On societies choosing social outcomes, and their memberships: internal stability and consistency

We consider a society whose members have to choose not only an outcome from a given set of outcomes but also a subset of agents that will remain members of the society. We study the extensions of approval voting, scoring methods and the Condorcet winner to our setting from the point of view of their internal stability and consistency properties.


Introduction
Classical social choice studies problems where a fixed set of agents have to choose an outcome from a given set of outcomes, and agents have preferences only over this set. However, there are settings where, depending on the chosen outcome, some agents might want to leave the society; and this, in turn, might be perceived by some agents that were initially willing to remain in the society as negative, and now they might also want to leave. For instance, in the case of an excludable and costly public good, agents' preferences may depend on the level of the public good and on the size of the set of agents consuming (and contributing to finance) it. Also, when membership is voluntary in a double sense, no agent can be forced to belong to the final society and any agent can be part of it, if the agent wishes to be. A prototypical example of this class of problems is a political party, whose membership may depend on the positions that the party takes on issues like the death penalty, abortion or the possibility of allowing a region of a country to become independent. A professor in a department may start looking for a position elsewhere if he considers that the recruitment of the department has not been satisfactory to his standards; and this, in turn, might trigger further exits. To be able to deal with such situations, the classical social choice model has to be modified to include explicitly the possibility that initial members of the society may leave it as the consequence of the chosen outcome and hence, preferences have to be extended to order pairs formed by the final society and the chosen outcome.
There is a large literature that has already considered explicitly the dependence of the final society on the choices made by the initial society. 1 Barberà et al. (2001), Barberà and Perea (2002), and Berga et al. (2004Berga et al. ( , 2006Berga et al. ( , 2007 study alternative models in terms of the voting methods used to choose the outcome and the timing under which members reconsider their membership. Jackson and Nicolò (2004) study the provision of excludable public goods when agents care also about the number of other consumers. In this note (as we also do in the companion paper Bergantiños et al. 2017), we look at the general setting without being specific about the two issues. We do that by considering that the set of alternatives are all pairs formed by a subset of the original society N (an element in 2 N , the subset of agents that will remain in the society) and an outcome in X . Then, we assume that agents' preferences are defined over the set of alternatives 2 N × X and satisfy two natural requirements. First, each agent has a strict preference between any two alternatives, provided he belongs to at least one of the two corresponding societies. Second, each agent is indifferent between any two alternatives, provided he is not a member of any of the two corresponding societies; namely, agents that do not belong to the final society do not care about neither its composition nor the chosen outcome.
We consider rules that operate on this restricted domain of preference profiles by selecting, for each profile, an alternative (a final society and an outcome). In Bergan-tiños et al. (2017) we characterize the class of strategy-proof, unanimous and outsider independent rules as the family of all serial dictator rules. 2 For applications where the profile is common knowledge (and hence, the strategic revelation of agents' preferences is not an issue), we focus on internally stable and consistent rules. 3 Internal stability says that nobody can force an agent to remain in the society if the agent does not want to do so. This is a minimal requirement of individual rationality, and it is a desirable property whenever membership is voluntary. A rule is consistent if the following property holds. Apply the rule to a given profile and consider the new problem where the new society is formed by the subset of agents chosen at the original profile. A consistent rule has to choose, at the subprofile of preferences of the agents that remain in the society, the same alternative. Thus, a consistent rule does not have to be reapplied after an alternative has been chosen. We want to emphasize that, in contrast with the standard notion, our consistency property requires to re-apply the rule only to the (non-empty) set of agents that has been selected at the original profile. We think that this is the relevant consistency notion because the new composition of the society is not just a hypothetical circumstance, it is a fact. Internal stability and consistency are desirable if we want to interpret the alternative chosen by the rule as being the final one, in a double sense. Members of the final society want to stay and if the rule would be applied again to the final society it would choose the same final society and the same outcome, so there is no need to do so.
We adapt well-known voting methods to our setting with the goal of making them either internally stable or consistent, or both. 4 We show that two prominent scoring methods, plurality voting and the Borda rule, do not satisfy consistency. However, approval voting not only satisfies internal stability and consistency but it also satisfies efficiency and neutrality. Finally, we show that the Condorcet winner is internally stable, consistent, efficient, neutral and anonymous at those profiles where an alternative beats all other alternatives by majority voting (namely, whenever it is a well-defined rule).
The paper is organized as follows. In Sect. 2 we describe the model. Section 3 contains the definitions of the properties of rules that we are interested in. Section 4 contains the analysis of well-known rules from the point of view of their internal stability and consistency properties. Section 5 has two final remarks.

Preliminaries
This section follows closely Bergantiños et al. (2017). Let N = {1, . . . , n}, with n ≥ 2, be the set of agents who have to choose an outcome from a given finite set X of possible outcomes. We are interested in situations where some agents may not be part of the final society, perhaps as the consequence of the chosen outcome. To model such situations, let A = 2 N × X be the set of alternatives and assume that each i ∈ N has preferences over A. Observe that for all x ∈ X , (∅, x) ∈ A; so we are admitting the possibility that the final society does not have any member. We will often use the notation a for a generic alternative (S, x) ∈ A; i.e., a ≡ (S, x), a ≡ (S , x ), and so on. Let R i denote i's (weak) preference over A, where for any pair a, a ∈ A, a R i a means that i considers a to be at least as good as a . Let P i and I i denote the strict and indifference relations over A induced by R i , respectively; namely, for any pair a, a ∈ A, a P i a if and only if a R i a and ¬a R i a, and a I i a if and only if a R i a and a R i a. We assume that each i does not care about any of those alternatives at which i does not belong to their corresponding final societies. Besides i is not indifferent between any pair of alternatives at which i belongs to at least one of the two corresponding final societies. Namely, we assume that i's preferences R i satisfy the following two properties: for all S, T ∈ 2 N and x, y ∈ X, The fact that agents' preferences satisfy (P.1) is the reason why our model cannot mechanically be embedded into the classical model. A specific analysis is required, partly because properties such as internal stability and consistency become specially meaningful under this domain restriction. We see property (P.1) as being a natural assumption for our setting, and it is a critical requirement for our results to hold. Let R i be the set of preferences of i satisfying (P.1) and (P.2), and let R = × i∈N R i be the set of (preference) profiles.
We denote the subset of alternatives with the property that i is not a member of the corresponding final society by [∅] i = {(S, x) ∈ A | i / ∈ S}. By (P.1), i is indifferent among them; i.e., By (P.1), (∅, x)I i (∅, y) for all x, y ∈ X and [∅] i can be seen as the indifference class generated by the empty society. Observe that [∅] i may be at the top of i's preferences. With an abuse of notation we often treat, when listing a preference ordering, the indifference class [∅] i as if it were an alternative; for instance, given R i and a ∈ A we write a R i [∅] The top of R i , denoted by τ (R i ) , is the set of all best alternatives according to R i ; namely, Note that by (P.1) and (P.2), τ (R i ) is either a singleton set or the indifference class To be explicit about the two components of the alternative chosen by f at R, we will often write f (R) To clarify the model, we relate it with two of the examples used in the introduction. The set of agents N corresponds to the initial members of the political party, the set of outcomes X to the set of choices that the political party has to make and the set S, if the chosen alternative is (S, x), to the set of final members of the party that stay after it supports outcome x. Similarly, N corresponds to the set of professors in the department, the set of outcomes X to all subsets of candidates and the set S, if the chosen alternative is (S, x), to the set of professors who remain in the department after the subset of candidates x has been hired.

Properties of rules
In this section we present several properties that a rule may satisfy. The first two impose conditions at each profile.
A rule is efficient if it always selects a Pareto optimal allocation.
The next property is related to the stability of a rule, and it captures the idea that agents are able to exit a society at their free will. Internal stability says that no agent belonging to the final society would prefer to leave it.

Internal stability For all R ∈ R and all
It is immediate to see that internal stability is indeed equivalent to the requirement of individual rationality (for all agents); i.e., for all R and all i, f (R)R i [∅] i . Individual rationality implies internal stability by their definitions and (P.2). Assume f is internally stable and let R be arbitrary.
The next three properties impose conditions by comparing the alternatives chosen by the rule at two different profiles. A rule is anonymous if the names of the agents are not relevant to select the alternative. To define it formally, let π : N → {1, . . . , n} be an ordering of N ( i.e., a one-to-one mapping). Given i ∈ N , π(i) (or π i to simplify notation) is the agent assigned to i after applying π to N . The set of all orderings π : N → {1, . . . , n} will be denoted by . Given S ∈ 2 N and π ∈ we denote by π(S) the subset of agents associated to S by π ; namely, π(S) = {i ∈ N | π( j) = i for some j ∈ S}. Given R ∈ R and π ∈ , we denote by R π the new profile where, for all i ∈ N , agent π(i) has the preference obtained from R i after replacing each A rule is neutral if the names of the outcomes do not play any role in selecting the social alternative. To define it formally, let σ : X → X be a permutation of X . Given x ∈ X , σ (x) is the outcome assigned to x after applying σ to X. The set of all permutations σ : X → X will be denoted by . Let Y ⊆ X be non-empty and σ ∈ . Denote by σ (Y ) the subset of outcomes associated to Y by σ ; namely, σ (Y ) = {x ∈ X | σ (y) = x for some y ∈ Y }. Given R ∈ R and σ ∈ we denote by R σ the profile where, for all i ∈ N , the preference R σ i is obtained from R i after replacing each (S, x) by (S, σ (x)).
A rule is consistent if the following requirement holds. Apply the rule to a given profile and consider the subset of agents that are members of the final society. Construct the new subprofile of preferences restricted to this new set of chosen agents. Then, the rule does not require to modify the chosen alternative because if it were applied to the new subprofile the new alternative would coincide with the alternative chosen at the original profile. To define the property formally, we first need additional notation.
Second, we specify how a given rule f can be applied to a subprofile by considering it as it were a family of rules, one for each non-empty subset of N .
In contrast with the standard notion, and as we have already said in the Introduction, our consistency property requires to re-apply the rule only to the (non-empty) set of agents that has been selected at the original profile. Hence, our notion is weaker since the choice of the rule f (R) is re-evaluated only at the profile R | f N (R) , instead of at all R |S (for S ⊂ N ). We think that this is the relevant consistency notion because the new composition of the society is not just a hypothetical circumstance, it is a fact. And indeed, the new set of agents might be willing to reconsider their membership and the chosen outcome; particularly because, in the choice of the later, preferences of members that are not anymore in the society may have played a relevant role. Consistency says that the original choice, if re-evaluated by the new society by means of the same rule, will continue to be chosen.
We say that a rule satisfies any of the above properties at R if the condition defining the property holds at R.

Internally stable and consistent rules
In Bergantiños et al. (2017) we characterize the class of all strategy-proof, unanimous and outsider independent rules as the family of serial dictator rules. 6 A serial dictator rule induced by π ∈ and x ∈ X , denoted by f π,x , proceeds (in up to n steps) as follows. Fix a profile R ∈ R and look for any alternative (S 1 , x 1 ) in the best indifference class of agent π 1 , the first agent in the ordering induced by π. If π 1 ∈ S 1 , set f π,x (R) = (S 1 , x 1 ). Otherwise, look for any alternative (S 2 , x 2 ) in the best indifference class of agent π 2 , the second agent in the ordering induced by π , only among those classes satisfying the property that π 1 / ∈ S 2 , If π 2 ∈ S 2 , set f π,x (R) = (S 2 , x 2 ). Otherwise, proceed similarly until the n−th step, if reached, by looking for any alternative (S n , x n ) in the best indifference class of agent π n , the last in the ordering induced by π, only among those classes satisfying the property that for each k ∈ {1, . . . , n − 1}, π k / ∈ S n . If π n ∈ S n , set f π,x (R) = (S n , x n ). Otherwise, and since no agent wants to stay in the society whatever element of X is selected, set f π,x (R) = (∅, x) . So, x plays the role of the residual outcome only when no agent wants to stay in the society under any circumstance.
It is obvious to see that any serial dictator rule satisfies efficiency but fails anonymity and neutrality. Consider a problem where the top alternative for agent π 1 is (N , x) but (∅, x) P π 2 (N , x) . Since the serial dictator rule selects (N , x) , it does not satisfy internal stability. We now see that the serial dictator rule satisfies consistency. Assume that f π,x (R) = (S, y) . Let π i be the first agent in S according with π. Notice that (S, y) is the best alternative of agent π i in which coincides with f π |S ,x R |S , where π |S is the restriction of π into S. Here, we consider situations where the strategic manipulation in the preference revelation game is not an issue and we will look for internally stable and consistent rules. To do so, we first ask whether two prominent procedures in classical social choice satisfy them. Recall that in the classical setting the goal is to select an outcome, from a given set X , taking into account (partially or fully) the strict preferences of agents over X. The procedures we consider are: 1. Approval voting. Each i ∈ N votes for a subset X i of X. For each outcome x ∈ X , compute the number of votes received by x; namely, |{i ∈ N | x ∈ X i }| .
The outcome with more votes is selected. A tie-breaking rule should be applied whenever two or more outcomes obtain the largest number of votes. Note that approval voting is not a rule because i's vote X i is not completely determined by P i 2. Scoring methods. Each i ∈ N strictly ranks all outcomes. Assign to each outcome a pre-established decreasing number of points depending on its position in i's ranking. 7 Compute the sum of the points obtained by each outcome. Select the one with more points. A tie-breaking rule should be applied whenever two or more outcomes obtain the largest number of points. We tentatively adapt the two procedures to our setting to deal with the indifferences generated by (P.1) and to define approval voting as a proper rule. 1. Approval voting. Each i ∈ N votes for all a ∈ A such that a P i [∅] i (if any). 2. Scoring methods. For each i ∈ N , assign a pre-established decreasing number of points to each outcome depending on its position in i 's ranking but considering [∅] i as a single alternative. For each (S, x) ∈ A and each i ∈ N \S , assign to (S, x) the score obtained by [∅] i .
Example 1 below shows that none of these extensions satisfy internal stability.
Example 1 Assume n ≥ 3 and fix x ∈ X . Let R ∈ R be any profile such that τ (R 1 ) = [∅] 1 and for all i ∈ N \ {1}, τ (R i ) = (N , x) and [∅] i P i (S, y) for all S = N , i ∈ S and y ∈ X . Then, each of the adapted procedures chooses (N , x) at R. Nevertheless, (N , x) is not internally stable because agent 1 prefers to leave the society.
Since we are interested in identifying rules satisfying internal stability, we modify the previous methods by considering only votes to alternatives (S, x) that are internally stable for each i ∈ S according to R i ; namely, only alternatives (S, x) with the property that (S, x) P i [∅] i for each i ∈ S can receive votes, not only from i but also from all other agents (we call these alternatives unanimously internally stable). In approval voting each agent votes, among the set of alternatives at which he is a member of the society, only for those that are unanimously internally stable. If no alternative receives a vote the rule selects a particular alternative (∅, x) by a tie-breaking rule that will be described later. In a scoring method we consider only the rank, given agents' preferences, among the unanimously internally stable alternatives. 8 Hence, at the profile of Example 1, each i votes for [∅] i and (∅, y) is selected according to some pre-established y. With these modifications, they satisfy internal stability by definition.
We now focus on two prominent scoring methods: plurality voting and the Borda rule, respectively denoted by f P and f B . Our first result is negative: they do not satisfy consistency (independent of the rule used to break ties). To see that, consider Example 2 below.
Example 2 Let N = {1, 2, 3, 4, 5, 6} and X = {y 1 , y 2 , y 3 , y 4 , y 5 } be respectively the set of agents and outcomes and consider the following profile R ∈ R. For each i ∈ N , (S, x) P i [∅] i whenever i ∈ S. Observe that for each i ∈ N , #{(S, x) | (S, x)P i [∅] i } = 2 5 × 5 = 160 is the number of alternatives that at R each agent i ranks strictly above [∅] i . In addition, R is one among all profiles satisfying the following properties, where the first column indicates the rank of each of the six preference relations.
Second, the Borda rule does not satisfy consistency since f B (R) = (N \{6} , y 4 ) but f B (R |N \{6} ) = (N \{6} , y 1 ) . To see that, in the Borda rule the first alternative receives 160 points, the second 159, the third 158 and so on, and each alternative in the class [∅] i receives zero points (since all alternatives in the indifference class [∅] i receive zero points). The alternatives ranked higher than (N \{6} , y 1 ) by some agent in N \{6} at R are: (N , y 1 ), (N , y 2 ), (N , y 3 ), (N \{6} , y 4 ), (N \{1} , y 4 ), (N \{2} , y 4 ) and (N \{3} , y 4 ) . Add to this list the alternatives (N \{6} , y 1 ) and (N , y 5 ) and observe that the remaining alternatives receive less points than (N \{6} , y 1 ). The table below shows the number of points (or an upper bound) assigned at R by each i ∈ N (in the columns) to each alternative in the list and the total number of points (or an upper bound) they obtain (in the last column). .
Similarly, for each i ∈ N \{6}, #{(S, x) | (S, x)(P |N \{6} ) i [∅] i } = 2 4 × 5 = 80 is the number of alternatives that at R |N \{6} each agent i ∈ N \{6} ranks strictly above [∅] i ; hence, in the Borda rule the first alternative receives 80 points, the second 79 and so on. The only alternative ranked higher than (N \{6} , y 1 ) by some agent at R |N \{6} is (N \{6} , y 4 ) . The remaining alternatives receive less points that (N \{6} , y 1 ) .The table below shows the number of points assigned by each i ∈ N \{6} (in the columns) to the two alternatives with the two highest number of total points received at R |N \{6} . . N \{6} , y 1 ). Thus, the Borda rule is not consistent.
Approval voting satisfies not only consistency (and internal stability by definition) but also other desirable properties. Before stating this result we need to specify a tie-breaking rule, to be used whenever two or more alternatives obtain the highest number of votes. The idea is simple. Suppose that several alternatives obtain the largest number of votes. We choose the alternative with coalition S in 2 N ranked higher following a specified monotonic and complete order ρ over coalitions. Since the order is monotonic, we choose an alternative associated with a coalition S which is not contained in other coalition T associated with an alternative with the highest number of votes. If it is unique, we are done. Suppose not, then we still have several alternatives with the highest number of votes, but now all of them coincide in having the same coalition S but together with a different outcome in X. We now choose the agent i in S ranked higher according to the order ρ. We select, among the alternatives with the highest number of votes having S as the final society, the one preferred by agent i. We now define this tie-breaking rule formally.
Let ρ be a monotonic and complete order over 2 N . Namely, for each pair S, T ∈ 2 N such that S T , Tρ S. Observe that Nρ S for all S = N . Denote by f AV,ρ the approval voting rule induced by ρ as follows. Let A = {(S k , x k )} K k=1 be the set of alternatives that have received the largest number of votes according to approval voting at R. If S k = ∅ for all k ∈ {1, . . . , K }, select (∅, y) where y is such that (N , y)P i (N , z) for all z = y and {i}ρ{ j} for all j = i. Assume S k = ∅ for some k ∈ {1, . . . , K }. First select the final society S ∈ {S 1 , ..., S K } ranked highest by ρ and consider the subset of alternatives (S k , x k ) ∈ A | S k = S . Select again the agent i ∈ S who is ranked highest by ρ (as a singleton set) and choose finally as f AV,ρ (R) the alternative most preferred by i among those in the family (S k , x k ) ∈ A | S k = S .
Proposition 1 below states that any approval voting f AV,ρ is internally stable, consistent, and additionally satisfies other desirable properties.
Proposition 1 Let ρ be a monotonic and complete order over 2 N . Then, the approval voting f AV,ρ satisfies internal stability, consistency, efficiency and neutrality. Moreover, in the subdomain of profiles where the tie-breaking rule is not applied, f A,ρ satisfies anonymity.
Proof Observe that if (S, x) is approved by i, then i ∈ S. This fact will be repeatedly used in the proof.
• Internal stability. By definition, f A,ρ is internally stable. • Consistency. Let R ∈ R be arbitrary and let f AV,ρ (R) = (S, x) be such that S = ∅. The set of agents approving (S, x) at R coincides with the set of agents approving (S, x) at R | f A,ρ N (R) . Hence, f AV,ρ (R | f A,ρ N (R) ) = f AV,ρ (R) and thus, f AV,ρ is consistent.
• Efficiency. Suppose otherwise, namely, there exist R ∈ R and (S, which implies that f AV,ρ (R) did not receive any vote and S = ∅. By (P.2), (S, x)P j f AV,ρ (R) for all j ∈ S and (S, x)I j f AV,ρ (R) for all j / ∈ S. But this means that (S, x) received more votes than f AV,ρ (R), a contradiction. Assume now that f  S, y). This means that the tie-breaking rule ρ has been used to select f AV,ρ (R), implying that there exists i ∈ S such that f AV,ρ (R) P i (S, x) which is a contradiction. • Neutrality. Let R ∈ R and σ ∈ . Observe that the number of agents approving (S, x) at R coincides with the number of agents approving (S, σ (x)) at R σ . We consider two cases. First, f AV,ρ (R) has been approved at R by more agents than any other alternative. Hence, ( f AV,ρ N (R) , σ ( f AV,ρ X (R))) has been approved at R σ by more agents than any other alternative, implying that f AV,ρ (R σ ) = f AV,ρ (R) . Second, it is necessary to apply ρ to select f AV,ρ (R) . Let {(S k , x k )} K k=1 be the set of alternatives receiving the largest number of votes at R. Thus, is the set of alternatives receiving the largest number of votes at R σ . Hence, (1) Now, let i ∈ f AV,ρ N (R) be the agent with the highest ranking, among singleton sets, according to ρ and let i ∈ f AV,ρ N (R σ ) be the agent with the highest ranking, among singleton sets, according to ρ. By (1) • Anonymity on the subdomain of profiles where the tie-breaking rule is not applied.
Let R be one of such profiles. Then, f AV,ρ (R) has been approved by more agents than any other alternative. Observe that the number of agents approving any (S, x) at R coincides with the number of agents approving (π(S), x) at R π . Thus, (π( f AV,ρ N (R)), f AV,ρ X (R)) has been approved at R π by more agents that any other alternative. Hence, f AV,ρ (R π ) = (π( f AV,ρ (R)), f AV,ρ X (R)), which means that f AV,ρ satisfies anonymity at R.
Remark 1 Proposition 1 also holds with other tie-breaking rules. For instance, consider an order π of the set of agents and assume that agents choose between the set of alternatives A = {(S k , x k )} K k=1 with the largest number of votes according to approval voting at R following such order. Namely, if S k = ∅ for all k, select (∅, y) where y is such that (N , y)P π 1 (N , z) for all z = y. Assume S k = ∅ for some k. Let (S 1 , x 1 ) in the best indifference class of agent π 1 in A . If π 1 ∈ S 1 , set f AV,π (R) = (S 1 , x 1 ). Otherwise, look for any alternative (S 2 , x 2 ) in the best indifference class of agent π 2 in A = (S k , x k ) ∈ A : π 1 / ∈ S k . If π 2 ∈ S 2 , set f AV,π (R). Otherwise, proceed similarly until the n−th step.
We end this note by applying the Condorcet winner to our setting. First, we recall the definition of the Condorcet winner in the classical setting. Fix a profile P of strict preferences over X and let x, y ∈ X be such that x = y. We say that x beats y if the number of agents preferring x to y is strictly larger than the number of agents preferring y to x. We say that x is a Condorcet winner at P if there is no y that beats x. There are profiles at which no Condorcet winner exists and others at which there are several Condorcet winners. Thus, the Condorcet winner is not a rule.
We adapt the notion of a Condorcet winner to our setting as we have already did for the previous two classes of rules. To ensure that the chosen alternative satisfies internal stability at R we only consider votes for unanimously internally stable alternatives at R. When several Condorcet winners exist we apply the tie-breaking (using a monotonic and complete order ρ) used to define approval voting.
We say that a profile R ∈ R is resolute if there exists a ∈ A such that a beats a for all a = a. Thus, the Condorcet winner selects a at R. Let f C,ρ (R) denote the Condorcet winner (if any) at R. If R ∈ R is resolute, then f C,ρ (R) is independent of ρ and f C,ρ (R) = 1. Proposition 2 states that the Condorcet winner at resolute profiles satisfies the same properties as Approval voting, at such profiles.
Proposition 2 Let R be a resolute profile. Then, f C,ρ (R) satisfies internal stability, consistency, efficiency, neutrality and anonymity at R.
Proof Fix a resolute profile R and set f C,ρ (R) = (S, x). We show that f C,ρ (R) satisfies the properties at R.
• Internal stability. By definition, f C,ρ (R) satisfies internal stability at R. • Consistency. We prove that f C,ρ R |S = (S, x) by showing that at R |S , (S, x) beats (T, y) for all (T, y) = (S, x) with T ⊂ S. Let (T, y) be an alternative with the above properties. Since (S, x) beats (T, y) at R, the number of agents in N preferring (S, x) to (T, y) is strictly larger than the number of agents in N preferring (T, y) to (S, x) . Moreover, each agent in N \S is indifferent between (S, x) and (T, y) . Thus, the number of agents in S preferring (S, x) to (T, y) (or (T, y) to (S, x)) coincides with the number of agents in N preferring (S, x) to (T, y) (or (T, y) to (S, x)). Hence, (S, x) beats (T, y) at R |S , implying that f C,ρ R |S = (S, x) . • Efficiency. Suppose otherwise, in particular, there must exist (T, y) such that (T, y) R i (S, x) for all i ∈ N and (S, x) = (T, y) . Let i ∈ S. Since (S, x) satisfies internal stability at R, (S, x) P i [∅] i . Hence, i ∈ T and (T, y) P i (S, x). Each agent in N \T is indifferent between (S, x) and (T, y) . Thus, (T, y) beats (S, x) , which contradicts that f C,ρ (R) = (S, x). • Neutrality. Observe that for each (T, y) = (S, x), (S, σ (x)) beats (T, σ (y)) at R σ . Hence, f C,ρ (R σ ) = (S, σ (x)) , which means that f C,ρ satisfies neutrality at R. • Anonymity. Observe that for each (T, y) = (S, x), (π(S), x) beats (π(T ), y) at R π . Hence, f C,ρ (R π ) = (π(S), x) , which means that f C,ρ satisfies anonymity at R.
Nevertheless, for non-resolute profiles the Condorcet winner, even when it is unique, may not satisfy consistency. To see that, consider the following example.
Example 3 Let N = {1, 2, 3, 4, 5} and X = {y 1 , y 2 } be respectively the set of agents and outcomes and let ρ be any monotonic and complete order over 2 N satisfying Consider any profile R satisfying the following properties, where the first column indicates the rank of each of the five preference relations.
Using arguments similar to those used in the proof of Proposition 1, we can prove that the scoring methods satisfy efficiency and neutrality in the general domain and anonymity when no tie breaking rule is used. We now present a table summarizing our main findings.

Rank
R .
Let ρ be arbitrary. Since f A,ρ (R) = ({123} , y) but f A,ρ R |{1,2,3} = ({123} , x) approval voting is not consistent and hence Proposition 1 does not hold if agents' preferences do not satisfy (P.1). Since f C,ρ (R) = ({123} , y) but f C,ρ R |{1,2,3} = ({123} , x) the Condorcet winner is not consistent and hence Proposition 2 does not hold if agents' preferences do not satisfy (P.1). Second, our domain of preferences satisfying conditions (P.1) and (P.2), yet restrictive, is still very large. And hence, one may ask about the possible extension of our results to meaningful subdomains for specific problems. First, for the case of an excludable public good (in a linearly ordered set X of outcomes) when agents have single-peaked preferences over X and also care about the size of the set of its users, as in Jackson and Nicolò (2004). Second, as in Berga et al. (2004), where a set of founders of a society are considering the possibility of admitting new members from a given set of candidates, and founders preferences on final societies are separable (or additive). Third, assume agent i has (S, x) as top-ranked and i ∈ S. If i ∈ T, then (T, x) should be ranked higher than (T \{i}, x); that is, (T, x) "lies" somehow between (S, x) and (T \{i}, x), and since (S, x) is the top-ranked alternative, agent i should prefer the alternative that is "closer" to the top. Each of those preference domains will require a specific analysis of the possibilities of designing individually rational and consistent rules, together with other desirable properties. However, we leave this analysis for further research.