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Stability of sincere voting under some classes of non-binary group decision procedures
JOURNAL
OF ECONOMIC
THEORY
8, 206-224 (1974)
Stability of Sincere Voting Under of Non-Binary Group Decision
Some Classes Procedures
PRASANTAK. PATTANAIK Delhi School of Economics, University of Delhi, Delhi 110007, India Received May 7, 1973
Murakami [S] and Pattanaik [9] have investigated the problem of stability of sincere voting for different classes of group decision procedures.l However, most of the group decision procedures considered by these writers are binary, i.e., they are based on pairwise comparison of alternatives. The problem of stability of sincere voting under non-binary group decision procedures has received much less attention though Murakami [S] does have some highly suggestive remarks on the subject and Pattanaik [9] considers a necessary condition for stability under any group decision rule. The purpose of this paper is to consider the stability of sincere voting-under three large classes of group decision procedures not based on pairwise comparison of alternatives: (1) simple finite ranking rules which have received considerable attention from voting theorists ever since the time of Jean-Charles de Borda, (2) finite ranking rules which constitute a generalization of simple finite ranking rules, and (3) methods of exhaustive voting based on simple finite ranking rules. Using the concept of stability introduced by Pattanaik [912 it is shown that every simple finite ranking rule violates stability for some sincere voting situation. Corresponding negative results are then established to cover the class of finite ranking rules satisfying certain reasonable restrictions and also the class of exhaustive voting methods based on simple finite 1 Gibbard [5] proves an important result on the stability of sincere voting under what he calls voting schemes. The structure of Gibbard’s voting schemes, however, is entirely different from that of the Arrowian concept of group decision rules with reference to which Murakami [8] and Pattanaik [9] discuss the stability of sincere voting. 2 For some closely related concepts of stability see Dummet and Farquharson [3], Farquharson [4], Gibbard [5], Murakami [8], and Shubik [ll]. See also Lute and Raiffa [7] for a survey of the stability concepts used in game theory.
206 Copyright All rights
0 1974 by Academic Press, Inc. of reproduction in any form reserved.
SINCERE VOTING
ranking rules. These results are of interest in so fa.r as the group decision procedures covered by these results constitute a very wide range an include many voting procedures used in actual practice (e.g., single ballot voting,3 dual ballot voting,4 exhaustive voting5). The results are also significant in so far as simple finite ranking rules seem to be a natural extension of the principle of majority voting to the context where social ranking is determined by overall comparison rather than by pairwise comparisons, and finite ranking rules seem to be the non-binary counterpart of our intuitive notion of binary group decision rules of the representative democracy type.
I. THE NOTATION
AND SOME PRELIMINARY
~E~~~~T~~NS
S is the set of all conceivable alternatives. S is assumed to have at least three distinct alternatives. A non-empty subset of S is called an issue. L = {I,..., N} stands for the set of individuals constituting the society. We assume that N 2 3. Let A be any given issue. We speak of various types of binary weak preference relations (wpr’s) defined over A and the corresponding strict preference relations (spr’s) and indifference reXations.6 We list them in Table I, together with the properties assumed for each of them. When there is no ambiguity about the issue A we omit the subscript A in RTA , PFA , RA , Ri, etc. and write simply Xi*, Pi*, R, Ri etc. Thus R, Ri* etc. are always to be interpreted with reference to the issue under consideration. A finite sequence (R, ,..., RN) of wpr’s expressed by in~viduals-tab wpr expressed by each individual figuring exactly once in the finite sequence-will be called a situation and will be indicated by CR,). A situation is sincere iff for all i E L, Ri = R, . For every issue A, and R*, C(A, R*) is the choice set generated by R*fodbr A and is defined as follows: x E C(A, R*) iff [(x E A) & (Vy)( y E A --f xR*y)]. For every issue A and every ordering R*, C,(A, R*) = C(A, R*) and for ail integers k > 1, &(A, R*) = C([A - C,(A, R*) - ‘.. - &,(A, DEFINITION 1. A group decision rule (GDR) is a f~~ction~which for every issue A and every situation (Ri) specifies exactly one social wpr R. 3 See Murakami [S, p. 621. 4 See Murakami [S, p. 631. 5 See Black [2, pp. 69-741. 6 Let R* be any wpr defined over an issue A. P* and I* are respectively the corresponding spr and indifference relation defined as follows: for all x, y E A, xP*y iff (xR*y & w yR*x); and xI*y iff (xR*y & yR*x).
of the society
wpr
of i-th
by i-th
GA1
RA
RA*
, RA
RA
&A
Symbol used for the wpr
-
-
Ordering
Ordering (i.e., reflexivity, connectedness, and transitivity)
Properties assumed for the wpr
I”
P,*,,PA”
pA,p,,
PiA
PzA
Symbol for corresponding spr
a Par all these relations, subscript A is omitted when there is no ambiguity about the issue A.
unspecified wpr (i.e., a wpr which is not necessarily the wpr of any individual or society)
wpr
true
or sincere individual
individual
wpr expressed
Type of wpr
TABLE
I* rAr
IA
I*A
, IA
&A
IdA
Symbol for corresponding indifference relation
SINCERE VOTING
We write R = f(A, (Ri)) or just simply R = f((&>) when there is no ambiguity about the issue A. Thus, given an issue A and a situation (Ri) the G specifies the social wpr R. R generates the choice set C(A, R) for A. A, R) is thus the set of socially best elements in A arrived at under the GDR given situation (Ri). C(A, R) may contain more than one element. In such cases we assume that the final social choice from C(A, R) is made by a random mechanism which assigns equal probability to all elements of C(A, DEFINITION 2. Letfbe a GDR. Let A be any issue and (I&) and (I?,?) be any two situations. Let R = f (( RJ) and R’ = f ((Ri’)). f is bim for all x, y E A, if [(xR*y iff X&‘JI) & (yRix iff yR,‘x)] for all i E L, [(X&J iff xR’y)&(yRx iff yR’$].’ Let A be any given issue and let B and D be any two subsets o B@iDiffthereexistxEBandyEDsuchthatxR,y;B@iDiff[B@i --D @!i B]; and B@< D if [B oi D & D @i B]. For every ordering defined over the issue A, Min(A, R*) = (x j x E A M,(A, R*) = Min(A, R*); and for all integers Min([A - M,(A, R*) - M,(A, R*) - **. - MkM1(A, It*)]? R*).
We now introduce the notion of stability of situations under a GDR as well as the notion of stability of GDR’s. For the sake of brevity we give just the formal definitions; for a discussion of the intuitive basis of these concepts the reader may refer to Pattanaik [9]. In all that follows subsequently f will stand for the GDR under consideration. For any given issue A, let WA be the set of all situations (R,) such that C(A, R) is non-empty where R =f((R$)). We assume that for any given issue A, we have for all j E L, a wpr TiA (with CQrrespondi~~ spr QjA and indifference relation J/) defined over WA. We assume that every individual follows a maximin rule, the class of maximin rules being defined as follows. DEFINITION 3. Let A be any issue, and let (I&) and ( WA. Let R = f ((R,)); R’ = f ((Ri’)); C = C(A, R); and C’ = C(A, R’). The individual j follows a maximin rule just in case (&) QjA(Ri’) if t exists a positive integer k such that for all integers 1 (k > 1 > O), Mt(C and AJ8(C’, aj) are both singletons, and ([M,(C, ([M,(C, R,) Qj M,(C’, WJ] V[M,(C,RJ # 63 an Note that Definition 3 introduces a class of m one unique maximin rule. Under Definition 3 the i~d~vidna~, say j, in 9 The property of binariness is essentially the same as Arrow’s Independence of Irrelevant Alternatives.
fl: condition
of
210
PRASANTA K. PATTANAIK
comparing (Ri) and (R,‘), compares the minima of C with the minima of C’. If the minima of C are better than the minima of C’, then (Ri) QjA(Ri’). Suppose, however, C has a unique minimum; C’ has a unique minimum; and the individual is indifferent between the two minima. Then the eliminates the two minima from C and C’ and considers the reduced sets. If the reduced set derived from C is non-empty and that derived from C’ is empty, then (Ri) QjA(Ri’>. If, however, the two reduced sets are non-empty, then the individual compares the minima of the two reduced sets and so on. It may be noted that if at any stage of successive reduction, there happen to be more than one minimum in any one set then the ranking of (R& and (Ri’) by individual j will not be specified by Definition 3. Thus Definition 3 gives us a class of maximin rules; the specific maximin rule introduced by Pattanaik [9] is only one element of this general class. In this paper we assume that each individual follows some maximin rule belonging to the class of maximin rules introduced in Definition 3: the maximin rule followed may vary from individual to individual. DEFINITION 4. Given an issue A, (Ri) E WA is stabk under the GDR f iff there do not exist (R,I) E WA and non-empty E(E _CL) such that for all i E (L - E), R, = Ri’ and for all j E E, [(Rj # R,‘) & ((Ri’) Qj”(Ri))].
A GDRfis stable iff for every issue A, every sincere situation belonging to WA is stable underf. Intuitively, what Definition 4 says is this: given an issue A, (RJ E WA is unstable under fiff there exists a non-empty coalition E of individuals with the will and the power to shift from situation (Ri) to some other situation (R,‘) E WA. If there exist such E and (Ri’) then we say that (Ri) is vulnerable to E and is unstable with respect to (Ri’>.s II. STABILITY
UNDER
FINITE PANKING
RULES
In this section we introduce an important class of GDRs to be called finite ranking rules,s and discuss the stability of sincere voting for this class of GDRs. DEFINITION 5. A finite ranking operator (FRO) is a function g which for every issue A and every finite sequence of orderings (RI*,..., R,*) specifies exactly one R* according to the following procedure: * See Farquharson [4, pp. 51-521. 9 As mentioned earlier, the literature on finite ranking rules goes as far back as JeanCharles de Borda. For a discussion of Borda’s work see Black [2, pp. 156-1591. For some recent discussions of finite ranking rules see Goodman and Markowitz [6], Murakami [S, pp. 64-671, and Sen [lo].
L-211
SINCERE VOTING
(i) A set of real numbers e, ,..., e, (which may vary with the issue 4 but which are invariant with respect to the finite sequence of orderings, given the issue A) is fixed such that n is the num.ber of elements in id; for all t’ (1 > i > y1- l), ei > ei+l ; and if y1 > 1, e, > e, ; (ii) For all x E A and all i E (l,..., v>, wia is specified such that wiJ: = ej iff x E C,(A, R,*); (iii) For all x, y E A, xR*y iff w, = i
wiz > w, = & u’iv ”
i=l
We write R* = g(R,*, R2*,..., R,*). We shall be concerned with complex structures where the image of a finite sequence CR,*) by an FRO g figures in another finite sequence (Rj*) the image of which by g in turn may figure in a third finite sequence, and so on up to a finite number of stages. For example consider an FRO g such that for any issue A, e, = 1 and e2 = ..* = e, = 0. Let the issue be (x, y, z]~ Let RI*, R,* and R,* be three orderings such that (xP,*yP,*z & zP,*yP,*x yl,*zP,*x). lo Consider the expression gUG*, g(&*,
R,*, g(R,*, &*)I,
&*)
In evaluating this expression we start with the innermost g and Then we evaluate evaluate g(R,*, R,*) which gives us xI,*y&*z. g(R,*, R3*, xIi*yIi*z) which gives us zP,*yP,*x. Hence the final ordering that emerges as the value of the given expression is zPi*xPi*y
= g(R,*, zPi*y
DEFINITION (7.1). A GDRf is a finite ranking rule (FRR) exists a finite string of symbols g(...) such that
(i) (ii)
g is an FRO; each symbol in the string is one of the following
lo Note that whenever Ri* is an ordering instead of writing (xPi*y & yp,*z (y&*z & zP,*x &yPi*x) etc. we shall write xPi*yPi*z, yI,*zP,*x etc. 64.2/8/z-8
iff there
& xp,*z&
212
PRASANTA K. PATTANAIK
(iii) the value of the string is unambiguously computable from inside out according to the method outlined earlier; (iv) for every issue A and every ( Rj) the value of the string so computed is equal to the value off((R,j); (v) the outermost g in the finite string has more than one entry and no single individual’s ordering constitutes half or more than half of the entries for the outermost g. DEFINITION (7.2). An FRR f is a simpleJinite ranking rule (SFRR) iff there exists a finite string of symbols g(...) containing only one g and fulfilling conditions (i) through (v) figuring in Definition (7.1).
The finite string of symbols g(...) referred to in Definition (7.1) will be called a representation of the FRR under consideration.ll We first explain the concept of SFRR which is simpler than the more general concept of FRR. Under an SFRR, each individual is given a certain number of ballots; a given individual may have more than one ballot but no single individual has half or more than half of the total number of ballots. Given the issueA, each individual indicates his ordering of the alternatives in A on each of the ballots given to him. Also, given the issue A, rank numbers are assigned to the first preference, second preference etc. that an alternative may receive in a ballot. The rank numbers specified for a given issue will differ depending on the SFRR; however, for every SFRR the rank numbers have to satisfy the restriction that e, > e2 3 *.m> e, and e, > e, (where II is the number of elements in the issue). The rank numbers for each alternative are then summed up over all the ballots and the social ordering decided on the basisof the aggregate number received by each alternative. An SFRR is a special case of FRRs which in general can have much more complex structures. Under an SFRR the aggregation of individual orderings takes place in a single stage. SFRRs are thus forms of direct democracy.’ Under an FRR, in general, aggregation may take place in several stages.This would correspond to a type of representative democracy where the aggregation at each stageis achieved through a given FRO. Consider the example given after Definition 5. Let L = (1, 2, 3) and let RI* = RI, R,* = R, and R,*. = R3. Then the expression given there becomesg(R, , g(R, , R3 , g(R, , R3)), R,) which is a representation of an FRR. The interpretation of this structure is fairly obvious. Individuals 1 I1 Note that an FRR may have more. than one representation. Consider an FRO g such that for every issue A, g specifies e, ,..., e, such that cl > e2 > ...> e, . Then g(&, R2, &) and &A&, &, &), g(R,, Rz, R3)) are clearly representations of the same FRR.
SINCERE VOTING
213
and 3 together constitute an electoral group at the lowest stage and have one ballot each at that stage; this electoral group sends a representative ordering, x~~*JJ&*z to the next higher electoral group (the second g from the left refers to this higher stage of aggregation). ~~divid~a~s 2 and 3 and the representative sent by the electoral group consisting of 1 and 3, together constitute this higher electoral group which sends an orderingzPi*yP,*x to the final or the highest electoral group. This representative together with individuals 1 and 2 constitutes the highest or final eiectoral group from which the social ordering zkPy emerges. At each stage, the aggregation is achieved through the FRO g. Note that in the perfectly general case, the ordering of a given individual may figure as more than one entry for the same g in the representation (i.e., an i~divid~~al may have more than one ballot in any given electoral group), and also it may figure as an entry for more than one g in the representation (i.e., the same individual may figure directly in more than one electoral group). Clearly FRRs constitute a very important class of non-binary G Not only many GDRs used in actual life are included in this class but also, as we shall see in Section III, many other decision procedures the structure of which may be different from that of GDRs are based on FRRs. The familiar single ballot voting is an SFRR under which every individual has exactly one ballot and the FRO is such that for al! issues, eL = 1 and ... = e, = 0. Similarly dual ballot voting discussed by ,Murakami 631 is an SFRR under which every individual has exactly one ballot he FRO is such that for all issues e, = e, = 1 and e3 = ..I = e, = 0, The FRRs satisfy certain very attractive properties such as “‘neutrality” and “monotonicity.” For a discussion of these properties of FRRs the reader can refer to Murakami [S, pp. 60-671. What is important for our purpose is that because of these properties one can consider the class of FRRs as the natural non-binary counterpart ur notion of binary democratic rules. For example, referring to SF which are based on s such that for every issue e, ,..., e, are all distinct, Murakami [S] p. 663 writes that these GDRs may be regarded as “the legitimate overallcomparison-type extension of simple majority voting.“12 Again, informally anticipating our notion of FRRs, Murakami ES,,p. 671 identifies the class of FRR.s with the class of non-binary democratic GD s: ““En a realm of two alternatives, we abstracted from simple majority voting, and conceived I2 Note that our SFRR corresponds to what Murakami calls a finite ranking rule. There is, however, a subtle difference. Under Murakami’s linite ranking rule (which like our SFRR involves a single stage aggregation) each individual has exactly one ballot but the numbers e, ,..., e, specified by the FRO can wry from individual to individual. Under our definition the numbers e, ,..., e, specified by the FRO are the same for all individuals but different individuals may have different number of ballots.
214
PRASANTA
K. PATTANAIK
a voting operator. In the present generalized setting, we might similarly conceive something like ‘finite ranking operator.’ Than we can define analogously, a democracy as a social decision function consisting of and only of ‘finite ranking operator’.” Thus it is of considerable importance to explore the stability of sincere situations under SFRRs as well as under FRRs in general. This is what we seek to do in Theorems 1 and 2 which follow. To prove these theorems we shall need some additional notation. Let g(...) be the representation of an FRR. The entries for the outermost g will be indicated by RIO, R,O etc. Let Z be the set of all entries for the outermost g in the representation and let Zi be any subset of Z. Then V(ZJ stands for the set of all the individuals whose orderings are involved directly or indirectly in arriving at the entries of the outermost g, which figure in Zi ; and Ni stands for the number of elements in Zi . An example will clarify our terminology. Let the representation be g(R, , R, , R2, R, , g(R, , R, , R5)). Then the outermost g has five entries R, , R, , R, , R, and g(R, , R, , R5) which will be indicated by RIO, R,O, R,O, R O and R,O respectively. Let Z, = {RIO}; Z, = {R,O, R,O}; and Z, = {RIO: R,O}. Then I’(&) = (I}; Y(Z,) = (2); and V(Z,) = (3,4, 5). Nr = 1 and N, = N3 = 2. We first prove Lemma 1 which we use in proving Theorem 1. LEMMA 1. Let f be an FRR with a representation g(...) such that no individual’s ordering figures as an entry in more than one g in the representation. Then either Z has exactly two elements, none of which is an indiltidual ordering, or Z has three non-empty subsets Z, , Z, , and Z, such
that [WI 2 N2 b N3) 8~(N, + N, > N&l and WI),
UZ,)
and V(ZJ
are pairwise disjoint. Proof. If Z has exactly two elements, then neither of them can be an individual ordering since no individual’s ordering constitutes half or more than half of the total number of entries for the outermost g. Let there be more than two elements in Z. Then partition Z (assumed to have N* elements) into subsetsZi such that: (1) for all i, either every entry figuring in Zi is an individual ordering or Zi contains a single entry; (2) no single individual’s ordering belongs to more than one Zi and no two individuals’ orderings belong to the same Zi; (3) for all Zh and ZL in the partition if h > k then Nh > Nk where Ni is the number of elements in Zi for all i. We assumethat there are ii subsetsof Z in the partition. Note that by construction and by the assumption that no individual’s ordering figures as an entry for more than one g in the representation, V(Zi) are all pairwise disjoint. Also, by the assumptions that N* > 2, that no individual’s ordering constitutes half or more than half of the
SINCERE VOTING
215
total number of entries for the outermost g, and that no individual’s ordering figures as an entry for more than one g, Ni < N”/2 for al Consider the partition of Z. Let j’ be the greatest j such that i
Ni < N*/2.
i=l
(Such a greatest j exists since for all i, Ni < N*/2). Now consider Nj’+l. Either Gil;:” Ni = N*/2 or C:Ly Ni > N*/2. If C:I=:” Ni > N*/2 then let 2, = 21 v 22 v . . . u Zj’; ST2= Zj’+l; and & = Z - (.& u &J. Let fii be the number of elements in zi for i = 1,2, 3. Then [(fil + N2 > N*/2 > &) & (N2 + & > N*/2 > &) & (& + f13 > fi2 + a3 > N*/2 > fi2j]. & , 2, and .& can now be suitably relabelled Z, , Zz and Z, such that ipr, 3 Nz > N, (note that after relabelling .?!I may become Z, or Z, ) gs vnuy become Z, or Z, and so on). Clearly, Z, , Z, and Z3 are non-empty subsets of Z such that [(N1 > N, > N3) & (Ng + N3 > NJ] and Y(Z,), V(Z,) and V(Z,) are pairwise disjoint. If C:l;’ Ni = N*/2 then consider 2, = Z1 u Z2 u ... u Zj’; 2, = Zj’+l u Z’; and Z, = Z - (Z, u 2,). Let Ni be the number of elements in Z, for i-1,2,3. Clearly fll+tz>W, and 7JJ,+nz>W1. m, + N3 > iv, then Z, , Z, , Z, can be suitabEy relabelled 2, , Z, , a 2, such that [(N1 > N, > N3) & (N, + N, > N1)] and V(Z,), V(Z,) and V(Z,) are all pairwise disjoint. Suppose, however, ?V, + N3 < ?V, . construction, (m, > N1 > Nj’fl); (w, > Nj’+2 > NE); and m, = Nj’+l + Hence, given 7J, f m3 < m, , we have ml = I\ij’+l = x3 = NE = N*/d. Then let Z, = 2, ; Z, = Zj’+l; and Z, = 2, . Then clearly
and V(Z,), V(Z,) and V(Z,) are all pairwise disjoint. This completes the proof. THEOREM
1.
Every SFRR violates stability.
ProoJ: Let f be any SFRR; let g(...> be its representation involving only one g; and let A = (x, y, z} be an issue with three distinct elements.13 The strategy of our proof will be to construct a sincere situation (R,j and an insincere situation (Ri) such that (Bi) is unstable with respect to I3Note that by assumption, S has three distinct elements.
216
PRASANTA
K.
PATTANAIK
(Ri) under J: Let W, , W, , and W, be the sum of rank numbers received by x, y, and z respectively when the situation is (R,). Similarly we have W w, , and w, for the situation (Ri).14 Let i? = f((&)) and R = f((R,)). Ni;e that since f defines a social ordering for every situation, every situation belongs to WA. By Lemma 1, and by the fact that the representation under consideration involves exactly one g, 2 has three non-empty subsets 2, , 2, and Z, such that [(N1 > N2 3 N3) & (N, + N, > N,)]; and V(Z,), V(2.J and V(Z,) are all pairwise disjoint. Let 2, = Z - (Z, u Z, u Z,). Given the issue A, g specifies e, , e2 and e3 such that e, > e2 > e3 and e, > e3 .15 There are two possibilities:
(9 Nlel + W2 + Nd e2 < W3 + W2 + NJ 6 (ii) Nlel + (N, + N3) e2 > Nle3 + (N, + NJ 6 I.
Suppose (i) holds. Then construct (Ri) and (Ri) for for for for
all all all all
such that
i e V(Z,),
XPi YPiZ & XPi ypiz
i E V(Z,),
ZPiXPi y & zP,xP, y
i 6 V(Z,),
YPiZPiX & ZPiXPi y
i E V(Z,),
xiiyiiz
8~ xI~YI,z
Then (w’, - ii&) = We1 + N2e2 + N,e, + N4el) - (he2 + N,e, + N,e, + N,eJ = Kk - e2>+ N2(e2 - 4 - N&s - 4 = We1 - e2>- N&3 - e2) + N2(e2 - 4 - N2(e2 - 4 Since N1 3 N, > N3 , W, - W, > 0. Hence xRy. Similarly it can be shown that xBz. Hence x E C(A, i?). Now consider situation (RJ. w, = Nlel + (N2 + NJ e2 + N4el and w, = Nle2 + (N2 + NJ e, + N4el . Hence given (i), w, > w, and ZPX. Hence -x E C(A, R). Since x E C(A, w) and -x E C(A, R) it is clear that
Suppose (ii) holds. Then there are two possibilities:
(4
e, = e,
(b)
el > ez
I4 See Dehition I5 See Definition
5. 5.
SINCERE
217
VOTING
11.1. Suppose e, = e2 . Then e, = e, > e3 . Then construct ( as follows:
(RJ
for for for for
all all all all
i E IV(&), i E V(Z,), i E V(Z,),
i E V(Z,),
x~S,J&Z 8L XP,YI~Z ZFiXi, y & ZP
zis,xiiy & ZPiyPiX Xii yig & Xii y&z.
Since e, = e2 it can be easily checked that W, = VGV= W, and hence Again, given e, = e2 > e3 it can be checked that C(A, R) = (z>. Hence (Ri) is vulnerable to V(Z,> u P’(Z,) and is unstable with respect to (&). C(A, R) = A.
II.2
Suppose e, > e, . Then construct (&) for all for all for all
and (RJ
such that
i E V(Z,), i E [V(Z,)
i E V(Z,),
U V(ZJ], xii
yIiz
&
Xii
y&z.
Since e, > e2 and since Nz + N3 > NI it can be checked that (zpx & zrjy). Hence C(A, R) = (z>. W, = N,e, + (N, + N3) e2 + N,e, 9 w, = 44 + N2 + N3) e2 + N,e, , and W, = NIe, f (N2 + NJ e1 + N4el . Given (ii) w, 3 w, and hence xRz. Also given that e, > e2 , MI, > W, 1 Hence xPy. Since (xRz & xPy>, x E C(A, R) and -y E C(A, R). (&) is vulnerable to V(Z,) and is unstable with respect to (Ri completes the proof. Theorem 1 is rather depressing in so far as it shows that some sincere situations are unstable under every SFRR. As we noted earlier, SF seem to be a natural extension of the notion of majority voting to the non-binary context. The corresponding notion of indirect democracy is given by the class of FRRs. We shall show that every FRR satisfying the following condition (which we call Condition A) violates stability. Condition A: Let f be an FRR. f satisfies Condition A iff there exists a representation g(...) off such that (AI) no individual’s ordering figures as an entry for more than one g in the representation; and (A.2) for every issue X with at least two elements g specifies e, ~Q ,... such that e, > e2 ~ Condition (A.2) stipulates that the rank number for the first ind~~erenc~ class should be higher than that for any subsequent indifference class.
218
PRASANTA
K. PATTANAIK
Condition (A.l) stipulates that no individual has a direct ballot in more than one electoral group. Both the parts of Condition A seem to impose fairly mild restriction on the FRR. THEOREM
2. Every FRR satisfying Condition A violates stability.
ProoJ: Let f be an FRR with representation g(...) satisfying Conditions A.1 and A.2. Let A = {x, y, z} be any issue with three distinct elements. By Lemma 1 and by the fact that the representation satisfies A. 1, either Z has exactly two elements neither of which is an individual ordering or Z has three non-empty subsets Z, , Z, and 2, such that [(N1 3 Nz > NJ & (N, + N3 > Nd] and V(Z,), V(Z& and V(Z,> are all pairwise disjoint.
(I) Suppose Z has three non-empty subsets Z, , Z, , and Z, such that [(N1 > N, >, N3) & (N, + N3 > &)I and I’(&), V(Z,), and V(Z,) are all pairwise disjoint. Let Z, = Z - (Z, U Z, u Z,). Given the issue A, g specifies e, , e2, and e3 such that e, 3 e, > e3 . By Condition A.2, e, > e2 . So there are two possibilities (4
el > e2 > e3
(P> e, > e2 = e3 If (a) holds then it follows that if all individuals belonging to V(ZJ (j = 1,2,3,4) have identical orderings then every Rio belonging to Zi coincides with that common ordering. Note that this is not necessarily true if (/3) holds as can be seen from the following counter example: Let the representation be g(g(R, , R,), g(R, , R4)) and suppose that e, = 1 and e2 = e, = 0. In this example, RI0 = g(R, , R,) and R,O = g(R, , R4). Suppose xPlyPlz and xP,yP,z. Instead of having xP~~~P~~zwe have XPIOyl,Oz.
If (CX)holds, then we proceed exactly as in the proof of Theorem 1. All the three steps I, 11.1 and II.2 given there can be transferred here exactly as they are. However, %, , W, and W, will now refer to the sum of rank numbers received by x, y, and z respectively at the final stage of aggregation (represented by the outermost g in the representation of FRRf) given situation (R,). Similarly W, , w, and w, for situation (R& have to be reinterpreted. Suppose @) holds. Then the following is true: for all a E A, for all situations (Ri), and for j E (1, 2, 3,4}, if {a} = C,(A, R,) for all i E V(ZJ,
SINCERE
219
VOTING
then {a> = Cr(A, Rko) for all Rko E Zj . Now construct (R,) and (&) follows for for for for
all all all all
i E V(Z1),
XP, yHg
i E V(Z,),
ZP
i E V(Z,), i E V(Z,),
yP,zP,x & z.P,xP, y,
y &
as
xP* yPgz, ZPiXPi
y,
xi, yiiz 22 ~1~Y~,z.
Consider the final stage of aggregation represented by the outermost g in the representation off. From what aid above it follows that given situation (i&), {x} = C,(A, Rio) .OE Z, ; (z> = C,(A, for all Rt E Z, ; ( y> = C,(A, Rio) for all 3 andix, Y, z> = GM for all Rio E Z, . It also follows that given situation (R,), {x) = C,(A, Rio) for all Rz E Zl, and (z} = C,(A, Rio) for all Rio E (2, u Z,) and (x, y, z} = C,(A, RF) for all RF E 2,. Since (Nr >, iti, 3 N3) and e, > e2 = e3 it follows that [(W, 2 Wz) & (W. 2 WY)] where W, , W, and E, refer to the aggregate numbers received by X, y> and z respectively at the final stage of aggregation given situation (R,).. Since (N, + N3 > NJ and e, > e, = e3 it also follows that [(w, > w,) & (w, > w,)] where W, I w, and w, refer to the aggregate numbers for x, y and z respe at the final stage of aggregation given situation (Rij. From what en said above it is clear that we have (xiiiy & xRz) which implies that x E C(A, a). Also we have (ZPX & zPy) which implies that (z> = C(A, R). Hence (R,) is unstable with respect to (&) and is vulnerable to V(Z,), (II) Suppose 2 has exactly two elements R,O and R,” neither of which is an individual ordering. Let V, = V(R,O) and V2 = V(&“). Clearly VI and V, are disjoint. Consider (Ri) such that for all i E V, , xPi yPiz an ZPi ypix. As before there are two possibilities: (a) e, > e, > e3 ; and @> 6 > e2 = e3 . If (a) holds then xPloyP,% and zP,*y clear that either {x, z} C C(A, R) or C(A, R) = ( y> where If (/I) holds then xP,~~I~~z and zP,~~&~x. In this ease (x, z> = C(A, R).
Thus either (x, z> C C(A, R) or C(A, R) = ( y>. Suppose (x, z> _CC(A, Then assume that (Ri) = (R,). Construct (&‘) ;such that for all 2 E VI , yPi’xPi’z and for all i E V, , yPi’zPi’x. Since e, > e2 > e3 ) C(A, R’) = (y] where R’ = f(A, (Rl)). Hence (Bi) (= (R,)) is unstable with respect to (R,‘) and is vulnerable to V1 u V, . Suppose C(A, R) = {y>. Then from what has been said above it is clear that (fi) does not hold. Then (a) holds. In that case we have ~~~Oy~~~z
220
PRASANTA K. PATTANAIK
and zP,OyP,Ox given (&). Hence, given (R,), we have wl/ = 2e, and w, = w, = e, + es . Since C(A, R) = {u} it follows that- -2e, > e, + e3 . Consider (i?,) and (Ri) such that for all i E V, , (xlyPiz & xPiyP:z) and for all i E V, , (zP,xI,y & zPj’xl:y). Clearly C(A, a) = A. Also, since 2e, > el + e3 , we have yP”z and hence -z E C(A, R”). Hence (&) is unstable with respect to (Ri) and is vulnerable to V1 . Q.E.D.
III.
STABILITYUNDERMETHODSOFEXHAUSTIVEVOTINGBASEDON
SFRRs
In this section we consider an important variation of SFRRs, namely, methods of exhaustive voting based on SFRRs. These group decision procedures have a somewhat different structure from that of GDRs and are frequently used in small committees. DEFINITION 8. A group decision function (GDF) is a functional rule T which for every issue A and every situation (Ri) specifies a non-empty subset T(A, (Ri)) of A.
We write T = T(A, (R,)). T(A, (Ri)) is called the set of outcomes and is to be interpreted as the set of socially best alternatives in A for the situation (R,).ls Definitions 3 and 4 can now be easily extended to cover GDFs; the formulations now have to be in terms of T = T(A, (R,)) and T’ = T(A, (R,‘)) rather than in terms of C = C(A, R) and C’ = C(A, R’). Since this extension is routine we omit the formal statement, and proceed to discuss stability of sincere situations under a class of GDFs. Let f be a given SFRR; and (Ri) be any situation. For all B _CA and all i E L, RJB] stands for the restriction of Ri to B. Let Ho = A and Hr = A - C,(A,f(A, (Ri))) where m is the number of indifference classes into which A can be partioned on the basis off(A, (Ri)). For all integers k(k > 1) let Hk = H,-, - C,,,(H,-, , f(H,-, , (R,[Hk-J))) where m’ is the total number of indifference classes into which Hkpl can be partitioned on the basis off (H,_, , (R,[Hk-J)). Hk is defined only if H,-, is non-empty. DEFINITION 9. A GDF T is a method of exhaustive voting based on an SFRRfiff for all issues A and every situation (Ri) there exists an integer k(k 3 0) such that T(A, (Ri)) = C(H, ,f(Hk , (RJH,]))) = Hk .
Under a method of exhaustive voting based on an SFRRJ;
we first
I6 Weuse the term “set of outcomes” rather than the more usual “choice set” so as to distinguish this concept clearly from the concept of a “choice set generated by R” which we have defined earlier.
SINCERE
VOTING
221
consider the ordering f(A, (IQ). The lowest indifference class in A defined in terms off (A, (R,)) is then eliminated from A and thus we get HI _ Then we take the orderingf(H, , (R&Y&). These successive rounds of elimination continue until we arrive at a subset of A such that aggregation through f yields a single indifference class.17 The type of exhaustive voting discussed by Black [2, pp. 69-721 constitutes a special case of the group decision procedures covered by our Definition !9~ls We now discuss the stability of sincere situations under methods of exhaustive voting based on SFRRs. We proceed to Theorem 3 via the following lemma. b?MMh 2. Let T be a method of exhaustiue voting based on SF and let A = (x, y, z> be any issue with three distinct elements = f (A, (&j) where (Ri) is any situation. Suppose (XPZ I%yPz) and supposexPiOy is true for more than half of the total mber of RiO in the representation off such that -(xl,“yIioz). Then T(A, .)) = (x).
Proof Let B = (x, y>. Consider the partition of A into indifference classeson the basis of R. Since (xPz & yPz) clearly (23 constitutes the lowest indifference class. Hence after the first round of elimination we shall be left with B. Let the rank numbers specified under be e, and e, . Clearly e, > e2. Consider w, and w, for sit Let 15’be the total number of &” such that
I7 Note that after each round of elimination we are essentially considering the preferences expressed by the individuals to start with. For example, if iR,) refers to the preferences we start with, we consider (R,[H,]) after the first round of elimination where for all i,
222
PRASANTA
K.
PATTANAIK
be the number of Rio such that (~l,~vl,~z). Since xP,Oy is true of more than N/2 of Rio such that +x~~~JJ~~~z), it is clear that w, > @J/2) 6 + @J/2) e2 + N’e, > w, Hence xPy. Therefore in the second round of elimination y will be eliminated and we shall be left with {x> which will emerge as T(A, (Rd)). Q.E.D. THEOREM 3. Every method of exhaustive violates stability.
voting based on an SFRR
Proof. Let T be a method of exhaustive voting based on an SFRR jY Let g(...) be a representation offinvolving only one g, and let A = (x, y, z> be an issue with three distinct elements. We shall consider a sincere situation (Bi) and two insincere situations (Ri) and (Ri’). Let W, , WY and W, be the sum of rank numbers received respectively by x, y and z under SFRRfin situation (R,) given the issue A. Similarly we have w, , w, and w, for (Ri) and We’, wy’, and wz’ for Nz 3 N3) & (N, + N3 > N,)], and V(Z,), V(Z& and V(Z,> are all pairwise disjoint. Let
z, = z - (Z, u z, u Z,). Construct (&), for for for for
all all all all
(R,),
and (Ri’)
such that
i E V(Z,),
XPi YPiZ & XPi ypiz & yPifzPi’x;
i E V(Z.J,
ZP,XPi y & zP,xP, y & ZPi’xPi’y;
i E V(Z,),
YPiZPiX &ZPiXP
i E V(Z,),
Xiiyiiz
&yPi'ZPi'X;
& XI~ ~~~~ & XI~'~JI~~'Z.
Let R = f(A, (K,)); R = f(A, e2 or e2 > e3) it can be checked that w, > w, and hence xPy. It can also be checked that since [(N, + N3 > N1) & (e, 2 e2 3 e3) & (e, > e2 or e, > e,)], w, > w, and hence zPy. Since (xPy & zPy); for all i E [V(Z,) u V(Z,)], zP,x; and N2 + N3 > N1 , by Lemma 2, T(A, (Ri)) = {z>. Next consider (Ri'). Again it can be checked that since (e, > e2 or e2 > es), w,’ > w,’ and hence zP’x. It can also be checked that since [(Nl + N3 > N,) & (e, 2 e2 > es) & (e, > e, or e2 > e,)] we have
SINCERE VOTING
223
wg’ > wz’ so that yP’x. Since (zP’x & yP’x); for all i E [V(Z,) v V(Z,)], yPi’z; and N, + N3 > N2 , by Lemma 2, T(A, (&‘)) = ( yj. Lastly, consider (Wi). From the proof of Theorem 1 we know that (xEy & xKz). Then there are the following possibilities - - -(1) xry1z (2) xPyPz (3) 3cPzPy - - (4) xfzFy (5) xrypz (6) xPyP2 If (1) holds then clearly T(A, (&)) = A. If (2) or (5) holds, then given that for all i E [ V(Z,) u V(Z,)], xpiy, and that N1 + N2 > N3 , we have T(A, (a,)) = (x> by Lemma 2. If (3) or (4) holds then given that for all i E [ V(ZJ u V(Z,)], zHix, and that N,+N, > N1, we have T((A, ( by Lemma 2. If (6) holds, then clearly T(A, (&)) = (xl. Thus either x E T(A, (&)) or T(A, (w,)) = (z>. Hf x E T(A then er T. (&) is unstable with respect to (Ri) and is vulnerable to V( If T(A, (&)) = (z>, then (R,) is unstable with respect to (&‘) and is vulnerable to V(Z,) under T. This completes the proof.
IV. CONCLUSION Using the notion of stability as defined in Pattanaik [9] we examined the stability of sincere situations for three very wide classes of group decision procedures: (1) simple finite ranking rules; (2) finite ranking rules (which include as a subclass the class of simple finite ranking rules); and (3) methods of exhaustive voting based on simple finite ranking rules. It was shown that under every group decision procedure belonging to the first and the third class, some sincere situations are unstable for some issues. It was also shown that under every group decision procedure which belongs to the second class and which satisfies some rather mild conditions, some sincere situations are unstable for some issues. These conclusions are disturbing in so far as the group decision procedures covered by these results constitute a very wide class of non-binary “democratic” decision procedures and include many decision procedures (e.g. single ballot voting, dual ballot voting, exhaustive voting) used in actual practice. REFERENCES “Social Choice and Individual Values,” 1st ed. (1951), 2nd ed. (1964) Wiley and Sons, Inc., New York. 2. D. BLACK, “The Theory of Committees and Elections,” Cambridge University Press,” Cambridge, England, 1958. 1. K. J. ARROW,
224
PRASANTA
K.
PATTANAIK
3. M. DUMMET AND R. FARQUHARSON, Stability in voting, Econometrica, 29 (1961). 4. R. FARQUHARSON,“Theory of Voting,” Yale University Press, New Haven, 1969. 5. A. GIBBARD, Manipulation of voting schemes: a general result, Econometrica (forthcoming). 6. L. GOODMAN AND H. MARKOWITZ, Social welfare function based on individual rankings, American Journal of Sociology, 60 (1952). 7. R. D. LUCE AND H. RAIFFA, “Games and Decisions,” Wiley and Sons, Inc., New York, 1957. 8. Y. MURAKAMI, “Logic and Social Choice,” Dover Publications Inc., New York, 1968. 9. P. K. PA-~TANAIK, On the Stability of Sincere Voting Situations, J. Econ. Theory 6 (1973), 558-574. 10. A. K. SEN, “Collective Choice and Social Welfare,” Oliver and Boyd, London, 1970. 11. M. SHUBIK, “Strategy and Market Structure,” Wiley and Sons, New York, 1959.
OF ECONOMIC
THEORY
8, 206-224 (1974)
Stability of Sincere Voting Under of Non-Binary Group Decision
Some Classes Procedures
PRASANTAK. PATTANAIK Delhi School of Economics, University of Delhi, Delhi 110007, India Received May 7, 1973
Murakami [S] and Pattanaik [9] have investigated the problem of stability of sincere voting for different classes of group decision procedures.l However, most of the group decision procedures considered by these writers are binary, i.e., they are based on pairwise comparison of alternatives. The problem of stability of sincere voting under non-binary group decision procedures has received much less attention though Murakami [S] does have some highly suggestive remarks on the subject and Pattanaik [9] considers a necessary condition for stability under any group decision rule. The purpose of this paper is to consider the stability of sincere voting-under three large classes of group decision procedures not based on pairwise comparison of alternatives: (1) simple finite ranking rules which have received considerable attention from voting theorists ever since the time of Jean-Charles de Borda, (2) finite ranking rules which constitute a generalization of simple finite ranking rules, and (3) methods of exhaustive voting based on simple finite ranking rules. Using the concept of stability introduced by Pattanaik [912 it is shown that every simple finite ranking rule violates stability for some sincere voting situation. Corresponding negative results are then established to cover the class of finite ranking rules satisfying certain reasonable restrictions and also the class of exhaustive voting methods based on simple finite 1 Gibbard [5] proves an important result on the stability of sincere voting under what he calls voting schemes. The structure of Gibbard’s voting schemes, however, is entirely different from that of the Arrowian concept of group decision rules with reference to which Murakami [8] and Pattanaik [9] discuss the stability of sincere voting. 2 For some closely related concepts of stability see Dummet and Farquharson [3], Farquharson [4], Gibbard [5], Murakami [8], and Shubik [ll]. See also Lute and Raiffa [7] for a survey of the stability concepts used in game theory.
206 Copyright All rights
0 1974 by Academic Press, Inc. of reproduction in any form reserved.
SINCERE VOTING
ranking rules. These results are of interest in so fa.r as the group decision procedures covered by these results constitute a very wide range an include many voting procedures used in actual practice (e.g., single ballot voting,3 dual ballot voting,4 exhaustive voting5). The results are also significant in so far as simple finite ranking rules seem to be a natural extension of the principle of majority voting to the context where social ranking is determined by overall comparison rather than by pairwise comparisons, and finite ranking rules seem to be the non-binary counterpart of our intuitive notion of binary group decision rules of the representative democracy type.
I. THE NOTATION
AND SOME PRELIMINARY
~E~~~~T~~NS
S is the set of all conceivable alternatives. S is assumed to have at least three distinct alternatives. A non-empty subset of S is called an issue. L = {I,..., N} stands for the set of individuals constituting the society. We assume that N 2 3. Let A be any given issue. We speak of various types of binary weak preference relations (wpr’s) defined over A and the corresponding strict preference relations (spr’s) and indifference reXations.6 We list them in Table I, together with the properties assumed for each of them. When there is no ambiguity about the issue A we omit the subscript A in RTA , PFA , RA , Ri, etc. and write simply Xi*, Pi*, R, Ri etc. Thus R, Ri* etc. are always to be interpreted with reference to the issue under consideration. A finite sequence (R, ,..., RN) of wpr’s expressed by in~viduals-tab wpr expressed by each individual figuring exactly once in the finite sequence-will be called a situation and will be indicated by CR,). A situation is sincere iff for all i E L, Ri = R, . For every issue A, and R*, C(A, R*) is the choice set generated by R*fodbr A and is defined as follows: x E C(A, R*) iff [(x E A) & (Vy)( y E A --f xR*y)]. For every issue A and every ordering R*, C,(A, R*) = C(A, R*) and for ail integers k > 1, &(A, R*) = C([A - C,(A, R*) - ‘.. - &,(A, DEFINITION 1. A group decision rule (GDR) is a f~~ction~which for every issue A and every situation (Ri) specifies exactly one social wpr R. 3 See Murakami [S, p. 621. 4 See Murakami [S, p. 631. 5 See Black [2, pp. 69-741. 6 Let R* be any wpr defined over an issue A. P* and I* are respectively the corresponding spr and indifference relation defined as follows: for all x, y E A, xP*y iff (xR*y & w yR*x); and xI*y iff (xR*y & yR*x).
of the society
wpr
of i-th
by i-th
GA1
RA
RA*
, RA
RA
&A
Symbol used for the wpr
-
-
Ordering
Ordering (i.e., reflexivity, connectedness, and transitivity)
Properties assumed for the wpr
I”
P,*,,PA”
pA,p,,
PiA
PzA
Symbol for corresponding spr
a Par all these relations, subscript A is omitted when there is no ambiguity about the issue A.
unspecified wpr (i.e., a wpr which is not necessarily the wpr of any individual or society)
wpr
true
or sincere individual
individual
wpr expressed
Type of wpr
TABLE
I* rAr
IA
I*A
, IA
&A
IdA
Symbol for corresponding indifference relation
SINCERE VOTING
We write R = f(A, (Ri)) or just simply R = f((&>) when there is no ambiguity about the issue A. Thus, given an issue A and a situation (Ri) the G specifies the social wpr R. R generates the choice set C(A, R) for A. A, R) is thus the set of socially best elements in A arrived at under the GDR given situation (Ri). C(A, R) may contain more than one element. In such cases we assume that the final social choice from C(A, R) is made by a random mechanism which assigns equal probability to all elements of C(A, DEFINITION 2. Letfbe a GDR. Let A be any issue and (I&) and (I?,?) be any two situations. Let R = f (( RJ) and R’ = f ((Ri’)). f is bim for all x, y E A, if [(xR*y iff X&‘JI) & (yRix iff yR,‘x)] for all i E L, [(X&J iff xR’y)&(yRx iff yR’$].’ Let A be any given issue and let B and D be any two subsets o B@iDiffthereexistxEBandyEDsuchthatxR,y;B@iDiff[B@i --D @!i B]; and B@< D if [B oi D & D @i B]. For every ordering defined over the issue A, Min(A, R*) = (x j x E A M,(A, R*) = Min(A, R*); and for all integers Min([A - M,(A, R*) - M,(A, R*) - **. - MkM1(A, It*)]? R*).
We now introduce the notion of stability of situations under a GDR as well as the notion of stability of GDR’s. For the sake of brevity we give just the formal definitions; for a discussion of the intuitive basis of these concepts the reader may refer to Pattanaik [9]. In all that follows subsequently f will stand for the GDR under consideration. For any given issue A, let WA be the set of all situations (R,) such that C(A, R) is non-empty where R =f((R$)). We assume that for any given issue A, we have for all j E L, a wpr TiA (with CQrrespondi~~ spr QjA and indifference relation J/) defined over WA. We assume that every individual follows a maximin rule, the class of maximin rules being defined as follows. DEFINITION 3. Let A be any issue, and let (I&) and ( WA. Let R = f ((R,)); R’ = f ((Ri’)); C = C(A, R); and C’ = C(A, R’). The individual j follows a maximin rule just in case (&) QjA(Ri’) if t exists a positive integer k such that for all integers 1 (k > 1 > O), Mt(C and AJ8(C’, aj) are both singletons, and ([M,(C, ([M,(C, R,) Qj M,(C’, WJ] V[M,(C,RJ # 63 an Note that Definition 3 introduces a class of m one unique maximin rule. Under Definition 3 the i~d~vidna~, say j, in 9 The property of binariness is essentially the same as Arrow’s Independence of Irrelevant Alternatives.
fl: condition
of
210
PRASANTA K. PATTANAIK
comparing (Ri) and (R,‘), compares the minima of C with the minima of C’. If the minima of C are better than the minima of C’, then (Ri) QjA(Ri’). Suppose, however, C has a unique minimum; C’ has a unique minimum; and the individual is indifferent between the two minima. Then the eliminates the two minima from C and C’ and considers the reduced sets. If the reduced set derived from C is non-empty and that derived from C’ is empty, then (Ri) QjA(Ri’>. If, however, the two reduced sets are non-empty, then the individual compares the minima of the two reduced sets and so on. It may be noted that if at any stage of successive reduction, there happen to be more than one minimum in any one set then the ranking of (R& and (Ri’) by individual j will not be specified by Definition 3. Thus Definition 3 gives us a class of maximin rules; the specific maximin rule introduced by Pattanaik [9] is only one element of this general class. In this paper we assume that each individual follows some maximin rule belonging to the class of maximin rules introduced in Definition 3: the maximin rule followed may vary from individual to individual. DEFINITION 4. Given an issue A, (Ri) E WA is stabk under the GDR f iff there do not exist (R,I) E WA and non-empty E(E _CL) such that for all i E (L - E), R, = Ri’ and for all j E E, [(Rj # R,‘) & ((Ri’) Qj”(Ri))].
A GDRfis stable iff for every issue A, every sincere situation belonging to WA is stable underf. Intuitively, what Definition 4 says is this: given an issue A, (RJ E WA is unstable under fiff there exists a non-empty coalition E of individuals with the will and the power to shift from situation (Ri) to some other situation (R,‘) E WA. If there exist such E and (Ri’) then we say that (Ri) is vulnerable to E and is unstable with respect to (Ri’>.s II. STABILITY
UNDER
FINITE PANKING
RULES
In this section we introduce an important class of GDRs to be called finite ranking rules,s and discuss the stability of sincere voting for this class of GDRs. DEFINITION 5. A finite ranking operator (FRO) is a function g which for every issue A and every finite sequence of orderings (RI*,..., R,*) specifies exactly one R* according to the following procedure: * See Farquharson [4, pp. 51-521. 9 As mentioned earlier, the literature on finite ranking rules goes as far back as JeanCharles de Borda. For a discussion of Borda’s work see Black [2, pp. 156-1591. For some recent discussions of finite ranking rules see Goodman and Markowitz [6], Murakami [S, pp. 64-671, and Sen [lo].
L-211
SINCERE VOTING
(i) A set of real numbers e, ,..., e, (which may vary with the issue 4 but which are invariant with respect to the finite sequence of orderings, given the issue A) is fixed such that n is the num.ber of elements in id; for all t’ (1 > i > y1- l), ei > ei+l ; and if y1 > 1, e, > e, ; (ii) For all x E A and all i E (l,..., v>, wia is specified such that wiJ: = ej iff x E C,(A, R,*); (iii) For all x, y E A, xR*y iff w, = i
wiz > w, = & u’iv ”
i=l
We write R* = g(R,*, R2*,..., R,*). We shall be concerned with complex structures where the image of a finite sequence CR,*) by an FRO g figures in another finite sequence (Rj*) the image of which by g in turn may figure in a third finite sequence, and so on up to a finite number of stages. For example consider an FRO g such that for any issue A, e, = 1 and e2 = ..* = e, = 0. Let the issue be (x, y, z]~ Let RI*, R,* and R,* be three orderings such that (xP,*yP,*z & zP,*yP,*x yl,*zP,*x). lo Consider the expression gUG*, g(&*,
R,*, g(R,*, &*)I,
&*)
In evaluating this expression we start with the innermost g and Then we evaluate evaluate g(R,*, R,*) which gives us xI,*y&*z. g(R,*, R3*, xIi*yIi*z) which gives us zP,*yP,*x. Hence the final ordering that emerges as the value of the given expression is zPi*xPi*y
= g(R,*, zPi*y
DEFINITION (7.1). A GDRf is a finite ranking rule (FRR) exists a finite string of symbols g(...) such that
(i) (ii)
g is an FRO; each symbol in the string is one of the following
lo Note that whenever Ri* is an ordering instead of writing (xPi*y & yp,*z (y&*z & zP,*x &yPi*x) etc. we shall write xPi*yPi*z, yI,*zP,*x etc. 64.2/8/z-8
iff there
& xp,*z&
212
PRASANTA K. PATTANAIK
(iii) the value of the string is unambiguously computable from inside out according to the method outlined earlier; (iv) for every issue A and every ( Rj) the value of the string so computed is equal to the value off((R,j); (v) the outermost g in the finite string has more than one entry and no single individual’s ordering constitutes half or more than half of the entries for the outermost g. DEFINITION (7.2). An FRR f is a simpleJinite ranking rule (SFRR) iff there exists a finite string of symbols g(...) containing only one g and fulfilling conditions (i) through (v) figuring in Definition (7.1).
The finite string of symbols g(...) referred to in Definition (7.1) will be called a representation of the FRR under consideration.ll We first explain the concept of SFRR which is simpler than the more general concept of FRR. Under an SFRR, each individual is given a certain number of ballots; a given individual may have more than one ballot but no single individual has half or more than half of the total number of ballots. Given the issueA, each individual indicates his ordering of the alternatives in A on each of the ballots given to him. Also, given the issue A, rank numbers are assigned to the first preference, second preference etc. that an alternative may receive in a ballot. The rank numbers specified for a given issue will differ depending on the SFRR; however, for every SFRR the rank numbers have to satisfy the restriction that e, > e2 3 *.m> e, and e, > e, (where II is the number of elements in the issue). The rank numbers for each alternative are then summed up over all the ballots and the social ordering decided on the basisof the aggregate number received by each alternative. An SFRR is a special case of FRRs which in general can have much more complex structures. Under an SFRR the aggregation of individual orderings takes place in a single stage. SFRRs are thus forms of direct democracy.’ Under an FRR, in general, aggregation may take place in several stages.This would correspond to a type of representative democracy where the aggregation at each stageis achieved through a given FRO. Consider the example given after Definition 5. Let L = (1, 2, 3) and let RI* = RI, R,* = R, and R,*. = R3. Then the expression given there becomesg(R, , g(R, , R3 , g(R, , R3)), R,) which is a representation of an FRR. The interpretation of this structure is fairly obvious. Individuals 1 I1 Note that an FRR may have more. than one representation. Consider an FRO g such that for every issue A, g specifies e, ,..., e, such that cl > e2 > ...> e, . Then g(&, R2, &) and &A&, &, &), g(R,, Rz, R3)) are clearly representations of the same FRR.
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213
and 3 together constitute an electoral group at the lowest stage and have one ballot each at that stage; this electoral group sends a representative ordering, x~~*JJ&*z to the next higher electoral group (the second g from the left refers to this higher stage of aggregation). ~~divid~a~s 2 and 3 and the representative sent by the electoral group consisting of 1 and 3, together constitute this higher electoral group which sends an orderingzPi*yP,*x to the final or the highest electoral group. This representative together with individuals 1 and 2 constitutes the highest or final eiectoral group from which the social ordering zkPy emerges. At each stage, the aggregation is achieved through the FRO g. Note that in the perfectly general case, the ordering of a given individual may figure as more than one entry for the same g in the representation (i.e., an i~divid~~al may have more than one ballot in any given electoral group), and also it may figure as an entry for more than one g in the representation (i.e., the same individual may figure directly in more than one electoral group). Clearly FRRs constitute a very important class of non-binary G Not only many GDRs used in actual life are included in this class but also, as we shall see in Section III, many other decision procedures the structure of which may be different from that of GDRs are based on FRRs. The familiar single ballot voting is an SFRR under which every individual has exactly one ballot and the FRO is such that for al! issues, eL = 1 and ... = e, = 0. Similarly dual ballot voting discussed by ,Murakami 631 is an SFRR under which every individual has exactly one ballot he FRO is such that for all issues e, = e, = 1 and e3 = ..I = e, = 0, The FRRs satisfy certain very attractive properties such as “‘neutrality” and “monotonicity.” For a discussion of these properties of FRRs the reader can refer to Murakami [S, pp. 60-671. What is important for our purpose is that because of these properties one can consider the class of FRRs as the natural non-binary counterpart ur notion of binary democratic rules. For example, referring to SF which are based on s such that for every issue e, ,..., e, are all distinct, Murakami [S] p. 663 writes that these GDRs may be regarded as “the legitimate overallcomparison-type extension of simple majority voting.“12 Again, informally anticipating our notion of FRRs, Murakami ES,,p. 671 identifies the class of FRR.s with the class of non-binary democratic GD s: ““En a realm of two alternatives, we abstracted from simple majority voting, and conceived I2 Note that our SFRR corresponds to what Murakami calls a finite ranking rule. There is, however, a subtle difference. Under Murakami’s linite ranking rule (which like our SFRR involves a single stage aggregation) each individual has exactly one ballot but the numbers e, ,..., e, specified by the FRO can wry from individual to individual. Under our definition the numbers e, ,..., e, specified by the FRO are the same for all individuals but different individuals may have different number of ballots.
214
PRASANTA
K. PATTANAIK
a voting operator. In the present generalized setting, we might similarly conceive something like ‘finite ranking operator.’ Than we can define analogously, a democracy as a social decision function consisting of and only of ‘finite ranking operator’.” Thus it is of considerable importance to explore the stability of sincere situations under SFRRs as well as under FRRs in general. This is what we seek to do in Theorems 1 and 2 which follow. To prove these theorems we shall need some additional notation. Let g(...) be the representation of an FRR. The entries for the outermost g will be indicated by RIO, R,O etc. Let Z be the set of all entries for the outermost g in the representation and let Zi be any subset of Z. Then V(ZJ stands for the set of all the individuals whose orderings are involved directly or indirectly in arriving at the entries of the outermost g, which figure in Zi ; and Ni stands for the number of elements in Zi . An example will clarify our terminology. Let the representation be g(R, , R, , R2, R, , g(R, , R, , R5)). Then the outermost g has five entries R, , R, , R, , R, and g(R, , R, , R5) which will be indicated by RIO, R,O, R,O, R O and R,O respectively. Let Z, = {RIO}; Z, = {R,O, R,O}; and Z, = {RIO: R,O}. Then I’(&) = (I}; Y(Z,) = (2); and V(Z,) = (3,4, 5). Nr = 1 and N, = N3 = 2. We first prove Lemma 1 which we use in proving Theorem 1. LEMMA 1. Let f be an FRR with a representation g(...) such that no individual’s ordering figures as an entry in more than one g in the representation. Then either Z has exactly two elements, none of which is an indiltidual ordering, or Z has three non-empty subsets Z, , Z, , and Z, such
that [WI 2 N2 b N3) 8~(N, + N, > N&l and WI),
UZ,)
and V(ZJ
are pairwise disjoint. Proof. If Z has exactly two elements, then neither of them can be an individual ordering since no individual’s ordering constitutes half or more than half of the total number of entries for the outermost g. Let there be more than two elements in Z. Then partition Z (assumed to have N* elements) into subsetsZi such that: (1) for all i, either every entry figuring in Zi is an individual ordering or Zi contains a single entry; (2) no single individual’s ordering belongs to more than one Zi and no two individuals’ orderings belong to the same Zi; (3) for all Zh and ZL in the partition if h > k then Nh > Nk where Ni is the number of elements in Zi for all i. We assumethat there are ii subsetsof Z in the partition. Note that by construction and by the assumption that no individual’s ordering figures as an entry for more than one g in the representation, V(Zi) are all pairwise disjoint. Also, by the assumptions that N* > 2, that no individual’s ordering constitutes half or more than half of the
SINCERE VOTING
215
total number of entries for the outermost g, and that no individual’s ordering figures as an entry for more than one g, Ni < N”/2 for al Consider the partition of Z. Let j’ be the greatest j such that i
Ni < N*/2.
i=l
(Such a greatest j exists since for all i, Ni < N*/2). Now consider Nj’+l. Either Gil;:” Ni = N*/2 or C:Ly Ni > N*/2. If C:I=:” Ni > N*/2 then let 2, = 21 v 22 v . . . u Zj’; ST2= Zj’+l; and & = Z - (.& u &J. Let fii be the number of elements in zi for i = 1,2, 3. Then [(fil + N2 > N*/2 > &) & (N2 + & > N*/2 > &) & (& + f13 > fi2 + a3 > N*/2 > fi2j]. & , 2, and .& can now be suitably relabelled Z, , Zz and Z, such that ipr, 3 Nz > N, (note that after relabelling .?!I may become Z, or Z, ) gs vnuy become Z, or Z, and so on). Clearly, Z, , Z, and Z3 are non-empty subsets of Z such that [(N1 > N, > N3) & (Ng + N3 > NJ] and Y(Z,), V(Z,) and V(Z,) are pairwise disjoint. If C:l;’ Ni = N*/2 then consider 2, = Z1 u Z2 u ... u Zj’; 2, = Zj’+l u Z’; and Z, = Z - (Z, u 2,). Let Ni be the number of elements in Z, for i-1,2,3. Clearly fll+tz>W, and 7JJ,+nz>W1. m, + N3 > iv, then Z, , Z, , Z, can be suitabEy relabelled 2, , Z, , a 2, such that [(N1 > N, > N3) & (N, + N, > N1)] and V(Z,), V(Z,) and V(Z,) are all pairwise disjoint. Suppose, however, ?V, + N3 < ?V, . construction, (m, > N1 > Nj’fl); (w, > Nj’+2 > NE); and m, = Nj’+l + Hence, given 7J, f m3 < m, , we have ml = I\ij’+l = x3 = NE = N*/d. Then let Z, = 2, ; Z, = Zj’+l; and Z, = 2, . Then clearly
and V(Z,), V(Z,) and V(Z,) are all pairwise disjoint. This completes the proof. THEOREM
1.
Every SFRR violates stability.
ProoJ: Let f be any SFRR; let g(...> be its representation involving only one g; and let A = (x, y, z} be an issue with three distinct elements.13 The strategy of our proof will be to construct a sincere situation (R,j and an insincere situation (Ri) such that (Bi) is unstable with respect to I3Note that by assumption, S has three distinct elements.
216
PRASANTA
K.
PATTANAIK
(Ri) under J: Let W, , W, , and W, be the sum of rank numbers received by x, y, and z respectively when the situation is (R,). Similarly we have W w, , and w, for the situation (Ri).14 Let i? = f((&)) and R = f((R,)). Ni;e that since f defines a social ordering for every situation, every situation belongs to WA. By Lemma 1, and by the fact that the representation under consideration involves exactly one g, 2 has three non-empty subsets 2, , 2, and Z, such that [(N1 > N2 3 N3) & (N, + N, > N,)]; and V(Z,), V(2.J and V(Z,) are all pairwise disjoint. Let 2, = Z - (Z, u Z, u Z,). Given the issue A, g specifies e, , e2 and e3 such that e, > e2 > e3 and e, > e3 .15 There are two possibilities:
(9 Nlel + W2 + Nd e2 < W3 + W2 + NJ 6 (ii) Nlel + (N, + N3) e2 > Nle3 + (N, + NJ 6 I.
Suppose (i) holds. Then construct (Ri) and (Ri) for for for for
all all all all
such that
i e V(Z,),
XPi YPiZ & XPi ypiz
i E V(Z,),
ZPiXPi y & zP,xP, y
i 6 V(Z,),
YPiZPiX & ZPiXPi y
i E V(Z,),
xiiyiiz
8~ xI~YI,z
Then (w’, - ii&) = We1 + N2e2 + N,e, + N4el) - (he2 + N,e, + N,e, + N,eJ = Kk - e2>+ N2(e2 - 4 - N&s - 4 = We1 - e2>- N&3 - e2) + N2(e2 - 4 - N2(e2 - 4 Since N1 3 N, > N3 , W, - W, > 0. Hence xRy. Similarly it can be shown that xBz. Hence x E C(A, i?). Now consider situation (RJ. w, = Nlel + (N2 + NJ e2 + N4el and w, = Nle2 + (N2 + NJ e, + N4el . Hence given (i), w, > w, and ZPX. Hence -x E C(A, R). Since x E C(A, w) and -x E C(A, R) it is clear that
Suppose (ii) holds. Then there are two possibilities:
(4
e, = e,
(b)
el > ez
I4 See Dehition I5 See Definition
5. 5.
SINCERE
217
VOTING
11.1. Suppose e, = e2 . Then e, = e, > e3 . Then construct ( as follows:
(RJ
for for for for
all all all all
i E IV(&), i E V(Z,), i E V(Z,),
i E V(Z,),
x~S,J&Z 8L XP,YI~Z ZFiXi, y & ZP
zis,xiiy & ZPiyPiX Xii yig & Xii y&z.
Since e, = e2 it can be easily checked that W, = VGV= W, and hence Again, given e, = e2 > e3 it can be checked that C(A, R) = (z>. Hence (Ri) is vulnerable to V(Z,> u P’(Z,) and is unstable with respect to (&). C(A, R) = A.
II.2
Suppose e, > e, . Then construct (&) for all for all for all
and (RJ
such that
i E V(Z,), i E [V(Z,)
i E V(Z,),
U V(ZJ], xii
yIiz
&
Xii
y&z.
Since e, > e2 and since Nz + N3 > NI it can be checked that (zpx & zrjy). Hence C(A, R) = (z>. W, = N,e, + (N, + N3) e2 + N,e, 9 w, = 44 + N2 + N3) e2 + N,e, , and W, = NIe, f (N2 + NJ e1 + N4el . Given (ii) w, 3 w, and hence xRz. Also given that e, > e2 , MI, > W, 1 Hence xPy. Since (xRz & xPy>, x E C(A, R) and -y E C(A, R). (&) is vulnerable to V(Z,) and is unstable with respect to (Ri completes the proof. Theorem 1 is rather depressing in so far as it shows that some sincere situations are unstable under every SFRR. As we noted earlier, SF seem to be a natural extension of the notion of majority voting to the non-binary context. The corresponding notion of indirect democracy is given by the class of FRRs. We shall show that every FRR satisfying the following condition (which we call Condition A) violates stability. Condition A: Let f be an FRR. f satisfies Condition A iff there exists a representation g(...) off such that (AI) no individual’s ordering figures as an entry for more than one g in the representation; and (A.2) for every issue X with at least two elements g specifies e, ~Q ,... such that e, > e2 ~ Condition (A.2) stipulates that the rank number for the first ind~~erenc~ class should be higher than that for any subsequent indifference class.
218
PRASANTA
K. PATTANAIK
Condition (A.l) stipulates that no individual has a direct ballot in more than one electoral group. Both the parts of Condition A seem to impose fairly mild restriction on the FRR. THEOREM
2. Every FRR satisfying Condition A violates stability.
ProoJ: Let f be an FRR with representation g(...) satisfying Conditions A.1 and A.2. Let A = {x, y, z} be any issue with three distinct elements. By Lemma 1 and by the fact that the representation satisfies A. 1, either Z has exactly two elements neither of which is an individual ordering or Z has three non-empty subsets Z, , Z, and 2, such that [(N1 3 Nz > NJ & (N, + N3 > Nd] and V(Z,), V(Z& and V(Z,> are all pairwise disjoint.
(I) Suppose Z has three non-empty subsets Z, , Z, , and Z, such that [(N1 > N, >, N3) & (N, + N3 > &)I and I’(&), V(Z,), and V(Z,) are all pairwise disjoint. Let Z, = Z - (Z, U Z, u Z,). Given the issue A, g specifies e, , e2, and e3 such that e, 3 e, > e3 . By Condition A.2, e, > e2 . So there are two possibilities (4
el > e2 > e3
(P> e, > e2 = e3 If (a) holds then it follows that if all individuals belonging to V(ZJ (j = 1,2,3,4) have identical orderings then every Rio belonging to Zi coincides with that common ordering. Note that this is not necessarily true if (/3) holds as can be seen from the following counter example: Let the representation be g(g(R, , R,), g(R, , R4)) and suppose that e, = 1 and e2 = e, = 0. In this example, RI0 = g(R, , R,) and R,O = g(R, , R4). Suppose xPlyPlz and xP,yP,z. Instead of having xP~~~P~~zwe have XPIOyl,Oz.
If (CX)holds, then we proceed exactly as in the proof of Theorem 1. All the three steps I, 11.1 and II.2 given there can be transferred here exactly as they are. However, %, , W, and W, will now refer to the sum of rank numbers received by x, y, and z respectively at the final stage of aggregation (represented by the outermost g in the representation of FRRf) given situation (R,). Similarly W, , w, and w, for situation (R& have to be reinterpreted. Suppose @) holds. Then the following is true: for all a E A, for all situations (Ri), and for j E (1, 2, 3,4}, if {a} = C,(A, R,) for all i E V(ZJ,
SINCERE
219
VOTING
then {a> = Cr(A, Rko) for all Rko E Zj . Now construct (R,) and (&) follows for for for for
all all all all
i E V(Z1),
XP, yHg
i E V(Z,),
ZP
i E V(Z,), i E V(Z,),
yP,zP,x & z.P,xP, y,
y &
as
xP* yPgz, ZPiXPi
y,
xi, yiiz 22 ~1~Y~,z.
Consider the final stage of aggregation represented by the outermost g in the representation off. From what aid above it follows that given situation (i&), {x} = C,(A, Rio) .OE Z, ; (z> = C,(A, for all Rt E Z, ; ( y> = C,(A, Rio) for all 3 andix, Y, z> = GM for all Rio E Z, . It also follows that given situation (R,), {x) = C,(A, Rio) for all Rz E Zl, and (z} = C,(A, Rio) for all Rio E (2, u Z,) and (x, y, z} = C,(A, RF) for all RF E 2,. Since (Nr >, iti, 3 N3) and e, > e2 = e3 it follows that [(W, 2 Wz) & (W. 2 WY)] where W, , W, and E, refer to the aggregate numbers received by X, y> and z respectively at the final stage of aggregation given situation (R,).. Since (N, + N3 > NJ and e, > e, = e3 it also follows that [(w, > w,) & (w, > w,)] where W, I w, and w, refer to the aggregate numbers for x, y and z respe at the final stage of aggregation given situation (Rij. From what en said above it is clear that we have (xiiiy & xRz) which implies that x E C(A, a). Also we have (ZPX & zPy) which implies that (z> = C(A, R). Hence (R,) is unstable with respect to (&) and is vulnerable to V(Z,), (II) Suppose 2 has exactly two elements R,O and R,” neither of which is an individual ordering. Let V, = V(R,O) and V2 = V(&“). Clearly VI and V, are disjoint. Consider (Ri) such that for all i E V, , xPi yPiz an ZPi ypix. As before there are two possibilities: (a) e, > e, > e3 ; and @> 6 > e2 = e3 . If (a) holds then xPloyP,% and zP,*y clear that either {x, z} C C(A, R) or C(A, R) = ( y> where If (/I) holds then xP,~~I~~z and zP,~~&~x. In this ease (x, z> = C(A, R).
Thus either (x, z> C C(A, R) or C(A, R) = ( y>. Suppose (x, z> _CC(A, Then assume that (Ri) = (R,). Construct (&‘) ;such that for all 2 E VI , yPi’xPi’z and for all i E V, , yPi’zPi’x. Since e, > e2 > e3 ) C(A, R’) = (y] where R’ = f(A, (Rl)). Hence (Bi) (= (R,)) is unstable with respect to (R,‘) and is vulnerable to V1 u V, . Suppose C(A, R) = {y>. Then from what has been said above it is clear that (fi) does not hold. Then (a) holds. In that case we have ~~~Oy~~~z
220
PRASANTA K. PATTANAIK
and zP,OyP,Ox given (&). Hence, given (R,), we have wl/ = 2e, and w, = w, = e, + es . Since C(A, R) = {u} it follows that- -2e, > e, + e3 . Consider (i?,) and (Ri) such that for all i E V, , (xlyPiz & xPiyP:z) and for all i E V, , (zP,xI,y & zPj’xl:y). Clearly C(A, a) = A. Also, since 2e, > el + e3 , we have yP”z and hence -z E C(A, R”). Hence (&) is unstable with respect to (Ri) and is vulnerable to V1 . Q.E.D.
III.
STABILITYUNDERMETHODSOFEXHAUSTIVEVOTINGBASEDON
SFRRs
In this section we consider an important variation of SFRRs, namely, methods of exhaustive voting based on SFRRs. These group decision procedures have a somewhat different structure from that of GDRs and are frequently used in small committees. DEFINITION 8. A group decision function (GDF) is a functional rule T which for every issue A and every situation (Ri) specifies a non-empty subset T(A, (Ri)) of A.
We write T = T(A, (R,)). T(A, (Ri)) is called the set of outcomes and is to be interpreted as the set of socially best alternatives in A for the situation (R,).ls Definitions 3 and 4 can now be easily extended to cover GDFs; the formulations now have to be in terms of T = T(A, (R,)) and T’ = T(A, (R,‘)) rather than in terms of C = C(A, R) and C’ = C(A, R’). Since this extension is routine we omit the formal statement, and proceed to discuss stability of sincere situations under a class of GDFs. Let f be a given SFRR; and (Ri) be any situation. For all B _CA and all i E L, RJB] stands for the restriction of Ri to B. Let Ho = A and Hr = A - C,(A,f(A, (Ri))) where m is the number of indifference classes into which A can be partioned on the basis off(A, (Ri)). For all integers k(k > 1) let Hk = H,-, - C,,,(H,-, , f(H,-, , (R,[Hk-J))) where m’ is the total number of indifference classes into which Hkpl can be partitioned on the basis off (H,_, , (R,[Hk-J)). Hk is defined only if H,-, is non-empty. DEFINITION 9. A GDF T is a method of exhaustive voting based on an SFRRfiff for all issues A and every situation (Ri) there exists an integer k(k 3 0) such that T(A, (Ri)) = C(H, ,f(Hk , (RJH,]))) = Hk .
Under a method of exhaustive voting based on an SFRRJ;
we first
I6 Weuse the term “set of outcomes” rather than the more usual “choice set” so as to distinguish this concept clearly from the concept of a “choice set generated by R” which we have defined earlier.
SINCERE
VOTING
221
consider the ordering f(A, (IQ). The lowest indifference class in A defined in terms off (A, (R,)) is then eliminated from A and thus we get HI _ Then we take the orderingf(H, , (R&Y&). These successive rounds of elimination continue until we arrive at a subset of A such that aggregation through f yields a single indifference class.17 The type of exhaustive voting discussed by Black [2, pp. 69-721 constitutes a special case of the group decision procedures covered by our Definition !9~ls We now discuss the stability of sincere situations under methods of exhaustive voting based on SFRRs. We proceed to Theorem 3 via the following lemma. b?MMh 2. Let T be a method of exhaustiue voting based on SF and let A = (x, y, z> be any issue with three distinct elements = f (A, (&j) where (Ri) is any situation. Suppose (XPZ I%yPz) and supposexPiOy is true for more than half of the total mber of RiO in the representation off such that -(xl,“yIioz). Then T(A, .)) = (x).
Proof Let B = (x, y>. Consider the partition of A into indifference classeson the basis of R. Since (xPz & yPz) clearly (23 constitutes the lowest indifference class. Hence after the first round of elimination we shall be left with B. Let the rank numbers specified under be e, and e, . Clearly e, > e2. Consider w, and w, for sit Let 15’be the total number of &” such that
I7 Note that after each round of elimination we are essentially considering the preferences expressed by the individuals to start with. For example, if iR,) refers to the preferences we start with, we consider (R,[H,]) after the first round of elimination where for all i,
222
PRASANTA
K.
PATTANAIK
be the number of Rio such that (~l,~vl,~z). Since xP,Oy is true of more than N/2 of Rio such that +x~~~JJ~~~z), it is clear that w, > @J/2) 6 + @J/2) e2 + N’e, > w, Hence xPy. Therefore in the second round of elimination y will be eliminated and we shall be left with {x> which will emerge as T(A, (Rd)). Q.E.D. THEOREM 3. Every method of exhaustive violates stability.
voting based on an SFRR
Proof. Let T be a method of exhaustive voting based on an SFRR jY Let g(...) be a representation offinvolving only one g, and let A = (x, y, z> be an issue with three distinct elements. We shall consider a sincere situation (Bi) and two insincere situations (Ri) and (Ri’). Let W, , WY and W, be the sum of rank numbers received respectively by x, y and z under SFRRfin situation (R,) given the issue A. Similarly we have w, , w, and w, for (Ri) and We’, wy’, and wz’ for
z, = z - (Z, u z, u Z,). Construct (&), for for for for
all all all all
(R,),
and (Ri’)
such that
i E V(Z,),
XPi YPiZ & XPi ypiz & yPifzPi’x;
i E V(Z.J,
ZP,XPi y & zP,xP, y & ZPi’xPi’y;
i E V(Z,),
YPiZPiX &ZPiXP
i E V(Z,),
Xiiyiiz
&yPi'ZPi'X;
& XI~ ~~~~ & XI~'~JI~~'Z.
Let R = f(A, (K,)); R = f(A,
SINCERE VOTING
223
wg’ > wz’ so that yP’x. Since (zP’x & yP’x); for all i E [V(Z,) v V(Z,)], yPi’z; and N, + N3 > N2 , by Lemma 2, T(A, (&‘)) = ( yj. Lastly, consider (Wi). From the proof of Theorem 1 we know that (xEy & xKz). Then there are the following possibilities - - -(1) xry1z (2) xPyPz (3) 3cPzPy - - (4) xfzFy (5) xrypz (6) xPyP2 If (1) holds then clearly T(A, (&)) = A. If (2) or (5) holds, then given that for all i E [ V(Z,) u V(Z,)], xpiy, and that N1 + N2 > N3 , we have T(A, (a,)) = (x> by Lemma 2. If (3) or (4) holds then given that for all i E [ V(ZJ u V(Z,)], zHix, and that N,+N, > N1, we have T((A, ( by Lemma 2. If (6) holds, then clearly T(A, (&)) = (xl. Thus either x E T(A, (&)) or T(A, (w,)) = (z>. Hf x E T(A then er T. (&) is unstable with respect to (Ri) and is vulnerable to V( If T(A, (&)) = (z>, then (R,) is unstable with respect to (&‘) and is vulnerable to V(Z,) under T. This completes the proof.
IV. CONCLUSION Using the notion of stability as defined in Pattanaik [9] we examined the stability of sincere situations for three very wide classes of group decision procedures: (1) simple finite ranking rules; (2) finite ranking rules (which include as a subclass the class of simple finite ranking rules); and (3) methods of exhaustive voting based on simple finite ranking rules. It was shown that under every group decision procedure belonging to the first and the third class, some sincere situations are unstable for some issues. It was also shown that under every group decision procedure which belongs to the second class and which satisfies some rather mild conditions, some sincere situations are unstable for some issues. These conclusions are disturbing in so far as the group decision procedures covered by these results constitute a very wide class of non-binary “democratic” decision procedures and include many decision procedures (e.g. single ballot voting, dual ballot voting, exhaustive voting) used in actual practice. REFERENCES “Social Choice and Individual Values,” 1st ed. (1951), 2nd ed. (1964) Wiley and Sons, Inc., New York. 2. D. BLACK, “The Theory of Committees and Elections,” Cambridge University Press,” Cambridge, England, 1958. 1. K. J. ARROW,
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3. M. DUMMET AND R. FARQUHARSON, Stability in voting, Econometrica, 29 (1961). 4. R. FARQUHARSON,“Theory of Voting,” Yale University Press, New Haven, 1969. 5. A. GIBBARD, Manipulation of voting schemes: a general result, Econometrica (forthcoming). 6. L. GOODMAN AND H. MARKOWITZ, Social welfare function based on individual rankings, American Journal of Sociology, 60 (1952). 7. R. D. LUCE AND H. RAIFFA, “Games and Decisions,” Wiley and Sons, Inc., New York, 1957. 8. Y. MURAKAMI, “Logic and Social Choice,” Dover Publications Inc., New York, 1968. 9. P. K. PA-~TANAIK, On the Stability of Sincere Voting Situations, J. Econ. Theory 6 (1973), 558-574. 10. A. K. SEN, “Collective Choice and Social Welfare,” Oliver and Boyd, London, 1970. 11. M. SHUBIK, “Strategy and Market Structure,” Wiley and Sons, New York, 1959.