Generic properties of simple Bergson-Samuelson welfare functions

Journal of Mathematical Economics 7 (1980) 175-192. 0 North-Holland Publishing Company

GENERIC PROPERTIES BERGSON-SAMUELSON WELF Norman

*

SCHOFIELD

University cf Essex, Colchester CO4 3SQ, UK

Received May 1977, final version received May 1979 Methods of transversality theory are introduced to determine the generic properties of a simple non-collegial preference function a, whose domain is the class of ordinal utility profiles on a smooth manifold W of policy alternatives. When the dimension of W is suffkiently large, it is shown that (i) the set of local optima of u is generically empty, and (ii) the local cycle set of G is generically dense.

1. Introduction ‘Arrowian’ social choice is concerned with the existence of a social welfare or preference function which preserves certain rationality properties of preference and satisfies the normative criterion of Pareto optimality. The stronger the rationality requirement, the more concentrated must power be. For example, if strong transitivity (i.e., transitivity of both strict preference and indifference) is to be preserved, tlhen there will be a dictator [Arrow (195111; if strict preference only is to be transitive then there will be an oligarchy [Hansson (1972)]; if strict preference is to be acyclic then there is a veto group or collegium [ For these results, prefe nted by a binary relation on the set of all conceivable states. F’urthermore the theorems assume as domain of the function the set of all preferences, which satisfy the appropriate rationality property. This is a rather strong assumption and makes interpretation of the results somew lat difficult. For example, it is quite conceivable that a particular social decision process would preserve a particular rationality property for all configurations of individual preference, other than a certain subset of preferens-- whic_h is smaE1, in some sense, relative to the set of all possible config\rrations. This perhaps is the motivation behind the work of Sen and Pattanaik (1969) ;2nd Inada (1969), elaborating the notion of singlepeaked preference configurations and showing that simple majority rule preserved transitivity on this set. *This material is based upon grant no. SOC-77-21651.

work supported

by the National

Science Foundation

under

176

N. Schofield,

Bergson-Smtueht

we!fart

_functions

If the domain of preference configurations may be given a topology, then a ‘solution’ to the Arrow problem would be to show the existence of a welfare function which was responsive (perhaps non-dictatorial or non-oligarchic), Paretian, and preserved some form of rationality on a dense set in its domain. A natural way to assign a topology to the domain is to assume that preferences are determined by smooth utility functions on a space of smooth alternatives, W (either Euclidean space or a smooth manifold). In this context we seek the existence of a Bergson-Sam&on we&re function, CL Various impossibility results have been obtained in this direction. With certain diversity assumptions on the preference configuration, Hammond (i976) and Parks (1976) have shown that the welfare function will be dictatorial if it preserves strong transitivity. On the other hand, Kemp and Ng (1976) have shown that, if the welfare function is neutral (i.e., roughly speaking, all alternatives are treated identically, but see below), W is of sufficiently high dimension and preferences are suitably diverse, the continuity of social preference must fail. In a most elegant result, Chichilnis’hy (1977) has shown that there is no welfare function o with domain smooth preferenceb i)n W (of at least two dimensions) which is continuous, Paretian, and anonymous (i.e., treats individuals alike). In all these results it is assumed that strong transitivity is preserved, although in Chichilnisky’s result this is phrased in terms of the requirement that I? maps smooth preferences (induced from utilities) to a smooth social preference (induced from a social utility). This rationality requirement may be regarded in terms of a social preference o(p) and ‘social utility’ function u,(p), where p is the preference configuration, satisfying xa(p)~~~,tp)(-~)>u,(P)(4!)

for any _u,J*in M!

These results indicate that this rationality requirement is too severe. A more reasonable requirement would be to seek a social preference o(p) compatible with some social utility function, 1.4,(p),i.e.,

In this paper we seek to determine whether an U*JYG social preference function 0 exists (since if acyclicity fails, there can be no compatible social utility function). Secondly the above results use certain diversity assumptions (which do appear reasonable nonetheless) or an unrestricted domain assumption. Instead we shall be concerned with the generic properties of a preference function (i.e., properties which hold for a dense set in the domain of the function).

177

We shall, however, assume that the preference function is simple (i.e., neutral crnd responsive in a certain sense). We shall show that a preference function which is simple and non-collegial will getzericnlly fail acyciicity when h dimension. One response to results is that consistency (i.e., strong a requirerrznt. Instead all one of the existence of natural equilibria (the core) as in modern economic theory [Arrow and Hahn (1971), Hildenbrand and Kirman (1976)3. We shall show that for simple non-colfegial preference functions the core is also empty generically, gi-den a dimensionality condition. 2. Social preference functions In this section we give a brief resumi of those aspects of social choice theory which are of relevance here. Dejitt it ion 2.1.

(i) A strict pr&rertce relation on W which is: irrq\?exit:e:

spx for n’o s in W, and

asymmetric:

xpJ*noa

(ypx),

p on a set of alternatives

W is a binary

any .u,y in WI

(ii) A pro#ik for a society N = (1,. . ., 11; is a list (pIa.. ., p,,) of strict preferences, bne to each member of N. (iii) Let P represent the class of strict preferences [or B(W) when attention is to be drawn to W]. and BS, or B(W)“, represent the class of profiles. (iv) A strict preference is w\~*lic iff for any sequence (x1.. . .,.v,: in M/t Sip-Xi+1 for i= I,.... r-- 1-not ~s,p.u, ). If p fails acyclicity then it is called cyclic. Dejinit iort 2.2.

A social preference function (SF) is a function

with the re _triction property (I*): Let I/c V’c W, and let p be a profile defined on I”, with restriction p! I/’ to v: If ,7:B( v’)*V--‘+3( I/‘), then @)lV =fl(P!I/‘) defines

N. Schofield, Bergson-Samuelson welfare functions

178

Definition 2.3.

(19

l

l

(i) An SF, 6, is anonymous

iff for any permutation

5 of

do,

(ii) Let p be a profile. Write xpM!?, for M c N, iff xpiy for all i E M. A set M cN is calEed a co&ion. (iii) Write xa, _Viff, any p E BN,

CxPn,tY--xa (P)Yl* If xflMy then M is said to be decisive on (xJ}. The set of coalitions which are decisive on (x, y} is written ~,(x,J). (iv) Write xc+(p)y iff xpMy and xoMy6 Obviously xc&(p)y, some M C N, *xo(p)y. (v) An SF is called decisitle iff

xb (p)Y-xoM (p)y

for some

M c N.

(vi) A coalition A4 is called decisive iff M is decisive on {x, v} for any x, r in M! Let gd be the class of decisive coalitions. An SF is called simple iff xo(p)yexcM(p)y

for some

M E gO’

[vii) An SF is called neutral iff for any two profiles p, q and any two pair sets {x, y), (MI, 2) : p on {x,y)=q *a(p)

on (x,y}=o(q)

on {WJ) on {WJ}.

(viii) An SF, 6, is collegial iff A,,. . ., A, all belong to .C@#, then A, n . . - n A, #@ The intersection. of all decisive coalitions, if non-empty, is called the collegium. If there is some finite sequence A,, . . ., A, all belonging to CBb,with empty intersection, then G is called non-collegial. (ix) A simple anonymous SF, 0, is called a q-game, if there is some integer qs YIsuch that any coalition M with ] M 1zq belongs to 9,. Write bp for a qgame. Rote that D is neutral and decisive iff it is simple. Any q-game with q
N. SchqJk&l,

Bergson- -Sumuelson

weljim

jimctions

179

made of the unrestrict,ed domain assumption. Consequently these results give no indication of the nature of the failure of acyclicity for a non-collegial SF. Secondly the results do not address the question of the existence of the core. For preferences induced frolm utilities the two questions can be dealt with in terms of generic properties. Let W be a smooth w-dimensional manifold (i.e., locally Euclidean with smooth coordinate transformations). A smooth utility profile for N on W is a smooth func ;PQsA

where INI = n and each tli: W+R is smooth and represents i’s preferences. Here if pi is i’s strict preference, then tdirepresents pi in case

If tl! and u: both represent i’s preference, then we may write u! -u,‘. If u’ and li2 are utility profiles such that u! -u’ for each iE N, then write ~1~-u’. A smooth prejkrence profile for N in W is a class of smooth utility profiles under this equivalence relation. Let u(VV)N be the space of smooth utility profiles endowed with the Whitney P-topology. Under this topology two profiles z11,u2are close if all derivatives dk(~I@-l), dk(uz$-*) are close, where &V-+R”’ is a coordinate chart [see Hirsch (1976, p. 35) or Golubitsky and Guillemin (1973, p. 43) for more details particularly for the definition of this topology when W is not compact]. A subset F of a topological space V is called ~vsich~d if it is the countable intersection of open dense subsets of K The space V is a Bairn spcrrv if every residual set is dense. Under the Whitney C” -topologjp, [I lw)” is a Baire space. A property K which can be satisfied by a utiiity profile IIE U (IV)” is called generic if the set {u satisfies K) is residual in U c WJ”. Since I/ ( W)” is a Baire space, a generic property is satisfied by a dense set in U ( W )“. Moreover if W is compact, lhen a generic property is open dense. For 11E u(w)” we shall write u,+, for the profile for &I induced by II. Obviously u-+ (UM, UN _ M \ induces an isomorphism

wwN “I.‘(W)”

x

U(W)‘-?

For this reason we shall call a property K, which c: ri he satisfied by a subprofile UM,generir for U ( W)N if it is generic for U ( A Bergson-Samuelson Preference Function (BSF) is ::+fur ct.ion (T:6/ (IV)” -+B(W) with the property that b(ul)=~(uz) whentiver tdp wti2, A property K JME-

C

which can be satisfied by Q(U), UE LJ(IJV)~, will be called generic for O, whenever (24E U ( W)N : c(u) satisfies K) is residual. Alternatively one could write P(W ? for the space U(W)” under the equivalence relation m . P(W)N may be thought of as the space of nreplicates of unit vector fields on W [See ;r hichilnisky (1976) for appropriate topologies on P(W)N.] It is not clear as jet that Abraham-Thorn transversality theory [Abraham and Robbin (196 I)] can be used directly in P( IJV)~, though Chichilnisky’s results seem to indicate that this can be done. In this paper we use the term generic to refer to properties of residual sets in U( w)N.’ In the following let CJrefer to a BSF Iwith domain U( W)N and image in B(W), and use 9 to represent the set of decisive coalitions of CL Dejinitiorz 2.4. (i) The globat cycle set GC’(q w N, u) of tI(ii), for WEU ( W)N, is the set of points {XE W:3aa(u)

cycle through x.1.

Thus x E GC@, Jib/: IV,u) iff there is some sequence {s,,

XjO(U)Xj+*,

. . ., s,;

with

j=l,.**,r-l,

x,0 (u )x. (ii) The cme or glohcd optima set GO@, W, N, u) of a(u), for UE Us,

is

the set

{XE w:gJT

Wst.

~‘cr(lf)X).

(iii) The global optimal set jhr M, GO( W, M, to, for UE V( W)N, where M is a coalition, is the set

Note that if CJis simple, then GO@, WA&u)=

n

hfE9

GO(W M,u).

N. .ichof~dd. Burgsort St~mut~~.w~n wlftrre

181

jirrtt-tions

If W,fV are fixed then we shall write without ambiguity GC(a,u), GO@, u), Our results below may be interpreted as showing that O@, u) is generically empty and GC(G, U) is generically dense. First of 8’ j 42 ive infinitesimal analogues of GO and GC, follo\ving Smale’s ( 1973) ideas. Suppose Ui: W-4 is a utility function for i. The derivative of a smooth curve u:( -- 1. + I)-* W at c*(Q) = .Y,we shall write as [C](X) a tangent vector in T, W! If dt~i(*Y)([c](S))>O, then the curve L’is (infinitesimally) optimizing for i at X. Define or even GC,GO.

to be the (infinitesimal) i-preference cone at s. If a smooth curve (1:( - 1, + 1) -+ W satisfies [c] (Z)E Hi(Z) for all z = c( t ), t E [0, 1) then call L’an i-preference curve from y=c(O) to .u=lim,,, c(t). For a fixed ui, define i’s (infinitesimal) preference pi by X/Iiy whenever there is’ an i-preference curve from y to x. For a utility profile u:W-+R” and coalition M, there is a natural Mpreference relation defined as follows. Let xp,,,~ iff the smooth curve c:( - 1, -I-1)-M satisfies [C](Z)EH,(Z)=

n

ieM

Hi(3)

for all

tE[O,

Z=C(l).

1).

Let sfkkr_r iff xy,. No*for some smooth curve LB. Dejhit

ion

If E U(W)“,

2.5. M

(i) The injhitesirnd a coalition, is

opthi

set

for

AA IO ( W, M. zr ), for

(XE w:H,(x)=qq. (ii) For a simple BSF, CT,define the kjinitesimcrl optima to be n

IO(W

set IO@,

W N, 14)

(0

Mdf).

ME 5’

(iii) For a simple BSF, 0, define the local cwl~~ set LCb, m; N, U) as # follows : x,

ELC(0, W,N,u)

iff for any open neighborhood xj+ 1

PCj*MjXi

V of x1, there is a sequence

for ”j=l,...,T-1,

{x2,. . .,?e,i s.t.

182

and

x 1 PC,.MrX,, where each M,, . . ., A$, . . ., M, belongs to 9, and each path Cj from Xj to Xi+ r, and the path c, from X, to x1, belongs to V Note that for a simple BSF, B, if ul, u2 are utility profiles satisfying u1 - u2, then IO(a,

w,N, 22) = IO@,

K N, u2),

and LC(s, W,N, 24’) ==LC(0, w: N, u2). Secondly, GC(a, W,N,u)cLC(o,

WJV,u)

for all

UE U(W)?

Note also that GO@, WJV, u&10(0, W;N, u) except possibly for points on the boundary 2 W of W However points in d W n GO seem ‘artificial’ equilibria and will be ignored. We may now state the theorems of this paper: Theorem 1. Let c be a simple, non-collegial BSF, J: U ( W)N43( there is an integer w(c) 5 n - 1, such that (i) if‘dim WOW, then

W). Then

(UE U( W)N:IO(~, W,N,u)=Q)} is residual, mzd (ii) if’dim W 3>w (a), the/l {uEU(W)~:LC(~,~N,U)

is dense;

is residual. Theorem 2.

If GTis a y-game, n/2
Discussion. To illustrate these results consider a voting game with rz=3 and LI= 2. By Theorem 2(i), IO@, W,u) is generically empty whenever dim IV>=2, and LC@ w U) is generically dense whenever dim Wz 3. For a voting game with n =4 and y = 3, dim IV2 3 (respectively dim Wz4) is sufficient for the generic emptiness of IO@, w u ), and the generic denseness of LC(a, W u). For simple majority rule ok+ 1, results by Plott (1967) and Matthews (1978) can be used to show that, for n(odd)= 2k + 1, then w(ok+ i)= 2, and for tl (even) ==2 k, then w (ck + 1 ) = 3 [Schofield (1978a)]. For example, suppose n = 5, 4 = 3, and dim W= 2. In general the core IO(a) l-C(a) need not be dense, it will generally be non-

N. Schofield, Bergson- Samuelson welJurefunctions

183

empty. Fig. 1, for example, presents a five-person situation where each individual has Euclidean preferences [ McKelvey (1976)] and the critical point of Ui is represented by IO(i). We show below that LC((T~) is the five pointed star. While the cycle set may be fairly small for large n in a simple majority game in a two-dimensional policy space, it will be dense when dim W is at -t 3 (or 4 when n is even). This observation extends a result

The direction gradtentsat x

The direction d"2

dU3

gradien:sat y

by Kramer (1973) which indicated that majority rule could be ‘badly behaved’. For weighted majority games or non-collegi,al simple preference functions, Theorem 1 indicates that with sufficient dimensions, not uniy is cyclicity a general feature, but the infinitesimal optima set, and thus the core (other than on the boundary of the policy space), is also empty. While we have taken dim IV>, n- 1, for a general simple non-collegial BSF, 6, it is clear that for a given such g, a dimension counting procedure can be used to obtain a precise estimate for w(a). The results indicate clearly that the strong ‘consistency’ properties implied by the existence of a Bergson-Samuelson preference Function can hardly ever be satisfied by non-collegial voting games when the decision problem is complex :md of high dimensionality. It also seems plausible that similar resuhs can be obtained for more general social preference functions.

3. Proof of the theorems

We make use of the following singularity theorems Golubitsky and Guillemin (1973) or Levine (1971)-j.

[see, for example,

Definition 3.1. For a coahtion M of size m, tV a smooth manifold of dimension IV,and u E U(W)“, define (i) the rth singularity set of (M, u) to be S,(M,u)== {XEW:rankdu(x)=z-rj. Here z=min(m, w) and du(x): rXkV-+Rmmay be regarded as a linear map R”’ -+R” whose rank is the dimension of its image. (ii) Let

(iii) Define /1(M,u)=Ix:(du,(~))~,~

are linearly dependent].

Note that if wzm, then S(M,u)=A(M,u). (iv) Let L(M,u)=VqA(A4,u). Singulurity

Theorem

A.

Generically

dimension s where s+(w-z++)(m-z+r)=w.

S,(M,u)

is a subnm@fii!d of W of

IV. Schofdd, Bergson-Samuelson wevare functions

185

P roo$ I.,et J’ (H(R*) be the jet space (i.e., space of l-jets or first derivatives of smooth functions kV+R”, with the appropriate Whitney topology). Let S, be the universal singularity set in J’ (r/t: R”) of l-jets of corank r. Fol. f.&r(W)M,

j’ (u): w-4

(lq R”)

assigns to XE CYthe l-jes of u at x. By Thorn transversality, (24~~ U(W)5j1(u) is transverse to S,)

is residual. Since

and S, is of codimension generically a submanifold Q.E.D.

{w- z + r)( m - z-0) in J’(I/t:R*ii, S,(M,u) is of W of codimension (w - z -t r)(cn - 2 + r).

Conzme~lt. For example suppose dimension

.

u72 171.Then

S, (M, 14) is penericahy

of

Moreover, S(M, u) is a stratified set and consists of S, (M,tr) together with the lower-dimensional but higher-corank singularities [Levine (196411. If IV > 2r - m, then some dimension counting, shotvs that generically S, (M, I() = Q, for all rg2. Our interest in the singularity set S(M,u) is that it ‘contains’ IO(M,U).

At points in JO(M,u), H&X)=@ For points on the boundary, dr/t: of W it may be the case that H&x) =$J, but such a situation is ‘accidental’ and will be ignored below. For x in the interior of H$ it is well known [Smale ( 1973)] that x E PO(M, u) iff there is a semipositive solution to the equation

Here serltipositive means that each ELi ~0 but not all are zero. Thus IO(b1, u) c A(iV+). If wznz, then IO(M,u)cS(M,u), since S(M,u) is precisely the set of points such that rank du(x) c nj. Indeed if HZ> 2~1-4 then IO(M,111 c S,(M,u) generically. As above, S(M, u) gpznerically has the structure of a stratified set, i.e., an (m- 1)-dimensional manifold with lower-dimensional strata. This structure will be inherited by , u) under various conditions.

For example in a pure exchange economy, with W an unrestricted commodity space where the utility functions have no critical points, IO(w U) will be a submanifold of W [Kand (19X), Smale (197411. Since we are only interested in the upper bound for the ‘dimension’ of IO(M, u) we will refer to IO(M, U) as a ‘submanifold’ of VWof dimension (at most) m - 1. Singularity Theorem B.

If

M”, M’dV, M’ n M2=&

then

is residual. Here u Mi is the restriction ofu

to

U

(W)Mi and mi = ] M’I.

Proof 0f Singularity Theorem B. IO(M’,u,i) and iO(M”,rt,~) will generically be closed ‘submanifolds’ of W of dimension at most m, - 1, m, - 1, respectively. Consequently IO( M’, u,@) will intersect IO( M2, ~~2) transversally with codimension at least w -- ~1, + 1 in 10(M2, ~~2). So if non-empty, the intersection will generically be of dimension at most m, + m2 - w - 2. Q.E.D. Comment. Since in what follows we seek only an upper bound on the intersection of the sets IO(M’, u&), 10(M2, u& \ve shall adopt the notation dim[IO(M1)nIO(M2)]=m,

+m,-w-2,

where this is to be interpreted as in the above discussion. Proof of Theorem 2 (i). Let 0 be a fixed simple, non-collegial RSF. Assume tz’= dim W 2 n - 1. Consider M ‘, M2 in ~3, of sizes ml, m2, respectively, both of size 5 n - 1. Two such coalitions may be found since cr is assumed to be non-collegial. Suppose tirst that M1 n M2 =@ A necessary condition for this is that 122, + ~2~ s n. Let ix(M’, M2) = ~12~+ m2 -w - 2. By Theorem B,

a(M1,M2)
generically. But then IO(0, WN,tr)=QI,

generically. We may therefore suppose that for dny pair ,‘M‘, M2 in 9, AI’ n M 2# (8. Let

~M1nM"~=ml,~O, and observe that ml2 2 m, + m2 - 12. Since

a(M',M')
with intersection of size m,,,>=m,,+m,-ti. Since ml,+m,-bv-2
[O(M")

is generically of dimension ml23 -- 1. By induction if there is a sequence Ml,..., M' with non-empty intersection n& 1 IO(Mj) isgenerically of dimension size ml,,..,,, then the intersection - 1. ml,...,, . . ., M' with Since 0 is non-collegial, there exists some sequence M ', common intersection C,. say, of size ml ,...,P which does not intersect

of

M'+1E9.

N. Schojk!d, Bergson-Samuelson wevaref‘inctions

A necessary condition for this is that ml,...,r

+m,+1

Q.

But dimly

IO(M’)]=m,,,__,.+m.,

1 -w-2,

generically, and this is strictly less than 0. Consequently IO@, WN,u)=Q), Q.E.D.

for all t: in a residual set in U(W)?

of Theorem 2 (i). Assume w = dim W>=q. Consider a coalition R of size (q + 1 ), and the set Q consisting of the q different subcoalitions of R, each of size q, each containing a fixed player i, say. For any member M’ in Q one can find a second coalition M2 in Q with Prn0j

IAIl nM2i=q-1. Since (M’(+(A+w-2sq-2, dim(IO(M’ ) r? 10(M2)) = generically. In the same way n&f’=

q-

2,

{1’1,so that

dim( Q IO(Mj))=Q, generically . Sow let Nq+ l = R - {i}, and observe that

Mq+l n

n M-++. Q

Consequently, =l+q-w-2<0,

q+l)

> and SQIO@, ti
QED.

N. Schofield, Bergson-Samuelson

To prove Theorems LC(cr, u: N, U). De$nitior 2 2. called hug qen

weware functions

l(ii) and 2(ii) we need a characterization

189

of the set

(i) Let V be some set of vectors in a vector space ‘7: I/ is iff there is a linear function 1: T-43 such that I(v) > 0 for all

UEK (ii) For a fixed u E U (IV)“, define the infinite+naI cycle set IC(q w N, u) to be the set of points such that

w700=

u

Meg(X)

H,(x)

does not belong to an open half space in TXIXHere

By Schofield (1978b), for 0 simple, --

IC(o,W,N,u)cLC(a,W,N,u)cICi~,M!N,u). where IC is the closure of the open set IC. Another interpretation of K(a) is as follows. For given UE U (IV)“, define du,(x) to be the closed cone in ‘&IV defined to be the convex span of {dUi(x)jM. I.e., dtl, (x ) =

UE

TxW:tl=

C Aidui(X) iEM

where all /zi20 but not all zero. As above, H,(X)=@

iff OEdti,v(x).

It is convenient to write Sz for T, W under the equivalence relation LI- h iff u =lb, RE R+, and S, = S,“\{O3.S, may be identified with a (w- 1 )-dimensional sphere. For ME 9(x), du, (x) may be regarded as a closed compact set in s X’

By Schofield (1978c), x E K(o) iff

dcr(x)=17dUM(X)+ Y(x)

Just to illustrate this, consider fig. 1, for the q-game with q= 3, n= 5. At x the minimal members of 6@(x) are {1,2,3}, { 2.3,4)., {3,4, 5fS ( 1,2,5). Blwiously

d,(x)=0 and by previous results x is locally cyclic. Note that y #IC, since d,(y)=du, (x). However, YE Ic, and is indeed locally cyclic. We seek to show that IC(a, W,N,u) is generically dense, by exploiting the relationship between eq. W,

IO(0, w,N,u)=(-pO(v(M,u), 9

and eq. (2), d,(x)-

n du,(x). 9(X)

Assume dim W 2 n. By Singularity Theorem A, /1(N, u) is generically a manifold of dimension (10- I ). By Milnor (1958), L(N, u) = w\/i (N, u) is generically dense. For ;a11 x E L(N, u), g(x) =g. Generically, there exists a dense subset v of L(N, u) with the following property: for each x E V and each M ~9, 1MI 5 12- 1, du, (x) is a (IM1- 1)-dimensional submanifcld of S,. For each M’, M2 E 9, let Proof of Theorem 1 (ii).

j3(M1, M2)==m1 +m, - (w- 1)-2.

Then x E q M, n M, = $I implies

since dim(&)&--

1.

But

50 that x

E

V-d,(x)

=

$9.

We may then proceed, precisely as in the proof of Theorem l(i), to show that if there is a sequence M’ , . . ., M’ with empty intersection, then generically for x E r/: d,(x) =(E!. But then K(c;;~ N, u) is generically open dense (neglecting behavior on the boundary 6W), and so LC(O, w N, u) is generically

Theorem 2 (ii )# Assume w = dim

1. As above, A(R,u) is let R be any coaiition in 33 o generically a manifold of dimension 4, and L(R, u) is generically dense. Let Q

N. Scltofield, Bergson--Sumuelsorl

welfare/unctions

‘be the set of q distinct subsets of R, a?1 of size player i. Bv the linear independence of {du&)),, n du&)=dui(x)

for

q

191

and all containing

a fixed

xd!,(R,u).

Q

Bl:t again by liltear independence,

so thsLt d,(x)=p on the dense set L(R, u ). Consequently genericSly dense. Q.E.D.

LC(a, M/:N, rr) is

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III: Pareto optirnn and price equilibria,

Journal