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Economic Theory 27, 657–677 (2006) DOI: 10.1007/s00199-004-0564-7

Candidate stability and probabilistic voting procedures 1,2 ´ Carmelo Rodr´ıguez-Alvarez 1 2

W. Allen Wallis Institute of Political Economy, University of Rochester, Rochester, NY 14627, USA Departamento de Econom´ıa, Universidad Carlos III de Madrid, Getafe (Madrid) 28903, SPAIN (e-mail: [email protected])

Received: February 4, 2003; revised version: September 14, 2004

Summary. We extend the analysis of Dutta, Jackson and Le Breton (Econometrica, 2001) on strategic candidacy to probabilistic environments. For each agenda and each proﬁle of voters’ preferences over running candidates, a probabilistic voting procedure selects a lottery on the set of running candidates. Assuming that candidates cannot vote, we show that random dictatorships are the only unanimous probabilistic voting procedures that never provide unilateral incentives for the candidates to withdraw their candidacy at any set of potential candidates. More ﬂexible probabilistic voting procedures can be devised if we restrict our attention to the stability of speciﬁc sets of potential candidates. Keywords and Phrases: Probabilistic voting procedures, Candidate stability, Random dictatorship. JEL Classiﬁcation Numbers: D71, D72.

This is a revised version of a chapter of my Ph.D. Dissertation submitted to the UniversitatAut` onoma de Barcelona. I am indebted to my supervisor Salvador Barber`a for his advice and constant support. I am grateful to Dolors Berga and an anonymous referee for their detailed comments and suggestions. I thank Jos´e Alcalde, Walter Bossert, Bhaskar Dutta, Lars Ehlers, Jordi Mass´o, Diego Moreno, Clara Ponsat´ı, Yves Sprumont, and William Thomson for many helpful comments and discussions. I thank the hospitality of the C.R.D.E. at the Universit´e de Montr´eal and the Department of Economics of the University of Warwick where parts of this research were conducted. Financial support through Research Grant 1998FI00022 from Comissionat per Universitats i Recerca, Generalitat de Catalunya, Research Project PB98-870 from the Ministerio de Ciencia y Tecnolog´ıa, and Fundaci´on Barri´e de la Maza is gratefully acknowledged.

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1 Introduction The decision of potential candidates whether or not to run for ofﬁce is crucial for the result of elections. Of course, you must run to win. But, even if you have no chance of winning, your presence as a candidate (or your absence) can affect the ﬁnal result. In a recent paper, Dutta, Jackson, and Le Breton [5] (henceforth DJL) provide a general analysis of candidates’ incentives to manipulate the result of an election by withdrawing.1 DJL consider deterministic social choice rules that select a winning candidate for each set of running candidates and each proﬁle of voters’ preferences over running candidates. They propose a stability condition called candidate stability. A social choice rule is candidate stable if no candidate would prefer to withdraw when all other potential candidates run. Candidate stability requires that standing for the election is a Nash equilibrium strategy for all potential candidates. Provided that candidates cannot vote, DJL show that in their deterministic framework, only dictatorships satisfy the joint requirements of candidate stability and unanimity. That is, the social choice is always determined by the preferences of a single voter.2 In this work, we generalize the analysis of DJL and study candidates’ incentives in a probabilistic framework. Thus, we model social choice rules as probabilistic voting procedures. A probabilistic voting procedure selects a lottery on the set of candidates for each set of running candidates and each proﬁle of voters’ preferences over running candidates. Probabilistic choices in a social context are sometimes criticized. However, the probabilistic framework provides many plausible voting rules, which provide scope for incorporating certain notions of fairness and reasonable compromise. For instance, when voters’ preferences conﬂict, it may be reasonable to use a lottery under which the different voters have equal probability of determining the social choice. Probabilistic voting procedures can be interpreted as a way to formalize candidates’ subjective beliefs about the ﬁnal resolution of a two-stage decision process. In a ﬁrst stage, potential candidates decide whether to run or not. In a second stage, voters choose among the running candidates using a voting mechanism. Assume that the second-stage voting mechanism admits multiple equilibria for some proﬁle of voters’ preferences. Assume also that candidates know the equilibria of the voting mechanism for each proﬁle of voters’ preferences, but candidates do not know the strategies that voters play. In this scenario, candidates cannot use backward induction arguments to focus on a speciﬁc equilibrium. However, they may assess a lottery assigning a probability to each possible equilibrium of the voting mechanism and, then, to each candidate to be the winner. Using the basic idea of candidate stability, we deﬁne two different requirements on probabilistic voting procedures. A strong one is to demand that candidate sta1

Other important articles analyze similar incentive issues for speciﬁc voting rules. Osborne and Slivinski [12] and Besley and Coate [4] concentrate on large elections with plurality rule and ideological positions of the candidates. Dutta, Jackson, and Le Breton [6] also falls into this branch of the literature. They examine the effects of strategic candidacy in the context of voting by successive elimination. 2 We postpone to the concluding section the discussion of the interesting case in which candidate are allowed to vote.

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bility is guaranteed for every possible set of candidates. Provided that candidates are Expected Utility maximizers and cannot vote, we show that only probabilistic combinations of dictatorial probabilistic voting procedures – random dictatorships – satisfy this strong requirement and unanimity. A weaker condition is to guarantee that a probabilistic voting procedure satisﬁes candidate stability for one speciﬁc set of candidates. This speciﬁc set can be interpreted as the set of candidates that the analyst knows that are actually in the run. We show that more ﬂexible rules satisfy this weaker stability requirement and unanimity. Yet, the power of decision remains concentrated in the hands of an arbitrary groups of voters. While random dictatorships play a crucial role in our characterizations, we do not view our results as negative or “impossibility" results. Random dictatorships have attractive features that are connected with intuitive concepts of fairness. Hence, it is not clear that a random dictatorship should be thought as undesirable in the same way in which a deterministic dictatorship is. Closely related to this article are Ehlers and Weymark [7], Eraslan and McLen´ nan [8], and Rodr´ıguez-Alvarez [14]. These authors study the implications of candidate stability for multi-valued voting procedures. Ehlers and Weymark [7], and Eraslan and McLennan [8] do not model explicitly candidates’ incentives. Instead, they propose a natural condition that captures the notion of candidate stability for multi-valued environments. Both papers show that only dictatorial rules satisfy their candidate stability condition and unanimity. Moreover, in Eraslan and McLennan [8], voters are allowed to express weak preferences over candidates. Their main result is that only serially dictatorial rules are candidate stable and unanimous. On ´ the other hand, in Rodr´ıguez-Alvarez [14], candidates’ incentives to withdraw are explicitly modelled since candidates are equipped with different domains of preferences over sets of candidates. Of course, the implications of candidate stability depend on the domain. For instance, negative results in the line of those of Ehlers and Weymark [7] and Eraslan and McLennan [8] are obtained when candidates’ preferences are consistent with Expected Utility Theory and Bayesian updating from some prior assessment. Additional positive results are obtained when candidates compare sets consistently with extreme attitudes towards risk. Finally, we refer the reader to Pattanaik and Peleg [13]. (Henceforth, PP.) Their main objective is to analyze the structure of the probabilistic voting procedures that satisfy probabilistic counterparts of the classical axioms of deterministic social choice.3 However, they do not consider candidates’ incentives. In spite of the differences, their set-up and results are closely related to ours. In fact, the set of axioms they analyze is stronger than the one studied here. Hence, as it becomes clear in the sequel, our theorems generalize theirs. The paper proceeds as follows. In Section 2, we introduce the set-up and basic notation. In Section 3 we present the implications of candidate stability in the probabilistic framework. We gather all the proofs in Section 4. In the concluding section, we discuss the case of voting candidates and other possible extensions.

3 Nandeibam [11] extends the analysis of PP by permitting voters to express indifference between alternatives.

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2 Deﬁnitions and notation 2.1 Voters, candidates and preferences Let N be a society formed by a countably inﬁnite set of candidates C, and a ﬁnite set of at least two voters V, N = C ∪V. Let 2C \{∅} denote the set of all non-empty ﬁnite subsets of C. We call A ∈ 2C \ {∅} an agenda.4 We focus on the case in which there is no overlap between the sets of voters and candidates, C ∩ V = {∅}. This assumption allows us to isolate candidates’ incentives to run, regardless of their interests as voters. A preference is a complete, transitive, and antisymmetric binary relation on C ∪ {∅}, where the empty set refers to no-candidate being elected. Let P denote the set of all preferences. Each individual i ∈ N is equipped with a preference Pi ∈ P. Hence, individuals are never indifferent between two candidates. A utility function is a mapping ui : C → R. A utility function ui represents the preference Pi ∈ P if for each a, b ∈ C, ui (a) > ui (b) if and only if a Pi b. For each i ∈ N , let P i denote i’s domain of admissible preferences. For each i ∈ N , each C ∈ 2C \{∅}, and each Pi ∈ P, top (C, Pi ) refers to the candidate in C that is ranked ﬁrst by Pi . We assume that for each a ∈ C, each i ∈ N , and each Pi ∈ P i , a Pi {∅}. Voters’ preferences over candidates are unrestricted, but each candidate is her own ﬁrst-ranked candidate. Hence, for each a ∈ C and each Pa ∈ P a , a = top (C, Pa ). Let P V = ×i∈V P i . Let P ∈ P V denote a preference proﬁle. For each A ∈ C 2 \{∅} and each P ∈ P V , P |A refers to the restriction of P to A. Finally, for each I ⊆ V, and each P ∈ P V , PI ∈ ×i∈I P i , refers to the restriction of P to the preferences of the members of I. Candidates’ preferences over lotteries. Let L denote the set of lotteries on the set C. For each C ∈ 2C \{∅}, let: LC ≡ {λ ∈ L such that for each b ∈ C\C, λ(b) = 0} . That is, LC contains all lotteries deﬁned on agenda C. Candidates are equipped with preferences over L ∪ {∅}. These preferences are complete, reﬂexive, and transitive binary relations that are consistent with the postulates of Expected Utility Theory. Additionally, candidates always prefer every λ ∈ L to the empty set. Hence, for each pair λ, λ ∈ L, a candidate a ∈ C equipped with preferences over candidates Pa ∈ P a and utility function ua representing Pa , prefers λ to λ , if and only if λ(b)ua (b) > λ (b )ua (b ). b∈C

b ∈C

Having deﬁned the strict component of candidates’ preferences over lotteries, the weak component is deﬁned in the usual way. For each pair λ, λ ∈ L, a candidate a is indifferent between them if neither she prefers λ to λ nor she prefers λ to λ. 4 We assume that the set of potential candidates is countably inﬁnite for technical reasons. This assumption implies that every agenda is a proper subset of another agenda that contains exactly one additional candidate. This assumption helps us to simplify the statement of our ﬁrst result, Theorem 1.

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2.2 Probabilistic voting procedures: random dictatorships We model social choice rules as functions that select a lottery on the running candidates for each agenda and each preference proﬁle over running candidates. A probabilistic voting procedure (PVP) is a function p : 2C \ {∅} × P V → L such that for each A ∈ 2C \ {∅} and each P ∈ P V : (i) p (A, P ) ∈ LA , (ii) for each P ∈ P V such that P |A = P |A , p (A, P ) = p (A, P ) . For each a ∈ C, p (a, A, P ) denotes the probability assigned to candidate a by the lottery p(A, P ). Part (i) states that the social choice selects a lottery on the running candidates. A candidate cannot win if she does not run. Part (ii) is usually referred to in the literature as Independence of Infeasible Alternatives. It requires that the selected lottery cannot depend on voters’ preferences over candidates who do not run. Our deﬁnition of PVP is less general than the deﬁnition proposed in PP since it includes Independence of Infeasible Alternatives. A single-valued voting procedure as deﬁned in DJL is a degenerate PVP. A voting procedure is required to select lotteries that assign strictly positive winning probability to a unique candidate. A PVP is more precise than a voting correspondence as deﬁned in Ehlers and Weymark ´ [7], Eraslan and McLennan [8], and Rodr´ıguez-Alvarez [14]. While a PVP assigns a winning probability to each running candidate, a voting correspondence selects a set of winning candidates but does not specify winning probabilities for each candidate. Finally, a PVP is a family of decision schemes as deﬁned by Gibbard [9], one for each agenda. We now deﬁne a family of PVPs that play a crucial role in the sequel. For each i ∈ V, the dictatorship associated to i is the PVP di such that for each a ∈ C, each A ∈ 2C \{∅}, and each P ∈ P V , 1 if a = top (A, Pi ) , di (a, A, P ) = 0 otherwise. A dictatorship is a degenerate PVP for which the preferences of an arbitrary voter always determine the result of the social choice. To present the probabilistic counterparts to dictatorships, we need additional notation. Let ∆#V−1 denote the (#V − 1)-dimensional simplex.5 That is: xi = 1}. ∆#V−1 ≡ {x ∈ R#V , such that for each i ∈ V, xi ≥ 0, and i∈V

The PVP p is a random dictatorship if it is a probabilistic convex combination #V−1 of dictatorships. That is, there is a list of weights {α , such that for i }i∈V ∈ ∆ C V each A ∈ 2 \{∅} and each P ∈ P , p (A, P ) = i∈V αi di (A, P ) . 5

For every set A, #A stands for the cardinality of the set A.

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Under a random dictatorship each voter may become a dictator. Each voter introduces in a hat a number of ballots with the name of her preferred candidate. Then, a ballot is drawn at random. Each candidate’s winning probability is the proportion of ballots that carry her name. Note that random dictatorships may assign different weights to different voters. Nevertheless, the weight assigned to each voter is invariant with respect to the agenda and voters’ preferences.

2.3 Candidate stability We are interested in devising PVPs for which the agenda can be considered as the result of candidates’ equilibrium decisions. An agenda C ∈ 2C \{∅} can be the outcome of a Nash equilibrium if: – The candidates who run for ofﬁce (c ∈ C) have no incentives to withdraw. – The candidates who do not run (b ∈ C\C) have no incentives to enter. It seems uncontroversial to introduce a stability condition for candidates who run. They should never beneﬁt by withdrawing. However, it is not clear what is an appropriate stability notion for non-running candidates. To ask for rules that never provide incentives to run for outsider candidates would lead immediately to impossibility results. Hence, we focus on the incentives of running candidates to withdraw. Indeed, our stability conditions are enough for obtaining the characterization results. We introduce two stability conditions. The weak requirement, is deﬁned as in DJL. For each C ∈ 2C \{∅}, a PVP is candidate stable at C if each candidate in C prefers the outcome when all candidates in C run to the outcome that she would obtain if she alone withdrew. The strong requirement applies the same condition to each agenda. A PVP is universally candidate stable if a candidate never beneﬁts from withdrawing, independently of which remaining candidates stay. Hence, candidate stability at C only refers to the stability of a speciﬁc agenda, while universal candidate stability implies the stability of every agenda. Let C ∈ 2C \{∅}. A PVP p is candidate stable at C if for each a ∈ C, each Pa ∈ P a , each utility function ua representing Pa , and each P ∈ P V : c∈C

p (c, C, P ) ua (c) ≥

p(c , C\{a}, P )ua (c ).

c ∈C\{a}

A PVP p is universally candidate stable if for each C ∈ 2C \{∅}, p is candidate stable at C. Our assumptions on candidates’ preferences have interesting consequences. Our stability conditions are automatically satisﬁed for agendas with less than three candidates. At an one-candidate agenda, the running candidate is selected with certainty and has no incentive to withdraw. At a two-candidate agenda, when a candidate withdraws, the remaining candidate is selected. By the self-preference of the candidates, the withdrawing candidate cannot be better off. Moreover, as we

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assume that candidates cannot vote, universally candidate stable PVPs are characterized by the following property. Whenever a candidate withdraws, the winning probability of every running candidate cannot decrease.6

2.4 Unanimity and efﬁciency We focus on PVPs that satisfy unanimity. Speciﬁcally, we require that a candidate who is ranked ﬁrst among the running candidates by every voter receives all the winning probability. A PVP p is unanimous if for each c ∈ C, each C ∈ 2C \{∅}, and each P ∈ P V , c = top(C, Pi ) for each i ∈ V implies p (c, C, P ) = 1. A condition closely related to unanimity is efﬁciency. Let C ∈ 2C \{∅} and P ∈ P V . A candidate c ∈ C is Pareto dominated at C and P if there is a candidate b ∈ C such that for each i ∈ V, b Pi c. For each I ⊂ V, each C ∈ 2C \{∅}, and each PI ∈ P I , Par (C, PI ) ≡ {c ∈ C | there is no b ∈ C, such that for each i ∈ I, b Pi c}. A PVP p is (ex-post) efﬁcient if for each c ∈ C, each C ∈ 2C \{∅}, and each P ∈ PV , c ∈ / Par(C, P ) implies p(c, C, P ) = 0. Evidently, unanimity is less stringent than efﬁciency. Unanimity does not preclude the possibility that Pareto dominated candidates receive positive winning probability.7 As candidate stability only applies to speciﬁc agendas, the requirements of the previous deﬁnitions can be too demanding. Hence, we provide restricted versions of unanimity and efﬁciency that are suitable for the study of the stability of speciﬁc agendas. Let C ∈ 2C \ {∅}. A PVP p is unanimous at C if for each a ∈ C, each A ∈ {C} ∪ {C\{c} : c ∈ C} , and each P ∈ P V , a = top(A, Pi ) for each i ∈ V implies p(a, A, P ) = 1. Let C ∈ 2C \ {∅}. A PVP p is efﬁcient at C if for each a ∈ C, each A ∈ {C} ∪ {C\{c} : c ∈ C} , and each P ∈ P V , a ∈ / Par(A, P ) implies p(a, A, P ) = 0. Of course, we think of unanimity as a weak and natural requirement. Nonetheless, it plays a crucial role in our analysis. We discuss the consequences of its relaxation in Section 5.2.

6 A mathematically equivalent property called regularity has been proposed by PP. We refer the reader to the preliminaries to the proofs of the theorems and Lemmata 1 and 2 in Section 4 for a precise deﬁnition of regularity and its relation to our stability conditions. 7 In probabilistic environments, it is possible to deﬁne another version of efﬁciency, ex-ante efﬁciency. Ex-ante efﬁciency requires that for each preference proﬁle there is no lottery that voters unanimously prefer to the selected lottery. It is easy to see that ex-ante efﬁciency is stronger than (ex-post) efﬁciency and unanimity.

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3 The implications of candidate stability The main message in DJL is that every unanimous and non-dictatorial single-valued voting procedures results in some situations where a candidate has an incentive to withdraw. The objective of this section is to analyze the extent to which this negative result still holds in a probabilistic environment. It is not difﬁcult to see that random dictatorships are universally candidate stable and unanimous. Under a random dictatorship, when a candidate withdraws, the voters who do not vote for her have no incentives to change their ballot. On the other hand, the voters for whom she is the ﬁrst-ranked candidate cast their ballot for another candidate. By withdrawing, a candidate transfers her winning probability to the remaining candidates. By candidates’self-preference, this transfer cannot represent an improvement for the withdrawing candidate. Our ﬁrst result, Theorem 1 shows that there are no universally candidate stable and unanimous PVPs other than random dictatorships. Theorem 1. A PVP p is universally candidate stable and unanimous if and only if p is a random dictatorship. Note that Theorem 1 is not a strengthening of DJL’s main result. Whereas DJL use candidate stability, Theorem 1 involves the stronger condition of universal candidate stability. Theorem 2 characterizes the class of PVPs that satisfy candidate stability and unanimity at agendas containing at least four candidates. To present it, we need additional notation. Let p be a PVP and let C ∈ 2C \{∅}. For each a ∈ C and each P ∈ P V , deﬁne: Lp (a, C, P )≡{λ∈LC , such that for each b ∈ C\{a}, λ(b) ≤ p(b, C\{a}, P )}. Let C ∈ 2C \ {∅}. A PVP p is a modiﬁed random dictatorship at C if there is a list of weights {αi }i∈V ∈ ∆#V−1 , such that for each a ∈ C and each P ∈ P V , p (C\{a}, P ) = αi di (C\{a}, P ) , and i∈V

p (C, P ) ∈

Lp (b, C, P ) .

b∈C

Note that for each C ∈ 2C \ {∅}, a random dictatorship is a special case of a modiﬁed random dictatorship at C. Then, every list {αi }i∈V ∈ ∆#V−1 deﬁnes a non-empty family of modiﬁed random dictatorships at C. Theorem 2. Let C ∈ 2C \{∅} be such that #C ≥ 4. A PVP p is candidate stable at C and unanimous at C if and only if p is a modiﬁed random dictatorship at C. A complete proof Theorem 2 appears in the following section. The intuition runs as follows. By exploiting the relation between candidate stable PVPs and voting correspondences, we show that candidate stability and unanimity at C imply

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efﬁciency at C. Then, we focus on preference proﬁles in which only two candidates are not Pareto dominated. On this restricted domain, each candidate’s winning probability only depends on the group of voters for whom she is the ﬁrst-ranked running candidate. Hence, we can associate to each group of voters a weight that is equal to the winning probability of a candidate who is ranked ﬁrst by all the members of the group. Finally, we show that the distribution of the weights associated to the different groups of voters is additive, which leads directly to our characterization. In the light of Theorem 2, we see that if we focus on speciﬁc agendas, random dictatorships are not the only candidate stable and unanimous PVPs. Nevertheless, the additional ﬂexibility is limited to situations in which no candidate withdraws. The following proposition makes precise the ﬂexibility that modiﬁed random dictatorships offer. Proposition 1. Let C ∈ 2C \{∅}. Let p be a modiﬁed random dictatorship at C with associated list of weights {αi }i∈V ∈ ∆#V−1 , and let S ≡ {i ∈ V, αi > 0}. Then, for each P ∈ P V : (i) There exists λ ∈ b∈C Lp (b, C, P ) such that λ = i∈V αi di (C, P ), if and only if there exists a ∈ C such that for each b ∈ C\{a} there is i ∈ S with a = top (C, Pi ) and a = top (C\{b}, Pi ) . (ii) For each two distinct a, b ∈ C, if a, b ∈ i∈S top (C, Pi ), then there is no λ ∈ b∈C Lp (b, C, P ) with λ (a) = λ (b) = 0. Part (i) characterizes the preference proﬁles for which a modiﬁed random dictatorship at C with weights {αi }i∈V may not coincide (for the relevant agendas) with the random dictatorship with the same list of associated weights. This situation may occur when there exists a candidate who, whenever another candidate withdraws, becomes the ﬁrst-ranked running candidate for some voter with positive weight. Such a candidate may receive a bonus of winning probability as compared to the winning probability that the random dictatorship would assign to her. Note that, for each C ∈ 2C \ {∅} such that #C ≥ #V + 2, no candidate satisﬁes the requirements of Part (i). In this case, every modiﬁed random dictatorship at C is equivalent (for the relevant agendas) to a random dictatorship. Part (ii) says that if no candidate withdraws from C, at most one of the candidates who are ranked ﬁrst by some voter with positive weight may receive null winning probability. The following example should clarify the notion of modiﬁed random dictatorship.8 Example 1. Let C = {a, b, c, d}, and V = {1, 2, 3, 4}. Let P ∗ ⊂ P V denote the domain of preference proﬁles such that each voter in {1, 2, 3} has different ﬁrstranked candidate, while the remaining candidate is the second-ranked candidate by each voter in {1, 2, 3} and the ﬁrst-ranked candidate by voter 4. Deﬁne the PVP p as follows. Let α1 = α2 = α3 = 13 , and α4 = 0, and for each A ∈ {C}∪{C\{c} : 8

This example resembles Example 5.6 in PP.

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c ∈ C} and each P ∈ P V :  1 1 1 1  ∗  , , ,   4 4 4 4 if A = C and P ∈ P , p (A, P ) ≡   αi di (A, P ) otherwise.   i∈V

Clearly, p is a modiﬁed random dictatorship at C. Note that for each ∗ P ∈ P , b∈C Lp (b, C, P ) contains more than one lottery and p(C, P ) = i∈V αi di (C, P ). Finally, we address the stability of three-candidate agendas. Interestingly, our next theorem shows that new possibilities arise. Let C ∈ 2C \{∅} be such that #C = 3. A PVP p is oligarchical at C if there is a list of non-negative weights {αT }T ⊆V such that for each a, b ∈ C, and each P ∈ PV : T = {j ∈ V, a = top (C\{b}, Pj )} ⇒ p (a, C\{b}, P ) = αT , and p (C, P ) ∈ Lp (b, C, P ) , b∈C

and furthermore, (i) for each T ⊆ V, αT = (1 − αV\T ), and αV = 1, (ii) for each T, T ⊆ V such that T ⊆ T , αT ≤ αT , (iii) for each T, T ⊆ V such that T ∩ T = {∅}, αT + αT ≥ α(T ∪T ) . Under an oligarchical PVP, when a candidate withdraws from a three-candidate agenda, the winning probability of each running candidate only depends on the group of voters for whom she is the ﬁrst-ranked running candidate. Part (ii) says that the weights associated to the groups of voters are monotonic. If we incorporate new members to a group of voters, the weight associated to the larger group is not smaller than the weight associated to the initial group. Part (iii) implies that these weights are sub-additive. The weight associated to the union of two disjoint groups of voters is not greater than the sum of the weights associated to each group. For each C ∈ 2C \ {∅} with #C = 3, every list of non-negative weights {αT }T ⊆V satisfying (i), (ii), and (iii) deﬁnes a non-empty family of oligarchical PVPs at C. In fact, (iii) implies that for each oligarchical PVP p, for each P ∈ P V , ∩b∈C Lp (b, C, P ) = {∅}. Note also that for each C ∈ 2C \ {∅} with #C = 3, a modiﬁed random dictatorship at C is oligarchical at C. Theorem 3. Let C ∈ 2C \{∅} be such that #C = 3. A PVP p is candidate stable at C and unanimous at C if and only if p is oligarchical at C. Theorem 3 is in line with the results by Barber`a and Sonnenschein [3] and McLennan [10] on probabilistic binary social choice (or stochastic social preferences). These authors analyze rules that map preference proﬁles to lotteries over preferences and provide probabilistic versions of Arrow’s [1] Theorem. They show

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that for small numbers of alternatives, the distribution of the decision power among the different groups of voters is sub-additive.9 The similarities of the results are not surprising. When a candidate withdraws from a three-candidate agenda, the choice between the remaining candidates is indeed a probabilistic binary choice. We conclude this section with an example of an attractive PVPs that satisﬁes candidate stability at agendas containing only three candidates. Example 2. Let p be such that for each a ∈ C, each A ∈ 2C \{∅}, and each P ∈ PV ,  1   if a ∈ Par (A, P ) , p (a, A, P ) ≡ #Par (A, P )   0 otherwise. Clearly, p is efﬁcient and but there is no agenda C ∈ 2C \ {∅} for which p is a modiﬁed random dictatorship at C. Hence, p is not candidate stable at agendas containing at least four candidates. However, for each C ∈ 2C \{∅} with #C = 3, p is oligarchical at C. Note that for each T ⊆ V, T ∈ / {V, ∅}, αT = 12 . 4 Proofs ´ We need some deﬁnitions and results from PP and Rodr´ıguez-Alvarez [14]. The ﬁrst condition that we introduce is called regularity. Regularity says that, given an agenda and a preference proﬁle, if some candidates withdraw, then the winning probability of each of the remaining candidates cannot decrease. A PVP p is regular if for each C, C ∈ 2C \{∅}, and each P ∈ P V , c ∈ C and C ⊆ C imply p(c, C, P ) ≥ p(c, C , P ). The main result in PP states that, provided there exists an agenda with sufﬁciently many candidates, every PVP that satisﬁes regularity and efﬁciency is a random dictatorship. Proposition 2 (Theorem 4.14, Pattanaik and Peleg [13]). Assume there exists an agenda C ∈ 2C \{∅} with #C ≥ #V + 2. If the PVP p satisﬁes regularity and efﬁciency, then p is a random dictatorship. A voting correspondence is a mapping v : 2C \{∅} × P V → 2C \{∅} such that for each C ∈ 2C \{∅} and each P ∈ P V , (i) v(C, P ) ⊆ C, and (ii) for each P ∈ P V such that P |C = P |C , v (C, P ) = v (C, P ). We present also several conditions for voting correspondences. No harm and insigniﬁcance incorporate the notion of candidate stability. For a given agenda C ∈ 2C \ {∅}, no harm at C requires that if candidate is selected when all the candidates in C run, she remains selected when other selected candidate withdraws from C. Insigniﬁcance at C implies that the withdrawal of a candidate who is not 9 An illuminating discussion relating the probabilistic binary framework and ours can be found in PP. (See Remarks 3.12, 4.15 and Lemma 3.13, PP.)

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selected when all the candidates in C run has no effect over the social choice.10 Finally, the interpretation of unanimity at C for voting correspondences is clear. Let C ∈ 2C \{∅}. Let v be a voting correspondence: – It satisﬁes no harm at C if for each a ∈ C and each P ∈ P V , a ∈ v(C, P ) implies v(C, P ) ⊆ v(C\{a}, P ) ∪ {a} . / v(C, P ) – It satisﬁes insigniﬁcance at C if for each a ∈ C and each P ∈ P V , a ∈ implies v(C, P ) = v(C\{a}, P ). – It satisﬁes unanimity at C if for each a ∈ C, each A ∈ {C}∪{C\{c} : c ∈ C}, and each P ∈ P V , a = top (A, Pi ) for each i ∈ V implies v (A, P ) = {a}. ´ Rodr´ıguez-Alvarez [14] shows that every voting correspondence that satisﬁes no harm at C, insigniﬁcance at C, and unanimity at C, concentrates the social decision power in the hands of an arbitrary group of voters. ´ Proposition 3 (Theorem 4, Rodr´ıguez-Alvarez [14]). Let C ∈ 2C \{∅} be such that #C ≥ 3. If the voting correspondence v satisﬁes no harm at C, insigniﬁcance at C, and unanimity at C, then there is a set of voters S ⊆ V such that for each A ∈ {C} ∪ {C\{c} : c ∈ C} and each P ∈ P V , (i) v (A, P ) ⊆ Par (A, PS ), (ii) for each a ∈ A, if there are i ∈ S and b ∈ A \ {a} with b Pi a, then v(A, P ) = {a} , (iii) for each a ∈ C, ∪i∈S top(C\{a}, Pi ) ⊆ v(C\{a}, P ). We now provide two crucial lemmata that relate our candidate stability conditions and regularity. Lemma 1. Let C ∈ 2C \{∅}. A PVP p is candidate stable at C if and only if for each two distinct a, b ∈ C, and each P ∈ P V , p (b, C\{a}, P ) ≥ p (b, C, P ) . Proof. Consider two distinct a, b ∈ C. Let P ∈ P V . Assume, to the contrary, that p satisﬁes candidate stability at C, but p (b, C\{a}, P ) < p (b, C, P ) . Let Pa ∈ P a be such that for each c ∈ C\{b}, c Pa b. There is a utility function ua representing Pa such that: c∈C\{b} (p(c, C\{a}, P ) − p(c, C, P )) ua (c) . (1) ua (b) < p(b, C, P ) − p(b, C \ {a}, P ) Rearranging terms, (1) yields p (c , C\{a}, P ) ua (c ) > p (c, C, P ) ua (c) , c ∈C

c∈C

which violates candidate stability at C. Let us prove the converse statement. Let a ∈ C and P ∈ P V . Assume ﬁrst that for each b ∈ C\{a} , p (b, C\{a}, P ) = p (b, C, P ). Then, 10 Rodr´ıguez-Alvarez ´ [14] also proves that a voting correspondence satisﬁes no harm at C and insigniﬁcance at C if and only if it is candidate stable at C when candidates compare sets of candidates according to leximin preferences over sets.

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p (C, P ) = p (C\{a}, P ) , and a cannot beneﬁt from withdrawing. Finally, assume that for some b ∈ C\{a}, p (b, C\{a}, P ) > p (b, C, P ) . Then, (p (b, C\{a}, P ) − p (b, C, P )) > 0, p (a, C, P ) = b∈C\{a}

and, as for each b ∈ C\{a}, each Pa ∈ P a , and each ua representing Pa , ua (a) > ua (b), (p (b, C\{a}, P ) − p (b, C, P )) ua (b) . (2) p (a, C, P ) ua (a) > b∈C\{a}

Finally, rearranging terms, (2) yields p (b, C, P ) ua (b) > b∈C

p (b , C\{a}, P ) ua (b ) ,

b ∈C\{a}

which proves p’s candidate stability at C.

Lemma 2. A PVP is universally candidate stable if and only if it is regular. Proof. This is a simple corollary to Lemma 1.

In the light of Lemmata 1 and 2, we see that our set of axioms is weaker than PP’s set. Provided that candidates cannot vote, universal candidate stability is mathematically equivalent to regularity. However, candidate stability at C is weaker than regularity, since it only refers to a speciﬁc agenda. Of course, unanimity at C is weaker than efﬁciency. Two remarks are in order. The ﬁrst remark highlights an implication of candidate stability that we use extensively throughout the proofs. The second remark is a convenient rephrasing of Lemma 1. Remark 1. Let C ∈ 2C {∅} and let the PVP p be candidate stable at C. Then, for each a ∈ C and each P ∈ P V , p(a, C, P ) = 0 implies p(C, P ) = p(C\{a}, P ). Proof. Let a ∈ C and P ∈ P V . Assume, to the contrary, that p(a, C, P ) = 0 but p(C, P ) = p(C\{a}, P ) . Then, there is b ∈ C \ {a} such that p(b, C\{a}, P ) < p(b, C, P ), which, by Lemma 1, violates candidate stability at C.

Remark 2. A PVP p is candidate stable at C if and only if for each P ∈ P V , p(C, P ) ∈ b∈C Lp (b, C, P ). We analyze now the relation between candidate stable and unanimous PVPs and voting correspondences. For each PVP p, we deﬁne the auxiliary voting correspondence vp as follows. For each c ∈ C, each C ∈ 2C \{∅}, and each P ∈ P V , c ∈ vp (C, P ) if and only if p (c, C, P ) > 0. By (i) and (ii) of the deﬁnition of PVPs, vp is well-deﬁned. Lemma 3. Let C ∈ 2C \{∅}. Let the PVP p be candidate stable at C and unanimous at C. Then, the auxiliary voting correspondence vp satisﬁes no harm at C, insigniﬁcance at C, and unanimity at C.

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Proof. Let a ∈ C and P ∈ P V . Assume ﬁrst that p(a, C, P ) > 0, so that a ∈ vp (C, P ). Let b ∈ C\{a} be such that p (b, C, P ) > 0. Then, b ∈ vp (C, P ). By candidate stability at C, p (b, C\{a}, P ) > 0. Hence, b ∈ vp (C \ {a}, P ), which proves no harm at C.Assume now that p (a, C, P ) = 0. Then, a ∈ / vp (C, P ). By Remark 1, p (C, P ) = p (C\{a}, P ). We therefore have vp (C, P ) = vp (C\{a}, P ) , which proves insigniﬁcance at C. Finally, by p’s unanimity at C, vp satisﬁes unanimity at C.

With the previous Lemmata and Propositions 2 and 3 at hand, we are in position to prove the theorems. Proof of Theorem 1. It is immediate to prove that random dictatorships are universally candidate stable and unanimous. Thus, we focus on the converse statement. Let p be a PVP satisfying universal candidate stability and unanimity. As C is an countably inﬁnite set, there exists C ∈ 2C \{∅} with #C ≥ #V + 2. By universal candidate stability, for each C ∈ 2C \{∅}, p is candidate stable at C. By Lemma 3 and Proposition 3, for each C ∈ 2C \ {∅}, there is a group of voters S ⊆ V such that for each P ∈ P V , vp (C, P ) ⊆ Par (C, PS ) . Thus, p is efﬁcient. Moreover, by Lemma 2, p is regular. Then, the result follows from Proposition 2.

Next, we prove Theorem 2. As for each C ∈ 2C \ {∅}, candidate stability at C is weaker than regularity, we cannot rely on the results of PP and we have to use more involved arguments. Proof of Theorem 2. Let C ∈ 2C \{∅} be such that #C ≥ 4. Let p be a modiﬁed random dictatorship at C. By Remark 2, p is candidate stable at C. Let us check unanimity at C. Let a ∈ C and P ∈ P V . Assume that for each i ∈ V, a = top(C, Pi ). Then, by the deﬁnition of modiﬁed random dictatorship at C, for each b ∈ C\{a}, p(a, C\{b}, P ) = 1. Finally, by the deﬁnition of Lp (b, C, P ), p(a, C, P ) = 1. Let p be a PVP satisfying candidate stability at C and unanimity at C. First, by Lemma 3 and Proposition 3, p is efﬁcient at C. We prove two claims: Claim 1. There is a list of non-negative weights {αT }T ⊆V such that for each two distinct a, b ∈ C, and each P ∈ P V with Par (C, P ) = {a, b}, T = {i ∈ V such that a Pi b} ⇒ p (a, C, P ) = αT , and furthermore, for each T ⊆ V, αT + αV\T = 1 . Proof. We prove this claim in two steps. In Step 1 we show that when only two running candidates are not Pareto dominated, the winning probability of each candidate does not depend on voters’ preferences over Pareto dominated candidates. In Step 2, we show a neutrality result. When only two running candidates are not Pareto dominated, the winning probability of each candidate only depends on the group of voters who rank her ﬁrst. Step 1. For each two distinct a, b ∈ C, and each two distinct each P, P ∈ P V , if P |{a,b} = P |{a,b} and Par(C, P ) = Par (C, P ) = {a, b}, then p (C, P ) = p (C, P ) .

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Let P ∈ P V be such that Par(C, P ) = {a, b}. Let p (a, C, P ) = α and, by efﬁciency at C, p (b, C, P ) = 1 − α. Let c, d ∈ C, and i ∈ V be such that top(C, Pi ) ∈ / {c, d}, and for each e ∈ C \ {c, d}, c Pi e if and only if d Pi e. (Hence {c, d} = {a, b}.) Let P¯ ∈ P V be such that Pi |C\{c} = P¯i |C\{c} , Pi |C\{d} = P¯i |C\{d} , while PV\{i} = P¯V\{i} , and Par(C, P¯ ) = {a, b} . Basically, P¯ is obtained from P by switching c and d’s positions in voter i’s preference. Note also that P |{a,b} = P¯ |{a,b} . Without / {a, b}. loss of generality, let c ∈ ¯ Then, c ∈ / Par (C, P ) and c ∈ / Par C, P . By efﬁciency

at C and Remark 1, p (C, P ) = p (C\{c}, P ). Analogously, p C, P¯ = p C\{c}, P¯ . By (ii) of the deﬁnition of PVP, p(C\{c}, P ) = p(C\{c}, P¯ ). Then, p (C, P ) = p C, P¯ . Repeating this argument as many times as necessary, switching the position of two contiguous candidates in the preferences of one voter at a time, we obtain the desired conclusion. Step 2. Let {a, b}, {c, d} ⊂ C. For each P, P ∈ P V , if Par(C, P ) = {a, b} , Par(C, P ) = {c, d} , {i ∈ V, a = top(C, Pi )} = {i ∈ V, c = top(C, Pi )} , and {j ∈ V, b = top(C, Pj )} = {j ∈ V, d = top(C, Pj )} ; then p(a, C, P ) = p(c, C, P ) and p(b, C, P ) = p(d, C, P ) . Let T ⊆ V. Let P ∈ P V be such Par(C, P ) = {a, b}, and for each i ∈ T , a = top(C, Pi ), while for each j ∈ V \ T , b = top(C, Pj ). Let p(a, C, P ) = α and p(b, C, P ) = 1 − α. Next, let Pˆ ∈ P V be such that for each e ∈ C \ {a, b, d}, for each i ∈ T , a Pˆi b Pˆi d Pˆi e ; while for each j ∈ V \ T , b Pˆj d Pˆj a Pˆj e ; and P |C\{a,b,d} = Pˆ |C\{a,b,d} . By Step 1, we have p(C, P ) = p(C, Pˆ ) . Let P ∗ ∈ P V be such that for each i ∈ T , d = top(C\{a}, Pi∗ ) ; for each j ∈ V \ T , d = top(C, Pj∗ ) ; while Pˆ |C\{b} = P ∗ |C\{b} , and Pˆ |C\{d} = P ∗ |C\{d} . Basically, P ∗ is obtained from Pˆ by switching b and d’s positions in voters’ preferences. By efﬁciency at C and Remark 1, p(C, P ∗ ) = p(C\{b}, P ∗ ) . By (ii) of the deﬁnition of PVP, p(a, C\{b}, Pˆ ) = p(a, C\{b}, P ∗ ) . By candidate stability, p(a, C\{b}, Pˆ ) ≥ p(a, C, Pˆ ) = α , and we get p(a, C, P ∗ ) ≥ α. Analogously, by efﬁciency at C and Remark 1, p(C, Pˆ ) = p(C\{d}, Pˆ ) = α . By (ii) of the deﬁnition of PVP, p(a, C\{d}, Pˆ ) = p(a, C\{d}, P ∗ ) . By candidate stability, p(a, C\{d}, P ∗ ) ≥ p(a, C, P ∗ ), and we get p(a, C, P ∗ ) ≤ α. Thus, p(a, C, P ∗ ) = α. Finally, by efﬁciency at C, p(d, C, P ∗ ) = 1 − α. Repeating the same arguments with candidates a and c, and applying Step 1, we obtain the desired result. Finally, deﬁne the weights {αT }T ⊆V in the following way. Let T ⊆ V. Let a, b ∈ C and P ∈ P V be such that T = {i ∈ V, a = top(C, Pi )}, and Par(C, P ) = {a, b} , and let αT ≡ p(a, C, P ) . By Steps 1 and 2, αT is welldeﬁned and, by efﬁciency at C, αV\T ≡ (1 − αT ) .

Claim 2. For each T, T ⊆ V such that T ∩ T = {∅}, αT + αT = α(T ∪T ) .

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Proof. The result follows immediately if T ∈ {V, ∅} or T ∪ T = V. Hence, let T, T ⊂ V. Let a, b, c, d ∈ C, and P ∈ P V be such that for each x ∈ (C\{a, b, c, d}), for each i ∈ T, a Pi b Pi c Pi d Pi x, for each j ∈ T , b Pj c Pj a Pj d Pj x, and for each k ∈ V \ (T ∪ T ) , c Pk a Pk b Pk d Pk x. Note that, by efﬁciency at C and Remark 1, p (C, P ) = p (C\{d}, P ) . Let P ∈ P V be such that for each i ∈ T , d = top (C\{a}, Pi ) , for each i ∈ V \T , d = top (C, Pi ) , and P |C\{d} = P |C\{d} . By Claim 1, p (a, C, P ) = αT . By (ii) in the deﬁnition of PVP, p(C\{d}, P ) = p(C\{d}, P ). By candidate stability, p(a, C\{d}, P ) ≥ p(a, C, P ) . Then, p(a, C, P ) ≥ αT . Let P ∈ P V be such that P |C\{b} = P |C\{b} and for each i ∈ T, a Pi c Pi b Pi d, for each j ∈ T , c Pj b Pj a Pj d, and for each k ∈ V \ (T ∪ T ) , c Pk a Pk b Pk d. Note that Par (C, P ) = {a, c}. Then, by Claim 1, p (a, C, P ) = αT . By efﬁciency at C and Remark 1, p(C, P ) = p (C\{b}, P ) . By (ii) in the deﬁnition of PVP, p (C\{b}, P ) = p (C\{b}, P ) . By candidate stability, p(a, C\{b}, P ) ≥ p(a, C, P ), and we get p(a, C, P ) ≤ αT . We therefore have p(a, C, P ) = αT . Using parallel arguments with b and c, we get p (b, C, P ) = αT , and p (c, C, P ) = αV\(T ∪T ) . By efﬁciency at C, p(a, C, P ) + p(b, C, P ) + p(c, C, P ) = 1 . Then, αT + αT + αV\(T ∪T ) = 1.

(3)

Finally, as αV\(T ∪T ) = (1 − αT ∪T ) , (3) yields αT + αT = αT ∪T .

Let us complete the proof of Theorem 2. By Claim 2, i∈V αi = 1. Hence, {αi }i∈V ∈ ∆#V−1 . Consider two distinct a, b ∈ C and P ∈ P V . Let T = {i ∈ V , b = top (C\{a}, Pi )} . Let P ∈ P V be such that for each top C, Pj , while i ∈ T , a = top (C\{b}, Pi ) , for each j ∈ (V \ T ), a = P |C\{a} = P |C\{a} . By Claims 1 and 2, p (b, C, P ) = i∈T αi . By can of PVP, didate stability, p(b, C\{a}, P ) ≥ i∈T αi . By (ii) in the deﬁnition α . Using p (b, C\{a}, P ) = p(b, C\{a}, P ) . Then, p (b, C\{a}, P )≥ i i∈T the same argument for each c ∈ ∪j∈V top (C\{a}, Pj ) , as i∈V αi = 1 , we get αi di (C\{a}, P ) . p (C\{a}, P ) = Finally, by Remark 2, p(C, P ) ∈

i∈V b∈C

Lp (b, C, P ).

We continue with the proof of Proposition 1. Proof of Proposition 1. Let p be a modiﬁed random dictatorship at C. Let {αi }i∈V be the weights associated to p, and let S = {i ∈ V, αi > 0}.

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(i) Let P ∈ P V be such that for each a ∈ C there is b ∈ C\{a} with {i ∈ S, a = top(C\{b}, Pi )} = {i ∈ S, a = top(C, Pi )}. Then, for each a ∈ C, there is d ∈ C \ {a} such that αi di (a, C, P ) = αi di (a, C\{d}, P ) = p(a, C \ {d}, P ). i∈V

i∈V

Thus, for ∈ C and each ∈ b∈C Lp (b, C, P ), each a λ λ(a)≤ i∈V αi di (a, C, P ) . Clearly, as α d (C, P ) ∈ L , for each i i i∈V λ ∈ b∈C Lp (b, C, P ) and each a ∈ C, λ(a) = i∈V αi di (a, C, P ). Next, let a ∈ C and let P ∈ P V . Assume that for each b ∈ C\{a}, there is i ∈ S with a = top(C, Pi ) but a = top(C\{b}, Pi ). Note that αi di (a, C, Pi ) < min αi di (a, C\{d}, P ) . d∈C\{a}

i∈V

i∈V

Hence, there exists λ ∈ L such that

αi di (a, C, Pi ) < λ(a) ≤

i∈V

min

d∈C\{a}

while for each b ∈ C\{a}, λ(b) ≤

i∈V

αi di (b, C, P ) ≤

min

d∈C\{b}

αi di (a, C\{d}, P ) ,

i∈V

αi di (b, C\{d}, P ) ,

i∈V

whichsufﬁces to prove that there exists λ ∈ λ = i∈V αi di (C, P ) .

b∈C

Lp (b, P ) such that

(ii) Let a ∈ C and P ∈ P V . Assume that a ∈ ∪i∈S top(C, Pi ). By the deﬁnition of Lp (a, b, c), for each λ ∈ Lp (a, C, P ), λ = p(C \ {a}, P ), λ(a) > 0. As for each b ∈ ∪i∈S top(C, Pi ) \ {a}, p(b, C \ {a}, P ) > 0, and for each a ∈ C, ∩b∈C Lp (b, C, P ) ⊂ Lp (a, C, P ), this sufﬁces to prove Part (ii).

Finally, we analyze the case in which there are only three potential candidates. Proof of Theorem 3. Let C = {a, b, c} and let p be oligarchical at C. By Remark 2, p is candidate stable at C. By (i) of the deﬁnition of oligarchical PVPs, and repeating the arguments we use in proving that modiﬁed random dictatorships at C are unanimous at C, we prove that p is unanimous at C. Let p be a PVP satisfying candidate stability at C and unanimity at C. By Lemma 3 and Proposition 3, p is efﬁcient at C. Note that, in the proof of Claim 1 of Theorem 2, we only assume the existence of three candidates. Hence, by Claim 1, there exists a list of weights of non-negative weights {αT }T ⊆V such that for each two distinct a, b ∈ C, each T ⊆ V, and each P ∈ P V , Par(C, P ) = {a, b} and T = {i ∈ V, a Pi b} imply p(a, C, P ) = αT and p(b, C, P ) = (1 − αT ) . By unanimity at C, αV = 1 . By efﬁciency at C and Remark 1, we have that

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p(a, {a, b}, P ) = αT and p(b, {a, b}, P ) = (1 − αT ) . Finally, by (ii) of the definition of PVP, for each P ∈ P V such that P |{a,b} = P |{a,b} , we get that p({a, b}, P ) = p({a, b}, P ). Let us check part (ii) of the deﬁnition of oligarchical PVP. It follows immediately if #V = 2. Hence, assume #V ≥ 3, let i ∈ V and T ⊂ V \ {i}. Let P ∈ P V be such that a Pi b Pi c ; for each j ∈ T , b Pj a Pj c ; and for each k ∈ V \ (T ∪ {i}), a Pk c Pk b . By Claim 1, p(b, C, P ) = αT . Let P ∈ P V be such that Par(C, P ) = {b, c} and P |C\{a} = P |C\{a} . Again, by Claim 1, p(b, C, P ) = α(T ∪{i}) . By efﬁciency at C and Remark 1, p(b, C \ {a}, P ) = α(T ∪{i}) . By (ii) of the deﬁnition of PVP, p(b, C \ {a}, P ) = α(T ∪{i}) . Finally, by candidate stability, we obtain p(b, C \ {a}, P ) ≥ p(b, C, P ) . Thus, α(T ∪{i}) ≥ αT . Repeating this argument as many times as necessary, we get that for each T, T ⊆ V such that T ⊂ T , αT ≤ αT . Let us check part (iii) of the deﬁnition of oligarchical PVP. Let T, T ⊂ V be such that T ∩ T = {∅} . The result follows immediately if T ∈ {V, ∅} or T ∪ T = V . Hence, let T, T ⊂ V . Let P ∈ P V be such that for each i ∈ T , a Pi b Pi c , for each j ∈ T , b Pj c Pj a , and for each k ∈ V \ (T ∪ T ) , c Pk a Pk b . Let P ∈ P V be such that Par(C, P ) = {a, c} , and P |C\{b} = P |C\{b} . By Claim 1, p(a, C, P ) = αT . By efﬁciency at C and Remark 1, p(a, C\{b}, P ) = αT . By (ii) of the deﬁnition of PVP, p(a, C\{b}, P ) = αT . By candidate stability, p(a, C \ {b}, P ) ≥ p(a, C, P ). Thus, p(a, C, P ) ≤ αT . Using parallel arguments, we get that p(b, C, P ) ≤ αT and p(c, C, P ) ≤ αV\(T ∪T ) . As p(a, C, P ) + p(b, C, P ) + p(c, C, P ) = 1 , αT + αT + αV\(T ∪T ) ≥ 1. Hence, as αV\(T ∪T ) = 1 − α(T ∪T ) , (4) yields αT + αT ≥ α(T ∪T ) . Finally, by Remark 2, for each P ∈ P V , p(C, P ) ∈ b∈C Lp (b, C, P ).

(4)

5 Concluding remarks 5.1 Overlap between candidates and voters This work only covers the case in which the sets of candidates and voters are disjoint. However, in many real-life social choice processes candidates are also voters. In this more realistic situation, the analysis of candidates’ incentives to withdraw becomes problematic. First, unanimity is empty of content when more than one candidate can vote because candidates are always supposed to support their own election. Moreover, if candidates can vote, candidates’ preferences can be used to generate punishments that prevent candidates from withdrawing.11 Whenever a candidate 11 In fact, our deﬁnition of candidate stability is unnecessarily strong when candidates can vote, since it does not takes into account the preferences reported by candidates who are also voters. A more appropriate statement that captures the notion of the stability of an agenda C ∈ 2C \ {∅} would be the following. For each P ∈ P V , for each a ∈ C ∩ V and each ua representing Pa , c∈C p(c, C, P )ua (c) ≥ c ∈C\{a} p(c , C \ {a}, P )ua (c ) , a , and each u representing P , ∈ C \ V, each P ∈ P while for each a a a a c∈C p(c, C, P )ua (c) ≥ c ∈C\{a} p(c , C \ {a}, P )ua (c ) .

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withdraws, we could assign all the winning probability to the candidate who is ranked last by the withdrawing candidate. Although, with this kind of punishment, candidates never have incentives to withdraw, the social choice may be against the preferences of the remaining members of society. An alternative approach consists of introducing “ad hoc” stability conditions and stronger unanimity as in Ehlers and Weymark [7], and Eraslan and McLennan [8]. The drawback of this approach would be that the strategic interpretation of the framework would not be clear.12 In our framework, regularity can be postulated as a reasonable stability condition. On the other hand, the following unanimity requirement is meaningful. A PVP p is strongly unanimous if for each c ∈ C, each C ∈ 2C \{∅}, and each P ∈ P V , if c = top (C, Pi ) for each i ∈ V\C, and c = top(C\{a}, Pa ) for each a ∈ (C ∩ V)\{c}, then p (c, C, P ) = 1. With regularity and strong unanimity at hand, we obtain the following result. Theorem 1 . Let C ∩ V = {∅}. A PVP p is regular and strongly unanimous if and only if p is a random dictatorship and for each j ∈ C ∩ V, αj = 0. Theorem 1 implies that when all voters are also potential candidates no PVP satisﬁes both regularity and strong unanimity. Note also that, when candidates are allowed to vote, voters’ preferences are restricted by the self-preference of the candidates. As PP consider unrestricted strict preferences, we cannot apply PP’s ´ results in the proof. Nevertheless, as the results from Rodr´ıguez-Alvarez [14] do cover the case of voting candidates, we can follow the same arguments of the proof of Theorem 2.13 5.2 Relaxing unanimity Unanimity plays a crucial role in our results. However, it would be interesting to know which candidate stable PVPs are ruled out by this requirement. Candidate stability has no bite at two-candidate agendas. It is clear that a probabilistic combination of two candidate stable PVPs is also candidate stable. Hence, we can construct non-unanimous and candidate stable PVPs by combining PVPs that only operate at two-candidate agendas. The following example presents an attractive PVP that can be devised in this way. Example 3. For each c ∈ C, each i ∈ V, let the mapping sci : (C \ {c}) × 2C \ {∅} × P i → {0, 1} be such that for each a ∈ C \ {c}, each A ∈ 2C \ {∅}, and each Pi ∈ P i , / A, 1 if a Pi c, or c ∈ c si (a, A, Pi ) ≡ 0 otherwise. 12 DJL also prove that candidate stability is incompatible with a weak unanimity condition that takes into account the self-preference of the candidates and a monotonicity condition that many common voting procedures satisfy. (Theorems 2 and 3, DJL.) 13 Theorem 4 in Rodr´ıguez-Alvarez ´ [14] admits voting candidates and proves that their preferences are not relevant for the social choice. This implies that for each j ∈ C ∩ V, αj = 0.

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Let C ∈ 2C \ {∅} and let the PVP p˜ be such that for each a ∈ C, each A ∈ {C} ∪ {C\{c} : c ∈ C}, and each P ∈ P V ,  2   sci (a, A, P ) if a ∈ A, p˜(a, A, P ) ≡ #V#C(#C − 1) i∈V c∈C\{a}   0 otherwise. Note that p˜ is a probabilistic version of the Borda Count rule.14 It is clear that p˜ is candidate stable at C since it is a combination of PVPs with only two alternatives in their range.

5.3 Reporting preferences over lotteries Our analysis admits an additional generalization. It would be natural that voters (and not only candidates) were Expected-Utility maximizers. Then, the social choice could be based on voters’ preferences over lotteries. Let D be the set of all complete, reﬂexive, and transitive binary relations over L ∪ {∅} that satisfy the postulates of Expected-Utility Theory. Let ∈ D denote a preference over lotteries. A preference proﬁle over lotteries ∈ DV is a #V-tuple of preferences over lotteries. Abusing notation, for each C ∈ 2C \ {∅} and each ∈ DV , |LC refers to the restriction of to LC . We deﬁne a voting procedure over lotteries as a map that selects a lottery over candidates for each agenda and each preference proﬁle over lotteries (π : 2C \{∅}× DV → L), such that for each C ∈ 2C \ {∅} and each ∈ DV , (i) non-running candidates cannot be elected ( π (A, ) ∈ LA ) and (ii) voters’ preferences over lotteries containing non-running candidates are irrelevant ( |LA = |LA implies π (A, ) = π (A, ) ). A PVP is a voting procedure over lotteries satisfying the following invariance requirement. For each pair of preference proﬁles over lotteries that coincide in their restriction to the set of lotteries that assign positive probability to a unique candidate, a PVP selects the same lottery. The deﬁnitions of (universal) candidate stability and unanimity can be straightforwardly translated to this general framework. Surprisingly, allowing for more ﬂexibility in the voting procedure to meet voters’ preferences does not generate new possibilities. Hence, we can state a more general version of Theorem 1. Theorem 1 . A voting procedure over lotteries π is universally candidate stable and unanimous if and only if π is a random dictatorship. The proof of Theorem 1 is parallel to the proof of Theorem 2. We provide here a sketch of the proof, a complete proof being available from the author. Note that in this general framework, as candidates cannot vote, Lemmata 1 and 2 remain valid. When there are only two running candidates, by (ii) in the deﬁnition of voting procedure over lotteries, a voting procedure over lotteries only uses the 14 This example adapts to a variable agenda setting the “supporting size” and “positional voting” methods proposed in Barber`a [2] for the study of non-manipulable decision schemes.

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information contained in voters’ preferences over candidates.15 Moreover, when only two candidates run, it is not difﬁcult to prove that each candidate’s winning probability only depends on the group of voters for whom she is the ﬁrst-ranked candidate. This fact allows us to associate to each group a weight that is equal to the probability of a candidate who is ranked ﬁrst by all the members of the group. Routine arguments apply to prove that this distribution of weights is additive, which leads to the random dictatorship result. In conclusion, the implications of (universal) candidate stability do not change in this more general framework. The results are essentially the same and the proofs proceed through similar (although more cumbersome) arguments. As the general framework does not provide new insights into the problem, we have preferred to highlight the less general one and to follow DJL’s original set-up.

References 1. Arrow, K.J.: Social choice and inividual values, 2nd edn. New York: Wiley 1963 2. Barber`a, S.: Majority and positional voting in probabilistic frameworks. Review of Economic Studies 46, 379–389 (1979) 3. Barber`a S., Sonnenschein, H.: Preference aggregation with randomized social orderings. Journal of Economic Theory 18, 244–254 (1978) 4. Besley, T., Coate, S.: An economic model of representative democracy. Quarterly Journal of Economics 112, 85–114 (1997) 5. Dutta B., Jackson, M.O., Le M. Breton,: Strategic candidacy and voting procedures. Econometrica 69, 1013–1038 (2001) 6. Dutta, B., Jackson, M.O., Le Breton, M.: Voting by successive elimination and strategic candidacy. Journal of Economic Theory 103, 190–218 (2002) 7. Ehlers, L., Weymark, J.: Candidate stability and non-binary choice. Economic Theory 22, 233–243 (2003) 8. Eraslan, H., McLennan, A.: Strategic candidacy and multivalued voting procedures. Journal of Economic Theory 117, 29–54 (2004) 9. Gibbard, A.: Manipulation of schemes that mix voting with chance. Econometrica 45, 665–681 (1977) 10. McLennan, A.: Randomized preference aggregation: additivity of power and strategy proofness. Journal of Economic Theory 22, 1–11 (1980) 11. Nandeibam, S.: Coalitional power structure in stochastic social choice functions with an unrestricted preference domain. Journal of Economic Theory 68, 212–233 (1996) 12. Osborne, M.J., Slivinski, A.: A model of political competition with citizen candidates. Quarterly Journal of Economics 111, 65–96 (1996) 13. Pattanaik, P.K., Peleg, B.: Distribution of the veto power under stochastic social choice rules. Econometrica 54, 909–921 (1986) ´ 14. Rodr´ıguez-Alvarez, C.: Candidate stability and voting correspondences. Discussion Paper #492.01 Dept. d’Economia i Hist`oria Econ´omica, Universitat Aut`onoma de Barcelona (2001)

15 Preferences over lotteries for which only two candidates receive positive probability are completely determined from the preferences over these two candidates. If candidate a is preferred to candidate b, for every two lotteries in L{a,b} , the lottery assigning the highest probability to the candidate a is preferred to the remaining one.

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