Mathematical Social Sciences 15 (1988) 307-3 12

North-Holland 307

NOTE

CHARACTERIZATION OF MONOTONICITY AND NEUTRALITY FOR BINARY PARETIAN SOCIAL DECISION RULES

Satish K. JAIN

Centrefor Economic Studies and Planning, School of Social Sciences, Jawaharial Nehru Univer- sity, New Delhi 110067, India

’ Communicated by B. Dutta

Received 29 September 1986

Revised 14 July 1987

It is shown that a necessary and sufficient condition for a binary Paretian social decision rule

to be neutral and monotonic is that it satisfies Pareto quasi-transitivity.

Key words: Binary social decision rules; Pareto-criterion; Pareto quasi-transitivity; monotonicity;

neutrality.

1. Introduction

In the context of binary social decision rules the properties of neutrality and monotonicity seem to be closely related to the Pareto-criterion and various rationali- ty conditions. Arrow (1963) showed that if a binary social decision rule (SDR) satis- fying the weak Pareto-criterion always yields transitive social weak preference relation then, if a group of individuals is almost decisive for some ordered pair of alternatives then it is decisive for every ordered pair of alternatives. In other words, satisfaction of transitivity implies both neutrality and monotonicity with respect to almost decisiveness relative to the entire society. Gibbard (1969) pointed out that this result holds even if the transitivity requirement is weakened to quasi-transitivity. Guha (1972) and Blau (1976) have shown that if the weak Pareto-criterion is replaced by the Pareto-criterion (also called the strict Pareto-criterion), then quasi-transitivity is sufficient to ensure neutrality and monotonicity everywhere.

In this paper we derive a necessary and sufficient condition for a binary Paretian

SDR to be neutral and monotonic. It is shown that a binary Paretian SDR is neutral and monotonic iff Pareto quasi-transitivity holds. Pareto quasi-transitivity requires that if x is judged to be socially better than y and y is Pareto-superior to E then x must be considered to be socially better than z, and if x is Pareto-superior to y and y is judged to be socially better than z then x must be considered socially better than

016S-4896/88/$3.50 0 1988, Elsevier Science Publishers B.V. (North-Holland)

308 S.K. Jain / Binary Parerian social decision rules

z. In the presence of the Pareto-criterion, Pareto quasi-transitivity is a much weaker condition than quasi-transitivity. We also obtain, for the class of binary SDRs, joint characterization for two properties which have played an important role in the con- text of the Arrow impossibility theorem and related results, namely (i) whenever a group of individuals is almost decisive for some ordered pair of alternatives, it is decisive for every ordered pair of alternatives and (ii) whenever a group of indivi- duals is almost semidecisive for some ordered pair of alternatives, it is semidecisive for every ordered pair of alternatives. We show that a binary SDR satisfies proper- ties (i) and (ii) iff weak Pareto quasi-transitivity holds.

2. Definitions and assumptions

We designate the set of social alternatives by Sand the set of individuals by N, and assume #S = n 13 and #N= L ~2. Each individual i E N has a binary weak prefer- ence relation Ri (at least as good as) defined over S. Corresponding to a binary relation R over a set S we define the relations P (better than) and I (indifferent to) in the usual way, i.e., for all x,y~S, [(xPy*xRy~-yRx)~(xZyoxRyr\yRx)]. We define a binary relation R over a set S to be (a) reflexive iff for all x E S, XRX; (b) connected iff for all x, y E S, (x#y-*xRy vyRx); (c) transitive iff for all x, y,zcS, (xRy~yRz-txRz); (d) an ordering iff it is reflexive, connected and transitive.

Throughout this paper we assume that each Ri is an ordering. For all X, y E S we define, xQr iff XRiy for all i E N and XPiy for some i E N, xTy iff xPiy for all i E N; and xJy iff xZiy for all ic N.

A social decision rule (SDR) f is a function which for every ordered L-tuple of indivjdual orderings (R ,, . . . , RL), to be written (Ri) in abbreviated form, specifies a unique reflexive and connected social binary weak preference relation R. An SDR satisfies (a) binariness (B) iff for all (Ri),(Rf), if (Ri) and (Rf) are identical over ACS then R and R’ are identical over A, where R=f((R,)) and R’=f((R!)); (b) weak Pareto-criterion iff for all x,ye S, (xTy*xPy); (c) Pareto-preference (P*) iff for all 4y E S, (xQy+xPy); (d) Pareto-indifference (P**) iff for all x, y E S, (x.Zy + xly); (e) Pareto-criterion iff P* and P** hold; (f) weak Pareto quasi-transitivity (WPQT) iff for all x, y, z E S, [(xPy AyTz-rxPz) A (xTyAyPz+xPz)]; (g) Pareto quasi-transitivity (PQT) iff for all x, y, z E S, [(xPy AYQZ+XPZ) A

(x@AyPz-,xPz)]. A binary SDR satisfies

(a) neutrality (N) iff for all (Ri),(RJ) and for all x,y,z, WAS, [[(ViEN)[(xRiy* zR;w) A CyRiX* ~Rfz)]] -+ [(xRY++zR’w)A (_,vRxH wR’z)]];

S.K. Juin I Binary Parerian social decision rules 309

(b) monotonicity (M) iff for all (R,), (RI) and for all x, YE S, [(ViElV)[(xP,y* xP;y ) A (x1; y * xR;y)] + [(xPy + _uP’y) A (xZy --t ,uR ‘y)] ] .

We define VCN to be (a) (N-A)-almost decisive for (x,y) [D,(s, y)] iff [(xl,y, Vie A)/\ (xP,y, Vie V)A (yP;x, ViEN-(AU V))+xPy], where x#y and ACN; (b) (N- A)-decisive for (x, y) [DA@, y)] iff [(xl;y, ViEA)A(xP;y, ViE V)+xPy], where xfy and ASN; (c) (N-A)-decisive iff it is (!I’-A)-decisive for b’(~7, b) E S x S, a# 6; (d) (N-A)-almost semidecisive for (x, y) [S,(x, y)] iff [(xl,y, ViEA)A (~Piy, Vie V)A(yPiXv ViEN-(AU v))-+xRy], where xfy and ASN; (e) (N-A)-semidecisivefor(_u, y) [S,(x, y)] iff [(xl,y, ViEA)A(XPiy, Vie V)-rxRy], where x#y and A GN, (f) (N- A)-semidecisive iff it is (N- A)-semidecisive for V(a, b) E S x S, a # b.

If A =0 then we drop the prefix (N-.4). (a) through (f) then give definitions of an almost decisive set for (x, y)[D(x, y)], a decisive set for (x, y)[b(x, y)], a decisive set, an almost semidecisive set for (x, y)[S(x, y)], a semidecisive set for (x, y)[S(x, y)], and a semidecisive set respectively.

3. Characterization theorems

Theorem 1. A binary social decision rule satisfies properties: (i) whenever a group of individuals is almost decisive for some ordered pair of distinct alternatives, it is decisive for every ordered pair of distinct alternatives, and (ii) whenever a group of individuals is almost semidecisive for some ordered pair of distinct alternatives, it is semidecisive for every ordered pair of distinct alternatives, iff weak Pareto quasi- transitivity holds.

Proof.

Necessity. SupposexPy and yTz; x, y, z E S. Let the configuration of individual prefer- ences over {x, y} in this situation (Rt) be characterized as follows; (xPiy, Vitz N,)A

(XZi_Y, ViEN2)A(vPtX, ViEN3)], where N~UN~UN~=N.~TZ+XP~Z, ViEN,UN2. Now consider the following configuration of individual preferences over {x, y,z}, [(xP,i?pz, Vi E NI) A (xZ;yP,!z, Vi E N2) A (yP,%;x, Vi E NJ)]. Suppose zR’x. Then Ns is S(z,x) and therefore by property (ii) a semidecisive set. But then in (Ri) situa- tion we must have yRx as YPiX for all ic NJ, contradicting the hypothesis xPy. Therefore - (zR’x), i.e., XP’Z must hold. XP’Z implies that Nr UN, is D(x,z). By property (i) then Nt UN2 is a decisive set. As in the (Ri) situation XPtZ for all ieN, UN,, it follows that XPZ holds. By an analogous argument it can be shown that xTy and yPz imply XPZ.

Sufficiency. Follows from the fact that in the original Arrow (1963) and Sen (1970) proofs, although transitivity (quasi-transitivity) and weak Pareto-criterion are assumed, only weak Pareto quasi-transitivity is used.

310 S.K. Jain / Binary Parerian social decision rules

Lemma 1. If a binary social decision rule satisfies Pareto quasi-transitivity then, whenever a group of individuals is (N- A)-almost decisive for some ordered pair of distinct alternatives, it is (N- A)-decisive for every ordered pair of distinct alter- natives, where A 5 N.

Proof. Let V be DA(x, y), x#y. Let z be an alternative distinct from x and y, and consider the following configuration of preferences, [(xZ;yZ,z, Vi E A) A (xPiyPjz, Vie V)/\((yP;x~yP~z), ViEN-(AU V))]. As Vis DA(x, y), we conclude xPy.

Now N-A # 0-+ yQz. xPy and yQz -xPz by PQT. As the preferences of individuals in N- (AU V) have not been specified over {x,z}, it follows that V is DA(x,z). Similarly by considering the configuration [(Xli_Yl;Z, Vi E A) A (zPixP,y, Vi E V) A

((yP,x~zp~x), Vie N- (AU V))], we can show that DA(x, y)+DA(z, y). By appro- priate interchanges of alternatives it follows that DA(x, y)*DA(a, 6) for all (a, 6) E {x, y, z> x {x, y, z}, where a # b. By spelling out in full the familiar Arrow- Blau argument one establishes that D,(x, y) implies DA (a, b) for all (a, b) E S x S,

a#b.

Lemma 2. If a binary social decision rule satisfies Pareto quasi-transitivity then, whenever a group of individuals is (N- A)-almost semidecisive for some ordered pair of distinct alternatives, it is (N- A)-semidecisive for every ordered pair of distinct alternatives, where A 5 N.

Proof. The proof is similar to that of Lemma 1 and is therefore omitted.

Theorem 2. If a social decision rule satisfies the Pareto-criterion, binariness and Pareto quasi-transitivity then it is neutral.

Proof. Consider any (Pi) and (Z?;) such that, for all i E Nl , (XPiy A zP;w);

for all iEN*, (XZiyAZZ,‘W);

for all i E NJ, cvP,XA wP;z); N1 UNzUNs = N. If Nr UN3 = 0 then xZy and ZZ’W follow by the Pareto-criterion. Suppose Nr UN, # 0. If xPy then Nr is (N- N2)-almost decisive for (x, y), and so by Lemma 1 an (N- N2)-decisive set. Therefore zP’w must hold as [(zZ;w, Vie N,) A

(zP,‘w, Vie Nl)]. Thus xPy+zP’w. By an analogous argument it can be shown that zP’w-+xPy. Next suppose yPx. Then Ns is (N- N,)-almost decisive for (y,x), and hence an (N- N2)-decisive set in view of Lemma 1. Therefore we must have wP’z as [(wZ;z, Vi E N2)A (wP~z, Vi E N3)]. So yPx+ wP’z. By a similar argument one obtains WP’Z + yPx. As xPy e, zP’w and yPx * wP’z, by the connectedness of social R we conclude xZy*zZ’w. This establishes that the SDR is neutral.

S.K. Jain / Binary Parerian social decision rules 311

Theorem 3. If a social decision rule satisfies the Pareto-criterion, binariness and Pareto quasi-transitivity then it is monotonic.

Proof. It suffices to consider the case when only one individual changes his prefer- ences. Consider any (Ri) and (Rf) such that for all i#k the restriction of R,: over {x, y} is identical with the restriction of Ri over {x, yj, and [(yP,x~xR;y)V (~l~y~xZ’~y)]. Let the restriction of (R;) over {x,y} be characterized as follows; [(XPi_Yl Vi E Nl) A (xZ~_Y, Vi E N2) A (_YPiX, Vi E NJ)], where Ni U Nz UN, = N. If N,UNs=O then xly and xP’y follow by the Pareto-criterion. Let N,UNs#0. If xPy then Nt is (N-Nz)-almost decisive for (x, y), and hence an (N-Nz)-decisive set by Lemma 1. So, if kens then xP’y. Now suppose k~Nz. If yR’x then Nj is (N- (Nz - { k}))-almost semidecisive for (y, x), and hence an (N - (Nz - { k)))-semi- decisive set by Lemma 2. As [(yPix, Vie N3)~ (ylix, ViE N2 - {k})], yRx is im- plied contradicting the hypothesis xPy. Therefore, -yR’x, i.e., xP’y must hold.

Next suppose xZy. Then Nt is (N-N,)-almost semidecisive for (x, y), and hence an (N- Nz)-semidecisive set by Lemma 2. So, if k~ Ns then xR’y. Suppose k E N2. If yP’x then N3 is (N- (Nz - {k}))-almost decisive for (y,x), and hence an (N- (Nz- {k}))-decisive set by Lemma 1. But then, as [cvP,x, Vie N,)A

cVZ,x, ViENz- {k})l, we get yPx contradicting the hypothesis xly. Therefore - yP’x, i.e., xR’y. Thus we have shown that (xPy*xP’y) and (xZy-*xR’y), which establishes the theorem.

Theorem 4. Zf a binary social decision rule is neutral and monotonic then it satisfies Pareto quasi-transitivity.

Proof. Suppose xPy and yQz. Let (Ri) over {x, y} be as follows; [(XPiy, Vie Ni)A (XZiy, ViENz)AQPiX, VicNs)], N,UNzUNs=N. Let N;, N; and N; designate the sets {i : XPiZ}, {i : xl,z}, and {i : ZPiX} respectively. yQz implies that N, C N; and N;CNs. Let (Rf) be any situation such that, [(xP,‘z, ViEN,)A(xZit, ViENz)A (Zp~x, ViEN,)]. As xPy we conclude xP’z by neutrality. XP’Z in turn implies XPZ in view of Ni C N; and N; c Ns, as a consequence of monotonicity. Thus xPy and yQz imply xPz. By an analogous argument it can be shown that (xQv~yPz-+xPz).

Combining Theorems 2, 3 and 4 we obtain:

Theorem 5. A binary social decision rule satisfying the Pareto-criterion is neutral and monotonic iff Pareto quasi-transitivity holds.

References

K.J. Arrow, Social Choice and Individual Values, 2nd ed. (Wiley, New York, 1963). J.H. Blau. Neutrality, monotonicity, and the right of veto: a comment, Econometrica 44 (1976) 603.

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P.C. Fishburn, The Theory of Social Choice (Princeton University Press, Princeton, 1973).

A. Gibbard. Intransitive social indifference and the Arrow dilemma, unpublished manuscript, 1969.

A.S. Guha, Neutrality, monotonicity, and the right of veto, Econometrica 40 (1972) 821-826.

A.K. Sen, Collective Choice and Social Welfare (Holden-Day, San-Francisco, 1970).

R.B. Wilson, The game-theoretic structure of Arrow’s general possibility theorem, J. Econ. Theory 5

(1972) 14-20.