnM ; yNnM h NnM nMxNnM ;yNnM ; X xM xMxNnM ;yNnM ;xNnMnMxNnM ;yNnM ; yNnMnMxNnM ;yNnM
Now, NnMnMxNnM ; yNnM NnMxN ; yN , andX xM xMxNnM ;yNnM
fzNnMxN ;yN jxMxNnM ;yNnM ; zNnMxN ;yN 2 X xMgfzNnMxN ;yN jxM ; xMxNnM ;yNnM ; zNnMxN ;yN 2 XgX xMxN ;yN :
Therefore f e; xN ; yN f er; xNnM ; yNnM , so that by INf I, xNReyN ,xNnMReryNnM . jProposition 10. 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 U, there exists h0N ;X 2 Esuch that Uh0N uUhN . A SOF R satises INVU and IOA if and only if itsatises BINV U.
On the informational basis of social choice 381
Proof. INVU and IOA jointly imply BINV/. Let e hN ;X ;e0 h0N ;X 2 E; uN UhN ; u0N Uh0N ; x; y 2 X , such that
9u 2 U; u0N x uuN x
u0N y uuN y:
By the richness assumption, there exists e00 h00N ;X 2 E such thatUh00N uuN . One therefore has
u0N x Uh00N x; u0N y Uh00N y:By IOA, this implies Re00jfx;yg Re0jfx;yg. In addition, INVU entails thatRe00 Re. Therefore Re0jfx;yg Rejfx;yg.
BINVU implies INVU. This immediately follows from the fact that:
9u 2 U; u0N uuN ) 8x; y 2 X ; 9u 2 U;u0N x uuN xu0N y uuN y:
BINVUg implies IOA. Let e hN ;X ; e0 h0N ;X 2 E; uN UhN ;u0N Uh0N , and x; y 2 X be such that uN jfx;yg u0N jfx;yg. The identity trans-formation belongs to Ug, and therefore
9u 2 Ug; u0N x uuN x
u0N y uuN y;
so that, by BINVUg;Re0jfx;yg Rejfx;yg.In order to prove that BINVU implies IOA, it is then sucient to show
that BINVU implies BINVUg. Let e hN ;X ; e0 h0N ;X 2 E;uN UhN ; u0N Uh0N ; x; y 2 X , such that
9u 2 Ug; u0N x uuN x
u0N y uuN y:
By Lemma 1, there exist u1; . . . ;um 2 U [ U1 such that u u1 . . . um.Let em hmN ;X 2 E be such that UhmN umUhN . Such em exists, by therichness assumption, since um 2 Ug. If um 2 U, then, by BINVU,
Rejfx;yg Remjfx;yg:If um 2 U1, then u1m 2 U, and
uN x u1m umuN xuN y u1m umuN y;
so that, by BINVU, one again obtains
Rejfx;yg Remjfx;yg:If one takes em1 hm1N ;X 2 E such that Uhm1N um1 UhmN
um1 um UhN , a similar reasoning leads to
382 M. Fleurbaey
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