An impossibility theorem with von Neumann-Morgenstern preferences

economics letters Economics Letters 56 (1997) 305-309

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An impossibility theorem with von Neumann-Morgenstern preferences Amrita Dhillon a, Jean-Franqois Mertens b'* aDepartment of Economics, Warwick University, Coventry CV4 7AL, UK bCORE, 34 Voie du Roman-Pays, B-1348 Louvain-la-Neuve, Belgium Received 23 April 1997; accepted 2 June 1997

Abstract We prove an impossibility theorem in the context of VNM preferences when Pareto and an analogue of Maskin's Monotonicity are used. We show the crucial weakening required to avoid this. © 1997 Elsevier Science S.A. Keywords: Axiomatisation; Social welfare function; Utilitarianism; Arrow's Impossibility Theorem; Expected utility JEL classification: D70; D71; D63; C71

I. I n t r o d u c t i o n I shall describe problems of this kind as conceptual or philosophical... They also arise in economics and in the other social sciences... Examples are.., how to define a social welfare function that would adequately reflect our intuitive notion of public interest... (Harsanyi, 1976, p. 63). Arrow's impossibility theorem provoked among others the question whether the theorem would continue to hold when information on preferences is tightened, e.g. by considering more restricted domains where preferences (individual and or social) are assumed to be cardinal. Thus Sen (1970) proved A r r o w ' s theorem when individual preferences are cardinal and non-comparable. Kala'i and Schmeidler (1977) proved in addition that A r r o w ' s theorem still held if individual and social preferences are assumed to be cardinal if a continuity axiom is added (however, Hylland, 1980, showed that in fact continuity is not needed). Chichilnisky (1985) shows that there is no aggregation function with cardinal domain and range which is anonymous, unanimous and continuous in a technical sense, that excludes the possibility of complete indifference for society. In this paper we narrow the focus even more to a framework where domain and range are preferences over lotteries satisfying the von Neumann-Morgenstern ( V N M ) axioms. We show in such a f r a m e w o r k two possible interpretations of A r r o w ' s Positive Association of Social and Individual Values. The first, 'rankmonotonicity', is the analogue for a social welfare function ( S W F ) o f the usual monotonicity axiom in *Corresponding author. Tel.: (32-10) 474309; fax: (32-10) 474301; e-mail: [email protected] 0165-1765/97/$17.00 © 1997 Elsevier Science S.A. All rights reserved. PI1 S0165-1765(97)00191-2

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implementation theory (Maskin, 1985). However, while Arrow (1963) derived the Pareto axiom from Positive Association, Independence of Irrelevant Alternatives (IIA) and Citizens' Sovereignty, we show that rank-monotonicity contradicts Pareto! So the result is strongly reminiscent of the wellknown result that there is no monotone social choice function (Moulin, 1988, Chap. 10). The axiom is of the type that if the 'rank' of p (in terms of the less preferred set) increases for all individuals, it does also for society. If one expresses a similar idea, that social preferences respond in a positive way to changes in individual preferences, but in terms of the more primitive concept of the preference relation itself, between pairs of lotteries, in particular, if we look at society's response to a change in preference between two lotteries p and q, we obtain the other interpretation of Positive Association, which does not contradict Pareto, and is the one leading to the monotonicity axiom in Dhillon and Mertens (1996). Section 2 proves those equivalences of Positive Association, and Section 3 the impossibility theorem.

2. Monotonicity and positive association As stated, the 'Positive Association' condition is meaningless in the context of lotteries. Indeed it requires that, if an alternative goes up in every individual's ranking, in the sense that both its weakly preferred set and its strictly preferred set decrease weakly, while his ranking over all other alternatives remains fixed, then every alternative which was socially strictly less preferred before, remains so. It is impossible to increase the rank of a single lottery while keeping the ordering of all other lotteries fixed. The condition was formulated in a framework of IIA and universal domain. In such a framework, we have the equivalent formulations, paraphrasing (Arrow, 1963, top of p. 97):

Proposition: For a SWF in Arrow's framework, satisfying IIA, the following are equivalent: (a) Positive Association: If an alternative a ~ A increases in every individual's ranking, in the sense that the preferences over C{a} remain unchanged and that both sets {bEAlb<__a} and {b E A[b - b. Then if initially one had, for the social preferences, a_~ b, one ha"s also a _>-*b after the change. n

n

,

.

Proof. ( a ) ~ (c). Using IIA, we can, without modifying the social preferences between a and b, change player n's preference orderings <__ and <_ * to corresponding orderings where a and b come on top of h~s list--without changing their order--and where the rest of the hst is put into some fixed order. If one has a < * b after the change, then by IIA n's preferences between a and b have changed, hence we have either a ___ b and a > * b or a-
.

.

.

.

n

tl

tl

n

n

n

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A. Dhillon, J. Mertens / Economics Letters 56 (1997) 305-309

going from _ * to ___ so positive association yields that if a<*b then a-b~a>- b. (c)~(b). Suffices to change a single individual n's preferences. If b <__a ~ b - < *a and b < a ~ b < *a were false, the change in social preferences between a and b implies by IIA that either a - b. Hence (c) yields a ~ b ~ a > - * b , our first relation. Similarly permuting 'a' with 'b' n and _ with <__* we obtain from (c) that b >- *a ~ b >--a, i.e. a ~ b ~ a > *b, our second relation. ( b ) ~ ( a ) is obvious. In Section 3 we will show that Rank-Monotonicity is not the right interpretation.

3. An impossibility theorem We assume individuals as well as society have VNM preferences over the set AA of all lotteries on the finite set of alternatives A. Thus a SWF is a map ~ from (all) profiles (N-tuples) of such individual preferences to such a social preference. For any preference relation <_, p - q stands for p<_q and q<_p, and p < q for p-
~o[(_),eu] by -<. n Pareto Axiom (PA): If p-
and q < p ~ q _ ~ * p , then Vq, q < p ~ q < _ * p .

Theorem. If #A-----#N+ 2, and # N - - 2 , there is no SWF satisfying PA and R M - - e v e n with complete indifference removed from the domain. Proof. Assume u k* =u k V k # n , and u*(b)=u.(b) Vbv~a, and u*(a) 0 by PA (Harsanyi, 1976, Chap. 2)] and U* ( = E . A.* u.* ) be the corresponding social utility functions. Then, any lottery p with p(a)= 0 satisfies the assumption of R M , so:

(U, p - q) > 0 ~ (U *, p - q) -->0 Vq, Vp: p(a) = O. Writing x for p - q

we get that

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x(b) = O, x(a) <- 0 ~ [(U,x) > 0 ~ (U *, x) -> 0] b~=A

i.e., the system Zx(b)=0, x(a)--<0, (U,x)>0, ( U * , x ) < 0 is contradictory. By Farkas' lemma, this means that 30--->0, ~----0, 0"+7->0, o'U *(b) - TU(b) = p Vb ~ a and o'U *(a) - 7"U(a) <- p So E. (0-A*~-~'A.)u.(b)=p V b ~ a . Letting y = 0 - A n -~-A., yo = - p , we get

Yo + ~

y,u,(b) = 0 V b # a .

ii

Hence, if the matrix of coefficients has rank N + 1, i.e., if {(un,p)n~Nlp(a ) =0} is full dimensional in •N the only solution is o-A* =~-A, Vn, and thus 0->0, r > 0 since / ~ n, > 0 and A,>0. So for # A > - - # N + 2 , consider a profile E with u l ( a ) > 0 V a E A and such that, V a E A , the set {{u,, p),EN[p(a)=O} is full dimensional. Find 8 0 > 0 sufficiently small such that, V a E A , V(Sb)b,, a with 0-----8b<--8o, the convex hull of {[(1- 8b)Ul(b ), (u,(b)),~,l]lb~a } is still full dimensional. • i i Let A={a~ ...ak}. Define profiles if' ( i = 0 . . . k ) by u , = u n for n ~ l , ul(a~)=(1-8o)Ul(aj) for i i i i j<-i, ul(aj)=u~(ai) for j > i . Let U = E , A,u, be corresponding social utility functions. Then E~ relates to E i-1 as E* does to E in the beginning of this proof. So 3/.ti>0 (/z,. = ~) such that i i-1 k 0 • A~=tz~A, Vn. Hence A =/zA, Vn, w i t h / z = I - I ~ / z ~ > 0 . Thus

luk

Ao(1- - 8o).1 + E

o

~nUn

.

n~l

But the profiles / ~ k and Eo represent the same preferences, hence social preferences should be the same--which means that U ° = A 0~u 1 + Y~m,,1 amUm o represents correctly the social preferences for u~k. By full-dimensionality, any other vector A representing those social preferences should be proportional--contradicting our finding that ( 1 - 8 o ) A °, A° . . . . . A° was also such a vector.

Acknowledgements

This text presents research results of the Belgian Programme on Interuniversity Poles of Attraction initiated by the Belgian State, Prime Minister's Office, Science Policy Programming. It was partly supported by NSF Grant # S E S 8922610. The authors assume scientific responsibility.

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References Arrow, K.J., 1963. Social Choice and Individual Values, Vol. 12 of Cowles Foundation Monographs, 2nd ed., John Wiley, New York. Chichilnisky, G., 1985. von Neumann-Morgenstern Utilities and Cardinal Preferences. Mathematics of Operations Research 10(4). Dhillon, A., Mertens, J.-F., 1996. Relative Utilitarianism, An Improved Axiomatisation, Discussion Paper 9655, CORE. Harsanyi, J.C., 1976. Essays on Ethics, Social Behavior, and Scientific Explanation, No. 12 in Theory and Decision Library, D. Reidel Publishing, Dordrecht, The Netherlands. Hylland, A., 1980. Aggregation procedure for cardinal preferences: A comment. Econometrica 48(2). Kalai, E., Schmeidler, D., 1977. Aggregation procedure for cardinal preferences: A formulation and proof of Samuelson's Conjecture. Econometrica 45 (6), 1431-1438. Maskin, E., 1985. The theory of implementation in Nash equilibrium. In: Hurwicz, L., Schmeidler, D., Sonnenschein, H. (Eds.), Social Goals and Social Organization Essays in Memory of Elisha Pazner, Cambridge University Press, pp. 173-204. Moulin, H., 1988. Axioms of cooperative decision making, No. 15 in Econometric Society Monographs, Cambridge University Press. Sen, A.K., 1970. Interpersonal aggregation and partial comparability. Econometrica 38 (3), 393-409.