The structure of exactly strongly consistent social choice functions

Journal

of Mathematical

Economics

8 (1981) 207-220.

North-Holland

Publishing

THE STRUCTURE OF EXACTLY STRONGLY SOCIAL CHOICE FUNCTIONS

Company

CONSISTENT

Ishai OREN* Stanford Uniuersity,

Received

August

Stanford, CA 94305, USA

1979, revised version

received

March

1980

For f an anonymous exactly strongly consistent social choice function (ESC SCF) and x an alternative, define b,= bif) to be the size of a minimal blocking coalition for x. By studying the correspondence between f and {by)}, we establish the existence, uniqueness and monotonicity of ESC SCF’s. We also prove the following conjecture of B. Peleg: A non-constant anonymous ESC SCF depends on the knowledge of every player’s full preference profile.

1. Introduction In 1978, B. Peleg introduced the class of exactly strongly consistent social choice functions (ESC SCF’s), those voting schemes for which manipulation of preferences by coalitions does not affect the final social choice. For such functions which are also anonymous, let the blocking coefficient, of each alternative be the minimal size of a blocking coalition with respect to that alternative. This paper is a study of the relation between the properties of these functions and their blocking coefficients. The paper is organized as follows: Basic definitions are provided in section 2. In section 3 we define the blocking coefficients and explore their properties. As a straightforward corollary, section 4 contains a proof of the following result (Theorem 4.1): If f is an anonymous ESC SCF which depends only on the ordering of the k most preferred alternatives of each player (where li 5 ~11 - 2, m being the number of alternatives) and there are at least (m - l)/(m1 - k) players, then f is a constant. In particular, there is no non-constant anonymous ESC SCF which depends only on the knowledge of every player’s first choice (assuming there are at least 2 players and 3 alternatives). Section 5 contains the main result, namely the construction and characterization of anonymous ESC SCF’s [which in addition satisfy a property called independently blocking, as do all the functions described by Peleg (1978)] in terms of their blocking coefficients (Theorem 5.2). Also *Present address: 19 Hasavion

0304-4068/81/000&0000/$02.50

St., Rehovot,

Israel.

0 North-Holland

208

1. Oren, Exactly strongly consistent social choice functions

included is a positive solution to the problem posed by Peleg regarding the monotonicity of these functions (Corollary 5.4).’ I am indebted to Professor Peleg for introducing me to this subject.

(1978)

2. Preliminaries Throughout, N will be the finite set of players (voters), denoted by 1 through n, X the finite set of alternatives, (XI = m, and L(X) the set of linear order relations on X. If CcN, then J!,~(X)=~~,~L(X), and if R’EL(X), ViEC, then RCeLC(X) is the c-tuple (Ril ,..., Ric), where C={i, ,..., i,]. Subsets of N are called coalitions, and if CcN is a coalition, then any RC ELM is called a preference profile for (the players in) C. A function f:LN(X)+X is called a social choice function (SCF), and its image will be denoted by X*. f is called anonymous if for every RN ELM and permutation rcn~S, we have f(Rfi) =f(RN), where Rt is defined by Rk =R”(“, i=l ,.-., II. To every SCF f and every preference profile of the players RN~LN(X) we associate the game G(f;RN), where L(X) is the strategy set for players 1 through n, f is the result function, and R’ the preference order over the results of player i. QN~LN(X) is called a strong equilibrium point (SEP) of the game G(f;RN) if for every 0# Cc N and TceLc(X) there exists an iEC such that f(QN)Rij(Q WC Tc)’ . An SCF f is called strongly consistent if for every RN E LN(X) the’ game G(f ; RN) has an SEP Q” E LN(X). Furthermore, f will be called exactly strongly consistent (ESC) if for every RN E LN(X), Q” as above can be chosen to additonally satisfy f (Q") =f (RN). For SCF’s in general there are two notions of a coalition Cc N being ‘decisive’ with respect to a set S c X of alternatives: Definition. A coalition Cc N will be said capable of forcing a set ScX of alternatives if there exists a preference profile TCg L’(X) of the members of C s.t. for every T N\C~ LN\‘(X) we have f(TC, 7N\‘)e S. TC will be called an Sforcing profile for C. Definition. A coalition Cc N is called winning with respect to a set S c X of alternatives if for every RN E LN(X) and x EX s.t. V s E S, sR’x V iE C (written SRCx), we have f (RN) $; x. That forcing and winning are observation in studying ESC SW’s:

actually

one

and

the

same

is a key

‘This work is concerned with SW’s which are defined for linear orders. Polishchuk 1978 has independently proved versions of the main results for less restricted orders. Polishchuk also discusses the non-independently blocking case, for which the general results are not known. where rIf CcN, U’eE(X) and V Mc~LMc(X), then f(U’, Pm’) is defined to be f(W”), W’=U’if ieC and W’=V’if ieN\C.

1. Oren, Exactly strongly consistentsocial choice functions Lemma

2.1.

Let f

be an ESC SCF.

SC X if and only if it can force

Then

CcN

is winning

209

with respect

to

S.

Proof WinningaForcing. This implication is true for all SCF’s, independent of the ESC condition. For assume that C is winning with respect to S, and let TCe L’(X) be s.t. STCX\S. Then f(TC, TWC)~ S, V TNiC E LN\’(X), so C forces S via TC. ForcingaWinning. Assume that C can force S, and let RN~ LN(X) and XEX be s.t. SRCx. Then let QN E LN(X) be an SEP of Gcf; RN), satisfying f(QN)=f(RN). Finally let TCe L’(X) be an S-forcing profile for C. Now if f(RN)=x then f(TC, Q’lC)~ S, while SRCf(QN), contradicting the fact that QN is an SEP of G(f; RN). Therefore f(RN)#x and C is indeed winning with respect to S. 1 That forcing implies winning is in fact the statement that for ESC SCF’s, a coalition need not exercise its power to get its demands, but rather, the potential ability to do so is already enough. We shall make use of the natural dual to that of forcing:

following

notion

of blocking,

which

is the

Definition.

A coalition Cc N will be said capable of blocking a set ScX of alternatives if there exists a TCe L’(X) s.t. VTNiCe L?(X), we have f (T’, TN\‘) $ S. TC will be called an S-blocking profile for C. Note.

C can block

S if and only if it can force X\S.

In view of the note above, Lemma Corollary ScX.

2.2.

Let

f

2.1 implies:

be an ESC SCF

Then any TC~ L’(X)

s.t. X\STCS

and CcN

be capable

is an S-blocking

of blocking

profile for C.

3. The blocking coefficients of an anonymous ESC SCF Let f be an anonymous ESC SCF which will be fixed throughout this section. Then a coalition Cc N being winning, forcing or blocking for a set ScX of alternatives depends only on the size (cardinality) of C (see note below), which leads to the following definition: Definition.

For ScX we define ws= WY) to be the minimal size of a winning/forcing coalition for S, and b,= bl,f’ to be the minimal size of a blocking coalition for S, unless SnX* =@ or X* c S, in which cases we

210

I. Oren, Exactly strongly consistent social choice functions

define w,=n+ 1 or b, =n + 1 respectively. If x~X and b, for b,,,.

we also write w, for xc,.,

Note.

Since f is anonymous it holds that a coalition CcN is winning (blocking) with respect to a set S c X of alternatives if and only if ICI2 ws (ICI2 b,). Here and throughout, (C( denotes the size of C. Also observe that

X*=(x~X~b,>O}={.x~X~w,jn}. Now if A, B cX, then clearly b AuBS b, + b,, since any coalition of b, + b, players can be broken up into two disjoint coalitions of b, and b, players, which can then block A and B independently (if b,+ b,>n then the inequality follows since bAuBSn+ 1). If A and B are disjoint we could hope that b,,, = b, + bg. While this is not generally true, it is ‘almost’ true in the following sense : Theorem 3.1.

For A,BcX

s.t. AnB=@

Before proceeding with the proof following lemma and its corollary: Lemma 3.2.

we have

of Theorem

3.1 we shall need the

Zf C c N cannot block S c X then D =N\C can force S.

Proof: Assume C cannot block S. Let TCe L’(X) be s.t. X\ST’S. Then by our assumption, 3 TD E L!‘(X) s.t. f( TC, TD) =f( TN) E S. Let Q” E LN(X) be an SEP of G(f; TN) satisfying f(QN)=f(TN)~ S. We claim that DD is an S-forcing profile for D. For let Vc E E(X). If f(V”, QD)$ S then f(VC,QD)TCf(QN), contradicting the fact that QN is an SEP of G(f; TN). There f(VC, QD)c S, V vc E L’(X), and D can force S as desired. 1 Corollary 3.3.

VScX

we have

w,+b,=n+l, or equivalently

ws+wV?,=n+l

or

b,+bWs=n+l.

Proof: We prove w,+ b,=n+ 1, the rest following from ws= b,,. Now ws + bs Ln + 1, otherwise S could be impossibly forced and blocked simultaneously. But let Cc N be s.t. (C(= 6, - 1 (if b, = 0 then SAX* = 0 and

I. Oren, Exactly

strongly

consistent

social choice functions

211

ws = n + 1 by definition, giving ws + bs = n + 1 as desired). By Lemmas 2.1 and 3.2, N\C is winning with respect to S, so IN\CI 2 ws or n - (bs - 1) 2 ws* w,+b,$n+l as desired. 1 Proof of Theorem 3.1. We have already seen that b, Us5 b, + b,. Now assume to the contrary that bAuB O, so there exists a partition (C,, Cz, C,) of N s.t. lC,l< bA, lCzl < b, and ]C, u C,] =\C, 1 [this is possible since (bA--l)+(bB-l)sbA-l+bAc-l=n-1 +(G(=km by Corollary 3.31. Let S = X\(A u II) and RN E LN(X) satisfy:

(1)

SRC’ BRC’ A >

(2)

ARC2 SRCZB,

(3)

BRC3 ARC3 S.

Then (C,uC,(=n-IC,I>n-b,=w,-1, so ICzuC3(2wwA and consequently C,u C3 is winning with respect to A. But ARC2 “c3S, whence f(R”)$ S. Similarly (Ci u CJ( 2 wg, and since BRC1 “CBA we have f (RN)+ A. Therefore f (RN) E B. But C, u C2 is blocking for Au B, and is therefore winning with respect to S, while SRC1 “‘*B, contradiction ! m The case when the blocking coefftcients are actually additive merits attention (all of section 5) and is therefore deserving of a name: Definition. We say that f is independently blocking AnB=o we have bAuB=bA+bB.

special

(ZB) if VA, BcX

ct.

f being independently blocking is equivalent to the fact that if A, B c X are disjoint, then any coalition Cc N which is blocking for A u B can be broken up into two disjoint subcoalitions that can block A and B independently. A quantitative notion blocking is provided by: Definition.

of how

deficient

The deficiency off is defined

def(f)=

f is from

being

independently

as

C b,--(n+l). XSX

That def(f) below. Proposition 3.4.

is actually

a measure

of how far f is from being IB is proved

f is ZB if and only $ def

(.f ) = 0.

212

ProoJ: Xj={xl

I. Oren, Exactly

strongly

consistent

TB*Deficiency 0. Assume that ,..., xj}, j=i ,..., m-l. Then

social choice functions

is IB. Let X= {xi,. . .,x,}

f

and

n+l=b,

=b,_, + b,,,,

=bx,-, +bx,,,_,+b,,,,

so defCf)=

f

b,,-(n+l)=O.

i=l

Deficiency O=>IB. Suppose def(f)=O and let A,BcX Assume to the contrary that b, us < b, + b,. Then

bA,B+

c

c

bx
c

Sxbb,+xb,+ XSB

XEA

AnB=@.

bx

x~(AuB)~

XE(ALJB)C

satisfy

xs(A

b,

u B)c

=x;xbx=n+l. But b AYB+~~~AC,~)~

(Corollary

bxZ’bA,,+b,A,,)c

3.3), contradiction

=n+l

1

!

1, we always have def(f)zO, which can be Since CxEx b,z b,=n+ interpreted as the impossibility of all the alternatives being simultaneously blocked. Theorem 3.1 can be used to obtain an effective upper bound for def(f ): Theorem

3.5.

The deficiency

Osdef(f)sm-2,

off

satisfies m=JX(.

I. Oren, Exactly

strongly

consistent

social choice functions

Proof We have already seen that Ogdef (f). As to the upper bound, application of Corollary 3.3 with the notation of Proposition 3.4 yields n+l=b,,_, Then repeated

213

an

+b,.

applications

n+l=b,mml

of Theorem

3.1 yield

+bxm

2(@xm_,+b+*-l)+b,__,

&l

+bxt-l+bJJ

-l)+b

-l+...+&_,

%I

-l+b+

=-T;,k- (m-2), SO def(f)=

1

b,-(n+1)5m-2,

X.5X

as desired.

1

in case b,=O, Note that since b AuB=bA+bs slight modification of the proof above yields

and

X*={x~X\b,>0},

a

O~def(J)~m*-2, where m*=(X*/

4. An anonymous alternatives

(assuming

of course that f is non-constant,

so M* >=2).

ESC SCF depends on every player’s complete order over the

We temporarily leave the main line of inquiry to give a now easily proved application (which was previously proved by the author in a direct manner) of the structure developed in section 3. For lsks’m-2 and REL(X) let RckI denote the order relation induced by R over its k largest (in the sense of R) alternatives; i.e., R,,=(x,>x,>...>x,). if R=(x,>

. ..>Xk>X.,,

>...>x,).

214

I. Oren, Exactly strongly consistent social choice functions

For RN E LN(X) we write R& for (R,&, . . ., Ryk,), and I&(X) for the set of all such R;‘. Peleg has conjectured that for 1 s k 5 m -2, an anonymous ESC SCF f cannot depend only on the kth truncation of its argument, i.e.,f(RN) cannot be determined from R& for every RN~LN(X) (provided that f is not a constant and the number of players is not exceedingly small). We prove the validity of this conjecture in the following theorem: Theorem 4.1. Let f be an anonymous ESC SCF which depends only on the ordering of the first k choices of each player (ksm-2), i.e., there is a function g: L&(X)+X s.t. VRN E LN(X),

If in addition, nz (m* - l)/(m - 1 - k), (m* = /X*1), thenf is a constant.

Assume to the contrary that f Proof s.t. VxeX there are at least b, players choices. The existence of such an RN n -2 (m* - 1 )/(m - 1 - k) by the following

is non-constant. Choose RN E LN(X) for which x is not among the top k is a consequence of the assumption construction:

There are n players, each one of whom has (m-k) choices after the first k, so there are (m- k).n ‘slots’ to fill. However, the number of required entries *-1 by Theorem 3.5. Then (m-k).n= is xx,.b,=def(f)+n+lsn+m n + (m - k - 1) . n 2 n + m* - 1 by the hypothesis, so there are at least as many ‘slots’ as there are required entries. Now starting with the first slot (the (k+ 1)th choice) of the first player and proceeding to fill in the first slots of the remaining players, then the second slots and so forth, we can for every x EX till b, slots with the choice x. The counting argument above assures us that we will not run out of slots, and since f is assumed to be non-constant, b, s n, Vx EX, so no choice will be allocated twice to the same player. Filling the unused slots and the first k alternatives of each player in an arbitrary manner yields the existence of an RN~LN(X) with the required properties. Now let x=f (RN). Choose QNe LN(X) s.t. Q& = RiV,,, but among the players that chose x among the last m- k choices in RN, x is their last choice in QN. Since the number of these players is at least b,, Corollary 2.2 implies f (QN)#x. But by the hypothesis, f (QN)=f (RN) = x, contradiction ! 1 Corollary 4.2. Let f be an anonymous ESC SCF which is independent of the ordering of the last two choices of every player, i.e., if R&_,, = Q&_2, for RN, QN E LN(X) then f (RN)=f (QN). If n L m* - 1 then f is a constant.

I. Oren, Exactly

5. Existence,

strongly

consistent

uniqueness and monotonicity

social choice functions

of anonymous

215

IB ESC SCF’s

It is our aim in this section to construct and fully characterize all the anonymous independently blocking exactly strongly consistent social choice functions. We obtain that for every sequence of non-negative integers {b,}xsx satisfying zEX b, = n + 1 there exists an anonymous IB ESC SCF f satisfying bj$= b,, Vx eX, and that this f is well determined up to a certain natural equivalence (Theorem 5.2). The second part of this statement means that an anonymous IB ESC SCF f is ‘essentially’3 determined by its blocking coefficients {b’s’} . It will also be shown that the f satisfying bif)= b,, VXEX, can bxe Thtsen to be ‘monotonic’ (yet to be defined), so every anonymous IB ESC SCF is ‘essentially’ monotonic (Corollary 5.4). It will be useful to introduce the concept of a social decision function (SDF), which is a function with domain E(X) and range the non-trivial power set 2’\{@}. An SCF f will be said to be selected from an SDF F, written ~EF, if f(RN)eF(RN), V RN ELM. An SDF F will be said to be exactly strongly consistent if every f~ F is, while the definition of anonymity is the same as for SCF’s. Two (ESC) SCF’s f and g will be called coconsistent if there exists an ESC SDF H s.t. fe H and g E H, or equivalently that any SCF h satisfying hi Cf(RN), g(RN)j, VRN E,?‘(X), is ESC. The restriction of the co-consistency relation to the class of anonymous IB ESC SCF’s will turn out to be an equivalence relation (a consequence of Proposition 5.1 and Theorem 5.2), so we are justified in calling two such functions f and g which are co-consistent equivalent up to independent selection. This is the aforementioned natural equivalence. As an example of the restrictiveness of the co-consistency relation we have: Proposition 5.1. Ifm>=3, and f and g are anonymous co-consistent (ESC) SCF’s then by’ = b(j), VA c X. Proof Suppose to the contrary that by’ < b(j) for some A cX. Let R, Q EL(X) be s.t. X\ARA and AQX\A. Let C cN satisfy (Cl = by) and define RN E LN(X) by R’=R for iE C and R’= Q for iE N\C. Define an anonymous ESC SCF h by h( VN) = g(V”) if VN = Rf for some z E S, and h(VN)=f(VN) for all other VN~LN(X). Then h(RN)=g(RN)EA by Corollary 3.2. We now differentiate Case 1.

between

two cases:

by’=0

Let Q’ E L(X)

be s.t. AQ’X\A but Q’ # Q (here we use the fact that

3By ‘essentially’ we mean up to the aforementioned independent selection (definition follows).

equivalence,

which

is equivalence

rnz 3). up to

216

I. Oren, Exactly strongly consistent social

choicefunctions

Defining Q’NE,?(X) by Q” = Q’, Vie N, we have h(Q’“)=f(Q’“), and since f(Q’N)$A(by’=O), h(Q’N)$A. Therefore bT’=n+ 1 -w(j)=0 [/z(Q’~)$A implies that w$‘)= n+ 11, contradicting the fact that h(RN)~A. Case 2.

by) > 0

Let R’ EL(X) be s.t. X\AR’A but R’ # R (again we must use the assumption rnz 3). Define R” EL’(X) by R” =R’, Vie C. Then h(R’C, TM’) =f(R’C, TNC) $ A for every T NC ELLS, so C can block A implying b$‘)s (C (= bsf). But h(R”) E A, while X\ARCA, contradiction ! 1 In view of Proposition 5.1 we can define the blocking coefficients of an anonymous ESC SDF F by by’ = by’, VA c X, where f is any anonymous ESC SCF selected from F. We say F is independently blocking if any (every) feF is. We can now state and prove the main result of this paper: Theorem 5.2. For every sequence of non-negative integers {bx}xsx satisfying xIEX b, = n + 1, the set function F = FtbXjXEX :LN(X)+2x defined by F(RN)=X\

i

XpAcX

s.t. /{i~Nj~f&)[~

1 b, ) XEAC

VR~EL~(X),

I

is an anonymous IB ESC SDF satisfying bkF)= b,, Vx EX, which is universal in the sense that if f is an anonymous ZB BSC SCF satisfying bLf’= b,, Vx EX, then feF.

In other words, for every sequence {b,},EX as above there exists an anonymous (IB) ESC SCF f with blJ) = b,, Vx E X, and all such f are equivalent up to independent selection. ProoJ Let {bX}XEX as above be fixed. For every Ac X define b, = xxsA b, and wA= n + 1 - bA. Then F = Ftb,)xeX is defined by

Let RN E LN(X) be fixed.

We claim the following are equivalent: (1) xeF(RN). (2) Let X\(X) = {zl,, . ., z,_ 1}. Then for i= 1,. . ,, m- 1 there exist Ci c N such that (a) CinCj=@, Vi#j, i=l ,...,m-1, (b) (Cil=bzi, i=l,...,m-1. (c) xRCi zi,

I. Own, Exactly

strongly consistent

217

social choice functions

To prove (l+(2) we use the following ‘Harem Lemma’, a generalization of Hall’s well-known Marriage Lemma [see Wilson (1972)]: Let B be a set of boys, and G a set of girls. For each beB let S,c G be the set of girls boy b knows, of whom b must choose a harem of exactly hb girls. Then a necessary and sufficient condition for the boys to be able to choose disjoint harems is that for every ScB, )UbsSSbl >=&hb. So assume xeF(RN). Set B=X\{x} and G= N. For each yeB let S,= {in N I&y} and h,= b,. Now let ScB. Since XEF(R~), we cannot have ({i~NlSR~x})~ws, therefore

IjsspI

=Iti+y~S

s.t. x~‘y}~=~{i~N(SR’x}‘(>~-~,,

so

as desired. We may therefore apply the lemma, and letting harem of zi, i = 1,. . . , m- 1, we have proved (l)+(2).

Ci be the chosen

The implication (2)*(l) is more straightforward. For let XEX, and assume CicN, i=l,..., m- 1, satisfy conditions (a), (b) and (c). Then for every A = X\{x>, I{ieNIAR’x}l=n-({iENIA@x}\=n-

snso xeF(RN)

and (2)*(l)

Zi;!{i~N(xRi~j} I

u

Cj=n-

c

IZj”A

I

Zj”A

b$
is proved.

We now prove the non-emptiness of F(RN), i.e., the existence of an XEX satisfying condition (2), by induction on m = 1x1. For m = 1 the assertion is trivial. Assuming the result for m - 1, we prove it for m (2 2). For every x E X let B,= {iE NIX is i’s last choice, i.e., (X\{x})R’x}. We claim that for at least one YEX, I&I>= b,. For assume to the contrary that JB,(< b,, VXEX. Then n=C,..IB,I~C,..(b,-l)=n+l-m
218

I. Oren, Exactly strongly consistentsocial choice functions

n’=IN’J, so we may use the hypothesis for m-l to obtain an x E X’ and coalitions C,, . , . , C,_ i c N’ satisfying

(a’)

CinCj=@,

Vi#j,

@‘I

JCil=bzi,

i=2

w

xRfC’zi,

i=2,...,m-1,

the existence

of

,..., m-l,

Setting z1 =y and C, =B we see that where X’={x}u{z,,...,z,_,}. conditions (a), (b) and (c) are met, so we have proved the hypothesis for m. The above is an adaptation of Peleg’s (1978) proof of the existence of ESC SCF’s. Since F(RN)#Q),VRN~LN(X), we may from here on call F a social decision function, F is clearly anonymous. We now prove that f E F implies that f is an ESC SCF. For assume fe F, and let RN ELM. Let x =f(RN)~ F(RN), and let z1 ,..., z,_i~X, Ci ,..., C,_, c N be as in (2). Choose QN E LN(X) s.t.

and (ii)

X\{zilQCizi,

i=l,...,m-1.

Since x E F(RN) we must have wx\(,.) > 0, so w, = II + 1 - w,~,) 5 n. Therefore (i) implies F(Q”)c {x}, and sincef(QN)E F(QN),f(QN)=x=f(RN). To show that f is ESC it remains to show that Q” is an SEP of the game G(f; RN). To prove this, let (b# C c N be a coalition and Tc E Lc(X) a preference profile for the players in C. Let y=f(Q NC, T’), i.e., y =f(SN), where S’ = Q’ for ie N\C and S’= T’ for ie C, and assume that y #x. Let zi* =y, where zi,. . ., z,_ 1 are defined as in (ii). Then C and C,=Czi* must intersect non-trivially, otherwise X\{y}QCy y implies X\{y} S’yy, so IC,( = b, = We{,,) would imply that y # F(SN) and f(SN) # y, contradiction ! Therefore 3 iE Cn C,. But C, was originally proving that Q” is indeed an chosen s.t. xRCyy, so f(Q WC TC)=yjtix=f(QN), SEP of the game G(f; RN),‘and f is ESC as desired. The observation that a Q” satisfying (i) and (ii) is an SEP was made by Peleg (1978). So far, we have proved that F is an anonymous ESC SDF. We have yet to prove that br)= b,, Vx E X, (and consequently, that F is IB). To this end let f~ F be an anonymous ESC) SCF. We prove that WY)= w,,V,A cX, which implies that br) = b$J)= w$&, = wWtxl= b,, Vx E X, as desired. So let A c X. Then WY’S wA since if RNeLN(X) and XEX satisfy [{~ENIAR’x}~ Zw, then x # F(RN), and consequently f(RN)#x. Therefore any coalition of w,_, WY)5 wA. Similarly members is winning with respect to A, implying U) 5 WAC, so by Corollary WAC 3 3 WY’+ w!$ =n+ 1= wA + wAc, showing tha; the inequalities must be equalitiei, and WY)= wA as desired.

1. Oren, Exactly

strongly

consistent social choice functions

219

to prove the determination up to independent selection Finally, equivalence of anonymous IB ESC SCF’s by their blocking coefficients, it remains to show that F is universal, i.e., that if f is an anonymous (IB) ESC SCF satisfying bif)= b,, VXGX, then f~ F. So let f be as such, and let RN~LN(X) and x=f(RN). Then for every AcX, ({i~N1AR’x}l
=n+l-

c

b,=wA,

XEA

and f (RN) = x E F(RN) as desired.

1

We conclude by answering a question posed by Peleg (1978) regarding monotonicity of certain anonymous (IB) ESC SCF’s (Corollary 5.4).

the

Definition. An SCF f is said to be monotonic if for every RNeLN(X), the following implication holds upon setting x =f (RN) : If QN E LN(X) satisfies yQ’z whenever yR’z, where yeX, z~X\(z} and ie N (i.e., QN is a copy of RN where just the position of x might be improved), then f (QN) = x. Definition. Let F be an SDF and f an SCF with f E F. f will be said to be selected independently of irrelevant choices (relative to F) if whenever RN,QN~LN(X) satisfy F(QN)cF(RN) and f(RN)~F(QN), we have f(QN) =f (RN). Lemma 5.3. Let f be an anonymous IB ESC SCF and F the appropriate universal SDF s.t. fe F. If f is selected independently of irrelevant choices (relative to F), then f is monotonic. Proof. Let RNeLN(X), x=f (RN), and let QNeLN(X) satisfy yR’z=>yQ’z for every i E N, y E X and z E X\{x}. The proof consists of verifying two facts: (i) XEF(Q~). VAcX, (iENIAQix}c{iENIARix}, and since xeF(RN), have I{iEN(AQix}~~I{i~NIARix}(
of irrelevant

VAcX,

choices

Corollary 5.4. Every anonymous IB BSC SCF is equivalent independent selection to a monotonic anonymous ZB ESC SCF.

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up

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to

220

I. Oren, Exactly strongly consistent social choice functions

Proofi Let f be an anonymous IB ESC SCF, and let F be the appropriate universal SDF s.t. f~ F. Let > be a linear order relation over X, and define an anonymous IB ESC SCF g by g(RN) = max,,F( R~j x, where the maximum is taken in the sense of >. Then g is selected independently of irrelevant choices with respect to F, g is monotonic by Lemma 5.3, and f and g are 1 equivalent up to independent selection.

References Dutta, B. and P. K. Pattanaik, 1978, On nicely consistent voting systems, Econometrica 46, 163170. Peleg, B., 1978, Consistent voting systems, Econometrica 46, 153-161. Polishchuk, I., 1978, Monotonicity and uniqueness of consistent voting systems, RM no. 32 (Center for Research in Mathematical Economics and Game Theory, The Hebrew University, Jerusalem). Wilson, R., 1972, Introduction to graph theory (Oliver & Boyd, Edinburgh).