The Liberal Paradox: A Generalisation

The Liberal Paradox: A GeneralisationAuthor(s): D. KelseySource: Social Choice and Welfare, Vol. 1, No. 4 (1985), pp. 245-250Published by: SpringerStable URL: http://www.jstor.org/stable/41105784 .

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The Liberal Paradox: A Generalisation*

D. Kelsey Department of Economics, Southern Methodist University, Dallas, Texas 75275, USA

Received November 21, 1983/ Accepted May 17, 1984

Abstract. A largely unsolved problem in economics is what criteria should an economic policy satisfy to be desirable. The Pareto principle is the most widely used normative criterion in economics. Some recent criticisms have suggested that it is not compatible with other objectives of economic policy, most significantly that the Pareto principle conflicts with the respect of individual rights. This paper argues that the Pareto principle is not a significant cause of this conflict. Our argument is based upon a simple generalisation of the Liberal Paradox.

I. Introduction

In this paper we present some generalisations of the Liberal Paradox. We show that a similar result can be proved with the Pareto principle replaced by non-imposition. Since non-imposition is a strictly weaker condition, our result is stronger than the Liberal Paradox. We shall argue that it has significant implications for the way in which the Liberal Paradox is interpreted. One consequence of this, is that if we wish to reject the Pareto principle in order to preserve individual rights, we shall have to go further and require that some choices be imposed irrespective of the preferences of the individuals.

II. The Liberal Paradox and Welfarism

A collective choice rule (CCR), is a function / which maps an n-tuple (R1,R2,..., Rn) of orderings of X onto a complete and reflexive binary relation R defined on X.

R=/(R1,JR2,...,Rn) Here R and jRf denote the weak preference of society and individual i respectively.

Similarly P and Pt will be used to denote the corresponding strict preferences of society and individual i.

* For Comments I would like to thank Ruvin Gekker, Kevin Lang, Yew-Kwang Ng, Prasanta Pattanaik, Amartya Sen, John Wriglesworth, the referee, and the editor of this journal



246 D. Kelsey

If R is always acyclic then / will be said to be a social decision function. Similarly if jR is transitive / will be said to be a social welfare function.

Here X is the set of alternative social states, H is the set of individuals and n is the total number of individuals. Unrestricted domain, U, says that any logically possible n-tuple of orderings is in the domain of / Independence of irrelevant alternatives, I, says that the social preference over a pair (x, y) depends only upon individual preferences over (x,y). The weak Pareto principle, P, requires that if x is strictly preferred to y by all individuals in society, then society should strictly prefer x to y. An individual is said to be decisive over the pair of alternatives (x,y) if xPty =>xPy.

Sen [9] proposed the following condition as a weak version of the liberties we would like to give to individuals. Minimal liberalism is satisfied if there are at least two individuals who are decisive (both ways) over at least one pair of alternatives each. (These pairs are supposed to represent choices which are personal to the individuals concerned.) Despite the apparent weakness of minimal liberalism Sen showed that it is not compatible with the Pareto principle. His result is as follows.

Theorem 1. There is no social decision function which satisfies unrestricted domain, weak Pareto principle, and minimal liberalism. ■

This result has been used as an argument for rejecting the Pareto principle by Sen [10]. A second criticism he made of the Pareto principle is that in the presence of other commonly used conditions in social choice it implies welfarism. We define Welfarism to be satisfied when any pair of social states (x,y) are judged solely in terms of individuals' utilities in x and y. Thus, in particular, welfarism prohibits consideration of the nature of the alternatives x and y. So a welfarist rule would not allow an individual a special say in the choice over a pair of alternatives which are personal to him. It follows that welfarism is incompatible with even minimal liberalism. Sen shows further that when utilities are ordinal and not interpersonally comparable (as in the Arrow framework of social choice) welfarism prevents any consideration of poverty or inequality.

Sen [10] showed that when combined with unrestricted domain and independence of irrelevant alternatives that the Pareto principle implies strict-ranking welfarism. This is a slightly weaker condition than welfarism which says that when individuals' preferences are strict, alternatives should be judged only in terms of individuals' utilities. We shall show that in this result, as well as the Liberal Paradox, the Pareto principle can be replaced by non-imposition. We define non-imposition as follows.

Non- Imposition. For any pair (x,y) of alternatives there is a possible profile of individual preferences over (x,y) which makes x socially strictly preferred to y.1

Non-Imposition seems a reasonable axiom. If it is not satisfied then the social preference will be insensitive to individual preferences, a situation which is scarcely better than dictatorship. Also it does not appear to suffer from some of the criticism which Sen has made of the Pareto principle. Consider the following quote:

1 Non-imposition as defined here is slightly stronger than the condition used by Arrow in [1] as we have incorporated some elements of independence into the definition. However, non- imposition is usually used with independence of irrelevant alternatives, and since we do not use the latter condition our assumptions are actually weaker than those of Arrow



The Liberal Paradox 247

Welfarism asserts that non-utility information is, in general, unnecessary for social welfare judgments. Paretianism makes non-utility information unnecessary in the special case in which everyone's utility rankings coincide. (It also makes the social-welfare judgment mirror the unanimous individual rankings, which is an additional feature, but that does not, of course, affect the redundancy of the non-utility information.) If everyone has more utility from x than from y, then it does not matter what jc and y are like in any other respect: the Pareto principle will declare x to be socially better than y without inquiring further, ([10], p. 549).

Non-imposition does not have such properties. While it requires that for any pair of alternatives there is a profile which gives rise to a strict preference over that pair of alternatives, it does not require the profile to be a unanimous one. More significantly it allows different profiles for different alternatives; hence, non-imposition does not appear to contain those aspects of the Pareto principle to which Sen objects. Thus a choice rule which satisfies non-imposition can take account of non-utility information by having different choice procedures over different pairs of alternatives. Such a choice procedure can depend on the characteristics of the pair concerned.

Although Non-Imposition is a weak condition we find it useful to consider a still weaker condition.

Weak Non-Imposition. For any pair (x, y) of alternatives there is a possible profile of individual preferences over (x, y) such that x is preferred or indifferent to y.

In the following result we extend Theorem 1 by replacing the Pareto principle with non-imposition.

Theorem 2. There is no social decision function which satisfies unrestricted domain, non-imposition and minimal liberalism.

Proof. Minimal liberalism requires that there be at least two individuals, 1 and 2 say who are decisive over at least one pair of alternatives each, say (x, y) and (u, v) respectively. As in Sen's proof of Theorem 1 we have to consider three cases depending on whether x, y, u, v are all distinct, consist of three distinct alternatives, or consist of two distinct alternatives. We shall only examine the case where y = u and the alternatives are otherwise distinct. The other cases can be handled similarly.

1 2 xv y

V A y xv

By unrestricted domain we may consider the preferences shown in the diagram. (In the diagram, a downward line denotes strict preference.) By minimal liberalism we have xPy and yPv. The diagram is compatible with any set of individual preferences over x and v. By non-imposition we know that there is one set of preferences over this pair which will give rise to vPx. In this case the social preference will contain a cycle. Hence / is not a social decision function.2 ■

2 Indeed it is the case that liberalism is incompatible with many social choice rules which do not even satisfy weak non-imposition. The above proof would work equally well if it were imposed that ü were preferred to x. This point is explored further in [7]



248 D. Kelsey

If we strengthen the collective rationality condition to transitivity, we may weaken the Pareto principle still further to weak non-imposition. This is done in Theorem 3. The proof is very similar to that of Theorem 2, and so we omit it.

Theorem 3. There is no social welfare function satisfying unrestricted domain, weak non-imposition and minimal liberalism. ■

I would now like to examine the claim that the Pareto condition forces social welfare functions to be welfarist. Sen shows that when combined with unrestricted domain and independence the Pareto condition does indeed imply strict-ranking welfarism. Theorem 4 shows that the Pareto condition does not play an essential role in this proof.

Theorem 4. Any social welfare function which satisfies unrestricted domain, independence of irrelevant alternatives and weak non-imposition must also satisfy strict-ranking welfarism.

Proof Let (x, y) be any pair of alternatives. Suppose that Hx , H2 is a partition of the set, H, of individuals. Consider the case where members of Ht strictly prefer x to y while the members of H2 strictly prefer y to x. To establish that strict-ranking welfarism is satisfied we must show that for any other pair (a, b) of alternatives when the members of Hl strictly prefer a to b while the members of H2 strictly prefer b to a, that we have aPb, b Pa, or alb, depending on whether the social preference over (x,y) was xPy, yPx or xly.

There are a number of different cases to consider. For simplicity we shall only consider the case where a, b, x, and y are all distinct and the social preference over (x, y) was xPy. To prove our result it is sufficient to show that the social preference over (a,b) is a Pb.

Hi H2 [x,a] [y,b] [y,b] [x,a]

Consider the preferences shown. Here [x, a] denotes that we have not specified individual preferences over x and a. Thus in this profile the members of Hl prefer both of x and a to both of y and b. By our assumption we have xPy. From weak non-imposition we know that there are pairwise profiles over (a,x) and (y,b) which give rise to the social preferences aRx and y Rb respectively. In this case that individuals have these preferences we can deduce from transitivity the social preference aPb. Since we are assuming independence of irrelevant alternatives this is sufficient to establish strict- ranking welfarism in this case.

The conclusion is false if non-imposition is dropped from the assumptions. In that case an arbitrary imposed strict order is possible, (which does not satisfy strict-ranking welfarism).

As with Theorems 2 and 3 a version of Theorem 4 may be proved with the collective rationality condition relaxed to quasi-transitivity provided that full non- imposition is satisfied.



The Liberal Paradox 249

Theorem 5. Any quasi-transitive social decision function which satisfies unrestricted domain, independence of irrelevant alternatives and non-imposition must also satisfy strict-ranking welfarism. ■

III. Conclusion

There are several ways to interpret the results in this paper. It is possible to build a defense of the Pareto principle against Sen's criticisms based on these results. Most of Sen's arguments against the Pareto condition are based on the two results, that it conflicts with minimal liberalism, and that it implies strict-ranking-welfarism. We have shown that versions of both these results can be proved without invoking the Pareto condition. This suggests that such results may be a consequence more of the other conditions they use, than of the Pareto principle.

In support of this view it can be noted that Gibbard [3] has shown that rights and unrestricted domain can cause a conflict in the absence of the Pareto condition. (Admittedly Gibbard's rights condition is stronger than that of Sen.) Wilson [11] has shown that even in the absence of the Pareto condition, unrestricted domain and independence of irrelevant alternatives can precipitate an impossibility theorem similar to that of Arrow. (Note also that our Theorem 4 is a corollary of Wilson's results.) Further, a number of recent impossibility results in acyclic social choice have been proved which do not use the Pareto principle or related conditions (see Kelsey [4], [5], [6]).

A second way of interpreting our result is that it tells us which aspects of the Pareto principle cause the trouble in Sen's results. Monotonicity and non-imposition together imply the Pareto principle. Thus this result can be seen as showing that it is the non-imposition component of the Pareto principle rather than the monotonicity component which is responsible for the Liberal Paradox.

Many of Sen's objections to the Pareto principle appear to be directed against monotonicity rather than non-imposition. His argument, in the passage quoted earlier appears to be saying that monotonicity with respect to utilities can lead to neglect of other important factors. Our results show that, whether or not monotonicity is desirable on ethical grounds it is not a major cause of the Liberal Paradox.

Theorems 4 and 5 seem to suggest also that it is the non-imposition component of the Pareto principle which is responsible for the Arrow Impossibility theorem. If after considering the results presented here it is decided that non-imposition is a objectionable condition, then this will automatically imply rejection of the Pareto principle.

Another possibility is that it is necessary to reject independence. All existing impossibility theorems use some form of independence, either explicitly as an assumption or implicitly contained in the definitions of their other assumptions. Thus in the Liberal Paradox, there are elements of independence contained in both the liberalism condition and the Pareto principle. This view is in many ways a revival of some of the early responses to Arrow's Impossibility Theorem which argued that the problem could be solved by rejecting independence. In certainly provides a logically valid way out of nearly all the impossibility theorems of social choice, though whether this solution is acceptable is an open question. In this context it should be noted that some



250 D. Kelsey

authors have argued against non-independent choice rules on the grounds that such rules are susceptible to strategic manipulation by the individuals. (See, for instance, Pattanaik [8].)

In Theorems 2 and 3 we show that giving two individuals rights over one pair of alternatives each means that choice over at least one pair of alternatives fails to satisfy non-imposition. If more extensive rights are given, then it is clear that more choices will fail to satisfy non-imposition. In Dasgupta [2] an explicit formula is given for the number of choices which have to be imposed under Gibbard's rights system.

References

1. Arrow KJ (1963) Social choice and individual values. Yale University Press, New Haven 2. Dasgupta M (1984) Non-imposition and liberal paradox. Working Paper, Southern Meth-

odist University, Dallas, Texas 3. Gibbard A (1974) A Pareto-consistent hberatanan claim. J Econom Theory 7:399-410 4. Kelsey D (1981) The structure of social decision functions. Working Paper, SMU, Dallas 5. Kelsey D (1982) Acyclic choice without the Pareto principle. Rev of Econom Stud (to

appear) 6. Kelsey D (1983) Anonymous acyclic choice. Working Paper, Southern Methodist Univer-

sity, Dallas, Texas 7. Kelsey D (1983) The consistency of rights. Working Paper, Southern Methodist University,

Dallas, Texas ». fattanaik FK (iy/ö) Strategy ana group cnoice. iNortn-tionana, Amsterdam 9. Sen AK (1970) The impossibility of a Paretian liberal. J Political Econom 72: 152-157 10. Sen AK (1979) Personal utilities and public judgments or what's wrong with welfare

economics? Econom J 84:537-558 11. Wilson RB (1972) Social choice theory without the Pareto principle. J Econom Theory

5:14-20



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The Liberal Paradox: A Generalisation