Gibbard 1974 a Pareto-consistent Libertarian Claim

JOURNAL OF ECONOMIC THEORY 7, 388-410 (1974)

A Pareto-Consistent Libertarian Claim*

ALLAN GIBBARD

Department qf Philosophy, University of Chicago, Chicago, Illinois 60637

Received May 21, 1973

1. INTRODUCTION

Anyone has the right to make certain decisions without outside inter- ference. So libertarians claim, but the claim is inconsistent-or at least, I shall show, it is inconsistent under one natural interpretation. Later, I shall develop and test two interpretations which make the claim consistent; one of them, I think, succeeds in expressing an important part of what many libertarians want to say.

A. K. Sen [3, 41 has already found one problem with this libertarian claim that everyone has a right to determine certain decisions by himself. In one simple form, he shows, the claim is inconsistent with the Pareto principle-the principle that an alternative unanimously preferred to another is preferable to it. Thus we must either give up the libertarian claim in that form or give up the Pareto princip1e.l Here I shall show that even giving up the Pareto principle is no sure protection; considerations behind the claim in Sens version also lead in a natural way to a version that is inconsistent by itself. Later, I shall consider two ways to modify this inconsistent libertarian claim. The first new version falls back into Sens paradox: the claim in that form is self-consistent but inconsistent with the Pareto principle. The second new version succeeds better: in that version the claim is not merely self-consistent, but consistent as well with the Pareto principle. A libertarian, in short, is neither entirely safe if he gets rid of the Pareto principle nor inextricably in trouble if he keeps it.

2. AN INCONSISTENT LIBERTARIAN CLAIM

What, more precisely, do I mean here by the libertarian claim? Begin with an example provided by Sen [4, p. 1521. I have a right, many

* This paper is revised in light of the incisive comments of an associate editor, for which I am grateful.

1 For further discussion of this, see [2] and [5].

388 Copyright 0 1974 by Academic Press, Inc. All rights of reproduction in any form reserved.

PARETO-CONSISTENT LIBERTARIAN CLAIM 389

people would hold, to choose the color of my bedroom walk. That means that if I prefer white bedroom walls to yellow, then no matter who wants my walls yellow, it is preferable, other things equal, for them to be white. The libertarian claim is that everyone has rights in this sense -in a very modest version, that everyone has at least one such right. Sen puts this modest version as follows: For each person, there is at least one pair of distinct states of the world, x and y, such that if he prefers x to y, then x is preferable to y, and if he prefers y to x, then y is preferable to x [3, p. 871. Suitably formahzed, the claim even in this form is inconsistent with the Pareto principle. If a right, though, is to mean something like the right to choose the color of ones own bedroom walls, then even the modest claim that everyone has at least one right can be put more strongly. For each person, the claim can be put, there is at least one feature of the world-such as the color of his bedroom walls-that is his business alone. That means that if two states of the world x and y differ only in that feature, and he prefers x to y: then x is socially pre to y. That, in turn, means at the very least that if x is attainable, ought not to be chosen. That is what I shall call the i~b~~~~r~~~ claim.

Here is a case where the libertarian claim breaks down. 4 am a perverse nonconformist: I want my bedroom walls to be a different color from Mrs. Grundys. Grundy, on the other hand, is a conformist: She wants her bedroom walls to be the same color as mine. Exactly four social states are available: a,LG, , azov , aszo , and ayl, ; they differ only in the color of our respective bedroom walls. The first index gives the color of my wails and the second that of Grundys, so that in a,, , for instance, mine are white and Grundys yellow.

Assume, then, the libertarian claim in a speciiic version: that if two states differ only in the color of someones bedroom walls, the one he prefers is preferable. In the case I have described, it follows for each attainable state that it ought not to be chosen. Take state a,, . I prefer a,, to a,, , for I want my walls to be different from Grundys, Since the two differ only in the color of my bedroom walls: the one 3 prefer Is preferable: a,, is preferable to aruE . I shall write this as 8 tow : then, ought not to be chosen, for avW is attainable and Similarly, Grundy prefers aluw to a,w ; therefore a&aWU, and not to be chosen. Again, I prefer a,, to aVs ; hence azOVPa.y!, ought not to be chosen. Finally, Grundy prefers a,, to .aYW , and so auuPa,, , and a,, ought not to be chosen. Pn short, with P, used for my preferences and P2 for Grundys, it follows respectively from aywPIaWW, awV2P2a,, , wuplav2/, and ayvP2a21w that a,Ja,, , awwf%uv, a,,P-a,, , ad

and each of the available alternatives ought not to be chosen. Here is the argument in its general form. First, the hbertarian claim

390 ALLAN GIBBARD

needs to be formulated more precisely. The libertarian claim is that each person has a right to determine certain features of the world by himself. To state this claim, we must represent each state of the world by a number of independent features: an alternative x, I shall say, is a p-tuple (Xl ,a.-, x,) of features. Each index from 1 to p, then, designates an issue: a choice among alternative possible features of some type. The set of all complete alternatives, available and unavailable, can now be defined as follows: For each issue i from 1 to ,u, let there be a finite set iV& of feature-alternatives for issue i, each with at least two members. The set &Z of alternatives, then, is the Cartesian product Ml x ... x MM ; it consists of every p-tuple


F: a nonempty set of possible combinations of additional relevant fact A.

The variables range as follows:

i and j: indices from 1 to pcL. designating issues;

a, w, x, y, and z, sometimes with subscripts or primes: alternatives in A. Each, then, is a p-tuple on the pattern w = (wl I..lY w,>;

b and c: integers from 1 to v, representing individuals;

P: preference v-tuples (PI ,,.., PJ, where each Pb is an ovdeving of A-that is, satisfies the axioms,

9: nonempty subsets of 4;

E: members of 8;

f: SCFS.

Finally, x&y means myPbx, and for given P, 9, and E, C =S(P, 9, c)!- The libertarian claim, then, is that for every person b there is an issue j

such that. in any alternative x, xj is his business alone. That means that for every pair of alternatives x and y with xi = yi whenever B + j, if xP,y, then x is preferable to y. What is it for x to be preferable to y? At the very least, it means that if x is available, then y is not optimal. The following definitions, then, lead to the libertarian claim in what is perhaps its most natural formulation.

DEFINITION 1. Givenf, P, Y, and e, we have xPy if and only if

XE.Y-+y$G.

DEFINITION 2. f accords b the right to x over y if and only if

w>P-~)(~obpbY + XPYI-

DEFINITION 3. x is a j-variant of y if and only if

(W)[i # j -+ xi = vi].

j-variants, then, are alternatives that differ from each other only in their jth feature.

CONDITION L ON A SCF f: For every individual b there is an isme .j

392 ALLAN GIBBARD

such that for every pair of j-variants x aHd y, f accords b the right to x over y.

THE LIBERTARIAN CLAIM. The just SCF satis$es Condition L.

That is to say, there is a SCF which, for each possible set of circumstances, gives the set of all alternatives which would be just under those circumstances, and that SCF satisfies Condition L.

It will now be shown that if there is more than one individual, then the Libertarian Claim cannot hold.

THEOREM 1. No SCF with number of individuals v > 1 satisfies Condition L.

The proof follows the pattern of the wall-color example at the beginning of this section. Suppose f satisfies Condition L and v > 1. Then consider people 1 and 2: For each, there is an issue j of the kind claimed in L. It makes no difference for the logic of the case which person has which issue, and so, without loss of generality, we can suppose that person 1 has issue 1 and person 2 has issue 2. Thus, if x and y are l-variants, then f accords person 1 the right to x over y, and if x and y are 2-variants, then f accords person 2 the right to x over y. Now let w, and y, be distinct members of M, , let w2 and yz be distinct members of M, , and let x3 E MS ,..., X, E MU . Define four alternatives as follows:

amw = , a t/w = , a 1121 =

??ARETO-CONSISTENT LIBERTARIAN CLAIM 393

3. A PARETO-INCONSISTENT LIBERTARIAN CLAIM

Modest though it seems, then, the Libertarian Claim as formulated sets a standard for justice which cannot be met. What has gone wrong? In cme respect at least, the way the claim was worked out looks insidious. The wall-color example started out with an apparent truism: that I have a right to choose the color of my own bedroom walls. That meant, I said, that if I prefer white walls to yellow, then no matter what anyone else prefers, it is socially preferable, other things equal; for my walls to white. That seems fair enough so far. My actual preferences, however, were nst, properly speaking, for white walls or for yellow, but for walls that differed from Grundys. That did give me a conditiona! preference for white walls-a preference for white walls on condition that Grundys walls were yellow -but that is not the same thing as an intrinsic preference for white walls. The libertarian claim as given, though, !ets this mere conditional preference govern: It says that if I want my walls to differ from Grundys, then it is better to have my walls white and Grundys yellow than to have both our walls yellow. In saying that, it may read too much into the truism it was supposed to formulate.

Perhaps, then, a reasonable libertarian claim would disregard such conditional preferences. I have an unconditional preference for white walls over yellow If other things equal, I prefer white walls ts yellow regardless of any other feature of the world. Perhaps what a libertarian ought to claim is this: If I have an unconditional preference for white walls over yellow, then regardless of what other people prefer, it is preferable-other things equal-for my walls to be white.

Such a claim could be put as follows.

DEFINITION 4. Let x and y be feature-alternatives in &fj . Then for a given I?, person 6 prefers x to y unconditionaily for issue j if and only if for every pair of j-variants x and y, if xj = x and yj = y, then S?,y.

CQNDITION k ON SCF J For each person b there is an imue j mch that for every pair of j-variants x and y, whenever b prefers xj to yj uncon- d~tiQ~~~~~y~~~ issue j, then xPy.

LIBERTARIAN CLAIM II. The just SCF satisjks IL.

Condition L, then, seems a plausible statement of what a libertarian should demand. There is a sense in which I have a right to walls of any color I want, without necessarily having the right to match Grundys walls. Condition L says that everyone has at least one right in that sense.

394 ALLAN GIBBARD

How does L fare? Unlike the old Condition L, as will be shown, Condition L can be satisfied. It thus escapes the paradox that besets Condition L in Section 2. L does, however, fall into Sens paradox: L is inconsistent with the Pareto condition:

CONDITION P ON SCF $ For every P, 9, .$, x, and y,

PM XPCYI - XPY.

That Conditions L and P are inconsistent will be shown in Theorem 2. In both these respects, L is like Sens libertarian condition (Condi-

tion L [3, p. 87]), which as a condition on SCFs, I shall call SL.

CONDITION SL ON SCFf: For each person b there are a pair of distinct alternatives x and y such that f accords b the right both to x over y and to y over x.

Even though L and SL are alike in some ways, the two are quite different conditions. SL guarantees each person a special voice on only one pair of alternatives, but the special voice is a strong one: the alternative he prefers is to be preferable, no matter what his other preferences. L guarantees each person a special voice on many pairs of alternatives -indeed for some issue j, it gives him a special voice on all pairs that dier solely in their jth feature-but the voice is limited. The one he prefers is to be preferable if indeed he prefers its distinguishing feature unconditionally; otherwise his preference may be overridden. L, then, grants a weaker special voice than SL does, but grants it for more pairs of alternatives.

Here is a case where rights in the sense given by L conflict with the Pareto principle. I presumably still have a right to choose the color of my bedroom walls, and that is now supposed to mean the following: If two states of the world differ only in the color of my walls, then if I prefer one color to another for my walls unconditionally, the state with the color I prefer is preferable. Mr. Parker presumably has a right in the same sense to choose the color of his walls. Now suppose I want my walls to be white, but care even more that Parkers be white; suppose Parker wants his walls yellow, but cares even more that mine be yellow. Then in the notation of Section 2, with my walls put first and Parkers second, I prefer the alternatives in order a,,a,,a,,a,, ; Parker, in order a,,a,d,,a,, . Now I prefer white to yellow for my walls unconditionally; that is to say, I prefer white walls to yellow whatever fixed color Parkers may be. Parker likewise prefers yellow to white for his walls unconditionally. Hence, we get a cycle as follows. au,V and awW differ only in the

PARETO-CONSISTENT LIBERTARIAN CLAIM 3%

color of Parkers walls, and since he prefers yellow to white unconditionally, we have aw,,Paww . aWW and a,, differ only in the color of my walls, and since I prefer white to yellow ~~condit~o~a~~y, we have hdJauw . Finally, we both prefer ayW to aGg , and so by the Pareto principle, we have ayzr;PaWl/ , and the cycle is complete. Sens paradox stands, even when conditional preferences are disregarded.

Now for the formal statement and proof of this result.

THEOREM 2. No SCF with number of people v > 1 satksjes boric Cotzdition L and Condition P.

Proof. Suppose ,f satisfies L and P, and u > I. Then as in the proof of Theorem 1, we can suppose without loss of generality that person I has a special voice on issue 1 of the kind stated in L, and person 2 has a like special voice on issue 2. Thus for any pair of l-variants x and y, if person I prefers x1 to y1 unconditionally for issue I, then xPy; hkewise for any pair of 2-variants x and y, if person 2 prefers xB to ~1~ unconditionally for issue 2, then xPy.

Now define .aUW , auu , agW , and avv as in the proof of Theorem 1, Let person 1 prefer each alternative x with x1 = wI to its l-variant x with x1 = y, , and in particular, let him prefer the four named alternatives in order a,,a,v,a,,a,, . Then he prefers w1 to y1 unconditionally for issue 4 ~ Thus since a,,, and aYW are l-variants with w1 as the first feature of and y, as the first feature of avW, we have awWPayW . Let person 2 prefer each alternative x with xz = yz to its 2-variant x s,uch that x2 = w2 9 and prefer the named alternatives in order avvavwaWvamW I Thm he prefers y, to wz unconditionally for issue 2, and since aug and aGW are 2-variants with yB as the second feature of aWv and w2 as the second feature of a,,,, 9 we have at-c7jPa,uW e

Finally, let everyone else prefer avzv to a,, . Then since both person i and person 2 do so as well, the preference for ayW over aWy is u~a~~rno~~. Since Condition P holds, we therefore have azlUPaWV . That gives a cycle a,,Pa,,, , a,,Pa,, , and a,,Pa,, ; hence if Y = (a,, , a,, s a,& then none of the members of 9 are in C. That contradicts the stipulation that C is always a nonempty subset of 9. The assumption thatf satisfies both L and P leads to a contradiction, and the theorem is proved.

Now for the result that I., is consistent by itself. I, can be satisfied, so long as there are enough issues to go around-so long as the number p of features in an alternative is at least as great as the number v of people.

THEOREM 3. Ifp > v, then there is a SCF which satisfies I..

The theorem is proved by constructing a SCF which gives each person b

396 ALLAN GIBBARD

an appropriate special voice on the identically numbered issue b. Let Q be the relation between x and y,

(3b)[x and y are b-variants & person b prefers xb to ya unconditionally for issue b].

Letfbe generated from Q in the following manner: For each P, 9, and [,

f(P, Y, f) = {x: x E Y & (Vy)[y E 9 --) -yQx].

Then whenever yQx, then yPx. For suppose -yPx. That means that y E Y but x E C. Hence from the construction offt since x E C, we have y E Y -+ -yQx, and thus since y E 9, we have -yQx. From -yPx, then, it followed that -yQx; hence if yQx, then yPx, as asserted.

The SCFJ; then, satisfies L. For by the construction of Q, if x and y are b-variants and b prefers x to y unconditionally for issue b, then xQy. Therefore, under those circumstances, xpy, and hencef satisfies L.

It remains to be shown thatfis really a SCF-that C is never the empty set. C will be empty only if for every x in 9,

In that case there will be a cycle 0,

x,Qx, ,...> x,-IQX, , x,Qx, ,

where x1 ,..., x, are members of Y. To prove the theorem, we must prove that such cycles do not occur.

Since xlQxz , from the way Q is defined, there is a b such that x1 and x2 are b-variants of each other. Take that b, and for any alternative y, let y* be the b-variant of x1 such that y,* = yb . The *-operator, then, has the following properties:

(i) For all y and z, y* and z* are b-variants of each other.

(ii) If ya = zb, then y* = z.

In addition, the *-operator satisfies the following lemma.

LEMMA. Let yQz. Then if y is a b-variant of z, then y*Pbz*, and otherwise y * = z*. In either case, then, y*R,,z*.

Proof. Let yQz. If y is not a b-variant of z, then by the way (2 is defined, there is a c such that y is a c-variant of z. Clearly b f c, and so y,, = zb , Therefore by (ii), y* = z*, and one part of the lemma is proved.

If, on the other hand, y is a b-variant of z, then from the way Q is defined, we can have yQz only if b prefers yb to z, unconditionally for

PARETO-CONSISTENT LIBERTARIAK CLAIM 397

issue b. Thus, since by the definition of y* and z* we have y, = yb andz, ==Zb, and since by (i) y* and z* are b-variants, by of ~unconditio~ally prefers it follows that y*P,z. That c proof of the lemma.

Now since x1 and x2 are b-variants, it follows from the lemma a from x1Qx2 in cycle 0 that x,*Pbx2*. Also, by the lemma and cycle -we have x2*&x3*,..., x,*Rbxl *. Hence we have a cyclic individual preference,

xl+PbXp*, X,*&X3*,..., X,RbX1*.

That violates the stip tion that individual preferences are orderings5 and hence such a cycle cannot occur. The theorem is proved.

4. A PARETO-CONSISTENT LIBERTARIAN CLArnl

A libertarian might, then, accept the Libertarian Claim II as an expression of what he wants to say and throw out the Pareto principle. In the wall-color example of Section 3, he could then say, the just outcome is the one in which my walls are white and Parkers are yellow. I, after all, prefer white unconditionally for my walls, and Parker prefers yellow unconditionally for his; and the color of ones own walls, the libertarian can say, is his own business: If he prefers one color unconditionally, then that is the color his walls ought to be. It should make no difference, on this account, that we both prefer the state in which my walls are yellow and Parkers are white. That preference stems from busybody interests which ought to be ignored. Each of us should choose the color of his own walls, however little he cares, and S&X the torments of the nosy at the choice the other makes.

To some libertarians, however, that way of thinking will seem too paternalistic: It keeps Parker and me from striking a bargain we would

e both prefer the outcome with my walls yellow and but on the account in the last paragraph, that outcome

is nonoptimal. Many libertarians would hold that Parker and I s be free to agree to that outcome. True, the preferences that would us agree to it stem from nosiness, but a persons motives, such a libertarian could say, are his own business. There is a strong libertarian tradition oE free contract, and on that tradition, a persons rights are his to use or bargain away as he sees fit. I shall not try here to formulate the notion of free contract, but it does seem important at least to find a version of the libertarian claim that is consistent with the Pareto principle, an permits at least those bargains to which everyone would agree.

398 ALLAN GIBBARD

How, then, should the libertarian claim be modified to accomodate the Pareto principle? Here is another case in which individual rights apparently conflict with the Pareto principle. The conflict here, as in the previous case, grows out of one persons taking a perverse interest in the affairs of another, and to my eye, at least, it is plain in this case how the conflict ought to be resolved. Later, I shall generalize the moral of the example.

Angelina wants to marry Edwin but will settle for the judge, who wants whatever she wants. Edwin wants to remain single, but would rather wed Angelina then see her wed the judge. There are, then, three alternatives:

wE : Edwin weds Angelina;

w, : the judge weds Angelina and Edwin remains single;

w, : both Edwin and Angelina remain single.

Angelina prefers them in order w,w,w, ; Edwin, in order w,,wEw, . Here, naive considerations of rights and the Pareto principle combine

to yield a cycle. First, Angelina has a right to marry the willing judge instead of remaining single, and she prefers wJ to w, . Hence w,Pw,, . Next, Edwin has the right to remain single rather than wed Angelina, and he prefers w, to w, , where the only difference between the two is in whether or not he weds her. Therefore w,Pw, . Finally, since all prefer w, to w, , by the Pareto principle we have wEPwJ . The cycle is complete: w,Pw,, w,,PwE, and wEPwJ.

At what point should the cycle be broken ? It could be broken at the step that invokes the Pareto principle-the step w,Pw, . On the account that breaks the cycle there, wJ is optimal, and Angelina should wed the judge. Here again, though, a libertarian may want to allow the parties to bargain. Angelina has every right to wed the judge, but she prefers Edwin; Edwin has every right not to wed Angelina, but if he wants her not to wed the judge, then Edwin must wed her himself. Left freely to bargain away their rights, then, Edwin and Angelina would agree to the outcome w, . * wedding each other. Hence, a libertarian may well hold that-deplorable though Edwins motives be-w, is a just outcome under the circumstances. If so, the cycle is to be broken at the step w,Pw,.

That means we must deny that Edwins preferring w, to w, automatically makes w, preferable to wE . He has the right to remain single, but the right is alienable: He can bargain it away to keep Angelina from marrying the judge. For the right is useless to Edwin: Although he prefers w, to w, , and could avoid w, by exercising his right to w,, over w, , Angelina


claims her right to w,, over wO , and Edwin likes wj no better than WE . If Edwin exercises his right to avoid wE , he gets something he likes no better. In such circumstances, even though Edwin has a right to wO over WE and prefers wO to wE , wE may still be optimal. It may be to Edwins advantage to waive his right to wO over wE in favor of the Pareto principle.

To say that Edwin has a right to wO over wE , then, is to say this. If

(1) Edwin prefers w, to w, , and

42) there is no z such that

(a> Edwin prefers wE to z or is indifferent between them, and

(b) others claim their rights to z over wO ,

then w,Pw, . This is not yet a definition of right, for the clause others claim

their rights to z over wO has not been given a meaning. The meaning it must be given is broad. The goal here, formally put, is to prevent cycies of the form

x$x, ,...) X,-$X, ) x,Px,

by breakmg them at some point. The problem is to accomodate both rights and the Pareto principle; the relation P, then, can be forced in two ways: by someones claiming his right on an issue and by unanimous preference. I shall say that others besides b claim their rights to z over x if z and x are connected by a chain of such Ps-if there is a sequence YI ,..I yn of alternatives in 9, with y1 = z and yh = x, and such that at each step at least one of the following holds:

(3c)[c f b & c has the right to yL over yL+I & ~JJ,+~].

That leaves a characterization of rights which is circular. has the right to has been defined in terms of itself. First, the expression Edwin has the right to wO over wE was defined by using the expression others claim their rights to z over w ,, ; now others claim their rights has been defined using C has the right to yL over yLT1 ~ How can this circularity be avoided ?

The best strategy, I think, is this: First take a complete list of everyones rights without saying what a right is; then define in one step what it is for a SCF to accord all of the rights on the list. First, then, the hst itself.

DEFINITION 5. A rights-system B is an assignment of ordered pairs of alternatives to individuals. Formally put, a rights-system will be a set of ordered triples of the form (x, y, b); if (x. y, b) E 92, it will be said that 9 assigns (x, y> to b.

400 ALLAN GIBBARD

If a rights-system assigns (x, y) to person b, then, in effect, it attributes to b the right to x over y without explaining what a right is.

To explain what a right is, we must now say what it is for a rights- system to be embodied in a SCF. Vaguely put, what needs to be said is this: If B assigns {x, y} to person b and xPBy, then under ordinary circumstances, xPy. If, in addition, however, for some z, yR,z and others besides b claim their rights under 9 to z over x, then bs right is waived, and xPy does not necessarily hold. All this can be said without circularity.3

DEFINITION 6. Let Z be a finite sequence yI ,..., yA of alternatives. Then for a given P and 5, others besides b claim their rights under W to z over x through 2 if and only if y1 = z, yn = x, and for every L from 1 to X - 1, at least one of the following holds:

(3c)k f b & (Y, , Y+I , c> E t@ & YJ~Y~

This will be written z $Q x[Z, W].

DEFINITION 7. For a given P, 9, and E, bs right to x over y is waived under g if and only if for some z, z # y, yRbz, and for some sequence Z of alternatives in 9, z >,b x[Z, 91. This will be written xW,y[9].

DEFINITION 8. Given P, Y, and (, person b determines issue (x, y) under 3? if and only if

(j) (x, Y, b) E 9,

(ii> xpby, (iii) NX wBy[99].

This will be written xD,y[9?].

DEFINITION 9. f realizes rights-system W if and only if for every P, 9, 6 b, x, and Y,

xDby[] + xPy.

We are now in a position to redefine what ,it is for a SCF to accord someone a right. Definition 2 of accord a right led to trouble, both for the libertarian claim and for ordinary beliefs about marriage rights. Here, then, is a new definition, shaped to eliminate cycles arising from marriage rights. To avoid confusion, I shall call rights in the new sense alienable rights.

3 The definitions which follow are revised in light of conversations with Thomas Schwartz.

PARETO-CONSISTENT LIBERTARiA?G CLAIM 401

DEFINITION 10. f accords b an alienable right to s over y if and only if for some rights-system W, f realizes 93 and (x, y, b) E 22.

Condition IL, then, will just be the old Condition I, with alienable right substituted for the old right.

CONDITION 1L ON f: For every b there is a j mch that for euery pair of j-variants x and y, f accords b an alienable right to x ouer y.

LIBERTARIAN CLAIM III. The ju,st SCF satis$es ~o~dit~o~ L.

5. Tm CONSISTENCY RESULT

Libertarian Claim III, it will now be shown, is consistent. Moreover, it is consistent with the Pareto principle, that the just SCF satisfies Condi- tion P. That is to say, the two conditions I, and P can be jointly sat&tie so long as there are enough features to go around-so long as the number p of features in an alternative is at least as great as the number Y of people.

THEOREM 4. I f p 3 v, then there is a SCF whicdz satis$es both Corzdi- tion k and Condition P.

ProoJ Let .SZ be the rights-system such that for every x, y, and 6, if and only if x and y are distinct b-variants. Then given

be the relation between x and y ,

As in the proof of Theorem 2, let f be generated by Q in the following manner: For each P, 9, and 5;

Then from the way Q is defined, it is clear that S satisfies Conditions % and P. It remains to be shown that f is really a SW-that C is never the empty set. G can be empty only if there is at least one cycle

where x1 ,..., x, are members of 9. To prove the theorem, we must show that such cycles do not occur.

Suppose, then, there is a cycle 0 of the kind indicated above. (For the subscripts, I shall use mod 7 arithmetic, so that 1 - 1 = 7 and Y + 1 = 1. Variables L and K will range from 1 to T.)

402 ALEAN GIBBARD

Case 1. For each b, there is at most one L such that ~,-~&x,[92] and -vc> XL-1pcx . Then there is an c such that (tic) x,-~P~x,. Otherwise the cycle 0 cannot be completed. For take a b and L such that x,-,D,x,[92] but -(Vc) x,-~P~x,. Then (x,-, , x, , b) E 92, and so from the way 9I? was defined, v-tuples x,-~ and x, differ in their bth place. To complete the cycle, there must be a K different from L such that x,+~ differs from x, in its bth place. Now x,-~D~x,[B] cannot hold for any c # b, since then x,_, and x, would have to be c-variants. By supposition, x,+~D~x, does not hold unless (Vc) x,-,P,x, . From the definition of Q, since x~-~Qx, , the only remaining possibility is that (Vc) x,_~P~x, .

Even though some step in the cycle must be one of unanimous preference, there cannot be a unanimous preference at every step in the cycle, because if there were, individual preferences would be cyclic, and thus not orderings. Therefore, there must be a transition from nonunanimous preference to unanimous preference: an L such that -(Vc) xl-$,x, , and therefore (lb) x,-~D~x,, but such that (Vc) x,P,x,+~ . By supposition, this L is the only one such that for that b, xLplDbx, and -(Vc) xLmlP,x, . Then let .Z be the sequence rightward from x,+~ to x,-~ in the cycle

- . . , X 7 , x1 )...) x, ) x1 ,... ;

at each step x, , x,+~ in 2, either (de) x,P,x,+~ must bold or else for some c f b, c) E 52 and x,P,x,+~ . Nence x,+~ >)b x,&Z, 91, and since x, # x,,, and x,R~x,+~, by Definition 7, x,_~ Wbx,[92]. Therefore by Definition 8, x,-~D~x, does not hold, and this case cannot arise.

Case 2. For some b, there is more than one L such that x,-,D,x,[SJ but -(Vc) x,_,P,x, . Then of the two or more such L, pick the one that makes x, highest in bs preference-ordering (or one of them in case of ties) andcallit~.Thenlet~bethefirsttotheleftof~inthecycle...,r,l,...,~,l..., such that xLd1Dbx,[9] and -(VIZ) x,J,x, . Then by the way K was chosen, x,Rbx, , and in the sequence C,

XL 3 x L+l 9*--s K-1 9

x, >b x,&Z, S?]. By Definition 7, then, since x, # x, , x,Rbx, , and x, >>b x,-JZ, W], we have x,-, W,x,[B?J. Therefore by Definition 8, we do not have ~,-~D~x,[i], and this case does not arise. That completes the proof of the theorem.

6. THE STRENGTH OF THE CLAIM

Libertarian Claim III is consistent with the Pareto principle, but is it really a libertarian claim? Are alienable rights, as defined, rights in any


sense strong enough to satisfy a libertarian? Having a right to something should give a person a decisive say on the matter, so that normally on that issue he can stand against the rest of the world. How decisive a say on an issue does an alienable right give? Unless it is waived, an alienable right lets a person decide an issue completely. The question, then, is how easily circumstances arise in which an ahenable right is waived. If alienable rights are to be rights in any recognizable sense, they must be waived only in exceptional circumstances, where there is manifestly an irrecon- cilable conflict of rights, or where rights conflict with the Pareto principle so that someone would be better off waiving his right than he would if the Pareto principle were sacrificed.

Now, it is plain from the definition that alienable rights will be waived only un such circumstances of conflict; the danger is that such circumstances ght not be exceptional. Could a rights-system 92 assign so many issues to so many people that a persons right on an issue would usually be waived under 92? If there were such a specious rights-system 95$ then even if a SCF f realized B and 32 assigned issue {x, y) to someone, his alienable right to x over y could not usually be exercised. Strictly speaking, according to Definition 10, f would accord him an alienable right to x over y, but the claim that f accorded him that alienable right would be empty.

The following theorem shows such extreme fears ~~a~~~l~s. It shows that ifs accords someone an alienable right to x over y, then at the very least, if y is his sole last choice from among the available alternatives, then no matter what others may prefer, f makes x preferable to y.

EHNHTION 11~ A SCF accords b a first-order right to x over y if and only if for every P, 9, and 5, if

(vz)rz E Y & z f y * zP,y],

and x # y, then xPy. Thus, what it is to accord a first-order right is defined without any

mention of realizing rights-systems. It will now be shown that any alienable right is a first-order right in this sense.

THEOREM 5. A SCF f accords b an alienable right to x over y only if,f accords b a$rst-order right to x over y.

fioof. Suppose x # y and (Qz)[z E Y & z f y + zP,y]. Then in the first place, x&y, and (ii) in Definition 8 is satisfied. In the second place, there is no z such that z E 9, z # y, and y&z. Thus, from Definition 7,

404 ALLAN GIBBARD

x W,y[L%] holds for no 9% and (iii) in Definition 8 is satisfied for every rights-system 92.

Now suppose f accords b an alienable right to x over y, Then there is an W realized by f which assigns (x, y) to b. That 2, then, satisfies (i) in Definition 8. Thus (i), (ii), and (iii) in Definition 8 of Db are satisfied, and hence XL&Y. Since f realizes ~22, we have xPy. We have shown that whenever (Vz)[z E 9 & z # y -+ zP,y] and x # y, then xPy. Therefore b has a first-order right to x over y, as the theorem asserts.

Alienable rights, then, are not empty. At the very least, they are first- order rights, and it is clear from Definition 11 that to have a first-order right is genuinely to have a special voice on an issue. First-order rights cannot be accorded indiscriminately. According me a first-order right to x over y precludes according anyone else a first-order right to y over x. For suppose I had a first-order right to x over y and someone else had a first-order right to y over x. Then if I preferred x to y and he preferred y to x, and only x and y were available, then no alternative could be chosen. That contradicts the definition of a SCF. In short, then, every alienable right is a first-order right, and a first-order right gives a person a special voice on an issue which cannot be accorded indiscriminately.

Rights of higher orders can be defined so that they are successively harder to override. A second-order right, roughly, is a say on an issue which can be overridden only by a conflict with the first-order rights of others. Already, then, a second-order right gives a person a strong say on an issue. For although first-order rights gave a person only a weak say on an issue, they could not, we saw, be accorded indiscriminately; hence, conflicts with first-order rights of others will, in all likelihood, be unusual. Second-order rights, then, will be overridden only in unusual circumstances. A third-order right can be defined as a say on an issue which is overridden only by a conflict with the second-order rights of others-and likewise for rights of higher orders. Rights of each order, then, are waived under successively more stringent conditions. An alienable right, it turns out, must be a right of every order.

Here, then, is the definition of &h-order rights put explicitly. First, let W,(f) be the system of first-order rights, so that ,. . . which assign rights of successively higher orders. Once 93,(f) is defined, Definition 8 automatically gives sense to the expres-

PARETO-CONSISTENT LIBERTARIAN CLAIM 40.5

INITION 12. %?',+l(f) is the rights-system such that for every x, y, (x, y, b> E W,+,(f) if and only if both (x, y, 6) E B,(f) and for , 9, and &

XDilY@%f 11 - XPY.

2!,(f) is now defined recursively for every n; I shall. often abbreviate it as g2, . For each II, we can now define an nth:order right.

DEFINITIQN 13. f accords b an n&-order rig,& to x over y if and only if

&gr,b)E%z.

Now for the crucial result of this section: alienable rights are rights of every order.

THEOREM 6. If ff accords b an alienable right to x over y, then for every n, f accords b an ath-order right to x over y.

This theorem is a corollary of the following theorem, which is proved in the Appendix.

THEOREM 6. Let f realize rights-system 9. Therz for every n, 92 _c .9?,(f).

heorem 6 follows from Theorem 4* is easy to see. Suppose S accords b an alienable right to x over y. Then there is a rights-system W realized by f such that (x, J, b) E W. For each n., then, by Theorem 4*, (x, y, b) E B,(f), and thus f accords b an yZth order right to x over y9 as Theorem 4 asserts.

The converse of Theorem 4 also holds: not only is an alienable right a right of every order, but, it can be shown, a right of every order is an alienable right. Thus, alienable rights can be completelly characterized as rights of every order.

~~~~~~~ 7. For any x, y, and b, f accords b a right of every order to x over y if and only jf f accords b an aliePz able right to x over y.

The part of the theorem that is still to be proved is that if f accords b a right of every order to x over y, then f accords b an ahenabble right to x over y. That follows from Theorem 7*, which is proved in the Appendix.

406 ALLAN GIBBARD

DEFINITION 14. SF = StYI n ~2~ n .-+.

It has been shown that if f realizes 98, then .9? _C 9; in the Appendix, it is shown:

THEOREM 7. f realizes 92.

From that, it immediately follows that a right of every order is an alienable right, and Theorem 7 is proved.

The new characterization of alienable rights as rights of every order shows that they are rights in a stringent sense-that they are waived only under exceptional circumstances. First-order rights cannot be accorded indiscriminately; a second-order right gives a person a say on an issue that can be overridden only when it conflicts with the first-order rights of others; hence a second-order right gives a person quite a special say on an issue. A third-order right gives a person a say on an issue that can be overridden only by a conllict with the second-order rights of others-a strong say indeed. An alienable right, it has been shown, is a right of every order.

Libertarian Claim III, then, is not only compatible with the Pareto principle, it is a claim that everyone has rights in a strong sense of the term. It is a sense which, despite its complexity, seems natural in the case of Edwin zis Angelina examined in Section 4. Libertarian Claim III, then, seems a tenable way to formulate what libertarians convinced of the Pareto principle want to say. A libertarian can consistently hold the Pareto principle and still claim that in a strong sense, everyone has rights.

THEOREM 6. Let f realize rights-system 92. Then for every n, 92 s_C 9fn .

Proof. From Theorem 5, we have 92 _C %?I . It remains to be proved that if 93 _C 92%) then 92 _C Wn+l .

LEMMA 1. Suppose 92 C 92% , and let 2 be the sequence yl ,. . ., yA . Then if z gfi x[Z, W], then z SD x[C, a,].

Proof. Suppose 2 2 an and z>>~ x[Z, 981. By Definition 6, yl = z, yA = x, and for every L with 1 < c < X, one of the following holds:

W) Y&PCY,l , (Wk =f b & (YL , Y+l , c> E @ & YP,Y,,l.

(1)

(2)

PARETO-CONSISTENT LIBERTARIAN CLAXM 407

Take any L from 1 to X - 1: for that L either (1) holds or (2) holds. If (2) holds, so that for some c, (y, , yLil , c) E 9, then since B _C 5% ) then for that C, (y, ) yL+I , c> E .%n, and hence

Wk f b 82 (Y 2 YL+l) c> E %z 82 YPCYLI1.

For each L from 1 to X - B, then, either (1) holds or (3) holds. Definition 6, z >,, x[C, W,], and the lemma is proved.

LEMMA 2. Suppose B? C W, . Then ifxW,y[kV], lhevl xWby[.G@,].

Proof Suppose B? C 2% and x W,y[g]. Then by ~e~~itio~ 7, for sequence Z of alternatives in 9, z f ys yRbz, and z x[Z, a]-

For that z and JY> then, by Lemma 1, z sb x[.Z, %?,I. By then, x W&Z,].

LEMMA 3. Suppose 9 c %, . Then if (x, y, b) E 9 and xD,y[9?,], then xD,ypq.

Proof. Suppose WC%, , (x, y, b) E B?, and x Then by Definition 8 of Db ,

y hypothesis,

and from (6) by Lemma 2, it follows that

-x W,y[Bq. (8)

Hence by Definition 8, from (4): (7j, and (8) we have XII,&%]. That proves the lemma.

Now the proof of the theorem: Suppose W C $Cn but A& 2 gn+l. Then for some x, y, and b, (x, y, b) E 9 but (x, y, b) $ 9?)n+l , Then for that x, y, and b, by Definition 12 of gnfl , for some P, Y, and 5 we have xD,J@!~] but not xPy. Now from Lemma 3, since (x, y, b) E ZZ, xD,y[CZ,], and 9 C 5%) we have x&y[Rj. By Definition 9, then, since xD,y[G?] but not xPy, f does not realize 9, contrary to supposition. Therefore, if .% C gn then %? C %!n+l , and the theorem is proved.

THEOREM 7 *. f realizes B.

408 ALLAN GJBBARD

Proof. Suppose, on the contrary, that f does not realize B. Then by Definition 9 of crealizes, for some P, 9, [, x, y, and 6,

XaYPu, (9)

-xPy. (10)

In that case, there is no y1 such that xD,y[B,]. For from (9), by Defini- tion 8 of D, , (x, y, b) E W. Hence by Definition 14 of W, for every n,

b y[Z, a]. Then no alternative appears more than once in Z. For if, on the contrary, where 2 = y1 ,..., yh , for some 9 and K, 8 < K and ye = yK , then let Z be the sequence yl ,..., ye , yKtl ,..., y,, . Then each step in the sequence 2: is a step in the sequence Z. Now by Definition 6 of sb, since

PARETCbG6)NSISTENT LIBERTARlAN CLAIM 409

x >>b y[-T;, ,%I, at each step y, , yL+1 in the sequence .6, at least one of the following holds:

WC? YLPCYLfl 3 (1)

(WC =+ b 22 (Y 3 Ye1 : c> c 22 & YPCYi-11. (9

Therefore at each step in the sequence .Z, (1) or (2) holds. Defmition 6, x >>b y[Z, W], and 2: is not the shortest s~bse~~~~~e of 2 such that x >b y[Z, 3]. That proves the lemma.

LEMMA 5. If -xWby[fP], then for every a with z f y and y&z and every sequence 22 of alternatives ipl Pv, there is a y such that for every n 2 y, -2 >>b x[& B,].

Pr55J Suppose -x W,y[W], take a z such that z # y and y&a, and take a sequence 2 = y1 ,..., yh of members of Y. It will first be shown that there is a y such that -2 >,b x[Z, 9q. Suppose otherwise, so that for every n, z >,h x[& $J. Then take any 6 from 1 to h - 1; for each yz, at least one of the following holds:

IV4 YLpcYL+l, (1)

WEc f b Jk (YL 3 Y+l , c> E gn & Y,P,Y,ll. Ia

Thus for that L, either (1) holds, or ifit does not, then (2) holds for every FZ. In the second case, for every n, (y&, yLil , c) E 93% ) and hence (y 9 yL+l , c) E 9P. From this and (2),

In either case, then, (I) or (17) holds for that L. (1) or (17) holds, then, for every L from il to h - 1, and so by Definition 6, z >Zb x[Z, a]. Since 2 # y and y&z, we have x Wby[Wo], contrary to assumption. Therefore for so1ne 8, -2. 2% XL% %J.

NOW let y be such an n. Then for every n > y, NZ >>b x[Z, 8& For by Definition 12, it follows immediately that Bm C 9,, , and hence from Lemma I, if z >,b x[Z, B,], then z >>b x[Z, B$,]. Thus since NZ >b x[Z9 &], for every y1 2 y, -Z >>b x[Z? 9,], and the lemma is proved.

NOW the proof of the theorem: On the assumption that the theorem was false, it was shown that

410 ALLAN GIBBARD

Assume (15) and (16). In (15), xWOy[gn] means that for some z and 2,

z # y & y&z & z >>b x[Z, W,] & 2 consists of members of Y. (18)

By Lemma 4, it follows that there is a z and a 2 which satisfy (IQ such that no alternative appears more than once in Z. The length of Z, then, is no greater than s, the number of alternatives in Sp. For every n, there are a z and a 2 such that

z and 2 satisfy (18) and 2 is of length < s. (19)

Consider all z and 2 which satisfy (19) for IZ = 1. Since there are a finite number of alternatives in P, there are a finite number of such pairs z, Z. Now by Lemma 5, for each such z and Z, there is a number y such that for every n 2 y, -z >b x[Z, J%,]. Since there are a finite number of such z and 2, there must be an n greater than any such y. For this IZ, from the way the 2s were defined, there are no z and 2 which satisfy (19). That contradicts the earlier conclusion that for every y1 there is a z and a 2 which satisfy (19). The assumption that the theorem is false has led to a contradiction, and thus the theorem is proved.

REFERENCES

1. K. J. ARROW, Values and collective decision-making, in Philosophy, Politics, and Society, Third Series, (P. Laslett and W. G. Runciman, eds.), pp. 215232, Basil Blackwell, Oxford, 1967.

2. C. HILLINGER AND V. LAPHAM, The impossibility of a Paretian liberal: Comment by two who are unreconstructed, J. Pol. Econ. 79 (1971), 1403-1405.

3. A. K. SEN, Collective Choice and Social Welfare, Chap. 6*, Holden-Day, San Francisco, CA, 1970.

4. A. K. SEN, The impossibility of a Paretian liberal, J. Pal. Econ. 78 (1970), 152-157. 5. A. K. SEN, The impossibility of a Paretian liberal: A reply, J. Pol. Econ. 79 (1971),

1406-1407.

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Gibbard 1974 a Pareto-consistent Libertarian Claim