pareto, interdependent rights exercising and strategic behaviour

Suppl. 5, pp. 79--98 (1986) Joumal of konomics Zeit~hrlW fOr National~conomle

�9 by Springer-Verlag 1986

Pareto, Interdependent Rights Exercising and Strategic Behaviour

By

Wulf Gaertner, Osnabrª F. R. G.*

(Received September 27, 1985; revised version received July 31, 1986)

1. Introduction

About fifteen years ago it was demonstrated by Sen (1970) that even a minimal amount of efficiency in social choice processes, as expressed by the Pareto principle, can heavily collide with a fairly weak libertarian requirement that guarantees a minimum of individual freedom to the member of society. This result which has become known as "the impossibility of a Paretian liberal ~ has caused a lot of concern among both economists and philosophers. The following illustration which is based on an example due to Sen (1976, 1983) depicts the trouble.

Consider a two-person society. A social decision is to be taken as to the employment of the two individuals. Imagine that there are only four alternative states available to the two persons: (1, 0) - - person 1 has a full-time employment while person 2 is without job (the first number in brackets indicates individual l ' s employment, the second number gives person 2's situation); (0, 1) m the exact opposite of the first state; (0, 0) m both individuals are jobless; (1, 1) ~ both persons have a full-time employment. Let us assume the following strict preferences for the two

* For helpful comments and suggestions, I am grateful to Friedrich Breyer and four anonymous referees of this joumal. The first version of this paper was written in the auturan of 1984 when I was a Visiting Fellow at AII Souls College, Oxford. The stimulating atmosphere in the College is gratefully remembered. My thanks also go to Deutsche For- schungsgemeinschaft for financial support.

80 W. Gaermer:

individuals (these are arranged in perpendicular order with the more preferred alternative arranged above the less preferred one):

Person 1 Person 2

(1, o) (o, 1) (o, o) (o, o) (1, 1) (1, ~) (o, 1) (1, o).

Ceteris paribus, both persons obviously prefer full-time employment to no employment at all. Given the job situation of the other person, each individual should - - according to the idea of a minimal amount of individual liberty - - be free to decide autonomously whether he (she) wants to work of stay without job. Once we follow this idea of "individual decisiveness over private spheres", our smaU society ends up in the state (1, 1) where both persons are employed. The unpleasant thing now is that the so-called liberal solution is Pareto-dominated by the state (0, 0). Our two individuals could bring about a Pareto improvement through common action which would take them to (0, 0) but it is important to see that (0, 0) is no point of equilibrium. Each individual alone would have an incentive "to p laya different strategy", and if both persons yield to this temptation they will end up in the Pareto-inferior state (1, 1) again. This dilemma can also be depicted in the following pay-off matrix:

1• 0 1

I 0 [ 3,3 1,4

1 I 4, 1 2, 2

Both persons have a dominant strategy which leads them to the Pareto-inferior pay-off (2, 2). The reader has already realized that our situation has the structure of the "prisoner's dilemma" game but it is worth emphasizing (Sen (1983), pp. 21--23) that the impossibility result of the Paretian liberal can hold without the corresponding game being a variant of the prisoner's dilemma 1. This will become clearer as we proceed.

1 The first author who mentioned the similarity of the impossibility of the Paretian liberal and the prisoner's dilemma apparently was Fine (197S).

Interdependent Rights Exercising 81

There have been many suggestions to circumvent Sen's impossibility theorem. In this paper we shall focus on two of them, Gibba rd ' s theory of alienable rights (1974) and the Sen-Suzu- m u r a approach (1976, 1978) of constraining the Pareto principle. In contrast to many other resolution schemes the Gibbard approach and the Sen-Suzumura approach require each individual to possess a lot of irtformation. In addition to his (her) own rights each individual has to know the preference orderings of all other individuals and the rights assignments to them. One may find this high degree of interdependence quite attractive within a libertarian system of collective choice. The problem, however, is that once the act of individual rights-exercising is made interdependent, the aspect of strategic behaviour comes to the fore.

The phenomenon of interdependent rights-exercising and strategic revelation of preferences within a system of libertarian social choice is what this paper is about. We shall, for example, show that within the Gibbardian framework of rights-waiving it may be profitable for an individual to obtain information on whether the other members of society intend to exercise their rights or intend to waive them. Though the number of contributions to the liberal paradox is quite large, only a few have focused on strategic aspects. Karn i (1978), G a r d n e r (1980) and Kr ª and G a e r t n e r (1983) have examined the manipulation of preferences in Gibbard's theory of alienable rights, the problem of rights-waiving in his system was discussed by Sen (1976), Kelly (1978), S u z u m u r a (1980) and Basu (1984). Cooperation and aspects of coalition formation were analysed by B e r n h o l z (1976), A ld r i ch (1977), and Breyer and G a r d n e r (1980).

The structure of the paper is as follows: Section 2 introduces some amount of notation and defines the basic concepts that will be used later on. Section 3 examines Gibbard's approach, section 4 considers strategic aspects in the Sen-Suzumura resolution scheme, and the final section 5 takes a look at some other schemes where the exercise of an individual right or its deprivation is not made dependent on society's total preference profile and the overall rights structure.

2. Notation, Definitions, and Basic Concepts

Let N = { 1 , 2 , . . . , n} denote a finite set of individuals (n>2) and let X={x, y, z , . . .} denote the set of all conceivable social states. 5a stands for the family of all finite non-empty subsets of X, and each S e 5 a denotes a set of implementable social states. R~ is

82 w. Gaermer:

person i's preference ordering on X and R=(R1, R~, . . . , R,) society's preference profile. We say that i E N weakly prefers a state x r X to another state y ~ X if and only ir (x, y) e R~. The strict preference relation corresponding to R~ will be denoted by P~: (x, y) e P~ ~ [(x, y) ~ R~ A (y, x) r Rd. The indifference relation will be denoted by h: (x, y) ~ I~ ~ [(x, y) ~ R~ A (y, x) ~ Rd. A collec- tire choice rule (CCR) is a function F which for each profile R generates a social choice C (S)=F (R, S), where C (S)4:fl for any S ~ S~' is the set of socially chosen states from S.

We assume that each individual i ~ N has a "private sphere" D~ consisting of at least one pair of personal alternatives over which he (she) can be decisive "both ways" in the social choice process, i. e., (x, y) ~ D~ ~-~ (y, x) ~ D~ with x 4= y; D~ is called symmetric in this case. A rights-system then is an assignment of ordered pairs of states to individuals, viz. an n-tuple D = (Dl, D~, . . . , Da) e t2 (n), ~2 (n) standing for the n-fold Cartesian product of se2, the set of all non-empty subsets of X x X. We require that the rights assignment be acyclical or "coherent". According to S u z u m u r a (1978, p. 331), the rights system D = ( D i , Dz , . . . ,Da) is coherent if and only ir for every n-tuple of orderings (R1, Rz , . . . , Ra) there exists an order-extension E of ta (D~ n Rd.

We now introduce two conditions from Sen's original set of conditions that led to his famous impossibility result.

Condition U (unrestricted domain). The domain of the CCR comprises aU logically possible profiles of individual orderings on X.

Condition P (weak Pareto principle). For all x, y ~ X,

(x, y) ~ n e~ --* [x e s ---, y r C (s)] for aU S E Se. i~N

3. To Waive of Not To Waive

The central idea of Gibbard's theory of alienable rights is the following: Given D~, it would, in some cases, not be reasonable for individual i to exercise his (her) right(s). The individual should rather waive bis (her) rights, or some of the rights, ir he (she) ex- pects to be better-off (at least not worse-off) by doing so. "There is a strong libertarŸ tradition of free contract, and on that tradition, a person's rights are his to use or bargain away as he sees fit" ( G i b b a r d (1974), p. 397).

G i b b a r d precisely defined those cases in which individual rights should be waived. Given a set S of feasible states and a


rights system D, a subset W, (R[S) of the set of private pairs Dz of person i, the so-called "waiver set" of i 2, is defined by the following condition: (x, y)~ W~ (RIS) ir and only ir there exists a sequence {yl, yg , . . . , ya} of states in S such that

and

yl ~ y,

(y, yl) ~ Re,

ya = x,

(1)

(2)

(3)

(4) V t ~ {1, 2 , . . . , ~t- 1}: (y~, yt+l) E (P u Q),

where P-- n P j a n d Q = u (DInPj). i~N i~N\{i)

With the help of this specification Gibbard formulated his libertarian daim.

Condition GL (Gibbard's libertarian claim). For every profile R, every S 6 5r every i 6 N , and every pair x, y ~ X, if (x, y) D, nP , and (x, y) r W, (RIS), then [x ES--*y r C (S)].

Gibbard obtained the following solution to the liberal paradox.

Theorem (Gibbard). There exists a CCR satisfying condiuons U, P, and GL.

Let us have another quick look at our introductory example. If both persons exercise all their rights, the only "candidate" for the social choice set is the state (1, 1). This alternative, however, is Pareto-dominated by (0, 0) so that the choice set is empty (what demonstrates Sen's impossibility result). According to Gibbard's scheme, the two persons waive their rights over the pairs ((1, 0), (0, 0)) and ((0, 1), (0, 0)) so that (0, 0) ends up to be the unique social choice. (0, 0) is Pareto better than the liberal solution (1, 1).

Therefore, our example illustrates that voluntary rights-waiving can be an advantage for the individuals concerned. But is this result generally true? Does it hold in all cases?

We think it is fair to say that Gibbard conceived his theory of alienable rights a s a general resolution scheme for the liberal paradox and no ta s a scheme that would be applicable only under rather special circumstances (i. e. particular preference profiles). Kel ly (1976), however, observed that there ate situations where it would not be reasonable for an individual to waive a particular right since there exists a so-caUed "repairing sequence". We can

This term was coined by Suzumura (1980); llz~ (R]S) c D~ n (S x S).

84 W. Gaertner:

illustrate his argument via a modification of our initial example. Let our two individuals have the following preferences"

Person 1 Person 2

(1, 0) (0, 0) (1/2, 1/2) (1/2, 1/2) (0, 0) (1, 1) (1, 1) (1, 0).

Furthermore, let person 1 be decisive over the pair ((1, 0), (0, 0)) and person 2 be decisive over the pair ((1, 1), (1, 0)). According to Gibbard's theory, person 1 would have to waive his (her) right. Kelly convincingly argued that there is no reason for person 1 to waive that right, since a repairing sequence from (1/2, 1/2) to (1, 1) exists which nullifies the damage to hito (her) caused by the exercise of person 2's right. Kelly proposed several refinements of Gibbard's theory but, contrary to the author's claim, not one of them leads to a new possibility result (see S u z u m u r a , 1980). K r ª and G a e r t n e r (1983) showed that one of Kelly's refinements does work under the requirement of unconditional preferences 3 of the individuals; S u z u m u r a and Suga (1986) proved that another one of Kelly's modifications 4 is successful if there is at least one "socially unconcerned individual" in the society.

Let us recapitulate: Our introductory preference profile has clearly shown that there ate situations where it is advantageous for the individuals to waive rights. Our second example was such that there exist cases where Gibbard's directive to give u p a right is completely unnecessary. We now show that under certain circumstances it may even be disadvantageous for some individual to waive a particular right. Sen (1976, p. 224) briefly mentioned that a person's decision to exercise his (her) right of better waive it very much depends on the other persons' decisions as to their rights. They, again, make their decision on whether to exercise or not dependent on what they expect the others to do with their respective rights. This interdependence phenomenon can be best illustrat- ed with the help of Sen's (1976, p. 222) "work-choice case" which

a For a definition of unconditionality, see sect. 5 below. 4 We apologize for being a bit vague here but Kelly's definitions ate

quite lengthy.


was the basis for our two initial preference situations. Again we have only two individuals with the following orderings"

Person 1 Person 2

(1/2, O) (0, i/2) (1, 1/~) (1/2, 1) (0, i/2) (1/~, O) (112, 1) (1, 112).

Let person 1 have a right over the pair ((1, 1/2), (0, 112)) and let person 2 be decisive over ((1/2, 1), (1/2, 0)). Due to condition P, the states (1, 1/2) and (1/2, 1) cannot be in the social choice set. Ir there is no rights-waiving at all, the choice set will be empty 5. According to Gibbard's scheme, however, both individuals must waive their rights. In this case, the choice set will contain both (1/2, 0) and (0, 1/2) and some random mechanism, let's say, will declare one of the two social states as the unique social outcome. Under these circumstances, it will be profitable for individual 1 n o t

to follow Gibbard's directive to waive for, then, the unique ele- ment in the choice set will be his (her) most preferred alternative (1/2, 0). Clearly, the given argument is completely symmetric with respect to person 2.

For the given preference profile we have shown that it pays f o r a single person not to comply with Gibbard's waiving instructions. Basu (1984, p. 418 419) describes this fact as the absence of "Nash consistency". In order to emphasize that this result does not hold under all profiles, let us have a third look at our introductory example. Here the phenomenon just described does not occur. If one of the two individuals alone does not follow Gibbard's waiving rule, i. e. does not waive the "upper" right, no personal benefit can be derived ffom this action. Each individual is, of course, always free to waive the "lower" right (which would be contrary to Gibbard's scheme) but this very act can only be prop- erly understood if that person is an altruist.

In Gibbard's Angelina-Edwin case (which has the same structure as Sen's famous Lady Chatterley's Lover case) Gibbard's

5 Note that this profile does not constitute a prisoner's dilemma game. In private communication, Breyer rightly pointed out that the given situation could, however, be converted into a prisoner's dilemma if additional individual preferences over additional social states were added to the profile above.

86 W. Gaertner:

theory determines that Edwin waive his right while Angelina can go along exercising hers. Only if Angelina decided to waive her right (why on earth should she do that?) would it be advantageous for Edwin to disregard Gibbard's waiving instruction.

We can summarize our findings above as follows. Let C a (S) be the social choice set according to Gibbard's resolution scheme and let ClVW~ (S) denote the choice set when some individual /�91 alone decides not to comply with Gibbard's waiving rule. We now claim that it is possible to derive an individual's preference ordering /~~ on choice sets from information on the same individual's preference ordering R~ on X 6. For example, ir w e S, v e S, (w, v) e P~, Ca(S) ={w, v} and C Nw~ (S)={w}, we claim that (C Nw~ (S), C a (S)) e q where/3~ is person/~'s derived strict preference relation with respect to choice sets (induced by /~~). Why does this strict preference hold for individual/~? We argue that it holds be- cause with respect to C G (S) there is a non-zero chance that the final outcome is v, and this state is inferior to w according to k's preference ordering R~.

We can now formulate the following

Proposition 1: Under conditions U and P, there exists a preference profile R such that some individual /~ ~{1 , . . . , n} wiU find it profitable not to comply with Gibbard's waiving scheme.

Proof. Let us suppose that there does not e x i s t a profile R such that for some individual ~ it will be beneficial to disregard Gib- bard's waiving instructions. Take the profile on p. 85 above which is clearly admissible under condition U. We obtain C a (S)= {(1/2, 0), (0,1/2)} according to conditions P and GL, C ~wl (S)={(1/2, 0)) and C Nw2 (S)= {(0, 1/2)}. Furthermore, we have (C ~w~ (S), C G (S)) e/3~ for each k e {1, 2} which leads to a contradiction of our supposition.

Quite distinct from the interdependence phenomenon we have just discussed is the aspect of strategic manipulation of preferences. In Gibbard's system of alienable rights there is a strong incentive for the individuals to lie or, as Ka rn i (p. 573) put it, Gibbard's CCR does not satisfy the condition of cheat-proofness v. An individual may achieve a better social outcome by stating untrue pref-

6 We are very sketchy here and admit that this claim is by no means trivial.

A CCR is said to be cheat-proof if it is nowhere manipulable.


erences than by revealing the true ordering s. Moreover, strategic manipulation will make other individuals alienate some of their rights which they would have exercised if everyone had stated the truth 9. Consider another variant of our introductory example (social states which are put on the same level are to be judged as equivalent to each other):

A~

Person 1

(1, 1) (0, 1), (1, 0) (0, 0)

T r u e P r e f e r e n c e s of B o t h P e r s o n s

Person 2

(o, o), (o, 1) (1, o), (1, 1).

Let person 1 be decisive over the pairs ((1, 1), (0, 1)) and ((1, 0), (0, 0)) while person 2 does not reveal a strict preference over his (her) pairs ((0, 0), (0, 1)) and ((1, 0), (1, 1)). The choice set will comprise (1, 1) and (1, 0).

B. U n t r u e P r e f e r e n c e s of 2

Person 1 Person 2

(1, 1) (0, 0), (0, 1) (o, 1), (1, o) (1, o) (0, O) (1, 1).

Now that person 2 exercises the right over the pair ((1, 0), (1, 1)), person 1 will have to waive the upper right according to Gibbard's scheme. The social choice set then contains (0, 1) and (1, 0) and this result is clearly better for person 2 according to/~~. Note that person l 's preferences ate exactly the same in the two situations. The phenomena just described certainly have ethical implications to which we will return in the last section.

4. Does It Pay To Be Liberal?

Sen (1976) and, following him, S u z u m u r a (1978) argued against a mechanical application of the Pareto rule. They proposed a resolution of the liberal paradox which restricts the use of the Pareto

s In order to achieve this, the individual, of course, has to gather a lot of information, a fact which we have mentioned before.

9 For various illustrations of this fact, see Karni, Gardner, and Krª and Gaertner .

88 W. Gaertner:

principle, thereby making a distinction between individuals' possess- ing preferences and wanting them to count in the social choice process.

Let (R1, R2, . . . , R,,) be any preference profile of society and let R~* be a transitive sub-relation of R~ which individual i wishes to be counted in the social decision. Then (x, y) ~ P~* means that person i wants his (her) strict preference for x over y to count in social choice. Consider now the following restriction of the Pareto rule.

Condition CP (conditional strong Pareto principle). Let

R ~ = n R~ ~. i ~ N

For all x, y ~ X,

(a) (x, y) › R* -o [x ~ S \C (S) --* y ~ C (S)]

for all S e ~ , and

(b) (x, y) ~ P ( n R~*) -o [x e S --~ y ~ C (S)] i E b l

for all S e ~ . The key to an escape from the libertarian dilemma obviously

is an appropriate restriction of the profile (Rx*, R2*,..., Rn*) in comparison with (Rx, R2, . . . , Rn). At this point the concept of a "liberal" individual comes in. An individual is said to be liberal ir and only if he (she) wishes only that part of the preference ordering to count which is compatible with others' preferences over their respective private spheres.

Given any preference profile R and given any coherent rights- assignment D, there exists an order-extension E subsuming all D~nR~ ( i~{1 ,2 , . . . , n}). For our arguments that follow it is important to notice that E need not be unique. In many cases there is a choice of several order-extensions. Therefore, let 8 denote the set of all logically possible order-extensions of u (D~ n R~).

lEN

More formally then, an individual j ~ N is said to be a liberal if and only ir Rj* = Rj n E for some E ~ 8.

The libertarian claim in this approach is

Condition CL (coherent libertarian claim). For any coherent rights- assignment D e/2 (n) and for each i e N,

(x, y) ~ D~n P~ --~ [x ~ S ~ y ~ C (S)] for all S e ~ .


Sen and Suzumura obtained the following possibility result.

Theorem (Sen-Suzumura). Ir there is at least one liberal individual in society, a CCR exists satisfying conditions U, CL and CP.

S u z u m u r a (1978, pp. 331--332) remarked that "a liberal need not really care very much how the order-extension is constructed . . . . An 'active' liberal would hold a clear idea of that part of his preference ordering which he wants to count in the collective decision . . . A 'passive' liberal, on the other hand, does not know his R J : instead, he knows only his R~ and that he knows he wants to be liberal". Suzumura of course observed that in the case of several order-extensions the final outcome crucially depends on which extension(s) is (are) being used. Should this fact worry us? We think, it should in the case of the passive liberal. A u s t e n - Smi th (1982) and G a e r t n e r and K r ª (1982) already expressed some misgivings. Suzumura's proposal that for the group of pas-

s i v e liberals somebody from outside ("a well-informed umpire") could determine which order-extension should be selected will be examined again by means of Sen's work-choice case from p. 85 above. Let person 1 once more have a right over the pair ((1, 1/2), (0, i/2)) and let person 2 be decisive o v e r ((1/2, 1), (1/2, 0)) . Note that we now have a multiplicity of order-extensions. Let us choose E' and E", defined as follows:

E' E"

(1, 1/2) (112, 1) (0, 1/2) (1/2, O) (1/2, 1) (1, 1/2) (1/2, 0) (0, 1/2).

We wish to examine the following situations:

(a) Individual 1 alone is liberal and chooses order-extension E'. The choice set is unique, consisting of (1, 1/2) only.

(b) Individual 1 alone is liberal and chooses order-extension E". The choice set is unique: (1/2, 1).

(c) Individual 2 alone is liberal and chooses E'. The choice set is unique: (1, 1/2).

(d) Individual 2 alone is liberal and chooses E". The choice set is unique: (1/2, 1).

(e) Both individuals are liberal; i chooses E', 2 chooses E". The choice set now comprises both (1, 1/2) and (1/2, 1).

7 Journal of Economics, Suppl. 5

90 W. Gaermer:

The foregoing analysis clearly shows that it pays to be an active liberal. If a liberal individual is careless enough to choose the "wrong" order-extension, he (she) will end up in a state low on the preference scale. Therefore, the liberal individual can be ex- pected to choose strategically the "right" order-extension 1~ Note that such a behaviour does not in any way collide with the Sen- Suzumura definition of a liberal person. Also, cases (a), (d) and (e) clearly demonstrate that it is advantageous for an individual to become a liberal if the other person has announced that he (she) will behave liberally. This again reveals some type of strategic behaviour within the proposed resolution scheme 1t. By the way, the non-liberal individual could also achieve a better social result by manipulating his (her) preference ordering instead of joining the group of liberals.

Let us again summarize our findings.

Proposition 2: In the Sen-Suzumura approach with a multiplicity of order-extensions, it is advantageous for an individual to be liberal provided that the order-extension is deliberately (strategically) chosen by him (her).

Proof. Due to the detailed analysis above we can be very brief here. From the perspective of individual 1, situations (a) and (d) and situations (e) and (d) ate being compared. In the first case we have (((1, 1/2)}, ((1/2, 1)}) ~/31, in the second case we can clearly infer (((1, V2), (V2, 1)}, {(1/2, 1)}) E P1.

From the second person's perspective, situations (d) and (a) and situations (e) and (a) ate being compared. We can state that ({(1/2, 1)}, {(1, 1/2)}) 6 q and ({(1, 1/2), (1/2, 1)}, {(1, x/s))) ~/32 respectively which proves our proposition.

5. Self-Promoting Rights-Exercising

In the last two sections we have discussed resolution schemes where the exercise of individual rights depends on the complete preference profile of society and the overall assignment of rights to

lo Within the ordinal framework, it is clearly impossible to say that from an ethical point of view some order-extension is better than some other. One of the referees suggested that with (rough) information on in- terpersonal cardinal utilities things may be different.

11 One can, of course, argue that this feature of the Sen-Suzumura scheme is a clever educational move.


the members of society. We have already mentioned that this fact can be considered as an attractive feature of those approaches. However, both the Gibbard scheme and the Sen-Suzumura scheme ate not proof against different types of strategic behaviour. These phenomena clearly have ethical implications. Does it appear de- sirable that a particular individual should lose the social protection over a pair of (private) alternatives solely on the ground that other people change their orderings, whatever those people's rea- sons? We have to ask whether it is justified to speak of an assignment of rights to individuals if the individual sphere of potential decisiveness is not shielded against intrusion from other people's decisiveness. Al1 this would perhaps not be too serious ir the interference by other persons improved the social outcome for the individual affected. Our last example in section 3 showed that this is not true in general. Furthermore, one would perhaps find less fault with Gibbard's waiving instructions ir the act of renouncing a particular right always yielded a preferred situation for the individual concerned.

Let us look once more at Sen's work-choice case. Each of the two persons prefers more employment to less, given the job situation of the other. On the other hand, both are "meddlesome" (Blau, 1975) in the sense that they attach greater importance to the other being unemployed than to their own job situation. Within the framework of rights and collective choice, the individuals are being granted an absolute protection of a particular part of their preferences from aU social interference. At the same time, they are being allowed to deviate from this very preference in other parts of their ordering, parts that they expect no less to be counted in the collective decision (may we remind the reader of Sen's famous Lady Chatterley case).

G a e r t n e r and Kr ª (1981, pp. 20--21) have argued that no individual should have it both ways, namely unconditional social protection of bis (her) private sphere and maximal gains from collective choice. Social protection has a price which the individuals are required to pay. For social protection means for all the other members of society that they accept and respectan individual's private preference, even when they all happen to have the opposite preference. Respecting an individual's preference implies that all the others ate willing to disregard their own. Yet, fairness then requires that the individual concerned equally adopt the protected- sphere preference regardless of any concomitant change in other people's private affairs. An individual who preserves an ordering of his (her) own personal features in order to secure social pro-

7*

92 W. Gaertner:

tection for his (her) choice between them is said to manifest self- supporting preferences.

In no case is it presupposed that some or even all individuals have to satisfy the (rather strong) property of showing self-supporting preferences. Such ah assumption would indeed amount to introducing domain restrictions. Rather, the concept can be viewed a s a condition for the recognition of a right within a collective choice processlL Each individual, i ndependen t l y of every other, has the option of either forfeiting his (her) right(s) through meddling or winning social decisiveness over private alternatives at the price of revealing self-supporting preferences. In other words, individuals should decide tbemse lves whether a particular potential right allo- cated to them becomes socially effective la. Both H a m m o n d ' s (1982) proposal and Gaertner and Krª approach - - though different from each other - - possess this characteristic. In order to discuss them briefly in our context we have to introduce the so- called issue approach which goes back to G i b b a r d (1974).

We consider a decomposition of each social state x E X into different components. Let X0 and X~ stand for the set of all im- personal features of the world and the set of all private features of person i e N respectively. The set of all social states is then given by X = X0 x (/7 i ~ n Xi) 14. X0 and Xi ate assumed to be finite with at least two elements each. Define

X ) i r . . . xX l - lXX l+ lX . . . xXn

for each i and x = (xo, xi . . . . . x~) ~ X. If xi ~ X~ and

z=(Zo, z l , . . . , Z~-l, Zl+l,. � 9 Zn) ~ X)l( , then

(xi; z) = (z0, zl, . . . . zl-1, x~, zl+l, . . . . z,,).

Individual i's private sphere can now be defined as 15

D~ ={(x, y) ~ X x X[x)~~=y)~~Ax~ ~y~}.

We first define Gibbard's not ion of uncondit ional preferences.

12 This approach, therefore, is more general than Breyer 's (1977) proposal to restrict the domain of preierence profiles appropriately.

13 We feel that one should not speak of a right ii there is no guarantee for the individual to safeguard it ii he (she) so wishes.

14 ir the whole set X is feasible, we have - - in Seidl's (1975) ter- minology - - the case of technological separability.

15 Contrast this definition with the one given in sect. 2 above.


Unconditional Pre[erences (UP): Individual i has unconditional preferences with respect to his (her) sphere of rights D, ir and only if for all (x, y) E D,, if (x, y) ~ P,, then ((x,; z), (y,; z)) e P, for all z ~ X)~ (.

We now introduce Gaertner and Krª concept of self-supporting preferences, though in a somewhat truncated version.

Sel[-Supporting Pre[erences (SSP): Individual i has self-supporting preferences with respect to his (her) sphere of rights D~ ir and only ir for alI pairs (xt, y~)eX~• with x~4:y~, ir ((x~;z), (y~; z')) ~ P~ for some z, z' ~ X) , , then ((x~; z), (y~; z')) e R~ for all g, Z' ~ X)~(.

Property SSP may appear as too restrictive when it is required for the total set X. Ir may, therefore, seem more appropriate to require SSP only with respect to each S, the set of implementable social states.

We formulate the following type of waiver set: (x, y) E Wt (R~IS) ir and only if (x, y) ~ D~ and person i violates property SSP.

The following libertarŸ condition is proposed:

Condition SSL (self-supporting libertarianism). For every profile R, every S ~ S~, every x, y ~ X and every i ~ N, if (x, y) ~ D~ t3 P~ and (x, y) ~ W~ (R~IS), then [x e S--*y ~ C (S)].

With this libertarian claim the following result was achieved by G a e r t n e r and K r ª (1981, p. 24).

Theorem. There exists a CCR which fulfils conditions U, P, and SSL.

From the above definitions it is clear that the social decisiveness of an individual is secured only ceteris paribus in other people's private affairs, since their rights are also protected if they comply with the rules. In the case of technological separability each individual will - - via a manifestation of self-supporting preferences - - attain his (her) preferred private feature. Ir the full cartesian product X is not feasible, this characteristic may not hold, however. Consider another modification of our introductory example where in terms of the issue approach we have X I = X 2 = ( 1 , 0).

Person 1 Person 2

(1, O) (0, 1) (1, 1) (1, 1) (o, o) (o, o) (o, 1) (1, 0).

94 W. Gaertner:

Technological separability does hold in this situation, each individual satisfies $SP and both persons achieve their preferred private feature. The Social choice is (1, 1).

Is the Gaertner-Krª approach proof against a strategic manipulation of preferences? Of course not. The question here is whether it always pays to reveal untrue preferences. The answer to this question is n o t a s simple as one would like it to be. To clarify the matter, we shall consider two situations. For the first situation, let the original true preferences be

Person 1 Person 2

(1, 0) (0, 1) (0, 1) (1, 0) (1, 1) (1, 1) (0, 0) (0, 0).

which satisfies SSP, while person 2 erences:

Both individuals do not manifest self-supporting preferences. According to the Gaertner-Krª resolution scheme, the choice set is C (S) = {(1, 0), (0, 1)}. Now person 1 (2) is considering whether a compliance with property SSP can yield a better social outcome for him (her). Let person 1 have the following ordering

sticks to the original true preZ-

Person 1

(1, 0) (1, 1) (0, 0) (0, 1).

For the social choice set we obtain C(S)={(1,0)}. This is clearly better for individual 1, for we see that ({(1, 0)}, {(1, 0), (0, 1)}) ~/;1 holds. We can construct ah analogous case in favour of person 2 alone. Let him (her) have the following manipulated ordering:

Person 2

(0, 1) (1, 1) (0, 0) (1, 0).

This time person 1 is supposed to stick to the original true preSerences. The choice set now is C (S)={(0, 1)}, and we observe


that ({(0, 1)}, {(1, 0), (0, 1)})~/;2. This shows that strategic manipulation can be advantageous to the person considering such a move. It occurs in the sense that compliance with property SSP is documented to the outside world while the true preferences say something else. In our example, both individuals apparently prefer states of the world with the allocation of labour being quite diverse to states with an equal share of labour. A manifestation of SSP preferences can then guarantee to the manipulating individual a final outcome that is his (her) most preferred social state.

In game-theoretic terms, the social outcomes (1, 0) and (0, 1) respectively ate equilibrium points. Ir one of the individuals decides to announce SSP preferences, the other person has no reason to follow as long as he (she) is rational. If, however, the second individual also decides to become insincere and manifests SSP preferences, the social outcome is (1, 1), and this state is Pareto inferior according to the true preferences of both. This again shows that in order to be successful manipulating behaviour requires a lot of information on other people's preferences and intentions.

The second situation is based on our very first preference profile in the introductory section 18. Both persons violate property SSP. The choice set is C (S)={(0, 1), (1, 0), (0, 0)} according to the Gaertner-Krª scheme. Let individual 1 now take over the ordering from person 1 of the first situation. The choice set becomes C (S)= {(1, 0), (1, 1)}. Ir both individuals ate extremely risk- averse what we wish to postulate here lv, person 1 wiU prefer the second choice set to the first one. Symmetrically, if individual 2 picks the ordering of the second person in the first situation above, the choice set is C (S)--{(0, 1), (1, 1)} and again, under extreme risk aversion, there is a clear preference for the second choice set over the first by individual 2. In contrast to the first situation, the isolated move of each of the two individuals does not yield an equilibrium point. If one of the individuals starts revealing SSP preferences - - and he (she) will necessarily do that under extreme risk aversion, the other person will follow. The result is that both persons end up in an outcome which is Pareto inferior with respect to the true preferences (C (S)= {(1, 1)}).

Under H a m m o n d ' s approach individuals are socially decisive only when their preferences satisfy property UP (unconditionality of preferences). Furthermore the Pareto principle is only effective where

18 i owe the following observations to Friedrich Breyer. 1~ This assumption is essential for the arguments that follow. We use

the pay-offs from the matrix in the introduction.

96 W. Gaertner:

the individuals' preferences are "privately oriented". For individual

i, P~ is a privately oriented strict preference relation if there exists ^

a strict preference relation p~0 on X0 x X~ such that for every

x, y ~ X:

(x, y) ~/'~ ~ ((x0, xd, (yo, y~)) E p~o.

P~ of individual i is privately oriented if the private features of all

other individuals are ignored; P~ is equivalent to a strict preference ^

relation p~o just on the product space X0 x X~. Hammond's con- strained Pareto rule reads

Condition PPP (private Pareto principle). For all x, y e X,

(x,y) e n P~--~[xeS-~y~C(S)] i ~ N

for all S e 6 a.

The possibility result is then captured in the following

Theorem (Hammond). There exists a CCR which satisfies conditions U and PPP and also respects individual rights condition- ally, i .e. , individuals are socially decisive over private pairs only ir their preferences fu[fil property UP.

Let us have a last look at our introductory example. Both persons manifest unconditional preferences but the unanimous preference for (0, 0) over (1, 1) is not converted into a social preference under condition PPP. Both Sen-Suzumura and Hammond obtain a solution to the liberal paradox via a restriction of Pareto's rule. The problems which we encountered for the case of multiple order-extensions under the first scheme do not exist under Ham- mond's approach. Each individual can decide for himself (herself) whether it is worth complying with the rules.

References

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98 W. Gaertner: Interdependent Rights Exercising

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Address of author: Professor Dr. Wulf Gaertner, University of Osna- brª Department of Economics, P.O. Box 44 69, D4500 Osnabrª Federal Republic of Germany.

pareto, interdependent rights exercising and strategic behaviour

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