Includes bibliographical references (p. 11)

Ramanujam, Vasudevan

Ritz, Zvi

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A note on nondictatorial conditions and the relations between choice mechanisms and social welfare fucntions

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S"l• •-^FACULTY WORKINGPAPER NO. 1134A Note On Nondictationai Conditions and the RelationsBetween Choice Mechanisms and Social Welfare FunctionsZvi RitzCcliege of Commerce and Business AdministrationBureau of Economic and Business ResearchUniversity of Illinois. Urbana-ChampaignSESRFACULTY WORKING PAPER NO. 1134College of Commerce and Business AdministrationUniversity of Illinois at Urbana-ChampaignMarch, 1985A Note on Nondictational Conditions and the RelationsBetween Choice Mechanisms and Social Welfare FunctionsZvi Ritz, Assistant ProfessorDepartment of Business AdministrationAcknowledgement: My thanks to Bezalel Peleg who directed my attentionto the topic discussed here.Digitized by the Internet Archivein 2011 with funding fromUniversity of Illinois Urbana-Champaignhttp://www.archive.org/details/noteonnondictato1134ritzA Note on Nondictational Conditions and the RelationsBetween Choice Mechanisms and Social Welfare Functionsby Zvi RitzAbstractIt is demonstrated that weakening the strong nondic-tatorship requirement stated by Gibbard and bySatterthwaite enables the construction of strategy-proofchoice mechanisms by shifting the difficulties from thephase of choice mechanism construction to the phase ofagenda formation.Ever since Arrow (1963), Gibbard (1973) and Satterthwaite (1975)proved their famous impossibility results, the relation betweenArrow's required axioms for social welfare functions and Gibbard' s andSatterthwaite 's stated axioms for strategy-proof choice mechanisms hasattracted a great deal of attention. Satterthwaite (1975) used his(and Gibbard's (1973)) results on the impossibility of constructingstrategy-proof, nondictatorial choice mechanisms to re-prove Arrow's(1963) impossibility theorem on constructing nondictatorial socialwelfare functions. Blin and Satterthwaite (1978) proved that whenpreferences are unrestricted the existence of a social welfare func-tion that satisfies the independence of irrelevant alternatives andmonotonicity conditions implies the existence of strategy-proof choicemechanisms. Maskin (1976) correctly observed that to prove completeequivalence between the two problems one must show equivalence on anygiven domain of restricted preferences. He gives an example (see alsoMuller and Satterthwaite (1983)) of a restricted domain of preferencesthat permits the construction of a nondictatorial strategy-proof choicemechanism but not the construction of an Arrow type social welfarefunction, thus refuting complete equivalence. He goes on to prove, as-2-did Kalai and Muller (1977), the equivalence over restricted domains ofpreferences of the class of Arrow type social welfare functions withthe class of rational strategy-proof choice mechanisms. For this pur-pose they used a weaker notion of nondictatorship as a requirement forchoice mechanisms than the nondictatorship condition used by Gibbardand by Satterthwaite. Blair and Muller (1983) replaced the nondic-tatorial requirement used by Maskin and by Kalai and Muller with thestronger requirement of essentiality, and they proved the equivalenceover restricted domains of preferences of the class of essential Arrowtype social welfare functions with the class of rational, essentialstrategy-proof choice mechanisms. This body of research was summarizedand discussed in detail in Muller and Satterthwaite (1983) and is anexcellent reference for the interested reader.The purpose of this note is to demonstrate that if the nondic-tatorial condition postulated by Gibbard (1973) and by Satterthwaite(1975) is replaced with the weaker nondictatorial condition required byMaskin (1976), Kalai and Muller (1977), Ritz (1983) and others, then itis possible to construct choice mechanisms which possess all therequired properties. These mechanisms depend heavily on the agenda atissue. Thus the difficulties with strategy-proof choice mechanismsshift from the choice mechanism construction phase to the agenda for-mation phase.Notations and DefinitionsThe notations and definitions used here follow those in Kalai andMuller (1977) and Muller and Satterthwaite (1983). Let N = {l,...,n}be a set (society) of n individuals or agents (|n| > 2). Let A denote a-3-finite set of feasible alternatives (|a| > 2), and let E denote the setof all transitive antisymmetric complete binary relations on A. (Forthe sake of brevity the discussion is restricted to the case of strictpreferences only.) An element of Z is called a preference relation.Let ft be a nonempty subset of Z; ft is the set of all the admissiblepreference relations in the society and referred to as the restricteddomain of preferences, ft represents the set of all n-tuples ofpreferences from ft, and element of ft , P = (p,,...,pn) is called ann-person profile . An n-person social welfare function (SWF) or ftis a function f:ft *£• The following are a number of properties aSWF f may possess.Unanimity : For every P in ft and x,y in A, if xp.y for i = l,...,nthen xf(P)y.Independence of Irrelevant Alternatives (11A) : For any x,y in A andP,Q in ft if [xp.y if and only if xq.y for i i,...,n] then [xf(P)y ifand only if xf (Q)y].Weak Nondictatorship ; There is no individual i in N such thatf(P) = p. for every P in ft .Essentiality : For every individual i in N there exists a P in ft andq. in ft and a pair of alternatives x,y in A such that xf(P)y andyf(P/qi)x (P/qi= (p{,. . .,pi _ 1 f q ± ,p i _ 1, .. . ,pn)) .Let II be the set of all nonempty subsets of A. II is the set ofall possible agendas. An n-person choice mechanism is a functionF:ft xII+A.The following is a list of properties a choice mechanism may possessFeasibility : For every B in H and every P in ft , F(P,B) is in B.-4-Unanimity (Pareto Efficiency) : For every P in ft and every B in II, ifx,y in B and xp.y for i = 1,2, ...n then y * F(P,JB).Independence of Nonoptimal Alteratives (INQA) : For every P in ft andevery B in II, if D C B and F(P,B) in D then F(P,D) = F(P,B).Manipulability : There exists a B in n, P,Q profiles in ft such thatfor some individual i, p. * q . , for every individual k * i, p, = q, andF(Q,B)p .F(P,B) . F is said to be strategy-proof (or nonmanipulable ) ifit is not manipulable.Rationality : There exists f, a social welfare function such that forevery B in I and P in ftn,F(P,B) = maxfi(f(P)).Mono tonicity : For every feasible set B in n and any element x inB, whenever (i) two profiles P,Q in ftn aSree on the set B"{ x }> and(ii) xp.y implies xq.y for all y in B-{x}, then F(P,B) = x impliesF(Q,B) = x.Weak Nondictatorship : No individual i in N exists such that, for allfeasible sets B in II, F(P,B) = max_.(p.) for all P in ft (maxD (p.) isa 1 o lthe alternative in B most preferred in p.).Full Agenda Nondictatorship : No individual i in N exists such thatF(P,A) = max.(p.) for all P in ftn .A 1Strong Nondictatorship : No individual i in N exists such that, for atleast one feasible set B in I, (|B| > 2) F(P,B) = maXgCp^ for all P inEssentiality : For every individual i in N there exists a feasible setB in II, a profile P in ft and a preference q. in Q such thatF(P,B) ± F(P/q.,B).Thus, a choice mechanism is weakly nondictatorial if no individualhas decisive powers over all possible agendas, it is full agenda non--5-dictatorial if there exists no individual who always possesses deci-sive powers when the largest possible agenda is considered, and it isstrong nondictatorial if no individual exists that has decisive powerson even one possible agenda.In the following representation we emphasize the different notionsof nondictatorship used by Arrow (1963) and by Gibbard andSatterthwaite (1975).Result 1 (Arrow (1963))If |a| > 3 and preferences are unrestricted (JJ = E) , then a socialwelfare function f cannot simultaneously satisfy unanimity, indepen-dence of irrelevant alternatives and weak nondictatorship.Result 2 (Gibbard (1973), Satterthwaite (1975))If |a| > 3 and preferences are unrestricted (ft = E) , then a choicemechanism cannot simultaneously be strategy-proof and satisfy unanim-ity and strong nondictatorship.Maskin (1976) and Kalai and Muller (1977) in their discussions ofthe equivalence between classes of strategy-proof choice mechanismsand Arrow type SWFs over restricted domains of preferences, relaxed theoriginal requirement of strong nondictatorship postulated by Gibbardand Satterthwaite, and used instead the weak nondictatorship require-ment. The same assumption is used in Ritz (1983) for the case analyzedthere. (In the following representation of the results derived byKalai-Muller and Maskin, we omit the part that deals with the charac-terization of the restricted domains of preferences.)-6-Result 3 (Kalal and Muller (1977), Maskin (1976))For n > 2 the following two statements are equivalent for every CI.1. Cl permits the construction of an n-person weakly nondic-tatorial, rational choice mechanism that is strategy-proof and alsosatisfies unanimity.2. CI permits the construction of an n-person weakly nondic-tatorial social welfare function that satisfies unanimity and indepen-dence of irrelevant alternatives.The difficulty in implementing stronger nondictatorship require-ments for choice mechanisms is demonstrated by the following trivialexample.Let A = {a,b,c}, N = {1,2,3} and CI = {p ,p~} such that ap bp.c andap,cp,b. Let's define a social welfare function f as follows: for3every x,y in A and P in CI , xf(P)y if and only if the majority in Nprefers x to y. Clearly f satisfies unanimity, independence of irrele-vant alternatives and is weakly nondictatorial. On the other hand, anychoice mechanism F that satisfies unanimity will choose for any P in3CI , F(P,A) = a, thus every agent in N is a full agenda dictator. Thisdemonstrates that the problem of how choice mechanisms satisfying theoriginal requirements of Gibbard and of Satterthwai te relate to nondic-tatorial SWFs is still an open one. Blair and Muller (1983) replacedthe weak nondictatorship requirement with the requirement that everyagent will have some decision powers— the essentiality requirement,and they proved:Result 4 (Blair and Muller (1983))The following two statements are equivalent.-7-a. The domain Q permits the construction of n-person essential,monotonic Arrow type social welfare functions.b. The domain ft permits the construction of n-person rational,essential and strategy-proof choice mechanisms.They also demonstrated that the largest restricted domain of pref-erences that admits essential, rational, strategy-proof choice mecha-nisms is the domain with "inseparable pair" defined in Kalai and Ritz(1979), which is of size | A | ! / 2 + (|A|-1)!. Kim and Roush (1981)demonstrated that this is also the largest restricted domain that per-mits the construction of stragety-proof , rational, weakly nondic-tatorial choice mechanisms which also satisfy unanimity. Muller andSatterthwaite (1983) observe that the question of the maximal size ofa domain that admits weakly nondictatorial , strategy-proof choicemechanisms which are not necessarily rational, is still an openquestion.In Kalai and Muller (1977), Maskin (1976), Blair and Muller (1983),Ritz (1983) and other works, the relaxation of the strong nondic-tatorship condition to the weak nondictatorship condition was con-sidered an acceptable modification. Here we will demonstrate that ifthe weak nondictatorship property is indeed considered to be a properreplacement for the strong nondictatorship property among the originalproperties defined by Gibbard and by Satterthwaite, and no additionalrequirements are specified, then it is possible to construct suchchoice mechanisms.-8-Theorem 1For every society N of at least two agents and every set of alter-natives A of at least two alternatives, when preferences areunrestricted there exist strategy-proof choice mechanisms that satisfyunanimity and weak nondictatorship.Indeed, it is even possible to replace the weak nondictatorshiprequirement with the essential requirement and still prove that:Theorem 2For any society N of at least two agents and every alternative setA of at least two alternatives when preferences are unrestricted,there exist essential, strategy-proof mechanisms that also satisfyunanimity.The following trivial corollary answers the question posed inMuller and Satterthwaite (1983).Corollary 1The largest domain admitting essential (therefore also weak nondic-tatorial) strategy-proof choice mechanisms is of size |A|!It is enough to prove theorem 2.Proof (of Thereom 2)Let us define a choice mechanism F as follows. Choose two alter-natives a,b in A. Then for any P in ft and B in H: if B # {a,b,}then F(P,B) = max (p ); if B = fa,b} and (i) n is odd, then F(P,B) isthe alternative preferred by the majority of all the agents in N; (ii)n is even but n > 2, then F(P,B) is the alternative preferred by the-9-majority among agents 2,3,...,n; (iii) n = 2 then F(P,B) is the alter-native preferred by the majority among agents 1,2 and the preference p ,where ap. t>.The proof that F is well defined, is essential and strategy-proof,and also satisfies unanimity can be derived easily and is omitted.Q.E.D.It is easy to verify that F is not a rational choice mechanism,which explains its sensitivity to changes in the agenda at issue.The above results do not eliminate the existence difficulties withchoice mechanisms. Rather these results demonstrate that by a properselection of requirements for strategy-proof choice mechanisms, thedifficulties can be kept either internal to the choice mechanism orexternal to it. By requiring either strong nondictatorship, fullagenda nondictatorship, or weak nondictatorship and rationality— thedifficulties are in the possibility that agents may take advantage ofthe specified choice mechanism and will misrepresent their preferences.This is a difficulty internal to the choice mechanism. On the otherhand, if only weak nondictatorship is required, the difficulties are inhow to decide on the agenda, prior to the choice process. Clearly,different agendas may give the decisive powers to different agents insociety. This can be considered as a difficulty external to the choicemechanism. The difficulties in establishing an agenda are widespreadand well known. For example, the president of the United States hasthe powers to decide on limited military operations, while he needsthe approval of Congress to wage war. The difficulty is then, as was-10-cl eraonst rated on a number of occasions, to decide whether an operationis of limited scale or an act of war.The results derived in this note suggest an interesting directionof research in the investigation of mechanisms that both select agen-das and choose outcomes out of these agendas.-11-REFERENCESArrow, K. J. (1963) Social Choice and Individual Values. 2nd ed. NewHaven: Yale University Press.Blair, D. H. and E. Muller (1983) "Essential Aggregation Procedures onRestricted Domains of Preferences," Journal of Economic Theory 33:pp. 34-53.Blin, J. M. and M. A. Satterthwai te (1978) "Individual Decisions andGroup Decisions: The Fundamental Differences," Journal of PublicEconomics, 10: 247-267.Gibbard, A. (1973) "Manipulation of Voting Schemes: A GeneralResult," Econoraetrica 41: pp. 587-601.Kalai , E. and E. Muller (1977) "Characterizations of Domains AdmittingNondictatorial Social Welfare Functions and Nonmanipulable VotingProcedures," Journal of Economic Theory 16: pp. 456-469.Kalai, E. and Z. Ritz (1979) "An Extended Single Peak Condition inSocial Choice," Discussion Paper No. 325, The Center forMathematical Studies in Economics and Management Science.Kalai, E. and Z. Ritz (1980) "Characterization of the PrivateAlternatives Domains Admitting Arrow Social Welfare Functions,"Journal of Economic Theory 22: pp. 23-36.Maskin, E. (1976) "On Strategy Proof ness and Social Welfare FunctionsWhen Preferences Are Restricted," Harvard University and DarwinCollege.Muller, E. and M. Satterthwai te (1983) "Strategy-Proofness : TheExistence of Dominant Strategy Mechanisms," Discussion Paper No.568, Northwestern University.Ritz, Z. (1983) "Restricted Domains, Arrow Social Welfare Functionsand Noncorruptible and Nonmanipulable Social ChoiceCorrespondences: The Case of Private Alternatives," MathematicalSocial Sciences 4: pp. 155-179.Satterthwaite, M. (1975) "Strategy-Proofness and Arrow's Conditions:Existence and Correspondence Theorems for Voting Procedures andSocial Welfare Functions," Journal of Economic Theory 10.D/299

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