A new informational base for social choice

Mathematical Social Sciences 17 (1989) 177-188

177

North-Holland

A NEW INFORMATIONAL Jerry

BASE FOR SOCIAL

CHOICE

S. KELLY

Department

of Economics,

Communicated Received

by E. Kalai

22 May

Revised

21 July

1987 1988

As soon as Arrow cedures

Maxwell Hall, Syracuse University, Syracuse, NY, 13210, U.S.A.

(1951) had shown us the impossibility

based solely on individual

ing collective preferences literature

choice by gathering and extended

ordinal

preference

and using better information

preferences

have been favored

is the idea that using only rankings

sities’ of preference

to improve

The get-more-information that,

for procedures

based only on rankings

available

the procedure

different

of improv-

than just those rankings.

alternatives.

Cardinal

One common

thread

in this

knowledge

about

‘inten-

keeps us from employing

has been unhelpful in detail,

of alternatives

This paper

in the sense that you still get impossi-

still bear a family

at each point of the domain

must operate.

‘good’ social choice pro-

the idea was developed

choices.

approach

bility theorems formation

though

of designing

rankings,

resemblance

is offset by the much larger

illustrates

to those obtained

(see Kelly, 1978). Roughly, these ideas again

the greater

domain

in-

over which

using a new informational

base. Key words: Set social choice

role; liberal

paradox;

strategyproofness;

nondictatorship.

1. Introduction This new informational base for aggregation to social choices is one that takes its origin from the literature on strategy-proofness for non-resolute social choice rules based on usual ordinal preferences over alternatives (see Kelly, 1977 and Girdenfors, 1976). By ‘non-resolute’ we mean that we are not required to select singleton choice sets from non-singleton agendas. In these circumstances, defining manipulability requires being able to order at least some sets of alternatives. Different authors have chosen different ways of getting partial orders on sets of alternatives from complete orders on the alternatives themselves. One such rule, where R; is the order on alternatives and >i is the correpsonding partial order on sets is: A>;B if xR,y for all xeA and ycB and xPjy for at least one xeA and y E B.

(1)

But suppose instead of starting with orders on alternatives and building partial orders on sets, we start with complete orders on sets. This might allow us yet another way of expressing intensity of preferences (e.g., x is intensely preferred to y if for 0165-4896/89/$3.50

0

1989, Elsevier

Science

Publishers

B.V. (North-Holland)

178

J.S. Kelly / New informational

base for social choice

all A disjoint from {x, y} we have {x} UA >; {y} UA). In this paper we will examine the nature of impossibility results that obtain when complete set orders form the informational base. (The nearest approach I have seen to working with complete set orders is von Wright (1963).) Let N(X) be the set of all non-empty subsets of the set X of all alternatives. A (strong) set order, >, is a complete, reflexive, transitive and anti-symmetric relation on N(X). As (1) suggests, not all logically possible set orders make sense as preferences. We let Q be the set of all set orders satisfying the Internal Rule: If, for all XEA and DEB, either x=y or (x}> {y}, and the latter holds for at least one pair, then A > B. The reader should be careful not to think of set orders as being based on maximizing expected utility with respect to some utility function on individual alternatives together with a probability distribution on those alternatives. Some set orders may be explainable in this fashion, but some may not. A set profile u=(>,,>~, . . . . >,) is a function from the set Z= { 1,2, . . . , n} of all individuals to Q, i.e., an element of Q’. An agenda u is a non-empty subset of X. Let I/C N(X) be the set of admissible agendas; a choice function C: V-N(X) selects from each admissible agenda V a non-empty subset C(u) of u. A set social choice rule f is a function on a subset of Q’ that takes each set profile u to a choice function CU.

2. Liberalism For our first impossibility result within this new framework, we will modify the Paretian liberal analysis of Sen (1970). He showed that there is no usual social choice rule that (i) works at all profiles of usual orderings; (ii) satisfies a Pareto condition; if xP,y

for all i, then x E u implies

y $ C,(u);

(iii) satisfies a ‘liberalism’ condition: there are individuals i and j and alternatives x, y, z, and w such that: (a) if xP,y then x E u implies y $ C,(u); (b) if YPiX then y E u implies x6 C,(U); (c) if ZPj w then z E u implies w $ C,(U); (d) if WPjZ then w E u implies z E @C,(U). With our new informational base, we can weaken liberalism so that an individual can exercise his right only when his preferences are intense in the sense alluded to in the Introduction. The Pareto condition also will only come into play if everyone intensely prefers one alternative to another.

J.S. Kelly / New informational

179

base for social choice

Theorem 1. There does not exist a set social choice rule f that: (i) works on ah profiles, Q’, and where all finite subsets of X are admissible; (ii) satisfies a Pareto condition subject to intensity: For all r,s EX, if (r} UA ?k {s) U A for all A disjoint from {r, s} and for all k E I, then r E v implies s @C,(v). (iii) satisfies a ‘liberalism’ condition subject to intensity: There are individuals i, j

and alternatives x, y, z and w such that: (a) if {x} U A >, {y} U A for all A disjoint from (b) if{y}UA>j{x}UAforallAdisjointfrom (c) if {z} U A >j { w}U A for all A disjoint from or all A disjoint from (4 if{w>UA>,(z}UAf

{x, y], then x E v implies y $ C, {v}. {x,y}, theny~vimpliesx$C,{v}. {z, w}, then z E v implies w $ C, {v}. {z, w}, then w E v implies z $ C, {v} .

Proof. We deal here only with the case where {x, y} tl {z, w} = 0. Consider file restricted to N({x, y, z, w}) as follows: i: {z) >, {x,z} >i {Y,z> >; {x> >, {z, w) >i {x,y,z)

>i {x,y>

a set pro-

>i {x,z, w>

>i (Y1 z3 w} >i {x3 Y9& w} >i {x3 Y, w} >i {Y} >r {x, w} >I {Y, w> >j Cw> 9

and all others: {Y} ‘k >k

{Y, w> >k {x, Y, z} >k

{Y, z> >k

iw}

{x, y, & w) >k

>k

ix,

Y} >k

(-6 z, w} >k

{Y, z, w> >k (z}

>k

{x,

iz, w}

w> >k >k

{x, Y, w>

{xv z} >k

{x}.

Looking at C,({x, y, I, w}), we see y won’t be chosen by condition (iiia), z won’t be chosen by condition (iiib), and neither x nor w will be chosen by condition (ii). Thus 0 C,({x, y,z, w}) = 0 violating condition (i).

3. Strategy-proofness

and decisiveness

To discuss the role of incentives and manipulation, we must introduce some terminology. A set profile u’= (>;, >;, . . . , >L) is an i-variant of set profile u=(>,,>,, ..*, >n) just when >j= >j for all j+i, i.e., when u’ differs from u at most at the ith ordering. Social choice rule f is manipulable by i at set profile u = (>I, >2, . . . , >n) and agenda such that G(u)>;

f is strategy-proof

v if there exists a set profile

u’, an i-variant

of u,

G(u).

if at every set profile u and agenda v it is manipulable by no individual. We will assume V= {X]. Let T(i,u) be the topmost set in >; at profile u. Then clearly the rule f with C,(X) = T(i, u) for fixed i is strategy-proof. It conveys dictatorial power on i. It is also possible to construct strategy-proof rules when the range is restricted, i.e., when very few sets A can be C,(X) for some u. As an extreme case, fix a set A and let

J.S. Kelly / New informational

180

C,(X)

=A

for

all u; this

{A/A = C,(X)

for We wish to find to those obtained strategy-proof set

defines

base for social choice

a strategy-proof

fule.

let Ef=

Henceforth,

some u}, the collection of sets that can be chosen from X. impossibility results, in the set social choice rule context, similar in Gibbard (1973) and Satterthwaite (1975). We will show that social choice rules with large Ef are both very rare and lead to

substantial concentration of power. First we will illustrate this idea by looking at two examples which have no great concentration of (exclusionary) power and show they fail strategy-proofness. These set social choice rule examples are closely related to ordinary rules (when orderings are only over alternatives) that are strategy-proof (see Kelly, 1977). Example 1. f(u)=C, where C,(X) choice rule fails strategy-proofness, 1: ix>>, z- { 1 >: {Y> >k

= T(l,u)U T(2,u). To see that examine the profile given by

{X~Z)>l{4>1 {x, Y) >k

this set social

{X,Y,Z1>1 {Y,Z1>l{X,Y)>,

{‘% Y, z> >k

{z>

>k

{x, z} >k

{Y, z} >k

{Y> (z}.

Here C,(X) = {x, y}. If individual # 1 lies and submits a set order with {z} on top, then at such a l-variant u’, C,,(X)={y,z}. But, at u, (Y,z)>~ {x,y}. Thus f is manipulable by 1 at u. A similar calculation works for any rule of the form C,(X)

= U T(i, u) rlzJ

when

JJJ > 1.

Example 2. f(u) = C, where C,(X) = {x 1there is no y such that {y} >; {x}, all i], the Pareto-optimals. To see that this set social choice rule fails strategy-proofness, examine the set profile given by

1: ix> >I

{XYYI >I {Y> >I

{x,z) >I

{y,z}>I

{x,y,z} >,

{z}

z-~1>~~~>~k~~~~}~k{~}~k{~,~}~k{~,~,~}>k{~,~}>k{~}. Here

= {x,Y,z}.

C,(X)

&+>I

If individual

{x,z)>,

cz1>1 {X,Y)>l

then, at the resulting l-variant U, showing the manipulability Spurred a theorem

# 1 lies and submits

u’, C,,(X) off.

{&Y,Z}>l

(Y,Z)>l

= {x, z}, which

{y},

# 1 prefers

to {x, y, z} at

on by examples like this, we will explore two routes by which to establish in the Gibbard-Satterthwaite family. Both depend on the following

Lemma. Zf f is strategy-proof and N(X) is partitioned into two classes V, and I/,such that V,nE,#0 then at anyprofile u=(>,,>,,...,>,) such thatfor all i, A >; B for all A E V, and BE V, we must have C,(X) E V, . Fundamental

Proof.

Before

we get into

the proof

proper,

it will be useful

here to establish

J.S. Kelly / New informational base for social choice

one

181

additional

piece of terminology. Let u=(>i, ~2, >3, . . . , >n_l, >,) and u’= >;_ ,, >;) be two set profiles. The usual sequence from u to U’ will e;, >;, >;, . *. , be the following sequence of n + 1 set profiles: u[Ol=(>,,>,,>,,...,>,-,,>,)=u U[11=(>;,>2,>3,...,>n-l,>n)

u[21=(>;,>~,>3,...,>,-,,>,)

u[n-l]=(>\,>i,>i

)..., >;_l,>n)

u[n] =(>;,>;,>;

,...) >b_,,>$=u’.

Notice that in the usual sequence, u[i] is an i-variant Now suppose the Lemma is false, i.e., suppose

of u[i - 11. there is a set profile

u=

>,_1,>,)withA>iBforalliandallAEV~,BET/2yetC,(X)EV2. Since Vi nE’#0 there is a set profile u’=(>;, >;, >;, . . . . >;1_,, >L) such that C,,(X) E Vi. Consider the usual sequence from u to u’. Since C,,,,(X) E V2 and such that C~li~il(X)EV2 and c,l,,&Y)~ Vi, there is an integer i, Ocisn, CUri, E Vi. Now u[i] is an i-variant of u[i - l] and at u[i - I], i has an ordering with (>1,>2,>3,*.*,

CU,,l(-V >i CU[i- l]V>. Hence f is manipulable This Fundamental

Lemma 1: ix)>1

I-

at u[i - l] by i.

{l}:

0

is weaker than a Pareto

{x,y)>,

{Y>>i {x,z1>1

condition. {Y,Z1>,

!z} >k {x,z} >k {x} >k (y,z> >k {x,y,z}

Consider {x,_v4>1

>k

{x,Y}

the profile (4

>k

{Y>.

Here, everyone strictly prefers {x} to {y}, but the Fundamental Lemma cannot be used to say (y} won’t be chosen. There do not exist sets, Vi, V2, with the requisite properties. If {x} E V,, everything above {x} in >2 must be in V,; so {z> E VI. But then everything above {z} in >1 must be in V, . But all other sets in N({x, y, z>) are above {z} in >, . Hence {y} $ V2. To begin the first route toward a theorem, suppose X contains a three element subset &= {x, y, z} such that N(z) c Ef. We are going to create from the set social choice rule f on X a standard social choice rule f on 8. Strategy-proofness off will ensure some Arrovian properties for f. f will associate with a profile u of strong orders on 8 a choice function CU. Actually, as an intermediate step, we will associate with u a binary relation R(u) on X and then define L;;, so as to satisfy rationality: C,(V)={xEuIxR@)y for all ~EU}. In order to use f, which works on set orders, to define a binary relation R(g) based on a profile of ordinary orderings on alternatives, we need to have a pattern for

182

J.S. Kelly / New informational base for social choice

building set orderings from ordinary ones. Given a pair of alternatives {r, s} c X and a profile u = (Pi, . . . , P,) of strong orders on X, define U(T,S) = (>,, . . . , >n) where >i is given by the following rule (here t is the third element of X): If rP;s, then at the top of >; appear

then fill in >, with the rest of N(X) Internal Rule.

the elements

at the bottom

of N(X)

in a manner

arranged

as follows:

consistent

with the

we define rR(g)s if and only if rE C t(cr,Sj(X). Before we go any further, we must make sure that R(g) is well defined by showing that C,(JX) does not depend on how the filling-in is done for the bottom part of u(i,s). Let u and u’ be two set profiles of orders satisfying the Internal Rule, each with the sets in N(X) ordered at top and such that the restrictions of the profiles to N(X) are identical. If C,(X)#C,,(X), there is an i, O
For the moment, we will assume R(u) satisfies acyclicity, thus ensuring &(o) is non-empty. If rP,s for all i at u, then N(X) c Ef together with the Fundamental Lemma (with I’, = {{r}}) give Cu(r,sj(X) = {r} so rP(u)s where P(g) is the asymmetric part of R(g) and so f satisfies the Pareto condition. If g and 4’ agree on the ordering of r and s, a proof like that of the well-definedness of R(g) ensures C,,,,,(X)

= C,,,,,(X)

so f satisfies Independence of Irrelevant Alternatives. What remains, of course, is establishing acyclicity there must be a profile 9 with say xP(tc)~, Mu)

of R(u).

Suppose

this fails;

and zP(u)x,

i.e., C u(xv)(X) = I.4 t C,cy,z~(X) = LY> and C,(x,z)(X) = Cd. From

_u we will construct

lf at g, individual

a set profile

i has rP;sPi t where

_u(x,y, z) = (>,, . . . , >,,) as follows: {r, s, t } = {x, y, z}, then >i has at the top

{r3>i{~~~}>i{~)>i{~,t}>i(~,f}>i{I;~,f}>,{t};

>i

is then filled in at the bottom

in such a way as to conform

to the Internal

Rule.

As before, Cu(x,Y,zj(X) is independent of this filling-in process. Lemma, it is an element of N({x, y, z}). What is C l((x,Y,zr(X)? By the Fundamental There are seven such. Using symmetry, we really only have to deal with three cases: C +,_&X) = (4, Cu~x,u,z~(~)= ix, y>, and Cll~x,Y,z~(X)= {x7Y, ~1.

J.S. Kelly

Case 1. C “(X, y,

/ New

informational

base for

social

choice

183

am ={xl f

Let ~(x,y,z)=(>,, . . . , >n) and 21(x,2)=(>‘,, . . . . >k) and consider the usual sequence from ~(x, y, z) to ~(x, z). Since C,,,,(X) = {x} and C,,,,(X) = {z}, there must be an integer i, 0 ; If cu[i]W) = {Yl or {KY1 or tY,Z> at u [i] than {x} so f would be manipulable by i at u[i]. If Culil(X) = {x, z} or {r}, these are ranked by i with respect to {x} the same way in >i as in >; so f would be manipulable by i at either u[i - l] or u[i].

Case2. Cu(x,v,,)(X)= {x99. Let g(x,y,z) =(>,, . . . . >,) and g(x,y)=(>;, . . . . >h) and consider the usual sequence from ~(x, y, z) to t_((x,y). CUrol(x) = {x, y} and C,,,,(X) = {x}. So there is an integer i, 0 < is n with Cuti- ,,(X) = (x, y} and C,,,,(X) # {x, y} . If C,til(X) = {z} or {XJ} or {y,z> or {x,y,z}, these *are all ranked lower in >i at u[i] than {x,y} so f would be manipulable by i at u[i]. If C,,&Y)= {x} or {y}, these are ranked with respect to {x,y} the same way in >i as in >; so f would be manipulable by i at either u [i] or u[i - 11. Case 3. C 2((X,Y,r)(X)= {x,Y, z}. This case would be excluded if we could appeal condition since everyone has all two-element sets above {x, y, Z} at ~(x,

Pareto

to a y, z).

So, if f is strategy-proof, if N((x, y,z})cEf and if a Pareto conditions holds, f is a (standard) social choice rule with full domain over strong orders on &satisfying the Pareto condition, Independence of Irrelevant Alternatives and acyclicity. Such rules have been subjected to prior study by Brown (1975) and by Blair and Pollak (1982). To apply one of the Blair-Pollak theorems, we must observe that f also satisfies monotonicity, i.e., for profiles g=(P,, . . . ,P,) and u’=(P;, . . . ,PA),-if for all i xP,y implies

xP;v

then xP(c)y implies xP(u’)y. Some of this is covered by Independence of Irrelevant Alternatives. We need only deal with the case where 9 and u’ are i-variants with YPiX and xP;r while xP@)y. Suppose yR(g’)x. Then we have set profiles ~(x, y) and ~‘(x, y) that are i-variants such that at g(x, y) i has l-4 >; {x9Y> >; {Y> on top. C,,,,,(X) = I4 while CuCx,vj(X) = (y} or {x,y}. But then f is mainpulable by i at g’(x, y). Now Blair and Pollak’s Theorem 3 tells us that if the number of individuals, n, equals the number of alternatives, (Y, then under a social choice rule f satisfying full strong domain, monotonicity, Independence of Irrelevant Alternatives and acyclic

184

rationality,

J.S. Kelly

there

/ New

inforrnaiional

exists an individual

base for

who is weakly

social

choice

decisive

over at least (a-

1)

pairs of alternatives. (The Blair-Pollak results are stated for weak orders, but the proofs work as well for the strong order case.) Suppose we partition our individuals into three coalitions of sizes as nearly equal as possible. Then our f yields a new social choice rule on coalitions to which the Blair-Pollak result may be applied. One of the coalitions is weakly decisive over at least 2 = 3 - 1 pairs of alternatives when each of the other coalitions has internal unanimity. But then by monotonicity, the coalition is weakly decisive even without the internal unanimity of the other coalitions. How large is the weakly decisive coalition? By making them of nearly equal size, it need be no larger then 1 + [(n/3)] or just n/3 if 3 divides n exactly. But what power does this coalition C have back in f rather than in just f? Clearly, if at u=(>t,..., >n) we have {x} >; {Y} for all ie C where C is weakly decisive for x against Y in _f, then C,(X) # (Y}. For suppose it were equal to {y}. Let U’ be a profile (>;, .., , >h) in which the members of N({x, y,z}) appear in >: above all other members of N(X) and are ordered as

{xl >: lx, Y>>: {Y>>: {Y,z}>;{x,z}>:(x, y, z}>; {z) if {x] >i (Y} and as {Y> >: {x9Y> X {XI >: {x7z> >: {Y, z} >: {x, y, z} >: {z} if {Y) >; IX>. It is easy to see that C,(X) = {y} and strategy-proofness C,,(X) = (.Y), but that violates C’s weak decisiveness on f.

imply

social choice rule and suppose there exists a triple {x, y, z} such that N( {x, y, z}) c E f and f satisfies a Pareto condition on profiles where everyone has the sets in N({x, y, z}) on top. Then there is a coaltion C no larger than 1 + U(n/3)! such that for at least two pairs of these three alternatives, say for example the pair x and y, if at u, {x} >i {y] for all i E C, then C,(X) # {y} . Theorem

2. Let f be a strategy-proof

Strategy-proofness (plus a kind of Pareto condition) leads to a substantial concentration of power in that some coalition not too large (about one third of the population) has the power to keep some particular alternatives from being the unique chosen element. Blair and Pollak have stronger results we could use to reduce the size of C or increase the number of pairs over which C is weakly decisive if we could extend acyclicity to more than three elements. They also point out that their analysis doesn’t imply asymmetry of power - everyone could be weakly decisive over every pair of alternatives. But that isn’t true here; such highly inclusive rules are manipulable as seen in Example 1 above.

4. Strategy-proofness We now try a second

and nondictatorship route toward

a theorem.

With somewhat

different

assump-

J.S. Kelly / New informational base for social choice

18.5

tions, we will get a stronger dictatorship conclusion. Our result will be conditional upon a conjecture just confirmed by Roush and Kim. Suppose we are able to find a three element restriction off

subset Y= {x, y, z} of X such that N(Y) fl Ef # 0. Let us look at the to the subset of Q’ in which every set order in a profile has all of

N(Y) ranked above all other sets. We are going to use this restriction to define a new set social choice rule f * on (Q*)’ where Q* is the set of set orders on N(Y) satisfying the Internal Rule. f* is to work as follows: Given a profile U* in (Q*)‘, extend each order in u * to an element of Q in such a way that all of N( Y) is on top. Combine these larger set orders in an n-tuple u that is in Q’. Apply f to get C,, and calculate C,(X). Then we say that f * at U* yields a choice C’,T*defined by C,**(Y)=C,(X). Two points must be made about this construction. First, the very last part makes sense because C,(X) EN(Y) by the Fundamental Lemma. Second, we should show that the construction is well-defined by showing that the result at u*, namely C,,(X), is invariant with respect to the extensions used to build up u from u*. This can be done in the same manner by which we proved the well-definedness of R@) in the previous section. Clearly, f * is isomorphic to a standard social choice rule that works on ordinal rankings on seven alternatives, x,, . . . , x7, and yields a choice function that picks a The isomorphism singleton set from the one agenda {x,, . . . , x7) of all alternatives. we use is XI + 1x1

x2 +

x4 -+ .(x9 Yl

X,“kZ)

x7+

{Y>

x3 --t {z) %-‘IYJI

{XYJ).

Thus we are back in the Gibbard-Satterthwaite

framework

where we might deduce

the presence of a dictator for f* and so some concentration of power for f, However, the Gibbard-Satterthwaite result uses an assumption of full (strong) domain and that is not true here. The requirement of the Internal Rule constrains individuals from expressing all possible strong set orders. So, for example, the isomorphic standard social choice rule must work on profiles of orders where x4 will always be ranked between alternatives x, and x2. We need to turn to the literature that tries to strengthen the Gibbard-Satterthwaite theorem by deriving a dictator result from strategy-proofness when the social choice rule only works on D’ where D is some subset of the set of all strong orders on result of this sort is provided by Kim and Roush cx ,, . . . ,x7}. One sufficiency (1980), but the D they use is wrong for our purposes (e.g., D must contain for each alternative r a strong order with r on top; that fails here as no two or three element subset of Y is on top of an order in Q* because of the Internal Rule). An alternative analysis is provided by Kalai and Muller (1977) who derive a necessary and sufficient condition for getting a dictatorship from strategy-proofness

186

J.S. Kelly / New informational

base .for social choice

of a social choice rule satisfying a Pareto condition. The condition on D is a technical constraint called indecomposability. How well does our f* fit into the Kalai-Muller analysis? First the set D* of strong orders on N( Y) satisfying the Internal Rule does satisfy indecomposability. does not follow easily, if at all, from

But secondly, the weak Pareto condition strategy-proofness. We can get a Pareto

result for some pairs of alternatives but apparently not for all. To illustrate, let’s strengthen our assumption N(Y) tl EJ# 0 to {x} E Ef. Suppose at u * = (>, *, . . . , >n*) in the domain of f * we have {x} >, {y} for all i but C,*.(Y) = {y}. Let u = r, . . . , >,) be a profile in Q’ such that for each i, >; has (x), {y), and {x,y} at (> the top ordered as they are in >;* and the rest of N(X) below in a way that satisfies the Internal Rule. Then by the definition of f*, C,(X)= {y}. Next, let u’= >‘,) be a profile with, for each i, {x}, {y}, and {x,~} at the top ordered as (>\,..., ix3

>i

tx,

Y}

>r

IYI

and the rest of N(X) below in a way that satisfies the Internal Rule. Then by the Fundamental Lemma, C,,(X) = {x). Constructing the usual sequence from u to u’, there must be an i, O or (4 y}, f ISmanipulable by i at u[i]. A more complicated argument would allow us to show, for example, that if at u*, {x} >;* (y,z,} for all i then C,.(Y) = {y,z}. But this style of argument seems to break down for some other pairs (e.g., {x, JJ> versus {x, y, z}). Accordingly, to get within the Kalai-Muller framework we will have to assume we can find a triple Y= {x,y,z} such that for f restricted to (Q*)’ a Pareto condition is satisfied. There remains another, more serious, barrier to using the Kalai-Muller theorem here which is best brought out by observing that their theorem does not contain a condition like the at-least-three-element-range requirement of the GibbardSatterthwaite theorem. In fact, suppose D satisfies indecomposability and let r be a new alternative (not in the set X of alternatives ordered by the elements of D) and consider the set D* of strong orders on X(r) which are extensions of the orders in D constructed by putting r on top of each order in D. Then D also satisfies indecomposability, but, assuming Pareto, the social choice rule on (D*)’ would always select r from XU {r}. This constant rule is clearly strategy-proof in the GibbardSatterthwaite sense. Since then, indecomposability does not force enough variety in the range of choice from the set of all alternatives, Kalai-Muller require the social choice rule to work for all subset agendas as well and then impose a consistency condition (Independence of Non-optimal Alternatives). For our f *,there is no natural extension of C,** tc other agendas - and even if there were, we could only use the Kalai-Muller theorem to derive from non-dictatorship a violation of strategyproofness for some agenda not necessarily (what we want) a violation of strategyproofness on the agenda of all alternatives. Elsewhere, Muller and Satterthwaite (1985) provide an example showing the need for all this extra agenda domain and rationality. Their example, however, contrasts with ours in providing relatively little variation in profiles restricted to three alter-

187

J.S. Kelly / New informational base for social choice

natives.

Call a set of three alternatives

a free triple if D restricted

to the set has all

strong preorders on the triple. In our case, every alternative belongs to some free triple while no alternative belongs to a free triple in the Muller-Satterthwaite example. Consider

the following

Let X= {x,, . . . , x,) and D be the set of all strong orders on X satisfying the restrictions: (1) x4 is between x, and x2; (2) x5 is between xl and x3 ; (3) x6 is between x2 and x3; (4) x7 is neither at the top nor at the bottom. lf f is a social choice rule on D’ such that each f(u) = C, has X in its domain, then f cannot satisfy all of (i) resoluteness (ii) xi, x2, x3 E Ef; (iii) Pareto condition ; (iv) Strategy-proofness (v) Non-dictatorship. Conjecture.

(A proof of this has been obtained by Roush and Kim.) Let us take this Conjecture and return to our original problem. Suppose there is a Y= {x, y,z} c X such that Y c Ef and, on the set social choice rule f restricted to profiles with N(Y) on top the Pareto condition is satisfied. Then this rule has a dictator,say #1,i.e.,ifAEEfnN(Y)andBEN(Y)withA>,Batu,thenC,(X)#B. This is a concentration of power result of the sort we sought, but it can be improved in several directions. We first show that l’s power within N(Y) spreads to more than just those profiles where the elements of N( Y) are on top. Let u = (>1, . . . , >n) be any set profile with A ,Ef fl N( Y) and A >, B yet C,(X) = B. Let u’= (>\I .*. 9>‘,) be another set profile such that >i, agrees with >; on N(Y) and has those sets on top. Consider the usual sequence from u to u’. C,,,,(X)=B and C,,,,(X)#B (by l’s dictatorship for f *). Let i, O;). If C,lil(X)~N(Y) then, since B and C,vl(X) are ordered the same by i at u[i- l] and u[i], f is manipulable by i at one of those profiles. Secondly, if Xc Ef and f satisfies a Pareto condition on all Q’, it is easy to see that l’s power on Y spreads to other sets. We note that just as 1 is a dictator on N({x, y,z)) so must someone i be a dictator on N({x, y, w)). But if if 1, look at a profile with N({x,y,z, w]) on top and where the restrictions to N({x,y,z, w}) are 1: {w) >, {w,x) >I 1-4 >1 {W,Y> >, {w,z) >, {X,Yl>,

>I IX,Zl>l {x,Y,z)>, ‘1 cz3

{Y,Z)>,

(w,x,z}>,

{W,.%Y) >, {VI

{w,y,z)>,

{w,x,r,z}

188

J.S. Kelly / New informational

base for social choice

‘- { ’ >: {z) >k {Y,z> >k b> >k ‘k

ix,z>>k {x,Y>>k 6%Y, z>>k {x} iw,z> {YYWI>k {x,w,w>>k {x,w}>k {x,z, w>

>k

iw,Y>z>>k {w,x,Y,z}>k (w}.

>k

After

confirming satisfaction N((x, y,z}) to see that C,(X) {u>, {Z>, {x,Y), Similarly

i’s dictatorship ix>,

of the Internal Rule, cannot be any of {x,zl,

{Y,z}

on N({x,y,

{w}, {x,y},

we use l’s dictatorship

on

or {x,y,z}.

w}) tells us C,(X)

cannot

be

{Y, w>, (4 w> or {x,x w).

That leaves only the four sets containing both w and Z. But everyone prefers {x} to each of these four. Hence i= 1. In like manner, l’s power is seen to cover all triples of alternatives. We will summarize these results in Theorem

3. Suppose that Weak Conjecture is true. There does not exist a set social

choice rule f satisfying all of the following: (1) V=(X); (2) Strategy-proofness; (3) {{x} IxEX} Mf; (4) Pareto condition ; (5) No dictator: there is no individual i such that 1x113 implies C,(X) singleton at the top of i’s order.

is the

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