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Weak independence and veto power
a , b
*
Donald E. Campbell
, Jerry S. Kelly
a
Department of Economics and The Program in Public Policy, The College of William and Mary, Williamsburg, VA23187-8795, USA
b
Department of Economics, Syracuse University, Syracuse, NY 13244-1090, USA
Received 17 May 1999; accepted 29 July 1999
Abstract
Weak independence (WI) prevents x from socially ranking above y at profile p if y ranks above x at profile p9
and each individual ordering of hx, yj is the same at p as at p9. If f is transitive-valued and satisfies WI and Pareto then someone has veto power. 2000 Elsevier Science S.A. All rights reserved.
Keywords: Pareto; Transitivity; Veto; Weak IIA
JEL classification: D71; D74
If f is a transitive-valued social welfare function satisfying the Pareto criterion and a weak form of independence of irrelevant alternatives (IIA) then there is an individual with veto power (Baigent, 1987). The claim is actually false if the set X of alternatives has only three members. This note provides a correct proof when X has four or more alternatives. We also show that if individual i is the one with veto power then x will rank strictly above y in the social ordering at some profiles at which everyone in the complementary coalition N\hij strictly prefers y to x. Moreover, we employ a much weaker domain assumption than Baigent, allowing us to extend the result to economic environments. The Borda rule illustrates the fact that abandoning IIA opens the door to rules that are far from dictatorial. But the Borda rule — as with almost all of the departures from IIA in the literature — requires the social ranking of x and y to depend on the entire feasible set. Not only is it typically far too costly to identify the feasible set completely, Bordes and Tideman (1991, p. 184) point out that the set of all candidates ‘‘is often not really defined.’’ We can still allow the social ordering of x and y to depend on at least a few additional alternatives. But on which additional alternatives should we condition the social ordering of x and y? For a given profile, the Borda rule can rank x above y when
*Corresponding author. 11-757-221-2383; fax: 11-757-221-2390. E-mail address: [email protected] (D.E. Campbell)
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the set X is used to generate the social ordering, but y can rank above x when the set X\hzj is used. Majority rule avoids such social ranking reversals because it satisfies IIA. Can we modify majority rule to eliminate cycles, without precipitating social ranking reversals? Yes. We can even satisfy transitivity of the social ordering if X has only three members. (See Example 2 below.) For any X, we can modify majority rule to satisfy quasitransitivity of the social ranking without precipitating reversals, as Example 4 at the end of the paper demonstrates. But if we require transitivity of the social ordering, WI and the Pareto criterion imply that some individual will have veto power. Before proving this, we state some basic definitions.
N5h1,2, . . . ,nj is the finite set of individuals. A complete and transitive relation on X is an
ordering; and it is a linear order if it is complete, transitive, and antisymmetric. Let P(X ) denote the
N
set of orderings on X and L(Y ) denote the family of linear orders on Y#X. If r#P(X ) , a social welfare function for domain r is a function f from r into P(X ). A member of r is called a profile.
p p
Profile p ascribes the ordering p(i ) to individual i. We will typically let K denote p(i ), with s
i i
representing its asymmetric factor. We let K and s denote f( p) and the asymmetric factor of f ( p) f ( p)
f( p), respectively. We let x|f ( p)y denote the fact that both xKf ( p)y and yKf ( p)x hold. The restriction of profile p to the subset Y of X, denoted puY, is the function that assigns the ordering p(i )uY to
arbitrary i[N, where p(i )uY is the restriction of p(i ) to Y. We say r has the free triple property if either (A) below holds for every three-element subset Y of X, or (B) holds for every three-element subset Y of X:
N
hruY: r[rj5L(Y ) , (A)
N
hruY: r[rj5P(Y ) . (B)
r has the free quadruple property if (A) holds for every four-element subset Y of X, or (B) is satisfied for every four-element subset Y of X.
for all i[H. H is weakly decisive if it is weakly decisive for all pairs. H is almost weakly decisive for
p p
(x, y) if we have xKf ( p)y for all p[r such that xsiy for all i[H and ysix for all i[N\H. H is almost weakly decisive if it is almost weakly decisive for all pairs. We say that individual i has veto power if hij is weakly decisive. One usually finds discussions of veto power in treatments of social choice that merely assume quasitransitivity or acyclicity of f( p), but we emphasize that transitivity of social preference is part of the hypothesis of our theorem.
Example 1. A person with veto power is not necessarily a dictator, even on the standard domain
N
r5P(X ) , and even with the Pareto criterion in force, as the following rule f shows: X has exactly
p p p
four members. Given p, re-label X5hw,x, y,zj so that ws1xs1ys1z. Set asf ( p)b for a[hw,xj p
and b[hy,zj. For ha,bj5hw,xj, or ha,bj5hy,zjwe set a|f ( p)b unless asib for all i[N, in which
1
p
case we set as b, or bs a for all i[N, in which case we set bs a. Individual 1 has veto
f ( p) i f ( p)
power, but no individual is a dictator: if i strictly prefers w to x and everyone else strictly prefers x to
w we will have w| x. The Pareto criterion is satisfied because the pairshw, yj,hw,zj,hx, yj, andhx,zj f ( p)
are ranked according to p(1) and the other pairs are also ranked so that Pareto is satisfied. Finally,
f( p) is transitive because aKf ( p)bKf ( p)c for three distinct alternatives a, b, and c implies a[hw,xj
and c[hy,zj. (This example can be extended to any number of alternatives greater than 1.)
Example 2 will show that Pareto, WI, and transitivity of f( p) are consistent when X is a N
three-element set andr5L(X ) . Examples 2 and 3 are based on majority rule, and we let xs y m( p) denote the fact that x ranks above y in more than half of the p(i ). Recall that Sen (1966) proves that
N
for p[L(X ) and n odd, m( p) fails to be transitive if and only if there is some three-element subset
hx, y,zjof X such that each alternative inhx, y,zjranks above the other two in p(i ) for some i[N, each
alternative ranks below the other two in some p(i ), and each alternative ranks between the other two in some p(i ).
2 N
Example 2. X5hx, y,zj, n is odd , andr5L(X ) . Set f( p)5m( p) if m( p) is transitive; otherwise set a| b for all a,b[X. It is obvious that f( p) will be transitive. WI is satisfied because as b and
f ( p) f ( p)
bs a implies that there is no majority rule cycle for either p or r. This in turn implies that as b
f (r) m( p) ranked at the top or bottom of p(i ) for any i[N. This domain has the free triple property but not the
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free quadruple property. Let f( p)5m( p) if m( p) is transitive; otherwise set a|f ( p)b for all a,b[X.
p WI is satisfied as shown in Example 2. We establish the Pareto property by showing that if as b for
i all i[N and some a,b[X (we say that a Pareto dominates b that case), then m( p) does not have a
cycle. It is easy to see that if m( p) has a cycle and some alternative Pareto dominates another at p, then there is a majority cycle involving three alternatives. Suppose that a Pareto dominates b. If
ss ts vs s forhs,t,vj,X, then by Sen (1966) there are three individuals, say 1, 2, and 3,
w must rank below each member ofhs,t,vj for p(1) or p(2) or p(3), another contradiction. Therefore,
w[hs,t,vj. We know that c5a or c5b. If w5s then p(1) must have c at the top, because w cannot
be the top ranked alternative, and p(2) has c at the bottom, because w cannot be at the bottom. In that case c does not Pareto dominate any alternative and it is not Pareto dominated either. We reach the same conclusion if w5t or w5v. Therefore, there is no three-alternative majority cycle, and thus no
four-alternative cycle either. In that case, f( p)5m( p) and thus as b. f ( p)
2
Examples 2 and 3 work for an even number of individuals if we discard one of them before applying the definition of f.
3 N
The key step in our proof of the theorem is the demonstration that the set W of weakly decisive
4
coalitions is an ultrafilter. First we prove a contagion lemma. It is implicit in the three lemmas to follow that f is a transitive-valued social welfare function satisfying WI and Pareto on a domain with the free quadruple property.
Lemma 1. If X has at least three members and H is almost weakly decisive for (x, y) then H is almost
weakly decisive.
Therefore, xs z by transitivity of f( p). Then xK z by WI. Similarly, we can show that H is
f ( p) f (r)
almost weakly decisive for (z, y) if it is almost weakly decisive for (x, y). A standard argument will show that H is almost weakly decisive for all pairs. (Alternatively, consult the general purpose contagion lemma, No. 2 on p. 44 in Campbell, 1992.) h
Lemma 2. If X has at least three members and H is almost weakly decisive, then H is weakly
decisive.
r
Proof. Suppose that profile r has xsiy for all i[H. Choose arbitrary z[X\hx, yj and p[r such
p p p p
that xsizsiy for all i[H, zsix and zsiy for all i[N\H, and puhx, yj5ruhx, yj. Then xKf ( p)z because H is almost weakly decisive, and zs y by Pareto. Therefore, xs y by transitivity of
f ( p) f ( p)
f( p). Therefore, xKf (r)y by WI. Lemma 1 completes the proof. h
Every majority coalition is weakly decisive for the rule of Example 2. But the intersection of h1,2, . . . ,(n11) / 2j and N\h1,2, . . . ,(n21) / 2j5h(n11) / 2j is not weakly decisive. X5hx, y,zj in Example 2, but if X has at least four members then the intersection of two weakly decisive coalitions is weakly decisive. In fact Lemma 3 below establishes that W is an ultrafilter. This is the point at which Baigent’s proof is flawed. Baigent uses a single profile p and a weakly decisive coalition H to
p
and thus xs z by transitivity. Baigent claims that hhj is almost weakly decisive for (x,z). But we f ( p)
p p
have xs ys z for all i[hhj<N\H, so the argument breaks down if H±N.
i i
Lemma 3. If X has at least four alternatives then the set W of weakly decisive coalitions is an
ultrafilter.
Proof. We know that N[W and[[⁄ W by the Pareto criterion. By definition of W, we have J[W if H[W and H,J.
Now we show that the intersection of two members of W belongs to W. Suppose that H and J are
4
r
weakly decisive. Choose arbitrary (but distinct) w,x, y,z[X. Suppose xsiw for all i[H>J and
r
Therefore, xKf (r)w by WI. Therefore, H>J is almost weakly decisive by Lemma 1, and hence is
weakly decisive by Lemma 2.
Finally, we show that either H or N\H is weakly decisive for arbitrary H#N. Choose arbitrary
p p p p
x, y,z[X and p[r such that xs zs y for all i[H, and ys xs z for all i[N\H. If xs y
i i i i f ( p)
then H is almost weakly decisive by WI and Lemma 1. If yK x we also have xs z by Pareto,
f ( p) f ( p)
and thus ys z by transitivity of f( p). In that case, N\H is almost weakly decisive. In light of this, f ( p)
Lemma 2 implies that either H or N\H is weakly decisive. h
Theorem. Suppose X has at least four members and r has the free quadruple property. If f is a transitive-valued social welfare function satisfying WI and Pareto then there is some i[N such that
N\hij is not weakly decisive and i has veto power.
Proof. Lemma 3 implies thathij[W for some i[N because N is finite. Thenhij is weakly decisive, and hence i has veto power. To show that coalition N\hij is not weakly decisive choose three
p p p p
alternatives x, y, and z and any profile p[r such that xsiysiz, and yshzshx for all h[N\hij. We have xK y because i has veto power, and ys z by Pareto. Therefore, xs z by
f ( p) f ( p) f ( p)
transitivity, and thus N\hij is not weakly decisive. h
Remark 1. Assuming that i has veto power, the proof that N\hij is not weakly decisive merely
requires WI, Pareto, and a profile p at which f( p) is transitive and for which an alternative y Pareto
dominates another alternative z, with a third alternative x ranking above y and z for individual i and
below y and z for everyone else. Recall that if f is oligarchical, neither xs z nor zs x can hold
f ( p) f ( p)
if one member of the oligarchy strictly prefers x to z and another member of the oligarchy strictly prefers z to x.
Remark 2. Finiteness of N is only used in showing that if the family of weakly decisive coalitions is
an ultrafilter then someone has veto power. We have proved that for any N, the family of weakly
decisive coalitions is an ultrafilter if f is a transitive-valued social welfare function satisfying WI and
Pareto and the domain has the free quadruple property.
2
Remark 3. The domain % of profiles of economic preferences on the space R of allocation vectors
1
to (3,3) to (2,2) to (1,1) at every profile in %. But if we let X be a convex downward sloping curve in 2
R then the restriction of % to X has the free quadruple property. (Lemma 2 in Campbell, 1995.)
1
Assume that f is transitive-valued and satisfies WI and Pareto. Then some h[N has veto power for
fuX, the restriction of f to X. Let X9be another strictly convex downward sloping curve that intersects
*
pairs by building up a network of intersecting curves. (If x$y±x then there is no downward sloping
curve passing through x and y, but monotonicity implies that everyone strictly prefers x to y at every
profile, and thus xs y by Pareto. Therefore, h is weakly decisive for (x, y).) Proving that some
f ( p)
individual hasveto power when we have private goods as well is more complicated, but the technique
is much the same.
Remark 4. The theorem does not go through if we replace ‘f is transitive-valued’ with ‘f is
quasitransitive-valued’, as the last example shows.
Example 4. N is odd. (If there is an even number of individuals, discard one of them before defining
N
f.) X is any set, and r5L(X ) . Given p, set xs y if and only if xs y and there is no
f ( p) m( p)
alternative z such that zs xs ys z. WI is satisfied for the same reason that it is satisfied m( p) m( p) m( p)
alternative in either case, so we have xs y because xs y. Finally, we show that s is
f ( p) m( p) f ( p)
transitive for arbitrary p. Suppose xs ys z. We cannot have zs x, by definition of f( p).
f ( p) f ( p) m( p)
Then xs z. Suppose that xs zs bs x. If ys b then we have xs ys bs
m( p) m( p) m( p) m( p) m( p) m( p) m( p)
x, contradicting xs y. Then bs y and now we have bs ys zs b,
contradict-m( p) f ( p) m( p) m( p) m( p) m( p)
ing ys z. Therefore, we know that xs z holds and that there is no cycle involving x and z and
f ( p) m( p)
one other alternative. Therefore, xs z. f ( p)
References
Baigent, N., 1987. Twitching weak dictators. Journal of Economics 47, 407–411.
Bordes, G.A., Tideman, T.N., 1991. Independence of irrelevant alternatives in the theory of voting. Theory and Decision 30, 163–186.
Campbell, D.E., 1975. Democratic preference functions. Journal of Economic Theory 12, 259–272. Campbell, D.E., 1992. Equity, Efficiency, and Social Choice, Clarendon Press, Oxford.
Kirman, A.P., Sondermann, D., 1972. Arrow’s theorem, many agents, and invisible dictators. Journal of Economic Theory 5, 267–277.
Saari, D.G., 1989. A dictionary for voting paradoxes. Journal of Economic Theory 48, 443–475. Saari, D.G., 1994. Geometry of Voting, Springer, Berlin.
