11
Introduction
This paper investigates the election model in which voters can select any subset from the set of candidates.  Examples are where the members of a club select new comers, or where the executive committee of a society select new members.  Barberà et al.  (1991) first study this model under the assumption that voters possess preferences with separable strict order- ings over the power set of the sets of candidates. In this paper, we extend their model so that not only voters but candidates have preferences and the actual running candidates is variable, and we investigate a rule satisfying "candidate stability".  Candidate stability requires that for any candidate, standing in the election with any set of running candidates is at least as de- sirable as withdrawing from it. Dutta et al. (2001) suggests the importance of this property as a basis of the study for any election model.  If an election rule satisfies candidate stability , all candidates may stand in the election. If an election rule does not satisfy candidate stability, candidates' strategies whether to run the election at t he pre-election stage may affect the rule and hence the rule designer has to take it into account. Dutta et al. (2001) first study this property in the single-winner election where preferences are unrestricted, and shows that a rule satisfying candi- date stability and unanimity is only dictatorial. 1 By comparison, our main result is a possibility result. We show that a rule satisfies candidate stability if and only if it satisfies the following independence condition. Suppose in the election with some set of running candidates and some preference profile of voters, a candidate wins. Then she also should win in the election with any set of running candidates she belongs to and any preference profile that is equivalent to the previous profile in the comparison of herself and the null outcome.  This is a quite parallel work to Barberà et al.  (1991).  They impose strategy-proofness, which requires non-manipulability by voters, and derive independence of the selection for each candidate as a part of voting by com- mittees that they characterize. Meanwhile, we impose non-manipulability of candidates and derive the similar independence condition. Related literature includes Berga et al.  (2004, 2006).  They consider similar models to ours in which voters are existing members of a society and have exit options. They study stability of the existing members while we focus on the stability of candidates as new entrants. Literature on candi- date stability includes Ehlers and Weymark (2003), Eraslan and McLennan (2004), Rodríguez-Álvarez (2005, 2006), and Samejima (2005, 2007). The structure of the rest of this paper is as follows. Section 2 states the model and the main result.  Section 3 gives a short discussion and conclusion. The appendix includes the proof of the main result.
2
The model and the result
Let C ≡ {1,2,… ,c} be the set of (prospective) candidates with c ≥ 1. We refer to a candidate who actually stands on the election as a running candidate.  Let V ≡ {c + 1,c + 2, . .  ,c + v} be the set of voters with v ≥ 1. For simplicity, we assume C ∩ V = ∅. For a voter or a candidate i ∈ C ∪ V , let P i denote her preference, which is a complete, transitive, and asymmetric binary relation over 2 . We assume that any preference P i satisfies the following restrictions. Separability: For all i ∈ C ∪ V , all x ∈ C , and all Y ⊆ C \{ x }, we have Y ∪ { x }  P i Y   ⇐⇒  { x }  P i Ø. Additivity: P i has an additive numerical representation. 2 We refer to a preference satisfying separability and additivity as an ad- ditively separable preference. Let Ds denote the domain of all such pref- erences. All preferences of voters are assumed to be in the domain D S . In addition to additive separability, we impose another restriction on the preferences of the candidates: for all x ∈ C , {x} P x Ø . Since without it, the concept of "candidate stability" is not meaningful, we refer to it as the min- imal restriction for candidates' preferences. Let D M denote the domain of all additively separable preferences satisfying the restriction.  A preference profile is a c + v tuple of preferences P = (P 1 , . . , P c, P c+1 , . . , P c+v) E D§M × D§. Let i,j ∈ CUV. Let (P i, P_i) ∈ D§M × D g denote the preference profile obtained from P by replacing P i with P i , (P j , P i , P { − i j } ) ∈ D i 4 × D i denote the profile obtained from ( P i , P_i ) by replacing P j with P j , and so on. Let P i | χ denote the preference relation over X ⊆ C induced by P i ∈ D S .
もし
Similarly, let P | x denote the preference profile over X  ⊆  C  induced by P ∈ D M × D y . A rule is a function φ : 2 ° × D M × D ξ → 2 ° .  Following Dutta et al.  (2001), a rule φ is assumed to satisfy the following three properties. First, winners should be chosen from the set of running candidates. Second, only voters' preferences matter.  Third, only preferences over the running candidates matter. Feasibility: For all X ∈ 2 C and al P ∈ D M × D ξ , we have φ(X,P) ⊆ X. Independence of nonvoters’ preferences: For all X ∈ 2 C and al P, P ′ ∈ Df × Dg such that for all i ∈ V, Pi = Pi, we have φ(X,P) = φ(X,Pi). Independence of irrelevant alternatives: For all X ∈ 2 C and al P, P ′ ∈ D M × D 5 such that P (x = P ′ (x, we have φ (X, P) = φ (X, P ′ ). The formal definition of "candidate stability" of a rule is given as follows. Candidate stability: For all X ∈ 2 C \{Ø}, all x ∈ X, and all P ∈ D M × D g, we have φ(X, P) P z φ(X\{x}, P) or φ(X, P) = φ(X\{x}, P).3 We claims that candidate stability is equivalent to the following inde- pendence condition.  Suppose that for some set of running candidates and some preference profile, a candidate wins. Then she also should win for any set of running candidates she belongs to and any preference profile that is equivalent to the previous profile in the comparison of herself and the null outcome. Independence:  For all X,Y ∈ 2 C \{Ø}, all x ∈ C such that x ∈ X and x ∈ Y, and all P, P' ∈ D M × D ξ such that P|{{x},Ø} = P'|{{x},Ø}, we have x ∈ ϕ ( X, P )  ⇐⇒  x ∈ ϕ ( Y, P ′ ). Proposition 1. A rule satisfies candidate stability if and only if it satisfies independence. 4 Proof of Proposition 1. See the appendix.
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The intuition behind the proof is as follows.  Since we impose the mini- mal restriction on candidate preferences, there exists a candidate who prefers another candidate to win no matter whether the other candidates, including herself, are elected. There also exists a candidate who prefers another can- didate not to win no matter whether the other candidates, including herself, are elected. If independence is violated, a candidate with such a preference may be better off by not running. This violates candidate stability.
3
Discussion and concluding remarks
Before concluding this paper, we note that if we drop the minimal restriction for candidates' preferences and impose a stronger but natural restriction for them, Proposition 1 fails. Let D M ⊆ D M be the domain of all additively separable preferences satisfying the following restriction for candidates. For all x ∈ C and al P x ∈ D x , { x } P x Ø and for all y ∈ C \{ x }, { x } P x { y }. This states that any candidate prefers herself to any other candidate. A counter example to the equivalent statement of Proposition 1 in this domain is the following. Example 1. Let C ≡ {1,2} and V ≡ {3}. Let ϕ be a rule such that (i) for any P ∈ D * 2 × D s , 1 ∈ ϕ(C , P 3 ) and 1 ∈ ϕ({1}, P ), (ii) if {1} P 3 Ø, 2 ∈ ϕ(C, P ) and if Ø P 3 {1}, 2̸∈ ϕ(C, P ), and (iii) for any P ∈ D* 2 × Ds, 2 ∈ ϕ({2}, P). Then φ satisfies candidate stability, however, it does not satisfy indepen- dence. The question how candidate stability works on the domain D*s is open. We hope that this paper will help further research on candidate stability.
Appendix
Before proving Proposition 1, we introduce one property and three lemmas. The next property is taken from Eraslan and McLennan (2004). Strong candidate stability: For all X ∈ 2 C \{Ø}, all x ∈ X and all P ∈ D M × D g , we have φ(X, P)\{x} = φ(X\{x}, P).5 Lemma 1. A rule satisfies candidate stability if and only if it satisfies strong candidate stability. Proof of Lemma 1. Since the if part of proposition is obvious, we only show the only if part. Note that strong candidate stability is equivalent to the requirement that for all X ∈ 2 C \{Ø}, all x ∈ X, and all P ∈ D M × D ξ, (i) φ(X,P)\{x} ⊆ ϕ(X\{x},P) and (ii) ϕ(X\{x},P) ⊆ ϕ(X,P)\{x}. At first, suppose, on the contrary, that φ satisfies candidate stability but not (i).  Then there exist X ∈ 2 C \{Ø}, x ∈ X, and P ∈ D M × D g such that φ(X\{x},X) ⊆ φ(X, P)\{x}. Let y ∈ φ(X, P)\({x} ∪ φ(X\{x}, P)). Let P z ∈ D M be such that for all Y, Z ⊆ X with y ∈ Y and y̸∈ Z, Z P z Y. Then ϕ(X\{x}, P) P z φ(X, P). By independence of nonvoters’ preferences, φ(X\{x}, P², P–x) P′ ϕ( X , P ′ , P – x ).  This contradicts candidate stability.  Hence our supposition is incorrect. Next, suppose, on the contrary, that ϕ satisfies candidate stability but not (ii). Then there exist X ∈ 2 C\{Ø}, x ∈ X, and P ∈ D M × D ξ such that φ(X,P)\{x} ⊆ φ(X\{x},P).  Let y ∈ φ(X\{x},P))\φ(X,P).  Let P x ∈ D M be such that for all Y, Z ⊆ X with y ∈ Y and y̸∈ Z, Y P x Z. Then φ ( X \{ x }, P ) P z φ ( X , P ). By independence of nonvoters' preferences, ϕ( X \{ x }, P z , P_z ) P z ϕ( X, P z , P_z ).  This contradicts candidate stability. Hence our supposition is incorrect. □ Lemma 2. If a rule φ satisfies candidate stability , then for all X ∈ 2 C with |X| ≥ 2, all x,y ∈ X and all P ∈ D M × D ξ, we have x ∈ ϕ(X,P)  ⇐⇒  x ∈ φ(X\{y},P). Proof of Lemma 2. This is direct from Lemma 1. ● Lemma 3. If φ satisfies candidate stability, then for all X ∈ 2 C with |X| ≥ 2, all x ∈ X, all P ∈ D M × D ξ, all i ∈ V and all P i ∈ D S such that for a candidate y ∈ X\{x}, Pi|{{y},0}̸= Pi|{{y},0} and for all other candidates z ∈ X\{y}, Pi|{{z},Ø} = Pi|{{z},Ø}, we have x ∈ φ(X,P)  ⇐→ x ∈ ϕ(X, P', P-i). Proof of Lemma 3. Let φ be a rule satisfying candidate stability. Let X ∈ 2C, x ∈ X, P ∈ D M × D S, i ∈ V and P i ∈ D S safisfy the hypothe- sis of Lemma 3.  Suppose, on the contrary, that x ∈ ϕ(X,P) but x ∈ φ(X, P ′ , P – i).  By strong candidate stability, φ(X, P)\{y} = φ(X\{y}, P)
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and φ(X,P i )\{y} = φ(X\{y},P i ,P i ). Thus, x ∈ φ(X\{y},P i and x & ϕ( X \{ y }, P i , P_i ). However, by independence of irrelevant alternatives, ϕ(X\{y},P) = ϕ(X\{y},Pi',P_i).  Thus, x ∈ ϕ(X\{y},Pi',P_i).  It is a contradiction. Hence our supposition is incorrect. ● Proof of Proposition 1. The only if part: By the iterated use of Lemma 2 on candidates and Lemma 3 on preference profiles, independence is obtained. The if part:  Let X  ∈  2 ′ \{Ø}, x  ∈  X  and  P  ∈  D ′ M ×  D ′ .  For all y ∈ ϕ(X, P)\{x}, independence implies y ∈ ϕ(X\{x}, P).  Also for all z ∈ φ(X, P)\{x}, independence implies z̸∈ φ(X\{x}, P). Thus φ(X\{x}, P) = ϕ(X,P)\{x} and ϕ satisfies strong candidate stability. Hence, it satisfies candidate stability. ●