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Federations and transitive group choice
Mathematical Social Sciences 2 (1981) 35-38 North-Holland
Publishing Company
FEDERATIONS
AND TRANSITIVE
GROUP CHOICE
B.G. MIRKIN institute of Economics and Industrial Engineering, Siberian Brmch of li’. S. S. R. .Ac*adem_rc?f Sciences, Novosibirsk,
USSR
Communicated by F. W. Roush Received 30 August 1980 Revised 1 November 1980
The following is a brief English summary of the paper Mirkin (1379). En \\ hich he discus3e\ social welfare functions where the relations involved are solely assu.lled to be reflesite. L:naer axioms of independence pnd neutrality, monotonicity, and two domain assumptions the auth\rr shows that such a social welfare function is determined by a ‘federatioq’, i.e., the set of coalitions such that if that each member considers a not worse than b so does the group. The author then characterizes transitive, completeness, definiteness, (a sort of non-indifference assumption), and majority rule in this framework.
He also characterizes those domains on which social welfare
functions determined by federations F satisfying I E F if and only if r’e Fe,are transitive. 15’~omit proofs, which are given in the Russian version, as wel! as the author’s discussion of these results. Key words: Social
welfare
functions;
Federation;
Transitive;
Completeness;
Definiteness:
Majority rule.
In this paper we generalize work of Arrow (1963), Guilbaud (1952), Monjardet (8978) and the author (1974). In particular we consider the case in which preference relations need neither be transitive nor complete. Let A be a set of m alternatives, sometimes denoted 111and N a set of n individuals, sometimes denoted n. The preferences of the ith individual is considered as a binary relation R’ CA x A such that (a, 6) E R1 if and only if the person considers a not worse than b. This relation is only assumed to be reflexive so that (a,a)ERj for all awl. We consider a domain D to be a set of reflexive binary relations, such that all individual preference relations belongs to D. A social welfare function (rule of consensus, group choice function) will be a function f: C+E where CC D” and E is a set of reflexive binary relations. An n-tuple (R I, R2, . . . , R”) will frequently be denoted 8. A coalition is a subset of n. Definition 1. A federation is a set F of coalitions of n, such that if k F then IUJE F for any coalition J1 ‘h an!v federation, we associate the group choice functionJ(B) = UIEF([lk EI Rk). oass-489~/8l/~-oooo/$o2.75
@ 1981 North-Holland
B.G. Mirkin / Federations arrd trunsitive
groupchoice
fhat is, 4 is considered not worse than 6 by the group if and only if there exists IE F such that mery member of I considers a not worse than b. Let Ml,a,lb,R) = (k : (a,h) ERk), that is, the set of individuab who consider a not worse tha,nQ. AIviorrr1, Independence (sometimes called independence and neutrality). For any &kCanda,b,c,d~A, ifM(a,b,R)=M(c,d,R’), then (a,b)qf(R) ifand only if (GdW(R’). A*#8
2. Monotonicity. For any R, R’ E C and a, b E A, if M(a, b, R) c M(a, b, R’)
and(a,Q)Ef(&
then (a,b)ef(R’).
A~iamro 3’. Simple Universality. For any distinct a, b E A and any coalition I there
a.&s R E C such that M(a,b, R) = 1. A-ODD37 Complicated Universality. For any distinct a, 6, CEA and any coalitions 1. I there exisls R E C such that M(a, b, R) = I, M(b, C,R) = J, M(a, C,8) = In J. Alriom 3# implies Axiom 3’.
T~~WPVL If Axiom 3’ holds then a social welfarefunction f arises from a federation # and only if it satisfies Axioms 1, 2. Tlwsnr 2. Let D be contained in the set of transitive relations and let Axiom 3” hold+Thenf(R) is transitive for all R E C if and only if the federation is closed under
intemections. CoWasy.
Let C be a domain of n-tuples of transitive relations satisfying Axiom 3”. Then any social weuare function satisfving Axioms 1, 2 is oligarchic, that is there extits a coalition I such that for all a, b E A, R E C we have (a, b) Ef(R) if and only if (a$]ER&forev~~kEI. m
3. Let CcDm satisfy Axiom 3’ and let D consist of complete binary relatiom. Then a social welfare function determined by a federation F satisfies (i) f(R) is completefw all R EC if and only if (ii) if Ie F, then the complement f of I
&longs in F. C~dhrgr. Arrow’s Paradox. Let Axiom 3”hold and let D be contained in the set of
weukorders, and let f(R) be a wc~gkorderfor all R E C. Then (i) f is a dictatorship (it coincides with the preferences of lilfued member of the group) or else for every 8, f(R) is the universalrelation if and only if (ii) f satisfies Axioms 1 and 2. Ahun 4. Definiteness,l-For any a, b E A, R E C ijf M(a, b, R)n M(b, a, 8) = 0, then
S&T (o;bgi@f(R) or (b,a)bf(R).
This means that if no individual is indifferent about any pair of alternatives, the group is not indifferent about that pair of alternatives. Theorem 4. Let C satisfy Axiom 3”and let f be a social welfare function determined by a federation. Then f satisfies Axiom 4 if and only if the federation F satisfies In J+B,for all I, JE F. Corollary. Let C satisfy Axiom 3” and let f be a social welfare function determined by a federation. Suppose D and f(C) are contained in the set of complet(Prelations. Then Axiom 4 holds if and only if IE F if and only if & F. Federations satisfying IE: F if and only if r@F (and the corresponding social welfare functions) are callerd simple, because of their relationship to the simple games of von Nt:umann and Morgenstern. For simple federations, we have f(R)=g.(fi,Rk)=;&Rk)* /
Axiom 5. Symmetry (Anonymity). For any permutation C, the profile (R 71(1), R n(Z!. . . , R =@I)E C and we have
n and any (R I,;
f(R’,R2 ,..., R”)=f(Rz(“,R”(2’ ,..., RR@‘). Theorem 5. Let n be odd and let C satisfy Axiom 3” and let D consist o,f complete relations. Then a social welfare function coincides with simple majority rule if and only [fit satisfies Axioms 1, 2, 4, 5 I1Here simple majority rule is treated in the sense of Mrkin (1974) as a social welfare function determined by the federation F=(I: IIl>+n}. Definition 2. A set of three orderings having the respective forms a2 &c, bz c~a, CL a= b is called a cyclic family if and only if (1) if all three elements are indifferent in some ordering then the two extreme elements (left and right ends) in this ordering are not indifferent to other elements in either of the other two orderings or (2) if all three orderings are dichotomous then not all three are of the same type (i.e., the single element class is not always first or always last). See Mirkin (1974). Theorem 6. Let D be a set of weak orders then any nondictatorial simple social welfare function defined on Dn is transitive for all l? E D” if and on& if D does not contain a cyclic family on any triple of obd;ects. The proof of Theorem 6 is analogous to the proof of Theorem 2.2..1 in the English version of Mir kin (1974).
38
8.G. Mirkin / Federahbns and transitive group choice
#;,I. Arrow, Social Choice and Individual Values (Wiley, New York, 1963). G‘I’. Guilbattd, Les th4ories de I’intCr%t gCnCral er Ie probl6mc Ic@que de I’agrt!gation, Econ. Appl. S (w52].
Mfrkin, Problems of Group Choice, in Russian, M. Nauka (1974). (English translation: Group (Winston, Washington, D.C., 1979).) B.G. Mirkin, Federations and transitive group choice, in Russian, in: Models of Socialist Economic and Socialist Planning, (Soviet Academy oi:’Sciences (Akademia Nauk), 1979). 13. Men&t&t, Dttali!y in the theov of social choice, isi: J.J. Laf’font, ed., Aggregation and Revelation af Preferences(Non h-Holland, Amsterdam, 1978).
KG.
Publishing Company
FEDERATIONS
AND TRANSITIVE
GROUP CHOICE
B.G. MIRKIN institute of Economics and Industrial Engineering, Siberian Brmch of li’. S. S. R. .Ac*adem_rc?f Sciences, Novosibirsk,
USSR
Communicated by F. W. Roush Received 30 August 1980 Revised 1 November 1980
The following is a brief English summary of the paper Mirkin (1379). En \\ hich he discus3e\ social welfare functions where the relations involved are solely assu.lled to be reflesite. L:naer axioms of independence pnd neutrality, monotonicity, and two domain assumptions the auth\rr shows that such a social welfare function is determined by a ‘federatioq’, i.e., the set of coalitions such that if that each member considers a not worse than b so does the group. The author then characterizes transitive, completeness, definiteness, (a sort of non-indifference assumption), and majority rule in this framework.
He also characterizes those domains on which social welfare
functions determined by federations F satisfying I E F if and only if r’e Fe,are transitive. 15’~omit proofs, which are given in the Russian version, as wel! as the author’s discussion of these results. Key words: Social
welfare
functions;
Federation;
Transitive;
Completeness;
Definiteness:
Majority rule.
In this paper we generalize work of Arrow (1963), Guilbaud (1952), Monjardet (8978) and the author (1974). In particular we consider the case in which preference relations need neither be transitive nor complete. Let A be a set of m alternatives, sometimes denoted 111and N a set of n individuals, sometimes denoted n. The preferences of the ith individual is considered as a binary relation R’ CA x A such that (a, 6) E R1 if and only if the person considers a not worse than b. This relation is only assumed to be reflexive so that (a,a)ERj for all awl. We consider a domain D to be a set of reflexive binary relations, such that all individual preference relations belongs to D. A social welfare function (rule of consensus, group choice function) will be a function f: C+E where CC D” and E is a set of reflexive binary relations. An n-tuple (R I, R2, . . . , R”) will frequently be denoted 8. A coalition is a subset of n. Definition 1. A federation is a set F of coalitions of n, such that if k F then IUJE F for any coalition J1 ‘h an!v federation, we associate the group choice functionJ(B) = UIEF([lk EI Rk). oass-489~/8l/~-oooo/$o2.75
@ 1981 North-Holland
B.G. Mirkin / Federations arrd trunsitive
groupchoice
fhat is, 4 is considered not worse than 6 by the group if and only if there exists IE F such that mery member of I considers a not worse than b. Let Ml,a,lb,R) = (k : (a,h) ERk), that is, the set of individuab who consider a not worse tha,nQ. AIviorrr1, Independence (sometimes called independence and neutrality). For any &kCanda,b,c,d~A, ifM(a,b,R)=M(c,d,R’), then (a,b)qf(R) ifand only if (GdW(R’). A*#8
2. Monotonicity. For any R, R’ E C and a, b E A, if M(a, b, R) c M(a, b, R’)
and(a,Q)Ef(&
then (a,b)ef(R’).
A~iamro 3’. Simple Universality. For any distinct a, b E A and any coalition I there
a.&s R E C such that M(a,b, R) = 1. A-ODD37 Complicated Universality. For any distinct a, 6, CEA and any coalitions 1. I there exisls R E C such that M(a, b, R) = I, M(b, C,R) = J, M(a, C,8) = In J. Alriom 3# implies Axiom 3’.
T~~WPVL If Axiom 3’ holds then a social welfarefunction f arises from a federation # and only if it satisfies Axioms 1, 2. Tlwsnr 2. Let D be contained in the set of transitive relations and let Axiom 3” hold+Thenf(R) is transitive for all R E C if and only if the federation is closed under
intemections. CoWasy.
Let C be a domain of n-tuples of transitive relations satisfying Axiom 3”. Then any social weuare function satisfving Axioms 1, 2 is oligarchic, that is there extits a coalition I such that for all a, b E A, R E C we have (a, b) Ef(R) if and only if (a$]ER&forev~~kEI. m
3. Let CcDm satisfy Axiom 3’ and let D consist of complete binary relatiom. Then a social welfare function determined by a federation F satisfies (i) f(R) is completefw all R EC if and only if (ii) if Ie F, then the complement f of I
&longs in F. C~dhrgr. Arrow’s Paradox. Let Axiom 3”hold and let D be contained in the set of
weukorders, and let f(R) be a wc~gkorderfor all R E C. Then (i) f is a dictatorship (it coincides with the preferences of lilfued member of the group) or else for every 8, f(R) is the universalrelation if and only if (ii) f satisfies Axioms 1 and 2. Ahun 4. Definiteness,l-For any a, b E A, R E C ijf M(a, b, R)n M(b, a, 8) = 0, then
S&T (o;bgi@f(R) or (b,a)bf(R).
This means that if no individual is indifferent about any pair of alternatives, the group is not indifferent about that pair of alternatives. Theorem 4. Let C satisfy Axiom 3”and let f be a social welfare function determined by a federation. Then f satisfies Axiom 4 if and only if the federation F satisfies In J+B,for all I, JE F. Corollary. Let C satisfy Axiom 3” and let f be a social welfare function determined by a federation. Suppose D and f(C) are contained in the set of complet(Prelations. Then Axiom 4 holds if and only if IE F if and only if & F. Federations satisfying IE: F if and only if r@F (and the corresponding social welfare functions) are callerd simple, because of their relationship to the simple games of von Nt:umann and Morgenstern. For simple federations, we have f(R)=g.(fi,Rk)=;&Rk)* /
Axiom 5. Symmetry (Anonymity). For any permutation C, the profile (R 71(1), R n(Z!. . . , R =@I)E C and we have
n and any (R I,;
f(R’,R2 ,..., R”)=f(Rz(“,R”(2’ ,..., RR@‘). Theorem 5. Let n be odd and let C satisfy Axiom 3” and let D consist o,f complete relations. Then a social welfare function coincides with simple majority rule if and only [fit satisfies Axioms 1, 2, 4, 5 I1Here simple majority rule is treated in the sense of Mrkin (1974) as a social welfare function determined by the federation F=(I: IIl>+n}. Definition 2. A set of three orderings having the respective forms a2 &c, bz c~a, CL a= b is called a cyclic family if and only if (1) if all three elements are indifferent in some ordering then the two extreme elements (left and right ends) in this ordering are not indifferent to other elements in either of the other two orderings or (2) if all three orderings are dichotomous then not all three are of the same type (i.e., the single element class is not always first or always last). See Mirkin (1974). Theorem 6. Let D be a set of weak orders then any nondictatorial simple social welfare function defined on Dn is transitive for all l? E D” if and on& if D does not contain a cyclic family on any triple of obd;ects. The proof of Theorem 6 is analogous to the proof of Theorem 2.2..1 in the English version of Mir kin (1974).
38
8.G. Mirkin / Federahbns and transitive group choice
#;,I. Arrow, Social Choice and Individual Values (Wiley, New York, 1963). G‘I’. Guilbattd, Les th4ories de I’intCr%t gCnCral er Ie probl6mc Ic@que de I’agrt!gation, Econ. Appl. S (w52].
Mfrkin, Problems of Group Choice, in Russian, M. Nauka (1974). (English translation: Group (Winston, Washington, D.C., 1979).) B.G. Mirkin, Federations and transitive group choice, in Russian, in: Models of Socialist Economic and Socialist Planning, (Soviet Academy oi:’Sciences (Akademia Nauk), 1979). 13. Men&t&t, Dttali!y in the theov of social choice, isi: J.J. Laf’font, ed., Aggregation and Revelation af Preferences(Non h-Holland, Amsterdam, 1978).
KG.