DISCUSSION

NOZICK

ON

SEN:

A MISUNDERSTANDING

ABSTRACT. This discussion examines Robert Nozick's claim in Anarchy, State, and Utopia (New York 1974) that his entitlement theory of justice avoids the paradox of collective choice shown by A. K. Sen in Collective Choice and Social Welfare (San Francisco 1970). Noziek argues his system is a stable principle of distributive justice. The author shows Nozick's principle of justice in transfer qualifies as a social decision function in Sen's sense because it is a collective choice rule and meets :necessary and sufficient conditions for the existence of a choice function. Next the author demonstrates Nozick's principle of justice in transfer requires Sen's conditions of unrestricted domain, the Pareto principle, and 'liberalism' which are the conditions of the Sen paradox Nozick claims to avoid. Thus, Noziek's principle of justice in transfer is shown not to be a stable principle of distributive justice.

In Anarchy, State, and Utopia ( N e w Y o r k : Basic B o o k s 1 9 7 4 ) , R o b e r t N o z i c k offers an e n t i t l e m e n t t h e o r y o f justice w h i c h h e believes avoids s o m e o f t h e difficulties o f collective c h o i c e discovered b y A. K. Sen in Collective Choice and Social Welfare. 1 This p a p e r e x a m i n e s N o z i c k ' s claim a n d argues t h a t S e n ' s results a p p l y t o N o z i c k ' s p r i n c i p l e o f justice in t r a n s f e r , w h i c h is a n integral p a r t o f his e n t i t l e m e n t t h e o r y . N o z i c k states his e n t i t l e m e n t t h e o r y as follows: 1. A person who acquires a holding in accordance with the principle of justice in acquisition is entitled to that holding. 2. A person who acquires a holding in accordance with the principle of justice in transfer is entitled to that holding. 3. No one is entitled to a holding except by (repeated) applications of 1. and 2.. The complete principle of distributive justice would say simply that a distribution is just if everyone is entitled to the holdings they possess under the distribution. 2 N o z i c k distinguishes his p r i n c i p l e s f r o m t h o s e w h i c h h e calls ' p a t t e r n e d ' . A principle o f d i s t r i b u t i o n is p a t t e r n e d " i f it specifies t h a t a distr:ibution is t o vary a l o n g w i t h some n a t u r a l d i m e n s i o n , w e i g h t e d s u m o f n a t u r a l d i m e n s i o n s , o r l e x i c o g r a p h i c o r d e r i n g o f n a t u r a l d i m e n s i9o n s . ,,a He e x t e n d s " t h e use o f ' p a t t e r n ' t o i n c l u d e t h e overall designs p u t f o r t h b y c o m b i n a t i o n s o f e n d - s t a t e principles.'4 F r o m a discussion o f ' p a t t e r n e d ' principles o f d i s t r i b u t i v e j u s t i c e , N o z i c k concludes

Theory and Decision 8 (1977) 387-393. All Rights Reserved. Copyright 9 1977 by D. Reidel Publishing Company, Dordreeht-Holland.

388

DISCUSSION

Any distributive pattern with any egalitarian component is overturnable by the voluntary actions of individual persons over time; as is every patterned condition with sufficient content so as actually to have been proposed as presenting the central core of distributive justice, s Nozick states explicitly that his system is not 'patterned' even though it has heavy strands of pattern running through it. 6 He also says "any patterning either is unstable or is satisfied by the entitlement system. ''7 Nozick is thus arguing that his entitlement system is a stable principle of distributive justice. Nozick asserts his conclusions are supported by Sen's Chapters 6 and 6", 'The Liberal Paradox', in Collective Choice and Social Welfare. The essence of these chapters, and the result to which Nozick is appealing, is Sen's Theorem 6"1. Loosely stated, Sen proves there is no social decision function (SDF) covering all logically possible alternatives, which satisfied the Pareto principle (that if each individual prefers x to y , society prefers x to y), P, and satisfies also the condition of minimal liberalism, L*, which specifies certain personal choices. Nozick maintains that Sen's result supports his conclusions because he holds that under a patterned principle, rights determine a social ordering, while under his entitlement theory rights are a constraint on but do not determine a social ordering. Nozick believes his system does not treat "an individual's right to choose among alternatives as the right to determine the relative ordering of these alternatives within a social ordering. ''a The question of the 'right to choose' is important to what Nozick calls 'stability', which is, strictly speaking, the problem of the existence of a social decision function which will generate a social ordering. It will be shown that Sen's Theorem 6"1 does not bolster Nozick's conclusions about the stability of the entitlement system as a set of principles of justice, but that Sen's result must hold (at least) for an important part of Nozick's system as well, the principle of justice in transfer. The two principles can be considered separately if they are viewed as weakly lexically ordered: the principle of justice in transfer cannot be satisfied unless the principle of justice in acquisition has been previously satisfied. This seems quite reasonable; one cannot be entitled to transfer something to someone else unless one acquired it in accordance with the principle of justice in acquisition. Sen's conditions will be stated formally. It will then be argued that Nozick's principle of justice in transfer meets the conditions for a social decision function and that it must require each of the conditions for which

DISCUSSION

389

Sen's result holds. The binary relations 'at least as good as' (or 'no worse than'), 'preferred to', and 'indifferent between' are represented formally as x R y , x P y , and x l y respectively. When used to represent the preferences of individuals in society, the relations will be subscripted, x R i y , x Piy, or x liy. Without subscripts, x P y will always be taken to mean society prefersx t o y . 9 Sen defines a social decision function as follows: DEFINITION 4"1. A soical decision function (SDF) is a collective choice rule f, the range of which is restricted to those preference relations R, each of which generates a choice function C(S, R) over the whole set of alternativesX. 1~ To say there exists a choice function C(S, R) over X is equivalent to saying there is a best element in every non-empty subset o f X . n An element x can be called a 'best' ('greatest', in the context of size relations) element of S if it is at least as good (great) as every other element in S with respect to the relevant perference relation R. 12 Clearly, the choice function C(S, R) does not have to generate a unique best element. If the subset S is the pair ( x , y ) and x R y andy R x, then the choice set C(S,R)=S=(x,y):/:d). A social decision function is a way in which society chooses by rules between alternative states of the world. Since to be a SDF in Sen's sense a collective choice rule must merely generate a choice function C(S, R ) over X, it will suffice to show Nozick's principle of justice in transfer is a collective choice rule and meets necessary and sufficient conditions for the existence of a choice function. Sen defines a collective choice rule as follows: DEFINITION 2" 1. A collective choice rule is a functional relation f such that for any set of n individual orderings R1 . . . . . Rn (one ordering for each individual), one and only one social preference relation R is determined, R = f(R1 . . . . , R n ) . 13 This says that society's preferences (the social ordering) are a function of individual orderings and that each set of individual preferences will give one, and only one social ordering. The principle of justice in transfer clearly meets these requirements: if transfer is voluntary, a change in the outcome (social ordering) could only come about through the change in some individual's preferences which satisfies the latter requirement. Nozick concedes the social

390

DISCUSSION

ordering as a function of the individual orderings for the principle of justice in transfer where he says: Ignoring acquisition and rectification, we might say: From each according to what he chooses to do, to each according to what he makes for himself (perhaps with the contracted aid of others) and what others choose to do for him and choose to give him of what they've been given previously (under this maxim) and haven't yet expended or transferred. 1, Nozick's principle obviously requires that the relation of justice be reflexive. Any alternative x must be at least as good as itself (x R x) unless justice is equivocally predicated, which Nozick surely would not want to say. Nozick also requires that the relation be complete: V x, y E X : ( x ~ y ) ~ ( x

R y V- y R x )

(For all alternatives x and y in X, such that x :/:y, either x is at least as good as y o r y is at least as good as x or both.) If an individual, i, is entitled to have q he is entitled to choose how he disposes of q, if at all. Society permitting i to have this choice entails a choice between states of the world: x, where i has the choice and y , where i does not. Similarly with possession if i chooses not to dispose of q. If q is any arbitrary thing to which any individual i (from among the n individuals in society) is entitled, then completeness is required. Nozick requires a further condition of transitivity: (xRy

& yR z)~xR

z

For Nozick, any state resulting from legitimate transfers must be at least as good as a state in which everyone is entitled to their holdings under the principle of justice in acquisition. But this is precisely transitivity. A weaker form of transitivity is what Sen refers to as acyclicity: if ( x l P x ~ , x 2 P x s . . . . , X n - 1 P Xn}, then xl R Xn ) s For Nozick, if society strictly preferred (moved to by a series of voluntary transfers) xi over x2, etc., x n, then xl must be at least as good as x n. Acyclicity is implied by transitivity. The conditions of reflexivity, completeness and acyclicity which Nozick's principle of justice in transfer has been shown to meet, together with it having been shown to be a collective choice rule, are sufficient to show that the principle of justice must fall under the rubric of a SDF as defined by Sen. This is clear when we consider Sen's lemma 1"1 which will be given without proof:

DISCUSSION

391

LEMMA 1"1. If R is reflexive and complete, then a necessary and sufficient condition for C(S, R ) to be defined over a finite X is that R by acyclical over X . 16

Having shown Nozick's principle of justice in transfer is a SDF, Sen's theorem 6"1 and its conditions can now be formally stated. It will then be shown that Nozick's principle requires each of the conditions of the theorem. THEOREM 6* 1. There is no SDF satisfying conditions U, P, L*.17 Condition U = the domain of rule f includes all logically possible combinations of orderings. ~8 Condition P = (Vi: x Pi Y) ~ x P y.19 Condition L* = 3 individuals 1, k (j :~ k) and disjoint pairs of alternatives ( x , y ) and ( w , z ) such that ] & k are decisive over (x~y) and (w, z)respectively. (x Ply) ~ ( x / ' y ) and (z Pk w) ~ (z P w) and vice-versa). 2~ Condition L* is the condition of minimal liberalism. That Nozick's principle of justice in transfer must satisfy U seems clear: the choice function must be generated for any non-empty subset. If the choice were in an area Nozick would 'constrain', the choice set would contain more than one element, i.e. society would be indifferent between (x, y): x l y . In other words, if the choice were 'none of society's business' the choice set C(S, R) = (x, y) and both elements are at least as good as each other. It also seems clear that condition P must hold for the principle of justice in transfer: if (Vi: x Pi Y) "-'>x P y . If everyone wanted x to obtain instead of y, and both were feasible, the move from y to x would be voluntary and would, therefore, satisfy Nozick's transfer principle. He could not object to the move. The most interesting condition is minimal liberalism, because it is the one Nozick considers to cause the trouble by "treating the individual's right to choose among alternatives as the right to determine the relative ordering of these alternatives in a social ordering. ''21 Minimal liberalism, L*, is implied by the stronger condition of liberalism, L. L - for each i, B ( x , y ) E X such that x P i Y ~ x P y & y P i x ~ y P x . 22 It will be shown Nozick's principle of justice in transfer requires condition L, and that Sen's result for U, P, & L* holds also for a SDF satisfying

392

DISCUSSION

U, P, & L. Thus Nozick's principle of justice in transfer does not get around Sen's result. Consider the following situation: Everything else in society being I2, an individual, i, prefers using $25,000 to purchase a Ferrari Dino (which will be called alternative x) over using the money to finance a Ph.D. in philosophy (alternative y). Nozick would have to say this is a personal choice with which society must not interfere. Stated more formally, with everything else held fZ, i must be decisive between (x, y). But this means (x Pi Y) ~ x P y and O' P/x) ~ y P x. Since the individual, i, is chosen arbitrarily, and one can find at least one such choice for any i, one can generalize and say that for each individual there exists some pair of alternatives over which he is decisive. But this is the statement of condition L, liberalism. Nozick cannot avoid this by arguing the choice is outside the social ordering. The SDF must cover all possible states of the world and the alternatives x and y are distinctly different states of the world. Since society must either prefer one or the other state of the world or be indifferent between them, and society acts on its preferences, then if one of the states obtains, society must prefer it. That is, i f x in fact obtains between (x: given everything else remains the same) and (y: given everything else remains the same), then x Py. I f n o t , i f y P x - that is, society choosesy - and a contradiction results. Indifference does not help Nozick here because if ( V i : x R~y and 3j such that x PiY), then x P y. If all individuals fred x at least as good as y and there is one individual who strictly prefers x, then society cannot be indifferent. The social ordering is the way in which society chooses which states of the world will in fact exist. If society prefers one state over another, it will move to the preferred state. If everyone save one were indifferent between two states, and the one could move to his preferred state via voluntary means then he would, and it would in fact exist. Society must prefer that state or move to another state. The example discussed above shows the individual's right to dispose of that to which he is entitled in accordance with Nozick's principle of justice in transfer is in fact a choice between alternative social states of the world, with everything else held the same. This has been shown to be equivalent to Sen's statement of condition L, liberalism. Nozick's principle of justice in transfer is thus subject to Sen's corollary 6"1.1 which states:

DISCUSSION

393

COROLLARY 6"1.1 There is no SDF satisfying conditions U , P , L . zs Nozick is thusclearly mistaken in maintaining that his principle of justice in transfer avoids the problems Sen's result poses for theories of distributive justice. Although the result obtained in this paper against Nozick's claim for the principle of justice in transfer does not directly affect his principle of justice in acquisition or his principle o f justice in rectification, one seriously wonders what is left of the claims for the unique stability of the entitlement theory of distributive justice without the principle of justice in transfer. C. R. PERELLI-MINETTI

Department of Ecomonics U n i v e r s i t y o f California a t S a n t a Barbara

NOTES i A.K. Sen, Collective Choice and Social Welfare (San Francisco, 1970). Robert Nozick, Anarchy, State, and Utopia (New York, 1974), p. 151. 3 1bid, p. 156. 4 1bid p. 156. s Ibid p. 164. Ibid p. 157. 7 Ibid p. 164. s Ibid p. 165. Sen op. cir., Ch. 2*. to Ibid p. 52. 11 Ibid p. 14. 12 Ibid p. 10. 13 Ibid p. 28. 14 Nozick, op. cit., p. 160. Sen, op. cit., p. 15. 1~ Ibid, p. 16. 1~ IbM, p. 87. Is Ibid, p. 41. 19 Ibid, p. 41. 2o Ibid, p. 87. 21 Nozick, op. cit., p. 165. Sen, op. cit., p. 87. Ibid, p. 88.