consistent voting systems with a continuum of voters

Soc Choice Welfare (2006) 27: 477–492DOI 10.1007/s00355-006-0140-0

ORIGINAL PAPER

Bezalel Peleg · Hans Peters

Consistent voting systems with a continuumof voters

Received: 19 December 2002 / Accepted: 10 July 2005 / Published online: 5 May 2006© Springer-Verlag 2006

Abstract Voting problems with a continuum of voters and finitely many alter-natives are considered. Since the Gibbard–Satterthwaite theorem persists in thismodel, we relax the non-manipulability requirement as follows: are there socialchoice functions (SCFs) such that for every profile of preferences there exists astrong Nash equilibrium resulting in the alternative assigned by the SCF? SuchSCFs are called exactly and strongly consistent. The paper extends the work ofPeleg (Econometrica 46:153–161, 1978a) and others. Specifically, a class of anon-ymous SCFs with the required property is characterized through blocking coeffi-cients of alternatives and through associated effectivity functions.

1 Introduction

We consider a classical voting system with many voters and finitely many alterna-tives. Such a system is representative of political elections on the local or nationallevel. As an, in our view, best approximation we model voters as elements of anon-atomic measure space. In particular, this approach allows us to accommodatethe fact that in such voting systems single voters have negligible influence on thefinal outcome, and to avoid potential combinatorial peculiarities of a model with alarge but finite number of voters.

The emphasis of the paper is on strategic aspects. If we talk about strategicaspects in this model, we necessarily deal with strategic voting by groups of voters

B. Peleg (B)Institute of Mathematics and Center for the Study of Rationality,The Hebrew University of Jerusalem, Feldman Building,Givat-Ram, 91904 Jerusalem, IsraelE-mail: [email protected]

H. PetersDepartment of Quantitative Economics, University of Maastricht,P.O. Box 616, 6200 MD Maastricht, The NetherlandsE-mail: [email protected]

478 B. Peleg, H. Peters

(coalitions). This does not have to imply that coalitions actually get together tocoordinate their voting behavior. Although single voters are negligible for the finaloutcome, they may nevertheless derive utility from voting and, thus, may also votestrategically, possibly resulting in strategic behavior of groups of equally-mindedvoters.

In this model the result of Gibbard (1973) and Satterthwaite (1975) can beshown to persist.1 In particular, the requirement of non-manipulability implies the(undesirable) existence of an ‘invisible dictator’ as in Kirman and Sondermann(1972). Since, therefore, we cannot hope to reach the sincere outcome since wecannot expect voters to reveal their true preferences, we ask whether this outcomeis at least attainable in an equilibrium of the voting game. Specifically, we con-sider social choice functions (SCFs) satisfying the weaker requirement of exact andstrong consistency (ESC). This means that for every given profile of preferencesthere is another profile which (1) is a strong Nash equilibrium (SNE) – no coalitioncan profitably deviate – in the strategic game in which each voter reports a pref-erence and the outcome is evaluated according to given ‘true’ preferences, and (2)results in the same alternative as the true preferences. ESC SCFs were first studiedin Peleg (1978a,b) for finitely many voters. See also Dutta and Pattanaik (1978),Ishikawa and Nakamura (1980), Kim and Roush (1981), and Holzman (1986a,b).

In order to investigate ESC SCFs we make extensive use of effectivity functions(EFs). EFs were first introduced by Moulin and Peleg (1982); see also Abdou andKeiding (1991) for an overview. For a given SCF the associated EF describes, moreabstractly, the power of each coalition in terms of subsets of alternatives into whichthe coalition can force the final alternative by an appropriate choice of preferences.

The main results of the paper concentrate on anonymous ESC SCFs. Theseinduce so called blocking coefficients on the alternatives. A coalition can block asubset of alternatives if the size of the coalition is either larger than or at least aslarge as the sum of the blocking coefficients of the subset. Depending on these twocases, we call a subset of alternatives either an i-set or an e-set. This distinctionis the source of major differences with the finite case as treated in Peleg (1978a,1991), Oren (1981), Polishchuck (1978), and Holzman (1986a,b); see also Peleg(1984, Chapt. 5). Conversely, we define EFs by specifying i-sets and e-sets andassociated blocking coefficients.

We show that for EFs with exactly one alternative that must be strictly blocked,i.e., exactly one i-alternative, every element of its core can be obtained by a socalled feasible elimination procedure (f.e.p) and conversely, and that any anon-ymous selection from the core is an anonymous ESC SCF and conversely. Thisway, we obtain an almost complete characterization of anonymous ESC SCFs inCorollary 3.16.

2 The basic model

Let (�, �, λ) be a non-atomic measure space. Here � is the set of voters or players,� is the σ -field of permissible coalitions, and λ is a non-atomic measure on �.The number λ(S) for a coalition S is interpreted as the size of S. By �0 = �\{∅}

1 A proof is provided in an earlier version of the present paper, see Peleg and Peters (2003).

Consistent voting systems with a continuum of voters 479

we denote the set of all nonempty coalitions, and by �+ we denote the set of allcoalitions S with λ(S) > 0. Throughout we assume � ∈ �+ and λ(�) < ∞.

Let A be a finite set of alternatives. We assume throughout that |A| ≥ 2. (For afinite set D, |D| denotes the number of elements of D. By P(D) we denote the setof all subsets of D, and by P0(D) = P(D)\{∅} the set of all nonempty subsets.)A linear ordering of A is a complete, transitive, and antisymmetric binary relationon A. The set of all linear orderings of A is denoted by L(A).

A profile (of preferences) is a measurable function R : � → L(A) (i.e., foreach R ∈ L(A), {t ∈ � | R(t) = R} is in �). Two profiles R1 and R2 are equiva-lent, written R1 ∼ R2, if λ({t ∈ � | R1(t) �= R2(t)} = 0. Let ρ denote the set ofall profiles.

A social choice function SCF is a surjective function F : ρ → A that satisfies

for all R1, R2 ∈ ρ, if R1 ∼ R2, thenF(R1) = F(R2). (1)

Condition (1) implies that SCFs do not depend on the preferences of coalitions ofmeasure 0. In particular, because of non-atomicity, single agents do not have anyinfluence at all.

3 Exactly and strongly consistent social choice functions

As mentioned in the Introduction, every SCF F is manipulable: unless there is aso-called ‘invisible dictator’ (see Kirman and Sondermann 1972), there is always aprofile of preferences and a coalition of positive size that can obtain a better alter-native by not reporting their true preferences (see Peleg and Peters 2003). Thus, wecannot guarantee sincere voting but we are still interested in optimizing somehowthe chance of reaching the sincere outcome F(R) for every profile R ∈ ρ. Oneway to do this is to look for SCF’s where the sincere outcome can be reached inan equilibrium of the voting game. To make this more precise, observe that forevery R ∈ ρ the pair (F, R) defines a game in strategic form in a natural way:each player t ∈ � has strategy set L(A) and preference R(t) on A for evaluatingany outcome F(R∗) ∈ A, R∗ ∈ ρ. For S ∈ �0, denote by ρS the set of all mea-surable functions RS : S → L(A). Let R ∈ ρ. The profile Q is an SNE of thegame (F, R) if for every S ∈ �+ and every VS ∈ ρS , there exists T ∈ �+ withT ⊆ S and F(Q)R(t)F(Q�\S, VS) for every t ∈ T . The SCF F is exactly andstrongly consistent (ESC) if for every R ∈ ρ there exists an SNE Q of (F, R) suchthat F(Q) = F(R). Thus, if F is an ESC SCF, then for every profile there is anSNE profile that results in the same outcome, and therefore F is not necessarilydistorted. ESC consistent SCFs were introduced in Peleg (1978a).

Before we proceed to an investigation of ESC SCFs, we first consider a simpleexample. In our model an SCF F : ρ → A is anonymous if for all R1, R2 ∈ ρ wehave: if λ({t ∈ � | R1(t) = R}) = λ({t ∈ � | R2(t) = R}) for all R ∈ L(A),then F(R1) = F(R2). For a profile R and a, b ∈ A, a �= b, we say that a Paretodominates b if λ({t ∈ � | bR(t)a}) = 0. We call an alternative a ∈ A Paretooptimal with respect to R if it is not Pareto dominated by some other element ofA, and denote by PAR(R) the set of Pareto optimal alternatives with respect to R.


Example 3.1 Let a ∈ A be a designated alternative, called the status quo, and letR0 ∈ L(A) be fixed. Define an SCF F : ρ → A by

F(R) =⎧⎨

⎩

a if a ∈ PAR(R)a if a /∈ PAR(R) and a is the R0 − maximum

of {b ∈ PAR(R) | b Pareto dominates a}for all R ∈ ρ. We show that F is an anonymous ESC SCF. Obviously, F is surjec-tive. Now let R ∈ ρ. We distinguish the following possibilities.

(i) a ∈ PAR(R).Let Q ∈ ρ satisfy aQ(t)a for all t ∈ � and a ∈ A\{a}. Then Q is an SNE of(F, R) and F(Q) = F(R).

(ii) a /∈ PAR(R).

Let q be the R0-maximum of B = {b ∈ PAR(R) | b Pareto dominates a}.Define Q ∈ ρ by qQ(t)aQ(t)a for all t ∈ � and a ∈ A\{a, q}. Then F(Q) =q = F(R) and Q is an SNE of (F, R). Indeed, � does not have a profitabledeviation from Q since q is Pareto optimal with respect to R. Now let S ∈ �+,λ(S) < λ(�), and VS ∈ ρS . Then F(Q�\S, VS) ∈ {a, q}. Hence, VS cannot be aprofitable deviation for S.

The main goal of this section and of the paper is to study and characterize anony-mous ESC SCFs. Corollary 3.16 below summarizes the main results.

We first study ESC SCFs through their associated EFs. EFs provide an alter-native description of the power that coalitions have as the result of using a specificSCF in the society. This description is independent of the preference profile. EFsprovide a convenient tool to study ESC SCFs.

3.1 Effectivity functions of ESC social choice functions

We start with introducing the concept of an EF in our model. Then we derive theEF associated with an ESC SCF, and collect some interesting properties of it.

Definition 3.2 An EF is a function E : � → P(P0(A)) that satisfies the followingconditions: (i) E(�) = P0(A); (ii) E(∅) = ∅; (iii) A ∈ E(S) for every S ∈ �0;and (iv) if S1, S2 ∈ �0 and λ(S1\S2) + λ(S2\S1) = 0, then E(S1) = E(S2).

If B ∈ E(S) we sometimes say that S is effective for B: the interpretation is thatthe coalition S can guarantee that the outcome (alternative) is in B. Condition (iv)in Definition 3.2 is specific for our model. It says that the EF does not distinguishbetween coalitions that differ only in a set of measure 0.

An EF E is super-additive if for all S1, S2 ∈ � with S1 ∩ S2 = ∅ and allB1 ∈ E(S1) and B2 ∈ E(S2) we have: B1 ∩ B2 ∈ E(S1 ∪ S2). The EF E ismonotonic if for all S, S∗ ∈ � and B, B∗ ∈ P0(A) with B ∈ E(S), S ⊆ S∗ andB ⊆ B∗, we have B∗ ∈ E(S∗). These two conditions are natural and satisfied bymany EFs.

An EF E is maximal if for all S ∈ �0 and B ∈ P0(A) we have: if B /∈ E(S)then A\B ∈ E(�\S).


We call an EF convex if for all S1, S2 ∈ � and B1 ∈ E(S1), B2 ∈ E(S2) wehave B1 ∩ B2 ∈ E(S1 ∪ S2) or B1 ∪ B2 ∈ E(S1 ∩ S2).

We now turn to cores of EFs. Let E : � → P(P0(A)) be an EF and let R ∈ ρ.Let B ∈ P0(A), x ∈ A\B, and S ∈ �. We say that B dominates x via S at Rif B ∈ E(S) and bR(t)x for all b ∈ B and t ∈ S. Also, x is dominated at R ifthere exists B ∈ P0(A) and S ∈ � such that B dominates x via S at R. If b is notdominated at R then b is undominated at R.

Definition 3.3 The core C(E, R) is the set of all undominated alternatives at R.The EF E is stable if C(E, R) �= ∅ for all R ∈ ρ.

Let F : ρ → A be a SCF. We associate with F an EF E F as follows. Let S ∈�0 and let B ∈ P0(A). Call S effective for B if there exists an RS ∈ ρS suchthat F(RS, Q�\S) is in B for every Q�\S ∈ ρ�\S . Define E F (∅) = ∅, and forS ∈ �\{∅}

E F (S) = {B ∈ P0(A) | S is effective for B}.In the following theorem we collect some useful properties of E F .

Theorem 3.4 Let F : ρ → A be an ESC SCF. Then E F is super-additive, mono-tonic, maximal, stable, and convex. Moreover, F(R) ∈ C(E F , R) for all R ∈ ρ.

Proof Super-additivity and monotonicity are straightforward from the definition ofE F (ESC is not needed for this). Maximality and stability, as well as the last state-ment in the theorem, can be proved analogously to the case of finitely many voters,see Peleg (1984). Finally, stability and maximality together imply convexity. Aproof of this fact is analogous to the proof of Theorem 6.A.9 in Peleg (1984). �

3.2 The blocking coefficients of an anonymous ESC social choice function

In the remainder of the paper we concentrate on anonymous ESC SCFs. Anonymityis a natural requirement for voting procedures. Moreover, imposing this conditionwill enable us to derive much more detailed results on both SCFs and EFs.

Let F : ρ → A be an anonymous ESC SCF, with associated EF E F . A centralconcept is that of a blocking coefficient.

For B ∈ P0(A)\{A}, the blocking coefficient is the real number

β(B) = inf{λ(S) | A\B ∈ E F (S)}. (2)

The number β(B) is well defined since F is anonymous. It represents the mini-mum size of a ‘blocking coalition’ of B. We call B an e-set (‘e’ from ‘equality’)if S ∈ � and λ(S) = β(B) imply that A\B ∈ E F (S); otherwise, B is called ani-set (“i” from “inequality”). The first observation is the following straightforwardimplication:

If Bis an e-set, then β(B) > 0. (3)

We now derive a number of other properties of β(·), in particular Theorem 3.5,which says that β(·) is an additive function.


If B1, B2 ∈ P0(A) and B1 ∪ B2 �= A, then

β(B1 ∪ B2) ≤ β(B1) + β(B2). (4)

To see this, note that if the right hand side of this inequality is equal to λ(�),then the inequality holds. Now assume it is smaller. Let ε > 0 be small andlet Si ∈ � with λ(Si ) = β(Bi ) + ε and A\Bi ∈ E F (Si ) for i = 1, 2, such thatS1∩S2 = ∅. By super-additivity, A\(B1∪ B2) ∈ E F (S1∪S2), hence β(B1∪ B2) ≤β(B1) + β(B2) + 2ε. By letting ε approach 0, (4) follows.

For every B ∈ P0(A)\{A} we have

β(B) + β(A\B) ≥ λ(�) (5)

because otherwise there would be disjoint coalitions S and T with B ∈ E F (S) andA\B ∈ E F (T ), contradicting the super-additivity of E F . We shall now show thereverse inequality. Assume β(B) > 0 otherwise there is nothing left to prove. Forevery 0 < δ < β(B) and S ∈ � with λ(S) = δ we have A\B /∈ E F (S). Henceby maximality of E F , B ∈ E F (�\S), so β(A\B) ≤ λ(�) − δ. This implies thereverse inequality of (5), hence

β(B) + β(A\B) = λ(�) (6)

for every B ∈ P0(A)\{A}. Super-additivity, maximality and (6) imply

B is an e-set ⇔ A\B is an i-set (7)

for every B ∈ P0(A)\{A}.Moreover, monotonicity of E F clearly implies monotonicity of the function

β(·):B1 ⊆ B2 ⇒ β(B1) ≤ β(B2) (8)

for all B1, B2 ∈ P0(A)\{A}.We now show that blocking coefficients are actually additive.

Theorem 3.5 β(·) is additive.

Proof Let Bi ∈ P0(A), i = 1, 2, with B1 ∩ B2 = ∅ and B1 ∪ B2 �= A. In viewof (4) it is sufficient to prove that β(B1 ∪ B2) ≥ β(B1) + β(B2). By (8) we mayassume β(Bi ) > 0 for i = 1, 2. Let S and T satisfy λ(S) < β(B1), λ(T ) < β(B2),and S ∩ T = ∅. Then by (6), B1 ∈ E F (�\S) and B2 ∈ E F (�\T ). By con-vexity of E F , B1 ∪ B2 ∈ E F (�\(S ∪ T )). Thus, by definition of β(·) and (6),β(B1 ∪ B2) ≥ λ(S) + λ(T ). Since, by (6) and (8), β(B1) + β(B2) ≤ λ(�), wecan choose λ(S) and λ(T ) as close to β(B1) and β(B2), respectively, as desired,which completes the proof. �We note that Theorem 3.5 is a substantial deviation from the case with finitelymany voters, see Theorem 5.2.16 in Peleg (1984).

In view of Theorem 3.5 and (6) it is useful to define β(A) = λ(�) and let Abe an i-set.

For e-sets we have the following theorem.

Theorem 3.6 If B1 and B2 are e-sets, then B1 ∩ B2 or B1 ∪ B2 are e-sets.


Proof Let B1 and B2 be e-sets. If B1 ∩ B2 = ∅ then take disjoint coalitions S1and S2 of sizes β(B1) and β(B2), respectively. Then A\B1 ∈ E F (S1) and A\B2 ∈E F (S2). By super-additivity, A\(B1 ∪ B2) ∈ E F (S1 ∪ S2). Since λ(S1 ∪ S2) =β(B1 ∪ B2) by Theorem 3.5, we conclude that B1 ∪ B2 is an e-set.

Next, assume B1 ∩ B2 �= ∅. Choose pairwise disjoint sets S1, S2, and S3 in�0 such that λ(S1) = β(B1) − β(B1 ∩ B2), λ(S2) = β(B2) − β(B1 ∩ B2), andλ(S3) = β(B1 ∩ B2). Define T1 = S1 ∪ S3 and T2 = S2 ∪ S3. Then λ(T1) = β(B1),λ(T2) = β(B2), λ(T1 ∩ T2) = β(B1 ∩ B2), and λ(T1 ∪ T2) = β(B1 ∪ B2).By assumption, A\B1 ∈ E F (T1) and A\B2 ∈ E F (T2). Since E F is convex,A\(B1 ∪ B2) ∈ E F (T1 ∪ T2) or A\(B1 ∩ B2) ∈ E F (T1 ∩ T2). Thus, B1 ∪ B2 orB1 ∩ B2 are e-sets. �Example 3.7 For the EF associated with the ESC SCF of Example 3.1 we haveβ(a) = λ(�) and β(a) = 0 for a ∈ A\{a}: any positive coalition can block anyset of alternatives not containing a by putting a on top of its preference profile.The i-sets are A and every B ⊆ A\{a}. In particular, if |A| ≥ 3 and a ∈ A\{a}then {a, a} is an e-set.

We conclude this subsection by generalizing the concepts of e-sets and i-sets. Letβ : P0(A) → [0, λ(�)] and let {i, e} be a partition of P0(A) satisfying

β is additive, β(A) = λ(�), and β(B) > 0 for all B ∈ e, (9)

for all B ∈ P0(A)\{A}, B ∈ e ⇔ A\B ∈ i, and A ∈ i, (10)

for all B1, B2 ∈ e, we have B1 ∩ B2 ∈ e or B1 ∪ B2 ∈ e. (11)

Properties (9), (10) and (11) summarize exactly all the properties of the e-sets andi-sets of the EF associated with an anonymous ESC SCF established above.

Next, for a system (β; e, i) satisfying (9), (10) and (11), we define an EF E byE(�) = P0(A), E(∅) = ∅, A ∈ E(S) for every S ∈ �0, and

for all B ∈ eandS ∈ �, if λ(S) ≥ β(B) then A\B ∈ E(S), (12)

for all B ∈ i and S ∈ �, if λ(S) > β(B) then A\B ∈ E(S). (13)

(Observe that condition (iv) in the definition of an EF follows from all this.)In the remainder of Sect. 3 we consider the following question: given a system(β; e, i) and associated EF E , is there an (anonymous) ESC SCF F such thatE = E F ? We tackle this problem through the study of so-called f.e.p (Subsect.3.3) and their relation with the core (Subsect. 3.4). A sufficient condition for anaffirmative answer to this question is given in Corollary 3.16 in Subsect. 3.4. Corol-lary 3.19 in Subsect. 3.5 shows that this sufficient condition is ‘almost’ necessary.

3.3 Feasible elimination procedures

In this section we describe a procedure that, in the end, will result in an anony-mous ESC SCF. Later, in Subsect. 3.4, we will see that in this way we obtain acharacterization of a class of ESC SCFs.

Let |A| = m ≥ 2. Let s ∈ A be a fixed element and let real numbers β(a) ≥ 0be given with

∑a∈A β(a) = λ(�) and with β(a) > 0 for all a �= s.


Definition 3.8 Let R ∈ ρ. A pseudo feasible elimination procedure (p.f.e.p.) is asequence (xi1, C1; . . . ; xim−1, Cm−1; xim ) that satisfies the following conditions:

A = {xi1, . . . , xim }; (14)

for all j, k = 1, . . . , m − 1 with j �= k, we have (15)

C j ∈ �, C j ∩ Ck = ∅, and λ(C j ) ≥ β(xi j );

for all j = 1, . . . , m − 1 and all t ∈ C j , (16)

yR(t)xi j for all y ∈ {xi j , . . . , xim }.In a p.f.e.p procedure, ‘bottom’ alternatives are eliminated consecutively. The des-ignated alternative s plays no special role. As

∑a∈A β(a) = λ(�), it is obvious

that for each profile there always exists at least one p.f.e.p. A more demandingprocedure is the following one; in this procedure, the designated alternative s canonly be eliminated if, during the procedure, it becomes a bottom alternative for acoalition of size strictly larger than β(s).

Definition 3.9 Let R ∈ ρ. A p.f.e.p. (xi1, C1; . . . ; xim−1, Cm−1; xim ) is a f.e.p. if itsatisfies the following condition:

xim = s or [xi j = s for some j < m and λ(C j ) > β(s)]. (17)

We shall now prove the existence of f.e.p.’s in our model and then relate them toESC SCFs. First a notation: for a profile R and a subset B of A, denote by R|B theprofile of preferences restricted to the set B. We start with the following lemma.

Lemma 3.10 Let R ∈ ρ and let x ∈ A satisfy

λ({t ∈ � | yR(t)x for all y ∈ A}) > β(x).

Then there exists a p.f.e.p. (x, Cx ; xi1, C1; . . . ; xim−1) with λ(Cx ) > β(x).

Proof The proof is by induction on m. The case m = 2 is obvious. Let m ≥ 3. Wedefine

A∗ = {y ∈ A | λ({t ∈ � | zR(t)y for all z ∈ A}) > β(y)}. (18)

By assumption, x ∈ A∗. We distinguish the following cases.

(i) |A∗| ≥ 2.Let y ∈ A∗\{x} and choose Cy ⊆ � such that λ(Cy) = β(y) and Cy ⊆{t ∈ � | zR(t)y for all z ∈ A}. Define the profile Q ∈ ρ as follows. If t ∈�\Cy with zR(t)y for all z ∈ A, then let xQ(t)A\{x, y}Q(t)y; otherwise,Q(t) = R(t). Consider the restricted profile Q1 = Q�\Cy |A\{y}. By theinduction hypothesis and by the construction of Q there exists a p.f.e.p.(x, Cx ; xi1, C1; . . . ; xim−2) with respect to Q1 such that λ(Cx ) > β(x) andCx ⊆ {t ∈ � | zR(t)x for all z ∈ A}. Then the p.f.e.p.(x, Cx ; y, Cy; xi1, C1; . . . ; xim−2) is as required.


(ii) A∗ = {x}.Let Cx satisfy Cx ⊆ {t ∈ � | yR(t)x for all y ∈ A} and λ(Cx ) = β(x).

Consider the profile R1 = R�\Cx |A\{x}. For y �= x let Cy = {t ∈ �\Cx |zR(t)y for all z ∈ A\{x}}. We distinguish two subcases.

(ii.1) λ(Cy) = β(y) for all y �= x .Choose y ∈ A\{x} such that

λ({t ∈ �\Cx | zR(t)yR(t)x for all z ∈ A\{x}}) > 0.

Let C ⊆ {t ∈ �\Cx | zR(t)yR(t)x for all z ∈ A\{x}} satisfy λ(C) > 0.(Observe that C ⊆ Cy , hence λ(C) ≤ β(y).) Let Cx = Cx ∪ C , and letA\{x, y} = {y1, . . . , ym−2}. Then (x, Cx ; y1, Cy1; . . . ; ym−2, Cym−2; y) isa p.f.e.p. as required.

(ii.2) There exists y �= x such that λ(Cy) > β(y).By the induction hypothesis there exists a p.f.e.p. (y, C y; xi1, C1; . . . , xim−2)

with respect to R1 such that λ(C y) > β(y). Note thatλ({t ∈ C y | zR(t)yR(t)x for all z ∈ A\{x}}) > 0 since y /∈ A∗. ChooseC ⊆ {t ∈ C y | zR(t)yR(t)x for all z ∈ A\{x}} such that 0 < λ(C) ≤λ(C y) − β(y). Then (x, Cx ∪ C; y, C y\C; xi1, C1; . . . ; xim−2) is a p.f.e.p.as required. �

Next, we establish the existence of f.e.p.s

Theorem 3.11 For every R ∈ ρ there is an f.e.p. with respect to R.

Proof Let R ∈ ρ. The proof is by induction on m. The case m = 2 is obvious. Letm ≥ 3. Define A∗ as in (18). We distinguish the following possibilities.

(i) A∗ = ∅.For a ∈ A let C(a) = {t ∈ � | yR(t)a for all y ∈ A}. Then λ(C(a)) = β(a)for all a ∈ A. Let A\{s} = {a1, . . . , am−1}.Then (a1, C(a1); . . . ; am−1, C(am−1); s) is an f.e.p.

(ii) A∗ �= ∅ and s /∈ A∗.Let y ∈ A∗ and let Cy ⊆ {t ∈ � | zR(t)y for all z ∈ A} satisfy λ(Cy) =β(y). By the induction hypothesis for R�\Cy |A\{y} there exists an f.e.p. (xi1,C1; . . . ; xim−1) for the restricted profile. Then (y, Cy; xi1, C1; . . . ; xim−1) isan f.e.p. for R.

(iii) s ∈ A∗.This case follows from Lemma 3.10. �

We shall use the existence of f.e.p.s established in Theorem 3.11 to derive theexistence of an interesting class of ESC SCFs. Let R ∈ ρ. Call x ∈ A R-maximalif there exists an f.e.p. (xi1, C1; . . . ; xim ) with respect to R such that x = xim .Further, denote

M(R) = {x ∈ A | x is R -maximal}.The following observation concerning M(·) will be very useful below.


Remark 3.12 Let R ∈ ρ and let x ∈ A\{s} satisfy

λ({t ∈ � | yR(t)x for all y ∈ A}) ≥ β(x).

Then x /∈ M(R). This is so since λ(⋃

y∈A\{x}{t ∈ � | A\{y}R(t)y}) ≤ λ(�) −β(x) and s has to be eliminated strictly in an f.e.p.

Theorem 3.13 Let the SCF F : ρ → A be a selection from M(·), that is, F(R) ∈M(R) for every R ∈ ρ. Then F is ESC.

Proof Let R ∈ ρ and x = F(R). Then there exists an f.e.p.(xi1, C1; . . . ; xim−1, Cm−1; x) with respect to R. Choose Q ∈ ρ that satisfiesyQ(t)xi j for all t ∈ C j , y ∈ A, and j = 1, . . . , m−1. We claim that F(Q) = F(R)and that Q is an SNE of the game (F, R). We distinguish the following cases.

(i) x = s.By Remark 3.12 F(Q) = s. Now assume, on the contrary, that Q is not an SNEof (F, R). Then there exist S ∈ �+ and VS ∈ ρS such that F(Q�\S, VS) = y,y �= s, and yR(t)s for all t ∈ S. Let y = xi j for some 1 ≤ j ≤ m − 1.Then S ∩ C j = ∅ because sR(t)xi j for all t ∈ C j . Hence, by Remark 3.12,F(Q�\S, VS) �= xi j , which is the desired contradiction.

(ii) x �= s.Then s = xi j0

for some j0 ≤ m −1. Hence, by definition of an f.e.p., λ(C j0) >

β(s). Hence, it is not possible to eliminate all x �= s in an f.e.p. with respect toQ, and therefore F(Q) �= s. Thus, by Remark 3.12 applied to all x ′ ∈ A\{x, s},F(Q) = x . The proof that Q is an SNE of (F, R) is analogous to that in case(i), observing that a profitable deviation from Q can never result in s sinceλ(C j0) > β(s). �

Let F be an anonymous selection from M(·). (E.g., for every R ∈ ρ select the max-imal element in M(R) according to a fixed order R0 ∈ L(A).) By Theorem 3.13 F

is an anonymous ESC SCF, and therefore its associated EF E F is characterized byblocking coefficients (say) β(B) for B ∈ P0(A). Since alternatives assigned by Fresult from f.e.p.s with weights β(a) (a ∈ A), it is easy to check that β(a) = β(a)for every a ∈ A, and that {s} is an i-set whereas all other singleton sets are e-sets.By the results established in Subsect. 3.2, it follows that a set B ⊆ A is an i-set

if and only if it contains s. Note that the EF E F is independent of the particularanonymous selection F chosen, and thus we can denote it by E . Since, for allR ∈ ρ and for every element x ∈ M(R) we can always find an anonymous selec-tion choosing that particular element, Theorem 3.4 implies that M(R) ⊆ C(E, R)for all R ∈ ρ. We can also state this as M(R) ⊆ C(E, R) for all R ∈ ρ, whereE is the EF associated with the system (β, e, i) as above (cf. Subsect. 3.2). In thenext section we shall establish the converse inclusion C(E, R) ⊆ M(R).

3.4 Core and feasible elimination procedures

In this section we prove that for any anonymous ESC SCF that has exactly onei-alternative, every element in the core of the associated EF can be obtained by anf.e.p.


Let (β; e, i) be a system satisfying (9), (10) and (11) with i containing exactlyone singleton {s} for some designated s ∈ A (hence β(y) > 0 for all y ∈ A\{s}).Let E be the associated EF and let R ∈ ρ and x ∈ C(E, R) (note that C(E, R) �= ∅,see the last paragraph of Subsect. 3.3). For every y ∈ A\{x} denote

S(y) = {t ∈ � | xR(t)y}.The fact that x ∈ C(E, R) is equivalent to

λ

⎛

⎝⋃

y∈B

S(y)

⎞

⎠ ≥ β(B) if s /∈ B (19)

λ

⎛

⎝⋃

y∈B

S(y)

⎞

⎠ > β(B) if s ∈ B,

for all B ∈ P0(A). In this setting we have the following result.2

Theorem 3.14 There exist disjoint measurable sets C(y) (y ∈ A\{x}) such that(i) C(y) ⊆ S(y) for every y ∈ A\{x}; (ii) λ(C(y)) ≥ β(y) for all y �= x andλ(C(s)) > β(s).

Proof We start by noting that, if x �= s, we may increase β(s) with a small ε > 0and decrease β(x) with the same amount (note that β(x) > 0). In this way, allinequalities in (19) still hold as weak inequalities and it is sufficient to prove (ii)in the theorem with only weak inequalities. Moreover, we may regard x as thei-alternative instead of s. For the rest of the proof we assume that this is the case.

We prove the theorem by induction on |A| = m ≥ 2. The case m = 2 isobvious, so we concentrate on the induction step for m ≥ 3. We first make thefollowing observation.

Remark Suppose there exists a set B∗ ⊆ A\{x} with ∅ �= B∗ �= A\{x}, suchthat λ(

⋃y∈B∗ S(y)) = β(B∗). Then, for every B ⊆ A\({x} ∪ B∗) we have

λ(⋃

y∈B S(y)\⋃y∈B∗ S(y)) ≥ β(B). Hence, we can decompose our problem

into two smaller problems, namely: (i) the problem with set of alternatives A\B∗,set of voters �\ ⋃

y∈B∗ S(y), blocking coefficients unchanged, and preferencesR(t) restricted to A\B∗; (ii) the problem with set of alternatives B∗ ∪ {x}, set ofvoters

⋃y∈B∗ S(y), blocking coefficients β(x) = 0 and β(y) = β(y) unchanged

for y ∈ B∗, and preferences R(t)|B∗ ∪{x} for t ∈ ⋃y∈B∗ S(y). Then we can apply

the induction hypothesis to both smaller problems and find sets C(y) (y ∈ A\{x})as required.

We now proceed to the induction step. Let m ≥ 3. We are done if there is adecomposition possible as in the Remark, so suppose there is none. Let b ∈ A\{x}and consider the set S = S(b)\⋃

y∈A\{x,b} S(y). We distinguish two cases.

2 Theorem 3.14 is a continuous version of the discrete ‘marriage theorem’ (cf. Halmos andVaughan 1950), suitable for our context, in particular for deriving Theorem 3.15 below. Fora proof of a slightly less general version of the continuous ‘marriage theorem’ see Hart andKohlberg (1974, p. 171)


Case 1 λ(S) ≥ β(b). Note that for every t ∈ S, yR(t)xR(t)b for all y �= x, b.Hence, since x ∈ C(E, R), 0 ≤ λ(S) ≤ β(x) + β(b). Now take C(b) equal to S,and apply the induction hypothesis to the problem with set of alternatives A\{b},set of voters �\S, blocking weights β ′ unchanged except β ′(x) = β(x)− (λ(S)−β(b)), and preferences equal to the original preferences restricted to A\{b}.Case 2 λ(S) < β(b). We also know λ(S(b)) > β(b) otherwise λ(S(b)) = β(b)by (19), and we would have a decomposition as in the Remark with B∗ = {b}.Now choose a measurable set S∗ satisfying S ⊆ S∗ ⊆ S(b) and λ(S∗) = β(b)(this is possible by Lyapunov’s theorem). Consider the set of vectors

⎧⎨

⎩λ(S∗ ∪ T ∪

⋃

y∈B

S(y))B⊆A\{b,x}, B �=A\{b,x} | ∅ ⊆ T ⊆ S(b)\S∗⎫⎬

⎭. (20)

For T = S(b)\S∗ and B = ∅ we have

λ(S∗ ∪ T ∪⋃

y∈B

S(y)) = λ(S(b)) > β(b) (21)

and for T = S(b)\S∗ and B ⊆ A\{x, b} arbitrary we have

λ

⎛

⎝S∗ ∪ T ∪⋃

y∈B

S(y)

⎞

⎠ ≥ β(b) + β(B) (22)

by (19). For B ⊆ A\{x, b} with B �= A\{x, b} and T ⊆ S(b)\S∗ consider theexpression

λ

⎛

⎝S∗ ∪ T ∪⋃

y∈B

S(y)

⎞

⎠ = λ(T ) + λ

⎛

⎝S∗ ∪⋃

y∈B

S(y)

⎞

⎠

−λ

⎛

⎝T ∩⎛

⎝S∗ ∪⋃

y∈B

S(y)

⎞

⎠

⎞

⎠ .

This is an affine function of two measures λ(T ) and λ(T ∩(S∗∪⋃y∈B S(y))). As B

varies on {B ′ | B ′ ⊆ A\{b, x}, B ′ �= A\{b, x}} we obtain an affine combination oftwo vector measures. Hence, its range (20) is compact and convex by Lyapunov’stheorem. By (21) and (22) we can choose T = T0 such that all inequalities in (22)are still valid but with at least one equality, say for B0. Now set S0 = S∗ ∪ T0, andset B∗ = B0 ∪ {b}. On S(b)\S0 change the preferences by shifting b over x . Theproblem with the new profile is decomposable according to the Remark. Applyingthe Remark, we obtain the desired sets: in particular, the resulting set C(b) is asubset of S0 and therefore of S(b). This concludes the proof of the theorem. �For every R ∈ ρ let M(R) be the set of alternatives attainable through an f.e.p., asin Subsect. 3.3. We have already established that these alternatives are in the coreC(E, R). We will now show the converse. Call b ∈ A a bottom alternative of R ifthe set S(b) = {t ∈ � | yR(t)b for all y ∈ A} has measure λ(S(b)) ≥ β(b), withstrict inequality sign for b = s.


Theorem 3.15 Let (β; e, i) be a system satisfying (9), (10) and (11) with i contain-ing exactly one singleton {s} for some designated s ∈ A. Let E be the associatedEF and let R ∈ ρ and x ∈ C(E, R). Then x ∈ M(R). In particular, if b is a bottomalternative of R, then there is an f.e.p. (b, Cb; yi1, C1; . . . ; yim−2 , Cm−2; x).

Proof Let b be a bottom alternative. If b = s we slightly increase the blockingcoefficient of b (as in the beginning of the proof of Theorem 3.14) so that we stillhave λ(S(b)) ≥ β(b).

The proof is by induction on m = |A|. For m = 2 the result is obvious again.Let m ≥ 3.

(i) First suppose that the problem is decomposable into two subproblems withsets of alternatives {x} ∪ B∗ and A\B∗ as in the proof of Theorem 3.14, andwith b ∈ B∗. Note that all voters in the problem with A\B∗ rank B∗ abovex . By the induction hypothesis, each of the subproblems has an f.e.p. leadingto x , with the one in the first subproblem starting with b. Let |B∗| = k, let(b, Cb; y1, C1; . . . ; yk−1, Ck−1; x) be an f.e.p. in the problem with {x} ∪ B∗and let (x1, C1; . . . ; xm−k−1, Cm−k−1; x) be an f.e.p. in the problem withA\B∗. Then

(b, Cb; x1, C1; . . . ; xm−k−1, Cm−k−1; y1, C1; . . . ; yk−1, Ck−1; x)

is an f.e.p. for the original problem.(ii) Next, suppose the problem is not decomposable in this way. As in the proof

of Theorem 3.14 let S = S(b)\⋃y∈A\{x,b} S(y) and distinguish two cases

as there. In Case 1, λ(S) ≥ β(b), we take again C(b) = S, observing thatS ⊆ S(b). Applying the induction hypothesis, we let (yi1, C1; . . . ; yim−2 ,Cm−2; x) be an f.e.p. in the problem with set of alternatives A\{b}, then(b, Cb; yi1, C1; . . . ; yim−2 , Cm−2; x) is as desired.

In Case 2, we proceed again as in the proof of Theorem 3.14 but we make surethat S0 there is chosen in such a way that λ(S0 ∩ S(b)) ≥ β(b). This is possiblesince S ⊆ S(b) ⊆ S(b) and so we can choose S∗ (which is a subset of S0 byconstruction) such that S∗ ⊆ S(b). We have now again a decomposition as in (i) ofthis proof: since b is eliminated first, shifting b over x in the original preferences ofvoters in S(b)\S0 does not change the restriction of these preferences to A\B∗. �We note that Theorem 3.15 may also be adapted to apply to the case with finitelymany agents and, thus, provides an alternative and shorter proof of Theorem 5.4.2in Peleg (1984).

The following corollary is a summary of the main results of Subsects. 3.3and 3.4.

Corollary 3.16 (i) Let F be an anonymous ESC SCF. Suppose that the associatedeffectivity function E has exactly one i-alternative. Then C(E, ·) = M(·) and Fis a selection from this set. (ii) Let (β; e, i) be a system satisfying (9), (10) and(11) such that i contains exactly one singleton. Then, for the associated effectivityfunction E, C(E, ·) = M(·), and any anonymous selection from these sets is ananonymous ESC SCF.


3.5 A further result on anonymous ESC social choice functions

A natural question is whether Corollary 3.16 can be extended to general systems(β; e, i). In this subsection we show that if an anonymous ESC SCF generates onlypositive blocking coefficients, then there can be at most one i-alternative.Let E be the EF associated with a system (β; e, i), satisfying (9), (10) and (11).

Definition 3.17 E satisfies D(k), where 1 ≤ k ≤ m −2, if there exist no partitions{x1}, . . . , {xk}, C1, C2 of A and S1, . . . , Sk, T1, T2 of �, such that

(i) λ(Si ) = β(xi ) for i = 1, . . . , k, and x1, . . . , xk are e-alternatives;(ii) λ(Ti ) = β(Ci ) for i = 1, 2, and C1 and C2 are i-sets.

In the Appendix we provide a proof of the following theorem.

Theorem 3.18 Let F : ρ → A be an anonymous ESC SCF, and let (β; e, i) be theassociated system. Suppose that β(a) > 0 for all a ∈ i. Then E = E F satisfiesD(k) for all 1 ≤ k ≤ m − 2.

We now have:

Corollary 3.19 Let F : ρ → A be an anonymous ESC SCF that generates onlypositive blocking coefficients. Then there is exactly one i-alternative.

Proof Suppose that there are two different elements x, y ∈ i in the associated sys-tem. Define C1 = {x}, C2 = i\{x}, and let e = {x1, . . . , xk}. Then 1 ≤ k ≤ m −2,and D(k) can be violated by choosing a partition of � as in Definition 3.17. Thiscontradicts Theorem 3.18. �The combination of Corollaries 3.16 and 3.19 yields an almost complete char-acterization of anonymous ESC SCFs. The case where there is more than onei-alternative – so that at least one i-alternative has zero blocking coefficient – isstill open. Example 3.7 shows that this can indeed occur.

Acknowledgements Financial support from the Research School METEOR of the Universityof Maastricht and from the Dutch Science Foundation NWO (grant no. B 46-476) is gratefullyacknowledged.

A Appendix: Proof of Theorem 3.18

We start with the following observation.

Lemma A.1 Let F : ρ → A be an ESC SCF. Then there exist no partitionsS1, . . . , Sp of � and B1, . . . , Bp of A (where p ≥ 2) such that A\Bi /∈ E F (Si )for all i = 1, . . . , p.

Proof Assume, on the contrary, that there exist partitions as in the lemma. Considerthe following profile:

S1 S2 · · · SpB2 B3 · · · B1B3 B4 · · · B2...

......

...Bp B1 · · · Bp−1B1 B2 · · · Bp


By maximality (Theorem 3.4) of E F , we have Bi ∈ E F (�\Si ) for everyi = 1, . . . , p, hence C(E F , R) = ∅. But this contradicts stability (Theorem 3.4)of E F . �

Proof of Theorem 3.18. The proof is by induction on k.(1) k = 1. Assume, on the contrary, that there are partitions {x1}, C1, C2 of A andS1, T1, T2 of �, satisfying (i) and (ii) in Definition 3.17. Let S1 = S1 ∪ S2 withS1 ∩ S2 = ∅ and λ(S1) = λ(S2). Consider the following profile R:

S1 S2 T1 T2C1 C2 x1 x1C2 C1 C2 C1x1 x1 C1 C2

.

Since S1 can block x1 (i.e., A\{x1} ∈ E(S1)), stability of E F (Theorem 3.4) impliesF(R) �= x1. Without loss of generality F(R) ∈ C1. Let Q be a SNE of (F, R)with F(R) = F(Q). We distinguish the following possibilities.

There exists y ∈ C2 such that λ({t ∈ S1 | x1Q(t)y}) > 0. (23)

Choose S3 ⊆ {t ∈ S1 | x1Q(t)y} such that 0 < λ(S3) < mina∈A β(a). Define theT1 ∪ T2-profile P by

x1P(t)yP(t)A\{x1, y}for all t ∈ T1 ∪ T2.

By considering the partitions S1\S3, S3, T1∪T2, and {x1}, {y}, A\{x1, y}, it followsfrom Lemma A.1 that T1 ∪ T2 blocks A\{x1, y}. Hence,F(QS1, PT1∪T2) ∈ {x1, y}. As T1 ∪ T2 ∪ S3 is effective for x1, andF(QS1, PT1∪T2) ∈ C(E, (QS1, PT1∪T2)), we have F(QS1, PT1∪T2) = x1. Thus,T1 ∪ T2 has improved upon F(Q), which is a contradiction.

C2Q(t)x1 for all t ∈ S1. (24)

Consider the T1 ∪ S2-profile P defined by

C2P(t)x1P(t)C1 for all t ∈ T1 ∪ S2.

Since λ(T1 ∪ S2) > λ(T1) = β(C1), T1 ∪ S2 blocks C1.Therefore, F(QT2∪S1

, PT1∪S2) /∈ C1. Suppose F(QT2∪S1

, PT1∪S2) = x1. Note

that, in QT2∪S1, PT1∪S2

, both T1 and S1 prefer C2 over x1. Moreover, λ(T1 ∪S1) = β(C1) + β(x1) and C1 ∪ {x1} is an e-set, because C2 is an i-set; therefore,T1 ∪ S1 blocks C1 ∪ {x1}. This contradicts F(QT2∪S1

, PT1∪S2) = x1 and, hence,

F(QT2∪S1, PT1∪S2

) ∈ C2. But this contradicts the fact that Q is a SNE in (F, R).

(2) Let 1 < k ≤ m − 2 and assume D(1), . . . , D(k − 1). We shall prove D(k).

Assume, on the contrary, that there exist partitions {x1}, . . . , {xk}, C1, C2 of A andS1, . . . , Sk , T1, T2 of �, satisfying (i) and (ii) in Definition 3.17. If C1 ∪ {xk} isan i-set, then we obtain a contradiction to D(k − 1). Otherwise, A\(C1 ∪ {xk}) isan i-set. Then consider the partitions {xk}, C1, A\(C1 ∪ {xk}) of A, and Sk , T1,�\(Sk ∪ T1) of �: this implies a contradiction to D(1). �.


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