Gibbardian Libertarian Claims Revisited

Gibbardian Libertarian Claims RevisitedAuthor(s): K. Suzumura and K. SugaSource: Social Choice and Welfare, Vol. 3, No. 1 (1986), pp. 61-73Published by: SpringerStable URL: http://www.jstor.org/stable/41105822 .

Accessed: 14/06/2014 22:03

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Springer is collaborating with JSTOR to digitize, preserve and extend access to Social Choice and Welfare.

http://www.jstor.org

This content downloaded from 62.122.79.31 on Sat, 14 Jun 2014 22:03:40 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=springer

http://www.jstor.org/stable/41105822?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


Soc Choice Welfare (1986) 3: 61-74 ^ • t C't- • Social ^ • t Choice C't- •

^Welfare © Springer-Verlag 1986

Gibbardian Libertarian Claims Revisited

K. Suzumura1 and K. Suga2 1 Institute of Economic Research, Hitotsubashi University, Kunitachi, Tokyo, 186 Japan 2 Department of Economics, Asia University, Musashino, Tokyo, 1 80 Japan

Received February 10, 1985 /Accepted February 6, 1986

Abstract. Two resolution schemes for the impossibility theorems on the Gibbard-Kelly claims of libertarian rights, which are rather contrasting with each other, are proposed and their implications discussed. The first scheme asserts that there exists a collective choice rule satisfying the Pareto principle and the Gibbard-Kelly libertarian claims if there exists at least one socially unconcerned individual. The second scheme asserts the existence of an eligible collective choice rule if there exists at least one liberal individual.

1. Introduction1

Among many attempts to find a way to circumvent Sen's [6; 7, Chap. 6*] Pareto libertarian paradox, Gibbard's [3] proposal clearly stands out in that it focusses on "a strong libertarian tradition of free contract," according to which "a person's rights are his to use or bargain away as he sees fit."2 In this scheme, an individual who is endowed with the rights to make certain "personal" decisions without outside interference exercises these rights only when he can thereby enhance his welfare. Despite its intuitive appeal, Gibbard's proposed resolution in a revised version presented by Kelly [4] (which supposedly causes "no significant changes in the [possibility] theorems that make up Gibbard's libertarian claim"3) strenuously brings back impossibility theorems as Suzumura [10; 11, Chap. 7] has demon- strated. The purpose of this paper is to propose two resolution schemes that apply to these impossibility results.

1 An earlier version of this paper was presented at the Summer Workshop on Public Choice Theory held at Halifax, Nova Scotia in 1984 and also at the seminars in Stanford University and the Ohio State University. Thanks are due to the participants for their helpful comments and discussions. We are also grateful to an anonymous referee of this journal for his incisive comments and to the financial help from the Japan Securities Scholarship Foundation 2 Gibbard [3, p. 397] 3 Kelly [4, p. 144]



62 K. Suzumura and K. Suga

Our first resolution scheme asserts that there exists a collective choice rule which satisfies the Pareto principle and a version of the Gibbard-Kelly claims of libertarian rights if only there exists at least one socially unconcerned individual in the society. In contrast, our second resolution scheme asserts that there exists a collective choice rule which satisfies the Pareto principle and a version of the Gibbard-Kelly claims of libertarian rights if only there exists at least one liberal individual in the society.

The relationship between our resolution schemes and the existing ones will be clarified as we go along.

2. Gibbard-Kelly Libertarian Claims

2.1. Let iV={l,2,. . . , n) denote the finite set of individuals, where n}>2, and let X denote the set of all conceivable social states. A social state is construed to be a list of impersonal and personal features of the world. Let Xo and Xx (i = 1 , 2, . . . , n) stand, respectively, for the set of all impersonal features of the world and the set of all personal features of the individual i. Then we have X=X0 x Xx x . . . x Xn, and a social state x e X is represented as x = (jc0 , jq , . . . , xn), where jc¿ e Xi (i = 0, 1 , . . . , n). It is assumed throughout that Xo and Xt (i = 1, 2, . . . , n) are finite with at least two elements each.

Ri denotes a weak preference relation held by the individual ieN. We assume that Ri is an ordering on X, that is, connected [for any x,yeX9 (x,y)eRi and/or (yix)eRi] and transitive [for any x,y,zeX, (x,y)eRi and (y,z)eRi imply (jc, z) 6 R¡] (i = 1 , 2, . . . , n). The strict preference relation P(R¡) and the indifference relation I(Rt) corresponding to Rt are defined as usual. A list of the individual preference orderings, one ordering for each individual, is called & profile. The set of all logically possible profiles will be denoted by s/.

A collective choice rule, CCR for short, is a function F that represents a method of amalgamating each profile a in the set s/F of admissible profiles into a social choice function Ca on the family Sf of all nonempty finite subsets of A": Ca=F(a). When a set Se ¡f is specified (as a set of realizable social states), Ca(S) denotes the nonempty subset of socially chosen states from S when a profile ae s/F prevails.

2.2. The following four conditions on the CCR needs no explanation.

Condition U (Unrestricted Domain). The domain of the CCR F consists of all logically possible profiles, viz., stF = st.

Condition IP (Inclusion Pareto). For every admissible profile aes/F and every x,yeX, if (x,y)ef) P(Rf), then [{xeS&yeCa(S)}->xeCa(S)] for all Se^, where Ca=F(a). Condition EP (Exclusion Pareto). For every admissible profile aes/F and every x,yeX, if (x9y)eÇ)P{Rf), then [xeS->ytCa(S)] for all SeS?, where

ieN Ca=F(a).



Gibbardian Libertarian Claims Revisited 63

Condition SP (Strong Pareto). For every admissible profile aesfF and every x, y g X,

if (x,y)ep( f] Rf' then [xeS-+yiCa(S)] for all Sg^, where Cfl = F(a).

Clearly, SP is stronger than EP which, in turn, is stronger than IP.

2.3. The following notational conventions facilitate our later discussion on the concept of libertarian rights-system. For each ieN, we let X)i{ = Xo x X1 x . . . x Xi-1 x Xi+i x . . . x Xn and, for each x = (xo,xl9. . .,xn)eX, we let X)H = (*o f *i > • • • > *¿ - 1 > *¿ + 1 > • • • > xn)- Furthermore, for each i g N, each x¡ g X{ and each z = (zo,z1,...,zi-1,zi+1,...,zn)GAr)l(, we let (xi;z) = (z0,z1,. . . , zi - 1 > *i 5 zi + 1 j • • • > ̂ n)-

Let us now define a set D¡c=i XxXby D¡ = {(x, ̂)elx X' jc¡ #>j & x)i( =^)l(} for every zgAT and call D = (Di,D2,. ,Dn) the rights-system. By construction, (x, >>) e Z)¿ holds true if and only if x and >> differ in the specification of f s personal feature. Suppose now that, given a profile a = (Rf,R.2,. . .,RZ)es/Fì a set of realizable states Sg«^, and jc, .yGS, (x,y)eDinP(Ri) holds true. A naive libertarian may claim, then, that the individual i should be empowered to exclude^ from Ca(S), where Ca = F(a). However, a sophisticated libertarian may think ahead and may deliberately let the individual / waive his right for (x, >>) if there exists a sequence {î,^,- • • ,yx} in S, to be called a nullifying sequence, satisfying:

yx = x&(yiyi)eP(R?) (1)

Vte{U29...,X-l}:(jt,yt + l)e(r'P(RJ))v( Ì (J [DjnP(RJ)]', )

(2) 'jeN Ì

'j€N'{i] )

because the mechanical exercise of right for (x, y) would then bring about a states , which is worse for him than >>. Let a set Wi*(a'S) c Di9 to be called the waiver set, be defined by (x,y)e W¡*(a'S) if and only if (1) and (2) hold true for some nullifying sequence {ylfy2,' • • ,yx) in S. We then have the following:

Condition KL(1) (Kelly's First Libertarian Claim). For every admissible profile aesfF, every SeSf, every ieN, and every x,yeX, if (x,y)eDtnP(Rf) and (x,y)tWi*(a'S)9 then [xg5->^^Cû(S)], where Ca=F(a).

2.4. To those who feel the rights-waiving in accordance with the waiver set Wi*(a'S) to be too conservative, the following alternative definition of the waiver set and a libertarian claim may be appealing. Given a profile a = (Rf9 R}, . . . , R¡) g s/ and a set of realizable states Se£f, an individual / waives his right for (x,y)eDt, viz., (x, y) g WPf** (a | S) if and only if (a) There exists a nullifying sequence {yi,y2,. ",yx' in S; (b) For any sequence {zl5z2,. . . ,zA*} in 5 such that

z»=y1&(zl,y)eP(R?) (3) and

Wg{1,2,...,A*-1}: (zf,z, + 1)G(V| />(*/)) u

j^lj [/),- n />(*/)]) (4)




hold true, which is to be called a rectifying sequence, there exists a corresponding nullifying sequence {wt , vv2, . . . , wx**} in S such that

w^=z1&(^,w1)€PW) (5) and

Wg{1,2,...,¿**-1}:

(w,,*,+1)e(n W)u( (J lA^W)]). (6)

Condition KL(2) (Kelly's Second Libertarian Claim). For every admissible profile aesfF, every SeS?, every /gJV, and every x,yeX, if (x,y)eDinP(Rf) and Cx,jO*J*¿**HS), then [jceS-^C^S)], where Ca = F(a).

It is clear that, for every profile a es/ and every set of realizable states 5 e £f, we have tff**(a|S) <z 0f*(a|S) for every / e N, so that KL(2) is a stronger libertarian claim than KL(1).

2.5. We are now in the stage of defining the concept of unconditional preferences. Given a profile a = (Rf ,i*2>- • • ,î)e j^ and a set of realizable states SeSf, let JV(fl|S)ciV be defined by ieN(a'S) if and only if (x,y)eDinP(R?)n(SxS) always implies that

Vz g JT)I( : [(*,. ; z) e S & (y i ; z) e S]->((*, ; z), (^ ; z)) e P(Äf) (7)

holds true. It makes obvious sense to call A^S) the set of individuals who have unconditional preferences over the set S at the profile a.

3. An Impossibility Theorem

To demarcate our arena, let us recollect several impossibility theorems on the revised libertarian claims. To begin with, it is unfortunately the case that KL(1) as well as KL(2) represents a standard for individual liberty that cannot be realized by any CCR with unrestricted domain.4

It might well be argued, however, that it is too much to ask for a rule which protects individual's claim of libertarian rights even when his preferences are conditional on others' personal choices. With this observation in mind, let us modify KL(1) and KL(2) as follows.

Condition KL(1 *). For every admissible profile a e s/F , every 5 e «9*, every i e N, and every x,yeX, if ie f] N(a'S), (x,y)el>lnP(Rf) and (x,y)tH{*(a'S), then

SeSf

[xeS-+ytCa(S)]9 where Ca = F(a).

Condition KL(2*). For every admissible profile ae^f, every S e 5?, every / e N, and every xêX, if ie f] N{a'S), (x,y)eDinP(R?) and (x,y)tfy**(a'S), then

[xeS->ytCa(S)l where Ca=F(a).

4 Indeed, there exists no CCR that satisfies U and KL(1 ) (resp. KL(2)). See Suzumura [1 0, Theorems 2 and 3]




It was nicely shown by Krüger and Gaertner [5, Theorem 3] that there exists a CCR which satisfies U, KL(1*) and EP. This comfortable result notwithstanding, the following impossibility theorem still holds true.

Theorem 1. Suppose that n^3 and *X^3 (z = l,2,3). Then there exists no CCR which satisfies U, KL(2*) and IP.5

Proof. Suppose that there exists such a CCR, say F, with séT = sä. Take any y0 e Xo and jêî for all /e7V'{l,2, 3} and fix them for the rest of this proof. Take xt, x[, x"eXi (/=1,2,3), which are all distinct, and define

x1 = (y0,x1,x2,x3,y4r1. . ,,yn' x2 = (y0,x'1,x2,x3,y4,. . .,;>„), x3 = (y0,x';,x'2'x'iiyA,. . . ,yn'

x*=(yo,xuX2,x3,y4>--,ynl x5 = (yo,xriixf2,xf39y4,. . . ,yn), x6 = (y0,xlix2,x'3,y4,...,yn),

and

Let5={x1,x2,. . .,.x7}e«ând let a profile a = (/??, R^- • . , R5) e s/ be such that

tf?^*7,*4,*5,*1,*6,*2,*3, R^Sy.x^x'^.x^x3^5^1, RSiSy.x^xKx^x2,^^1^*,

and, forall/eM{l,2,3}, RfiSy.x2^3,

where Rf(S) = R?n(SxS). Note that (x'x2)eDlnP(Rf), (x5,x6)eDínP(Ríl (x'x1)eD2nP(RÏ), (x'x5)eD2nP(R^ (x6, x1) s D3 n P(R¡)9 (¿,x*)eDz nP(R%), and (x2, x3)e f] P(R?) hold true. Note also that N(a'S) = N holds true.

ieN Consider Ca(S), where Ca = F(a). There exists no nullifying sequence for the

exercise of (x1ix2)eD1, so that we have

[(x'x2)6D1nP(RÏ)&(x'x2)ÎWr(a'S)]^x2tCa(S). For the exercise of (x5, x6) e D^ , there exist two nullifying sequences {x3, x*, x5}

and {x2,^,^:4,^5}. However, corresponding to {jc3,*4,*5} (resp. {x2,x3,x4',x5}), there exists a rectifying sequence {x1,*2,*3} (resp. {x1,^2}), which cannot be nullified. Therefore,

[(x^x6)€Z)1nP(^&(^^^6)^^**(û|5)]-^x6^Cû(5). 5 As a matter of fact, the full force of the condition U is nowhere needed. We may assert that there

exists no CCR satisfying KL(2*) and IP on the restricted domain sf* = 'ae s/'N= C' N(a'S)> I Sey '




Consider (x79xL)eD2. There exists no nullifying sequence for this exercise of right, so that we have

[(x' x1) eD2n P(R¡) & {x' x1) ¿ Wf* H^H*1 * Ca{S). Turn now to (x4, x5) e D2 . There exists a nullifying sequence {x1, x2, x3, x4} for

this exercise of right. However, there exists a rectifying sequence {x^x1}, which cannot be nullified. Therefore,

[(x4, x5)eD2n P(R$ & (x4, x5) t WÌ* (a'S)] -x5 $ Ca(S). Consider (x6,x7)e/)3. The only nullifying sequence for this right-exercising is

{x4,*5,*6}, which can be rectified by {x3,*4}. Since there exists no nullifying sequence for {x^x4}, we obtain

[(x'x1)eD3nP(RS)&(x6,x1)ÍW?*(a'S)]-+x1tCa(S). Since there exists no nullifying sequence for (j¿9x?)eDZ9 we have

[(x3, x4) 6 D3 n P(RI) & (x3, xA) i Wr (a'S))->x* ¿ C°(S).

Finally, suppose that x*eCa(S) happens to be the case. Since we have (xz,x3)e f] P(R?) and F satisfies IP by assumption, we then obtain x1eCa(S)i

i€N a contradiction. Therefore, we must admit that x3$Ca(S), which implies that Ca(S) = 0. This is enough to show the non-existence of a CCR satisfying KL(2*) and IP on stF = s/. D

It follows from Theorem 1 that depriving an individual of his libertarian right when he holds conditional preferences is not a sure-fire escape route from the impossibility of Pareto libertarianism. How, then, should we proceed?

4. Socially Unconcerned Individual : First Possibility Theorem

Our first resolution scheme hinges on the concept of a socially unconcerned individual. We say that an individual ieN is socially unconcerned at the profile a = (R* , R2 , . . . , R%) e sé if and only if the following two conditions are satisfied : (a) If (x, y) g P(R?) and xt *>'- for some x,yeX, then ((x¿ ; z), ( >>¿ ; w)) e P(Rf) for every z,we X)i{ ; and (b) ((x¿ ; z), (x¿ ; w)) e I(R°) holds true for every x¡ € Xx and every z,weX)i{. In other words, a socially unconcerned individual is the one whose preferences over social states are regulated exclusively by his personal features specified by these states. If there exists at least one socially unconcerned individual in this sense, the impossibility result formulated in Theorem 1 may be turned into the possibility result.

Theorem 2. Suppose that there exists at least one socially unconcerned individual at every profile. Then there exists a CCR satisfying U, KL(2*) and EP.

To facilitate our proof of this theorem, we need an auxiliary lemma, which is due essentially to Gibbard [3, pp. 396-7]. Let R be a binary relation on a set Z. What we call an R-cycle is a finite sequence {jc^-i (/^2) in Z such that (/,/+1)eÄ




(/i = l,2,. . . , /-hi) and (xi,xi)eR hold true. We say that R is acyclic if there exists noP(R)-cycÌQ. We may then assert the following:

Lemma 3. For every profile a = (R° , R¡ > • • • > ̂n) e ¿^ and ez;ery 5 6 Sf, let a binary relation Qa be defined by

ß«= (J [AnP(J?O]. (8) /e n N(*'s)

Se?

Then there exists no Qa -cycle.6

Proof of Theorem 2. Given any a = (R° , R5 , . . . , i*") e sé and any Se ¿f, let a binary relation Qa be defined by

ß' = ( H P(*fl) u / U {[AnPW)]'^.**(a|5)}' . (9) ^•^ / 1/6 n ^(«is) I

' 5e^ / Using ßfl, we define another binary relation Ra by Ra = {(x,y)eSxS'(y,x)$Qa}.

To begin with, let us prove that Äfl is connected. Suppose, to the contrary, that there exist x, y e X such that (x, y)$Ra and (y, x) i Ra. We then obtain (x, y) e Qa as well as (y,x)eQa, which may happen in the following four alternative ways:

n (i) q- • n vr I« f(*^)6AnF(J?f)&(x,y)^»?*(«|S) n (i) awe^âlS). q- • n vr I«

jù>>Jc)e2)jnp(Jif)40,,je)#B7.(fl,S) f(*^)6AnF(J?f)&(x,y)^»?*(«|S)

(¡i) (x, >») € n /»(a?) & (>-, x) e n ^(^ ; jeN jçN

(3ieQN(a'S):(x,y)eDinP(Rn&(x,y)tWi**(a'S) (iii) |(>,,x)Tn/W); Ux,y)ef) P(RJ)

(iv) K } < jeN (iv) K } ¡lie Ç) N(a'S):(y,x)eDinP(Rn&(y,x)iWi**(a'S).

The case (i) contradicts the asymmetry of P(R?) if z=y, whereas it contradicts Lemma 3 if /*/ The cases (ii), (iii) and (iv) are also impossible in view of the asymmetry o{P(Rf). Therefore, we cannot but to accept the connectedness of Ra.

Next, we show that P(Ra) = Qa holds true. For any x, y e X, (x, y) e P(Ra) holds true only if (>>, x) $Ra, which yields (x, y) e Qa by definition. Therefore, P(Ra)cz Qa must be true. To show the converse, suppose that there exist x,yeX such that (x9y)eQa and (x,y)$P(Ra). Ra being connected, (y,x)eRa is implied by (x, y) $ P(Ra), which in turn implies (x, y) $ Qa, a contradiction. Therefore, we must accept Qa cz P(Ra)9 and we are home.

6 It may be of some interest to notice in passing that a binary relation ££ defined by

i*N(a'S)

is not necessarily acyclic




Third, we prove the acyclicity of Ra. Assume, to the contrary, that there exists a sequence {x1,*2,. . .,**} such that (xT,xx+1)eP(Ra) (t = 1,2,. . .,/-l) and (xt,x1)eP(Ra) hold true. Since P(Ra) = Qa, this cycle can happen only when

(xx,xx+1)eQa(T = l,2,...,t-i)&(x',x1)eQ* (10)

holds true. The cycle (10) can happen in several alternative ways, among which

(Jc'xt + 1)6Q^W)(T=1,2,...,/-1)&(JCÍ,^6Q^W) (H)

can be discarded straightforwardly, because (11) contradicts the asymmetry of P(Rf). Similarly, the case where

Vre{l,2,...,i-1), 3izef] N(a'S): Sey

(X'xt+l)eDunP(R0 & (xl,xt+1)# W¿*(a'S) (12)

Hi.eß N(a'S): (x',x1)eDitnP(R0 & (Ax1)* W¿*(a'S)

can be excluded, since (12) contradicts the asymmetry of P(R°) if {^ , i2 , . . . , /,} is a singleton set, whereas it contradicts Lemma 3 if {¡i , i2 , . . . , i, } is not a singleton set. Therefore, the cycle (10) must contain an ordered pair, say Oc*"1,**), such that

((xk-i,xk)€f) P(RJ)

3ikef) N(a'S): (xk,xk^)eDiknP(Rfk) &(x*,x*+1)^ JV¿*(a'S) V '

holds true. Let /* be a socially unconcerned individual at the profile a. Define:

x; = (^;4-i)(T = A:-l,Â:,...,M,...,/:-2). (14)

Since i * is socially unconcerned, we obtain (x£~ i, x£) € P(R°*). On the other hand, it follows from (x'xx+l)eQa(i = k, * + l,. . . ,/, 1,. . . ,A:-2) that {x',xx^l)eR^ (T = Ar, Ä: -h 1 , . . .,1,. . .,A:-1), which yields (x^.j^'êRfi by the transitivity of Äf», a contradiction. Therefore, we must admit that Ra is acyclic on S.

It is clear, then, that Cfl(5) = {xe5|Vje5: (x,y)eRa} for every S e 6? is a well-defined choice function on «9* for each and every a es/. To show that this CCr satisfies EP, suppose that a es/, x,yeX9 and SeSf are such that (x,y)ef) P(R°), xeS and ̂ €^(5) hold true. By definition, we then have

(y,x)eRa, hence (xiy)$Qa. But this contradicts (jc.^ëH W) in view of (9).

Finally, to show that this CCR satisfies KL(2*), suppose that a es/, SeSf, ief] N(a'S) and x,yeX are such that (x9y)eDtnP(Rf), (x,y)fWi**(a'S),

ssy xeS and yeCa(S). Then we arrive at the same contradiction as we encountered above. D




5. Liberal Individual : Second Possibility Theorem

Let us now turn to our second resolution scheme. Although this scheme may seem to be rather complicated analytically, the gist thereof is quite simple as well as intuitive. The key concept here is that of a liberal individual due to Sen [8] and Suzumura [9], which may be defined as follows.

As an auxiliary step, take any profile a = (R°,R2,. . . ,RZ)esé and define a binary relation Qa by (8). By virtue of Lemma 3, Qa contains no cycle. A fortiori, there exists no finite sequence {*"}£« i (/^ 2) satisfying (xi,x2)eP(Qa), (xM,^+1)Eßfl(/i = 2,3,. . .,/-l) and (^xêß0, so that Qa is consistent in the sense of Suzumura [11, p. 8]. It then follows from Theorem A(5) in Suzumura [11, p. 16] that Qa has at least one ordering extension7. Let 0ta denote the set of all ordering extensions of Qa. Let Rf be those parts of/??, which the individual i wants to be counted in social choice. An individual / is said to be liberal if and only if, for every profile a = (J?f, /?2a, . . . , R¿) e sé,

Rf = RlnRf for some Rle3ta (15) holds true, whereas / is said to be illiberal if and only if, for every profile a = (/??, R}9 ...,/?nfl)€ sé, Rf = Rf holds true.

Let Nx and N2 stand, respectively, for the set of all liberal individuals and the set of all illiberal individuals in the society. Note that this demarcation is in accordance with the individual's attitude toward others. In particular, it does not depend on what de facto preferences individuals happen to express. In what follows, we assume that Nl and N2 are jointly exhaustive, viz., N1kjN2 = N.

For each profile a = (R°,R2,. ..,RS)esé, Nt can be subdivided into Nn(a) = Nin(f] N(a'S)) and A^M^Wufa) for z = l,2. In words, Nn(a) (resp.

N21(a)) is the set of liberal individuals (resp. illiberal individuals) who have (universal) unconditional preferences at a, whereas N12(a) (resp. N22{a)) is the set of all liberal individuals (resp. illiberal individuals) who have conditional preferences at a.

In this revised context, a natural reformulation of the strong Pareto principle would read as follows.

Condition CSP {Conditional Strong Pareto). For every admissible profile aeséF and

every x,ysX, if (x,y)ep(f) Rf), then [jc e S-+y $ Ca(S)] for all Se^, where Ca = F(a). veN '

Turning to the libertarian claims of rights, we define a new waiver set ̂ +(<z|S) for each profile a = (Rf , R2 , . . . , /?") e sé and each S e ¥ as follows. For each ieNl9 who regards the libertarian rights to be deeply rooted in the sovereignty of individuals, we let Wi+(a'S) = 0, whereas for each ieN2i to whom the libertarian rights are "his to use or bargain away as he sees fit," we assume that (x, y) s W* (a'S) holds true if and only if either i e N22(a), or the following conditions are satisfied for ieN21(a):

7 Recollect that an ordering R is an ordering extension of a binary relation Q if Q c R and P(Q) <=■ P(R) hold true




(a) There exists a nullifying sequence {yi,y2i- • • ,yx' in S such that yx=x, (y,yi)eP(JRf)Bnd

We{l,2,...,A-l}:

(y<,y<+i)e( H P(R?))U U [DjnP(RJ)]); (16) 'jeN / 'j€[Nn(a)uN2l(a)J'{i} I

(b) For any rectifying sequence {z1,z2,. . . ,zA*} in 5 such that zk.=yly {zuy)eP{Rf)zná

We{l,2,...,A*-l}:

(zt,zt+1)e(f) W)W U [/>,n/>(*;)]' (17) 'j€N J 'jeNu(a)uN2i(a) J

there exists a corresponding nullifying sequence {w1 , vv2, . . . , h>a~} in S such that m/a- =zx , 0>, wj e P(Ä/0 and

We{l,2,...,¿**-1}:

(wt,ìvf+1)Gf H ̂ W)Vf U [DjnP(Rf)]'. (18) 'j6iV / '>6[Ar„(fl)uAr21(fl)l'{l} /

In terms of waiver set W*(a'S' we may now formalize the following:

Condition CLC {Compound Libertarian Claim). For every admissible profile a e sfF, every Se£f, and every x,yeX, if

B/eAûAâ): (xìy)eDinP(Rì) & (*,>>)* ̂ +(a|S), then [xsS-+yi Ca(S)]9 where C^F^).

The concept of a liberal individual plays a pivotal role in resolving the Pareto libertarian paradox as the following theorem crystallizes.

Theorem 4. Suppose that there exists at least one liberal individual in the society. Then there exists a CCR which satisfies U, CSP and CLC.

Proof. For any a = (Ä1fl,Ä23,. . .fR^)es/ and any SeS?, let Q§ be defined by

QS = '(x,y)eXxX'(x,y)ep(f) Rf)v( f] PiR1))

J (J {[AnP(^)]'^+(a|5)}]j, (19)

where Rt=RlnRf for some Rle&a if ieN^ and Rf = Rf ifieN2. Recollect that Wi+(a'S) = 0 in (19) if ieNn(a). Let Ä| be defined by R§ = {(x,y)eSxS'(y,x) $ ÔI}, which we make use of in defining Ca by Ca(S) = {jc € S'Vy € S : (jc, j>) € RS} for every 5 6 Sf.

We must verify that (a) RS is connected on 5, (ß ) Q$ = P(Ä|) AoWs írwe, and (y) ̂ is acyclic on S.



Gibbardian Libertarian Qaims Revisited 71

(a) Suppose to the contrary that there exist x,yeS satisfying (x,y)$R§ and (y, x) $ R§ . It follows that (jc, y) e QS and (y, x)eQs, which occur in various ways. It is easy, if tedious, to verify that each and every case yields a contradiction as far as Ni is non-empty. We may safely skip the details.

(/?) may be proved along the same line as in the corresponding step in the proof of Theorem 2.

(y) Suppose to the contrary that Q% = P(R$) contains a cycle {*"}{,= t (t ̂ 2) in S. We then obtain

V/xe{l,2,...,r}:

C-.»«').J.(£l#)u(ûW>) J U [An/>W)]'^+(a|5)Y (20) ' ieNn(a)uN2i(a) /

where we define ¿+1=x1. Since Nn(a)uN2i(a)ci f] N(a'S) for any aes/, SeST

yfie{i,2,...,t}:(^,^+1)ep(^ßRt)u^P(Ri))j, (21)

holds true, where use is made of the definition of R'eât" (ieNi). Since

p( f) rAczR1 holds true by definition for every ieNi9(2i) yields a contradiction 'ieN )

with the transitivity of RîeN^). Now that Rs is connected and acyclic, Ca is a well-defined choice function on Sf.

Associating this Ca with a es/, we obtain a well-defined CCR i7 on stf . What remains is the proof that F satisfies CSP and CLC.

To show that F satisfies CSP, suppose that there exist a es/, S e SS and x,yeS

satisfying (x,y)ep(P' R? ), x e S and y e Ca(S), where Ca = F(a). It follows from 'ieN )

y e Ca(S) and x e S that (y, x)eR§, viz., (x, y)$Q§, which is a contradiction in view

of {x9y)ep(j^*t^QS. Finally, to show that F satisfies CLC, suppose that there exist a es/,

ieNn(a)uN21(a),Sey2indx,yeSssitisfying(x,y)eDinP(RnAx,y)iWi+(a'S), xeS and yeCa(S). It follows from _y e Ca(S) and xeS that (y,x)eR$, viz., (x,y)tQ§. However, (x,y)e[DinP(R?)]'Wi+(a'S)czQ$, a contradiction. D

6. Concluding Observations

Several concluding remarks are in order.

6.1. Scrutinizing the proof of Theorem 2 and that of Theorem 4, we may recognize that the exact structural specifications of the waiver set - W^**(a|S) in the case of Theorem 2 and W+ (a'S) in the case of Theorem 4 - are nowhere actually invoked.




Therefore, a possibility of generalizing these theorems naturally suggests itself. Given a profile aesé and an S e Sf, let ̂ ++(a|S) be any waiver set for the individual ieN. The only constraint we require of W** (a'S) is that it is a subset of D¡, which is well-defined in terms of the profile a e sé and the set of available states SeSf. In particular, ̂++(a|S) = 0 is an admissible possibility. Let KL(2+) (resp. CLC+) be defined by replacing W¡**(a'S) (resp. Wi"(a'S)) in KL(2*) (resp. CLC) by Wi++(a'S). We then obtain the following generalization of Theorem 2 and Theorem 4.

Theorem 5. (a) Suppose that there exists at least one socially unconcerned individual at every profile. Then there exists a CCR satisfying U, KL(2 +) and EP. (b) Suppose that there exists at least one liberal individual in the society. Then there exists a CCR satisfying U, CEP and CLC+.

6.2. It is worthwhile to highlight the similarity and dissimilarity between our resolution schemes. Let us begin with the dissimilarity. In contrast with the first resolution scheme, which ensures the consistency of the pure claims of libertarian rights, our second resolution scheme ensures the consistency of the compound claims of rights, which we feel is a rather interesting feature thereof. Note that, in this scheme, individuals are divided into liberals and illiberals in accordance with their attitudes towards others. Individuals are also divided into those who hold unconditional preferences and those who hold conditional preferences in accordance with their personal wishes. Only those who hold unconditional preferences are qualified to claim their libertarian rights if they so wish. For liberals, these sacred rights should not be subjected to such "mean" acts as bargaining, whereas illiberals use or bargain away their rights as they see fit. Let us stress that, whatever rights- waiving calculus illiberals happen to indulge in, a liberal may serve to make all actual claims of libertarian rights compatible with the Pareto principle. Turn, now, to the similarity. Both schemes generate the eligible collective choice rules, which are not rational. By rationality of a collective choice rule we here mean that the social choice function generated by the rule corresponding to any given profile is rationalizable by a social preference relation. It is an open question, which is not altogether without interest, if our possibility theorems may be generalized by proving the existence of the eligible collective choice rules which are rational.

6.3. Our concept of socially unconcerned individual is related to, but is distinct from the concept of an extremely liberal individual aus to Breyer [1 ], on the one hand, and that of an individual with self supporting preferences due to Gaertner and Krüger [2], on the other. An individual ieN is said to be extremely liberal (resp. to have self- supporting preferences) if and only if, for every profile a = (R*,R2>. . .,RS)esf, (x,y)eP(R?) and x¿*^ for some x,yeX imply ((*¿;z), (>>,.;*)) e />(*?) (resp. (fa ; z), (yt ; h>)) e Rf) for every z,we X)i( . Clearly, our concept is stronger than theirs in that we require additionally ((xr;z), (Xi'9w))eI(Ri) for every x^Xi and every z, we X)i{, but is weaker than theirs in that we do not stipulate an individual to be socially unconcerned uniformly for every profile. These differences reflect them- selves in the assertions they entail : Breyer [1 ] (resp. Gaertner and Krüger [2]) could assert the existence of an eligible CCR by assuming that all but one individuals (resp.




all individuals) are extremely liberal (resp. have self-supporting preferences), whereas our Theorem 2 requires only that, for each profile, there exists at least one socially unconcerned individual.

6.4. Back to back this nice feature lies a weakness of our first resolution scheme, which may be crystallized in the following: Theorem 6. Even if there exists at least one socially unconcerned individual for each profile, a CCR satisfying U, KL(2+) and SP does not exist*.

Suffice it to quote an example where n = 3 and W¡++ (a'S) = 0 (i = 1 , 2, 3) for all ¿zejând Se£f. Let x = (xOìxi,x2,x3), y = (xo,x/lìx2ìx3) and z = {xQix'ux'2,x3). Consider the following profile :

Ri:y,z,x Ri'- [x,y,z]9

where R§: [x,y,z] indicates that individual 3 is indifferent among x, y and z. In this case, all individuals have unconditional preferences and individual 3 is socially unconcerned at a. Nevertheless, if a CCR F satisfies KL(2+) with Wi++(a'{xiy,z}) = 0(i = i,2,3) and SP, we have Cfl({;c,j>,z}) = 0, where Ca = F(a).

It follows that a seemingly thin barrier separating EP and SP is in fact a sharp watershed between possibility theorem and impossibility theorem.

6.5. In contrast, our second scheme generates a CCR that makes a strong Paretian ethics compatible with the libertarian claims, which is worthwhile to be emphasized, so far as there exists at least one liberal individual.

8 Here again, the full force of the condition U is nowhere invoked. Indeed, there exists no CCR satisfying KL(2+) and SP on the restricted domain s/*

References

1. Breyer F (1977) The liberal paradox, decisiveness over issues, and domain restrictions. Z Nationalökon 37:45-60

2. Gaertner W, Krüger L (1981) Self-supporting preferences and individual rights. Economica 48: 17-28

3. Gibbard A (1974) A Pareto-consistent libertarian claim J Econ Theory 7:388-410 4. Kelly JS (1976) Rights exercising and a Pareto-consistent libertarian claim. J Econ Theory 13:

138-153 5. Krüger L, Gaertner W (1983) Alternative libertarian claims and Sen's paradox. Theory Decis 15:

211-229 6. Sen AK (1970) The impossibility of a Paretian liberal. J Polit Econ 78:152-157 7. Sen AK (1970) Collective choice and social welfare. Holden-Day San Francisco 8. Sen AK (1976) Liberty, unanimity and rights. Economica 43:217-245 9. Suzumura K (1978) On the consistency of libertarian claims. Rev Econ Stud 45:329-342. A

Correction (1979) Rev. Econ. Stud. 46:743 1 0. Suzumura K (1 980) Liberal paradox and the voluntary exchange of rights exercising. J Econ Theory

22:407-422 1 1 . Suzumura K (1983) Rational choice, collective decisions and social welfare. Cambridge University

Press New York



Documents

Gibbardian Libertarian Claims Revisited