Mathematical Social Sciences 2 (1982) !V9-208 North-Holland Publishing Company

199

ENVY-FREE RIGHTS ASSIGNMENTS AND SELF-ORIENTED PREFERENCES

Wulf GAERTNER Department of Economics, University of Osnabriick, D-4500 Osnabriick, West Geman!

Communicated by F.W. Roush Received 30 January 198 1 Revised 10 April 1981

Very recently a new solution to Sen’s “Impossibility of a Paretian liberal” has, been suggested where the focus is on the rights assignments per se (Austen-Smith, 1979). Ir was shown that the

concept of fairness, when applied to rights, admit; the existence of sociai! decision functions which satisfy Sen’s original conditions. Unfortunately this result collapses u hen ;:ndividuals have rights over more than one pair of alternatives.

In order to obtain possibility results for this more genera! case the present paper proposes to

restrict individuals’ preference orderings. It is proved that envy-free collective choice rules exist if individual preferences are self-oriented and if, in addition, people attach primary importance to their own private sphere alternatives. These restrictions are quite severe, but they may be justified if one values the absence of envy in rights allocations very highly.

Key words: Liberal paradox; envy-free rights assignment; self-oriented preferences.

1. Introduction

In a novel approach to circumvent Sen’s (1970) famous “impossibility of a Paretian liberal” Austen-Smith (1979) suggested that individual rights be allocated in such a way that there is no envy among the members of society. More precisely, it was proven that under a so-called K-fair rights assignment a social decision function satisfying Sen’s original conditions U, P and L exists if each individual is decisive over one pair of (private) alternatives. The focus on the rights assignment per se is quite interesting, but unfortunately Austen-Smith also provided an example which demonstrated that this possibility result collapses when individuals have rights over more than one pair of alternatives. We think that this last result is rather devasta- ting, for a moment’s reflection shows that in real world situations individuals normally are allowed to be socially decisive over more than just one pair of (private) alternatives (compare the bill of rights).

One way to obtain possibility results for this more general case - and this is the road that will be taken in the present paper - is to abandon the requirement of unrestricted domain. We readily admit that there are no convincing a priori reasons for restricting societies’ preference profiles, but if one wants to achieve a state of

01654896/82/0000-0000/$02.75 0 1982 North-Holland

200 W. Gaertner /Envy-free rights assignments

envy-free rights assignments one obviously has to demand that individuals order wiat states ‘in certain ways’.

Fait assignments of rights will be the more likely the more people focus on their own personal affairs, It will be sieen that, roughly speaking, individuals have to rank the prefered elements of their private choices at the top of their respective orderings. For the general case of rights over more than one pair of alternatives, however, this requirement is not sufficient for the existence of a social decision function. Therefore, we additionally introduce the concept of self-oriented prefe- rences. Furthermore, we will propose a weaker version of ,husten-Smith’s notion of K-fairness.

Austen-Smith compared his results to those of Blau (1975) who developed the notion of meddlesome preferences and proved the existence of a social decision function for societies with two persons. For more general cases Blau also obtained impossibility results. His requirements proved to be too weak and therefore had to be strengthened in order to handle larger preference profiles more successfully (cf.

aertner and Kruger, 1981). This paper will be formulated in terms of Gibbard’s (1974) issue representation.

In Sectio;n 2, we introduce the bulk of notation and the necessary definitions. In ion 3, we discuss Austen-Smith’s preference situation for two persons which led is .tegative result in the presence of K-fairness. Section 4 gathers our possibility

ults for self-oriented preferences and envy-free assignments of rights, and Section 5 contains some concluding remarks on the relationship between fairness and indivi- dual rights over private spheres.

2. Notation and definitions

Let X= (x,y, z, . ..) be a finite set of all conceivable social states and let .“J’ denote the family of non-empty finite subsets of X. Each and every A e .d denotes a set of available social states. Let N be a finite set of n individuals, each of whom has a preference ordering Ri on X, together forming a profile (RI, Rz, . . . , R,) = (R) of individual orderings. A collective choice rule CCR is a function F which for each profile (R) generates a social choice function C on .d/: C= F( (R),A), where C(( R),A)#B for any A E s/ represents the set of socially chosen states from A when the individuals have orderings R,, R2, . . . , R,.

Following G;bbard (1974) we define a personal issue for an individual i as a set of personal feature-alternatives Xi = {Xi, yi, . . .} containing a finite number of elements (at least two). For notational sim;ulicity we shall assume that there is just one perso- nal issue per individual over which the question of individual rights may arise. Assu- ming furthermore that the issues are all mutually independent, we can define the set of ali corxeivable social states X as the Cartesian product X = X1 x X, x l x X, , Mth x== <xj1x2 ) ,. .,x,) being a typical element from it. Basic to the following arqjumetats is the concept of j-variants; these are alternatives which differ from each

W. Gaertner / En by-free rights assignments 201

other only in their jth feature. More formally, x is aj-variant of y if and only if Vi, i#j, Xi=Yi and Ti++Yj. For convenience we write the social state cbz2 I s*.,Zi-- j,XirZi+ 19 l **9 z, ) more briefly as (Xi, E) .

The set of all personal pairs over which individual i will have rights is

Si= {(X*y)EXXXI X= (Xi,Z), _Y= (_Yi,t’), Xi*Yiand z=z’l*

FW ._a& i, Si is a set of doubleton sets and describes person i’s private splrere. S=U iEN Si, the union of all private spheres, may be considered as a global

indicator for the range of private decision in society.

Definition 2.1. Given X and Ri, person i is decisive over ihe pair of social states x,y under S if and only if

(i) (%Y) 45 Si9 (ii) (X, y) E Pi*

This will be denoted by (x,y) E Die Deckiveness means for all iE A’: [(x,y) E D, =+ (x,y) E P], where P is the asymmetric part of the social preference relation R on X. Di is called the set of decisive pairs of person i, or i’s decisive set. Di also is a set of doubleton sets, and for each k N, elements from Di will be denoted by QI, D,?, . . . . Di,n. In the following we assume that the cardinality of every Di is the same; in other words: all individuals are decisive over an identical number of pairs of social a1te;rnatives.r We call D = U ie N Di society’s rights assignment.

Following Austen-Smith, we now define an ordering on D for each in N. For any DjhEDj, Dkh~Dk, j, kN, he{1 ,..., m) and any kN,

(Djhv Dkh) E R? e [ZIOEDjh/ t/wEDkh:(O,W)ERi]m

P& the asymmetric part of R,?@, is defined obviously. Furthermore, for any in N, for any j, k E N,

(Dj, Dk) E J?i* e [there exists an injective mapping /I:D’+O, 1 (Djh,,CL(Djh))E Ry, hE { 1, .*.,m}].

2’;* obviously is an ordering on D, since Ri was assumed to be an ordering on X, and

U itg.vDi EX- We are now able to formulate Austen-Smith’s notion of K-fairness.

Definition 2.2. A rights assignment D is said to be K-fair if and only if (i) [(Dig Dj) E 2i*, Vi, je NJ and (ii) [Z.-C NU { 0) 1 (Dk, 0,) E yk*, Vk E K, some (not necessarily fixed) /E N]. Here (Dk, 0,) E C&*, the asymmetric part of .&.*, meaning that for at least one

&ED,+, (&,,&)EP$ for SOme Dlh’~DI, h,h’E{l,..., nr).

1 This assumption is acceptable as long as there are no convincing arguments saying that some indivi-

duals should be decisive over a larger number of pairs than others.

W. Gaertner / Envy-free rights assignments

According to this definition, K-fairness is given if and only if no individual envies molther person his (her) private sphere alternatives over which that person is &l&e, while B subset K of all the individuals strictly prefers at least one of its private pairs DkA to come private pair Dlh8 of some person I (for an illustration see

iorb 3). When K =0 (which is, of course, admissible) we have an allocation which would be called envy-free according to the original terminology by Foley (1967). Each person finds that his (her) own rights allocation is at least as good as that of any other individual. We should mention in brackets that Pazner (1977) called an envy-free aDocation fair, while for Varian (1974) for example, an allocation is fair if it is envy-free and Pareto-efficient (on these terminological differences, see also Sumnura, 1981).

AS we do not see a convincing argument for requiring K #0 in the above definition of K-fairness, we simply work with the notion of O-fairness in the follo- wing. We prefer to call it fairness and thereby are in conformity with Pazner’s terminology.

D&&Ion 2.2’. A rights assignment D is said to be fair if and only if

[(D,,D,,E Jy, Vi,jEN].

A we&r concept is the foEowing which we call minimal fairness. For any ieJV,

!D,,UD,)E*~;o(3xeDih,somehE(l,..., m} 1

VW u JEN \ li) {Djl 9 l ee, Dj,q}: (x,Y)ERi and (X,Z)ERi)*

D..fla/tion 2.3. A rights assignment is said to be minimally fair if and only if, for t~~tr i&V, (O,, UDj)E .%,.

Minimal fairness is-satisfied if and only if for each individual there is one personal alternative which he (she) weakly prefers to any private alternative pertaining to any other individual.

We now turn to the proposed domain restriction conditions. We have assumed that each individual i has an ordering Ri on X. This means that for each pair (Xi, yJ

with X, #tyi from Xi, there are combinations of other private features z, z’ from

such that the vectors (Xi, t), (vi. z’) are ordered under Ri. TO have preferences is now defined as a property of this ordering Ri.

self-oriented

M&ion 2.4. An individual i&V has self-oriented preferences (SOP for short) if aad only if for all pairs (Xi, yJ E Xi X Xi with Xi # yt: if ((Xi, z), ( yi, z’)) E Pi for some (& E’) E Z,, X Z,,, t&‘I (<Xi, Z), (_J’i, 2’)) E Ri for a!! (2, Z') E Zlic X Zpt.

W. Gaertner /Envy-free rights assignments 203

Self-orientation means that an individual who has revealed a strict preference for his private feature xi over another personal feature yi in a given situation is not permitted to reverse his preference for xi over y,, irrespective of changes that the other persons mike concerning their own private affairs. Conditional preferences, for example, i.e. preferences which are conditional on what other persons choose, are excluded (cf. Gibbard, 1974, p. 393). Self-orientation is a weaker requirement than Breyer’s (1977) restriction of extreme liberalism. IIowever, condition SOP alone is too weak to guarantee the existence of fairness or at least minimal fairness (for an illustration, see our example in the following section).

In a particular choice situation, let A be the set of available social states. For each ie N, Di= (Dil, . . . . Di,) was defined as person i’s set of decisive pairs. We now consider the set of ‘winning’ private sphere alternatives with respect to Di, i.e. those social states which individual i prefers within each DihB h E { 1, . . ., m}. This set of preferred alternatives will be denoted by Ai with Sib, h E { 1, . . . . m}, representing a typical element of Ai. According to our assumption that for each i E N, Rj is an ordering on X, there clearly exists a transitive subrelation Bi on ;i3i.

Definition 2.5. (Top Ranking of All Preferred Elements). An individual i E N is said to rank all his preferred private sphere alternatives at the top of his ordering Ri (TRAPE for short) if and only if for each h E { 1, . . . . m), 3dih E Ai such that VeA \ Ai, (aihsY)ERi*

Definition 2.5’. (Top Ranking of One Preferred Element). An individual i E N is said to rank one of his preferred private sphere alternatives at the top of his ordering (TROPE.) if and only if for each h E { 1, . . . , m}, ZaihEAi such that VZEA \ {a/h}, (si,, Z) E Ri l

We are now able to formulate tw$ conditions of domain restriction. Condition DR (Domain Restriction). All possible sets of individual orderings

satisfy the properties SOP and TRAPE. Condition MDR (Minimal Domain Restriction). All possible sets of individual

orderings satisfy the properties SOP and TROPE. We finally add Sen’s conditions P and L. Condition P (Weak Pareto). For all x,-v E X,

(X,Y)E nPi =$ [XEA and yEC(( R),A)] for no A E .Y: icN

Condition L (Libertarianism). There exists 3 rights assignment D.

3. An example

It is high time that we present an example to illustrate the various concepts intro-

204 W. Gaertner / Envy-free rights assignments

duced above. We have chosen Austen-Smith’s preference profile for two persons which Jed to his impossibility result under a K-fair rights assignment. The example is such that n=2; A - (o,w,x,Y,E}; DI = ({wJ), {.GY)}; & = { (u,x), (_YJ)}. The indi- v$-;lal o&rings are, for individual 1: w txy o, for individual 2: oy zx w, where the mare p@zred alternative is to the left of the less preferred alternative. Fpr this situation we obtain the following relationships. (0, l9 Dzz) E P pand (&, DzI ) E R 1’,

so that (D,,tr,) E l y, . + Furthermore, (Dtl,DI1) E Pz* and (L&, &) E Rz*, so that (D&)E ‘+$+. From this follows that the rights assignment is 2-faii according to Austen-Smith’s definition. We also have d , = {wJ}, 4= {QY), OvM,, (V,Y)E & It is easily seen that person 2 satisfies TRAPE, for (v, E) E P2, (VJ) E P2, (v, w) E Pz and (JZ)E PD (~,x)E P2, (y, w) E P2. Individual 1 satisfies TROPE, but not TRAPE, since (ZJ) E PI. We now translate our example into the issue representa- rion and assume that there is one issue for each individuaL2 The social states v, w, X, Y and z will then be written as (vlrQ, (Wl,w2), (x1,x2), (yl,y2), and (E,,z~), respc,tively. From (w,z) E D, it follows that w 2 = 22 (remember the definition of Sip ie (1.2)). Analogously, from (x,y)~ DI we have x2 =y2; from (0,~) E D2 we obtain 01 ==xl; from (y, z) E D2 we have y1 = zl. This, however, shows that both individuals violate property SOP.3

Itiividual 1, for example, prefers (tr,z2) to (x1,x2) to (xl,z2) to (xl, v2)‘. A aeimilac result is obtained for person 2.

Aunren-Smith showed that although the righ:s assignment is 2-fair, a social prefe- rence cycle exists, comprising alternatives x, y and z.

Qrc the other hand, simple examples demonstrate that condition SOP alone does not guarantee the fulfilment of any of our fairness properties (this is quite obvious since ~$3 fairness characteristics enter the definition of SOP). Consider the following profi!r::

(0 0wxyt, (2) 0 wyxz,

with D, = ((0, w), (~,y}} and D2 = ({ w,x), {y,z}}, the social states being defined as in our first example. Clearly, property SOP is fulfilled, but person 2 envies person 1.

4. The existence of fair collective choice rules

We now state and prove our rules.

; ‘&e could equally have assumed that

existence tlheorems for envy-free collective choice

here are two issues for each person. The translation would have kome more cumbersome wrth no extra insight.

3 Gibbard’s cmdition of unconditionality is not violated. We should like to mention in brackets that preference profile which led to Blau’s impossibility result for societies with more than two individuals

if* our M&Won) also violates property SOP once it is appropriately translated into the issue approach.

W. Gaerrner / Env_v-free rights a-i; ;mrnmts 205

Theorem 4.1. There exists a fair collective choice rule which satisfies conditiom DR,

P, and L.

Proof. Let (I?,, Rz, . . . , R,,) be any profile of individual orderings, We define P= {(x,y) 1 (x,y)~ f&.2: Pi} and Q= U,E,vDi.

We then define a binary relation R0 as a function of (R) and D,

One possible social choice function, for example, is

C=F((R),A)={x!xwl and b’y:y~A=(x,y)~R~}

for any A E .d. Associating this C with the given profile (R) we obtain a ClCR if R. is reflexive, complete, and acyclic. Reflexivity and completeness of R0 are obvious so that for brevity’s sake we restrict ourselves to the proof of acyclicity.

Assume to the contrary that there is a sequence of states {xp} E .Y’ such that (x~,x~+~)EP~ for ,u=lJ,...,t, with xl+] =x1 by definition, where PO is the asym-

metric part of RO, for which obviously PO = PU Q. We now distinguish the following

cases. Case A. (xfi, xp + I) E P for all p, CaseB. (xp,~p+~)~Qfor alIp, Case C. (xp,xj’+ l) E Q and (xp,xp+ ‘) $ P for some p =po, while (.~“,.I-“+ I) E P for

some v fpO.

Case A: cannot occur because of the assumed transitivity of all individual preference relations.

Case B: Let the individual who is decisive for the first step of the cycle be denoted by K, i e. (x1,x2) E DK. K cannot be decisive just in this one step. From tt>e definition of privare spheres Si it is clear that whenever an individual other than K is decisive the personal components xi, xi’ ’ of states x jr ,x p + 1 remain unchanged, i.e. xi = xi := . . . =Xi = Xi. However, (xl, x2) E SK implies xi# XL. K must therefore be decisive for two or more pairs in the cycle. Let these (and only these) pairs be denoted by (xk,xk + *), where k runs over the appropriate subset {k} of integers { 1,2, . . . ( t). Let ‘k denote the predecessor, k’ the successor of a given k within (k). Then (~kxk+1)~p~andx~+‘=x~+~=~~. =x$ by assumption. Hence, thanks to property

SO’& (xk,xfl)cRK fotallp with k+2spsk’. Moreover, because of (-6s + ‘kh,

also (xk xk’+ 1 ) E PK by transitivity of K’s preferences. Now,‘&&+1 l IS impossible. For this would imply that

((X~“,t”‘), (x:,2”‘+ ‘))E Pk_

which, given ((xi, zk), (xi-’ *, kk + 1 )) E P,+ contradicts SOP. Hence .I-; #xi * ‘.

SOP can now be applied to the step (xk,xk’+ l) E Pk’. xg + ’ = l . . =si’ leads to (xk,xfl) E RK for k’+ 21~ 5 k” and by transitivity to (x”,x”” + r) E Pli. Once again, Xi#XK k”+l thanks to SOP.

Thus the argument can be repeated through the entire cycle of k- and /c-values. In

2a W. Gaertner / Envy-free rights assignments

particular, therefore, we obtain xi+ xz + ‘. But on the other hand, from the con- struction of K’s decisjve steps, we have x$+ ’ =x$+~ = l = xi, which leads to a contradiction, and the proof is complete.

Case C: The assumed cycle must contain pairs frokm both P and Q. Let the pairs for which a particular jndividual K is decisive again be (xk,xk+ *) and the Pareto pairs be (xP,xP+ I ). The index sets (k} and {p} may overlap. Let p’ (k’) be the uccessor and ‘p (‘k) the predecessor of a given p within {p} (of a given k within

(k)), From the proof of Case B we know that (xk,xP) E RK for the first xp which follows xk in the cycle. Since (xP,xP+ t) E P implies (xP,xP+ I) E PK, the transitivity of K’s ordering leads to (#,xP+ t) E PK. As in Case B, xi= xf’ ’ can be seen to contradict prcqwty SOP. One therefore obtains &#xg+ ‘. Thus the argument can be repeated through the entire cycle resulting in x~#$’ and x~#x~+‘. But due to our construction either$‘=$‘= •~~=x~ or xi+’ =x$+~= l *m =xk, according to whether ‘k or ‘p is closer to k in the order of the cycle (‘k =‘p being, of course, admitted). This is a contradiction and the proof is complete.

Now that the acyclicity of RO has been established, it has to be shown that the proposed CCR satisfies conditions P and L and that it meets the fairness require- ments from Mbition 2.2’. That our CCR fulfils conditions P and L is immediately ckar r”rom the construction of Ro.

Assume that for some j, ke Iv, -[(D’&)E $*). This means that there exist at 1-t two decisive pairs Djh’E Dj, p(Djh*)E Dk, h’E { 1, . . ..m}. such that -[~D,~‘,~cI(D~~‘))ER/JI. Let (x,_~)=Djh’, (w,~)=p(Djh#), and XEA~ without 10s~ of gentrafity. Then (w,x) E Pj or (z,x) E Pj or both. Since x E dj also w E dj or z E dj, or both thanks to property TRAPE. Obviously x is ranked lower in Rj than w, or 2, or both. Then, however, there exists Djh*tE Djp h”+ h’, such that [(Djhw,(w,z))E R,?]. This argument can be iterated for all pairs Djh, p(Djh), h E (1, . . ., m}, such that - I( D’k, /((D,h)) E Rj?J l Th ere ore, f under property TRAPE, there always exists an injective mapping &: Dj + Dk such that (Djh,fi(Djh)) E Rj*, h E { 1, . . . . m}, and this is true for all j, k&v. Therefore, a contradiction to our supposition has been established. This completes the proof of the theorem.

Tkonm 4.2. There exists a minimally fair collective choice rule which satisfies conditions MDR, P, and L.

Proof. After the proof of Theorem 4.1 it pnly has to be shown that property TROPE secures the existence of minimal fairness. This, however, follows imme- dktely from Definitions 2.3 am, 2.Y

Et might be conjectured that one could do without property SOP if one strengthened requirement TRAPE, i.e., if one required the subrelation i& on di to h strict for all ieN. The following example, however, shows that this is not true. Consider the individual orderings

(I) wxzyrp,

W. Gaertner / Envy-free rights assignments 207

(3YVZXW, with D, = ((w, z), (x, y}} and DZ = {{y, z), {v,x)}. The subrelations 8, are strict, and there exists a preference cycle. This is due to the fact that for the subset {x, y, v)., individual 2 strictly prefers the Pareto optimal state y to the preferred alternative in his (her) pair (v,x).

5. Concluding remarks

We have seen that envy-free collective choice rules exist if individuals’ preferences are self-oriented and if, in addition, people attach primary importance to their own private sphere alternatives. Will these requirements be met in real life? Probably not.

Reading Blau’s (1975) paper and re-examining the many examples circling around Sen’s ‘liberal paradox’ one gets the impression that nosiness is a wi&s;pr& pheno- menon among human beings. But what is nosiness and what is unselfish interest in the private affairs of others? It there a clear line of demarcation?

Caught in this dilemma the following question may be legitimate: why do we want to have fair assignments of rights? The absence of envy has frequently been viewed as an important factor in the construction of a ‘stable’ society. This is probably true, but how can this type of stability be achieved in the presence of nosiness or envy?4 By a large-s tale educational program directed towards meeting our resricted domain requirements? Isn’t it more probable that society will agree on a system of rights which is to protect the private spheres of individuals in Ihe presence (or should we say: in spite) of envy and meddlesomeness? The members of society nay decide that the protection of individual rights depends on certain requirements that the individuals in question are to fulfil (cf. a suggestion by Gaertner and Kruger, 1981), or - some people may view this as a more liberal approach - society may hope that there always exist individuals who are willing to renounce parts of their preferences (cf. the proposals by Sen (1976) and Suzumura (1978)).

For a given preference profile it is possible to avoid cyclical social choices by reassigning individual rights. In Sen’s famous “Lady Chatterley Case”, for example, consistent social decisions are obtained when the prude is made decisive over the private sphere of the lude, and vice versa. This redistribution of rights not

only prevents social cycles, it also converts a non-fair situation into a fair (even l- fair) one. At the same time, however, a fundamental libertarian principle has been upset, viz. if social decisiveness is at all assigned to single persons, it should be confined, for each individual, to his (her) own private sphere. Therefore, a mere switching of rights does not appear to be too promising as a way out of the liberal paradox.

4 It should be mentioned that within a somewhat different framework Suzumura (198 1) also investigated

the existence of fairness and obtained results that are largely negative.

W. Gaertner /Envy-free rights assignments

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