This paper examines the implications for social welfare functions of restricting the domain of individual preferences lo type-one preferences. Type-one preferences assume that each person has a most preferred alternative in a euclidean space and that alternatives are ranked according to their euclidean distance from this point. The result is that if we impose Arrow's conditions of collective rationality, IIA, and the Pareto principle on the social welfare function, then it must be dictatorial. This result may not seem surprising, but it stands in marked contrast to the problem considered by Gibbard and Saiterthwaite of finding a social-choice function. With unrestricted domain, under the Gibbard-Satterthwaite hypotheses, choices must be dictatorial. With type-one preferences this result has been previously shown not to be true. This finding identifies a significant difference between the Arrow and the Gibbard-Satterwaite hypothesis

Border, Kim C.

Caltech Authors - Main

DIVISION OF THE HUMANITIES AND SOCIAL SCIENCES 
CALIFORNIA INSTITUTE OF TECHNOLOGY 
PASADENA, CALIFORNIA 91125 
AN IMPOSSIBILITY THEOREM FOR SPATIAL MODELS 
Kim C. Border 
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SOCIAL SCIENCE WORKING PAPER 437 
August 1982 
AN IMPOSSIBILITY THEOREM FOR SPATIAL MODELS 
Kim C. Border 
INIRODUCTION 
Arrow's impossibility theorem (Arrow [l]) concerns the 
inability to find a "satisfactory" procedure of aggregating individual 
preferences over social alternatives into a social ordering. There 
are five basic requirements for satisfactoriness imposed by the 
theorem. The first is what has become known as collective 
rationality, that is, the requirement that the social ordering so 
derived be transitive. The second is that of unrestricted domain, 
namely that the procedure ought to work for a society whose members 
may have any transitive preference ordering. The third requirement is 
known as the Pareto principle, and states that if the members of 
society are unanimous in preferring one alternative to another, then 
the social ordering must also prefer it. The fourth condition, known 
as the requirement of independence of irrelevant alternatives, allows 
the social ordering of any pair of alternatives to reflect only the 
individual orderings over the pair, without reference to any other 
alternatives. The fifth condition is that of nondictatorship, that 
is, the procedure may not select an individual in advance and use his 
or her preferences for every realization of individuals' preferences. 
The impossibility theorem asserts that if there are at last thre� 
alternatives, then it is impossible to satisfy all five requirements 
2 
simultaneously. 
In an effort to define "satisfactoriness" in a way that admits 
the existence of satisfactory procedures, various authors have 
weakened or eliminated some of the above conditions. The 
contributions in these areas are too numerous to catalog. The 
approach taken in this paper is to weaken the requirement of 
unrestricted domain. Social scientists frequently make assumptions 
about the nature of individual preferences in formulating their 
models. For instance, economists view the set of alternatives as a 
set of allocations and assume that individuals have convex, monotonic, 
selfish preferences. Restricted domains of this sort have been 
studied by Kalai, Muller and Satterthwaite [7], Maskin [10] and Border 
[3]. The conclusions of these papers indicate that the sorts of 
restrictions on.preferences used by economists do not vitiate the 
result of the impossibility theorem. 
Another class of preferences are those used by political 
scientists in modeling electoral competition. In thes'e models, the 
set of alternatives is an m-dimensional space of policy positions and 
voters are assumed to have type one preferences (Riker and Ordeshook 
[12]), i.e., preferences with a favorite point such that points 
further away are less preferred. (This is a real restriction in that 
not every triple is free. Given three colinear points, no type one 
preference ranks the middle point last.) In the case of a one­
dimensional policy space, the preferences are "single-peaked" and it 
is well-known (Arrow [l], Black [2]) that simple majority rule 
provides a satisfactory procedure. In a space of more than one 
3 
dimension, Kramer [9] pointed out that the single-peakedness condition 
is quite restrictive and is not the same as a restriction to type one 
preferences. McKelvey [11] demonstrated the rather dramatic failure 
of simple majority rule to be transitive in more than one dimension. 
The failure of simple majority rule to be satisfactory does not 
however rule out the existence of other satisfactory procedures. We 
prove below that even with a restriction to type one preferences there 
are no satisfactory procedures for aggregating individual orderings. 
Another important paper dealing with restricted domains is 
Kalai and Muller [6]. They characterize all the domains which admit 
satisfactory procedures, provided that individuals or society are 
never allowed to be indifferent. This condition makes it impossible 
to ap�ly their result to type one preferences which have spherical 
indifference surfaces. It is also not apparent how to carry their 
proof over to the case where there may be indifference. The proof 
presented below makes use of their notion of decisiveness 
implications, but is more closely related in structure to the proofs 
of Hansson [5], Kirman and Sondermann [8] and B order [3]. 
NOTATION AND DEFINITIONS 
The set of alternatives is an m-dimensional euclidean space 
m 
It m with norm Ix I = [ [ (x�)]fs. A preference is a total transitive 
1=1 
ordering on 1R m, the set of all preferences is denoted �. Given a 
preference R e �. the strict preference associated with R is denoted 
4 
P. A preference R is of type one if there is some point p such that 
x R y  if and only if Ip - xi� Ip - yl. Such a point p is unique and 
is called the ideal point of R. The preference with ideal point p 
will be denoted RP, and its strict preference will be denoted pP. The 
class of all type one preferences will be denoted by D. 
The set of voters in society will be denoted by N. An N-tuple 
<R. > e DN is called a profile of individual preferences. A social 1 ��� 
welfare function for type one preferences is a function f : r}J--7 �. 
This definition embodies the domain restriction and collective 
rationality requirement. Note that society itself is not required to 
have type one preferences. The preference f((Ri)) is the social 
ordering of alternatives for the profile where each voter i e N has 
preference Ri. When it is not likely to lead to confusion, we will 
refer to the social ordering f((Ri)) as R and its strict preference 
simply by P; f(<R'. >> will be denoted R1, etc.) 
1 
An Arrow-type social welfare function is a social welfare 
function satisfying the following two conditions: 
The Pareto Principle. If x Pi y for all i e N, then x P y. 
Independence of Irrelevant Alternatives. If for all 
i e N, x Ri y <�> x R� y, then x R y <�> x R1 y. 
A social welfare function is dictatorial if there is some 
j e N such that for every profile <Ri) and every pair of alternatives 
(x, y), if x Pj y, then x P y. Voter j is called the _!lictator. 
THEOREM AND PROOFS I 
Theorem. If the dimension m of the space of alternatives satisfies 
m 2 2 and the set N of voters is finite, then every Arr01l""""type social 
welfare function for type one preferences is dictatorial. 
Proof of the theorem, The proof of the theorem is broken down into 
several lemmas. In order to facilitate the proof we first introduce 
some more definitions. 
5 
A coalition is a subset of N. A coalition V is coercive for x 
over y if x f y and whenever x P. y for all i e V and y P. x for all 
1 1 
i e v° , then x P y. Coalition V is decisive for x over y if x f y 
and whenever x Pi y for all i e V, then x P y. A coalition is 
decisive (resp, coercive) if it is decisive (resp. coercive) for x 
over y for every ordered pair (x,y) of distinct alternatives. Observe 
that if there is a voter who is a decisive coalition by himself, then 
he is a dictator. 
The strategy of proof will be to characterize the class of 
decisive coalitions and show that one decisive coalition consists of a 
lone voter. 
The following notion is borrowed from Kalai and Muller [6]. 
Let S be a collection of ordered pairs of distinct alternatives, We 
say that S is closed under coerciveness implication if the following 
condition C. holds. 
Condition C, If there are type one preferences � · Rz e D with 
x P 1 y P 1 z and y P 2 z P 2 x, then • 
a), (x,y)e S => (x,z)e S 
and 
b), (z,x) e S => (y,x) e S. 
Lemma 1. Let V be a coalition and let S = {(x,y) : Vis 
coercive for x over y}. Then S is closed under coerciveness 
impl ica ti ons. 
Proof: Let � · Rz e D satisfy x P 1 y P 1 z and y P 2 z P 2 x. 
a). Suppose (x,y) BS, i.e., Vis coercive for x over y. Define the 
profile <Ri> by 
Ri 
11.
�
i 8 v 
i 8 v°. 
We can represent the preferences for members of coalitions V 
and v0 schematically as follows. 
y r.
x y 
y z 
z x. 
Since V is coercive for x over y we have x P y. It follows 
from the Pareto principle that y P z (as y Pi z for all i). Thus by 
transitivity of the social ordering x P z. But x Pi z for all i s V 
and z P. x for all i e v0, so by definition V is coercive for x over 1 
6 
z, i.e.• (x, z) e S. 
b). Supvose ( z,x) e S and define the profile <Ri> by 
Schematically we have 
R. 
1 
! 
y 
z 
x 
�
11.
i £ v 
i £ v:. 
r.
x 
y 
z. 
From the Pareto principle y P z and since V is coercive for z over x. 
z P x. Thus y P x, so V is coercive for y over x. i.e., (y,x) e S. 
7 
q.e.d. 
The next step is to prove that if S is any set of pairs which 
is closed under coerciveness implications, then S is either empty or 
contains all pairs. It then follows from Lemma 1 that if a coalition 
is coercive for x over y, then it is coercive for any pair and hence 
coercive. Before proceeding we introduce some more notation. 
Let �(y) = {z : Ix - zl < Ix - ylJ. 
Lx(y) = {z : Ix - zl > Ix - yl} and H(x,y) = {z : Ix - zl < ly - zl}. 
Then Ux(y) is just the inside of the sphere centered at x passing 
through y, Lx(y) is everything outside the sphere, and H(x,y) is the 
half-space of everything closer to x than to y .  See Figure 1. 
8 
If x pZ y, then z e H(x,y); if z e ux (y), then z pX y; and if 
z e Lx (y), then y pX z.
[Figure 1 about here] 
Lenuna 2 .  Let S satisfy condition C., i.e., let S be closed under 
coerciveness implications. If (a,b) e S and c e La(b), then 
(a,c) e S. 
Proof. First consider the case where c B cl = [U
b(a) nLa(b)]. See 
Figure 2 .  Then a Pa b Pa c and b Pb c Pb a. By condition C.a. (with 
x = a, y = b, z = c, 11_ = Ra. � = Rb) we conclude (a,c) e S. Thus 
starting with one pair (a,b) e S we now have a multitude. The next 
step is to expand the set even further. This part of the argument 
uses the fact that the space of alternatives is at least two-
dimensional. 
Choose a point c 1 e c 1 as in Figure 2, that is, c 1 should 
nearly maximize the distance from b and minimize the distance from a. 
(Since c 1 is open, we cannot actually maximize or minimize these 
distances.) Repeating the above argument with c 1 in place of b y ields 
(a,c) £ s for all c 8 c2 
cl 
[La(c 1) n U (a)]. 
Continue in this fashion to construct regions 
ck -1 a Ck = [U (a) n L (ck_ 1)1, which encompass the point a. We can also 
choose points like ct, which is colinear with a and ck for some k, to 
find a region Ct+l which includes points far away from a and such that 
c £ ct+l implies that (a,c) 8 s. 
9 
Thus for any x e La(b) we can find a finite sequence of 
regions c 1, • • •  ,�, constructed iteratively as above, such that x e Ck' 
and hence (a,x) B S. 
[Figure 2 about here] 
Lemma 3. Let S satisfy condition C. If (a,b) e S and 
c s  [La(b) n H(b,a)], then (b,c) e S. 
q.e. d. 
Proof. Put w = (l/2)a + (l/2)b + o(b-a), where 6 ) 0 is chosen small 
enough so that b Pw a Pw c. See Figure 3. We also have that 
c Pc b Pc a, as c s  H(b,a). Then from condition C.b. (setting 
x = c, y = b, z c w) a, 11_ = R ,  � = R we have that (b,c) £ S. 
[Figure 3 about here] 
q. e. d. 
Lemma 4. Let S satisfy condition C. If (a,b) B S, then (b,a) e S. 
Proof. Let (a,b) e S and choose c e [La(b) n H(b,a) n Ub(a)]. See 
Figure 4. Then by Lemma 3, (b,c) e S. Since c e Ub(a) we have 
a e Lb(c), so by Lemma 2, (b,a) e S. 
[Figure 4 about here] 
Lemma 5. Let S satisfy condition C. Then S is either empty or 
includes all ordered pairs of distinct points. 
q.e. d. 
Proof. Suppose S is nonempty and let (a,b) e S. By Lemma 4, 
(b,a) e S. Set b' = a +  a(a - b), a > 1. See Figure 5. Lemma 2 
implies (a,b') e S so by Lemma 4 (b',a) e S. But by construction 
b' b m [L (a) n H(a,b')] U [L (a) n H(a,b)] = lR \{a}, so by Lemma 3, for 
any x f a, (a,x) e S and so by Lemma 4, (x,a) £ S. 
10 
q. e.d. 
[Figure 5 about here] 
So far we have shown that if a coalition V is coercive for a 
over b for some (a,b), then it is coercive. Next we show that a 
coercive coalition is decisive. 
Lemma 6. If Vis coercive, then V is decisive. 
Proof. Let <R.> s DN be a profile with x P. y for all i s V. Put �� 1 1 
w0 = (l/2)x + Cl/2)y, w1 (5/S)x + (3/S)y, w2 = (3/S)x + (5/8)y, and 
z = x +o/z(x - y). See Figure 6. The ordering of x,y,z for the 
w. 
preferences Rx, R 1,i = 0,1,2 are then: 
iC WO .ll.. wl B.....:. w2 .B.....: 
x x y x y 
z y x 
y z z z 
[Figure 6 about here] 
11 1 2
I 
Define the profile <R/ by 
Rx i e V 
wl 
ia Vand x Pi y R I 
R. = 
1 w 2 
R i s V and y P i x 
WO R otherwise. 
I 
Then x Ri y <==> x Ri y and so x R y <==> y R x. By the Pareto 
I • I 
principle x P z. By construction z Pi y for i e Vand y Pi z for 
I I 
i 8 -r. Since V is coercive, z P y. By transitivity, x P y and 
hence x P y. Thus V is decisive. 
q. e. d. 
It follows from the Pareto principle that the coalition N is 
decisive. The following lemma shows the existence of other decisive 
coalitions. 
Lemma 7. If V is not coercive, then Ve is coercive. 
Proof. Suppose V is not coercive and let <Ri> be a profile with 
x P. y for all i B Vand y P. x'for all i s  Ve and y R x. Put 
1 1 
z = x + (l/ 2)(x - y). By the condition of Independence of Irrelevant 
Alternatives we can assume without loss of generality that 
R. 
1 
Rx i e V 
Ry i e Ve. 
Then x Pi z for all i e N so x P z by the Pareto principle. Since 
y R x, we have by transitivity that y P z. By construction z Pi y for 
i s Vand y P. z for i e Ve • Thus Ve is coercive. 
1 
q. e. d. 
The last piece of information we need to know about the 
collection of decisive coalitions is that the intersection of decisive 
coalitions is decisive. This argument relies heavily on the fact that 
the s�ace of alternatives is multi-dimensional. 
Lemma 8.  If V1 and V2 are decisive, then V1 n V2 is decisive. 
Proof. Choose alternatives a, b, c to be three vertices of an 
equilateral triangle. Put 
x = CV:a + e)a + CV:a -e)b 
y = CV:a + a)b + (V:a -e) c 
z = CV:a + s )c + CV:a -a)a 
where a > 0 is small enough so that the following orderings hold: 
r � Rz 
a b c 
b c a 
c a b 
[Figure 7 about here] 
See Figure 7. This is the classic paradox of voting situation. 
Define the profile <Ri> by 
Schematically we have 
R. 
1 
v1 \V2 
b 
c 
a 
Ry i e v1 w2 
Rx i s v1 n v2 
Rz i e v2 w1 
Rc otherwise. 
v1 n v2 v2 \V1 Others 
a c c 
b a a 
c b 
13 
Since V1 is decisive b P c. and since v2 is decisive a P b. Thus by
transivity a P c. By construction a Pi c for i e V 0 V2 and c Pi a 
for i e CV1 0 V2)
c. Thus V1 n V2 is coercive and hence decisive. 
q.e.d. 
Proof of the Theorem. Since N is finite, enumerate the voters from 1 
to n. Assume that voters 1 to n-1 are not dictators. That is {j} is 
not a decisive coalition for j = 1 •• • • •  n-1. Then by Lemma 7. {j}c is 
decisive. It follows from Lemma 8 that {n} 
voter n is a dictator. 
n-1 
n {j}c is decisive so 
j=l 
Q.E.D. 
Remarks. It is clear from the proof that any domain of preferences 
larger than type one preferences admits no nondictatorial Arrow-type 
social welfare function. Also. we have shown that the class of 
14 
decisive coalitions is an ultrafilter and so for infinite sets of 
voters there are nondictatorial Arrow-type social welfare functions 
(Fishburn (4]. Hansson (5]. Ki:cnan and Sondermann [8]). 
REFER.F.NCES 
1. K.J. Arrow, Social Choice and Individual values, 2nd ed., Yale 
University Press, New Haven, 1963. 
2. D.J. Black, "The Decision of a Committee Using a Special 
.Majority," Econometrica 16 (1948). 
15 
3 .  K.C. Border, "Social Welfare Functions for Economic Environments 
With and Without the Pareto Principle," forthcoming in Journal of 
Economic Theory. 
4. P.C. Fishburn, "Arrow's Impossibility Theorem: Concise Proof 
and Infinite Voters," Journal of Economic Theory 2 (1970). 
S. B. Hansson, "The Existence of Group Preference Functions," 
Public Choice 28 (1976). 
6. E. Kalai and E. Muller, "Characterization of Domains Admitting 
Nondictatorial Social Welfare Functions and Nomnanipulatable 
Voting Procedures," Journal of Economic Theory 16 (1977). 
7. E. Kalai, E. Muller and M'. Satterthwaite, "Social Welfare 
Functions when Preferences are Convex, Strictly .Monotonic, and 
Continuous," Pub lie Choice (197 9) • 
8 .  A. Kirman and D. Sondermann, "Arrow's Theorem, Many Agents and 
Invisible Dictators," Journal of Economics Theory S (1972). 
9. G. Kramer, "On a Class of Equilibrium Conditions for Majority 
Rule," Econometrica 41 (1973). 
10. E. Mashi, "Social Welfare Functions for Economics," Cambridge 
and Harvard University, mimeo, 1976. 
16 
11. R. McKelvey, "General Conditions for Global Intransitivities in 
Formal Voting Models," Econometrica 47 (1979). 
12. R. Riker and P. Ordeshook, Introduction to Positive Political 
Theory Englewood Cliffs, N.J.: Prentice-Hall, 1973. 

La(b) 
a 
,U
b (a)
Figure 2b 
Figure 2a 





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An Impossibility Theorem for Spatial Models

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