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Article Excerpt
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.
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. (3) 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. (4)
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 stability 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 McLennan [8], and Rodriguez-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 Rodriguez-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. (5) 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.
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[union]V. Let [2.sup.C]
{[empty set]} denote the set of all non-empty finite subsets of C. We call A [member of] [2.sup.C]
{[empty set]} an agenda. (6) We focus on the case in which there is no overlap between the sets of voters and candidates, C [union] V = {[empty set]}. 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 [union] {[empty set]}, where the empty set refers to no-candidate being elected. Let P denote the set of all preferences. Each individual i [member of] N is equipped with a preference [P.sub.i] [member of] P. Hence, individuals are never indifferent between two candidates. A utility function is a mapping [u.sub.i] : C [right arrow] R. A utility function [u.sub.i] represents the preference [P.sub.i] [member of] P if for each a, b [member of] C, [u.sub.i] (a) > [u.sub.i](b) if and only if a [P.sub.i] b.
For each i [member of] N, let [P.sup.i] denote i's domain of admissible preferences. For each i [member of] N, each C [member of] [2.sup.C]
{[empty set]}, and each [P.sub.i] [member of] P, top (C, [P.sub.i]) refers to the candidate in C that is ranked first by [P.sub.i]. We assume that for each a [member of] C, each i [member of] N, and each [P.sub.i] [member of] [P.sup.i], a [P.sub.i] {[empty set]}. Voters' preferences over candidates are unrestricted, but each candidate is her own first-ranked candidate. Hence, for each a [member of] C and each [P.sub.a] [member of] [P.sup.a], a = top (C, [P.sub.a]).
Let [P.sup.V] = [x.sub.i[member of]V][P.sup.i]. Let P [member of] [P.sup.V] denote a preference profile. For each A [member of] [2.sup.C]
{[empty set]} and each P [member of] [P.sup.V], P [|.sub.A] refers to the restriction of P to A. Finally, for each I [??] V, and each P [member of] [P.sup.V], [P.sub.I] [member of] [x.sub.i[member of]I][P.sup.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 [member of] [2.sup.C]
{[empty set]}, let:
[L.sub.C] [equivalent to] {[lambda] [member of] L such that for each b [member of] C
C, [lambda](b) = 0}.
That is, [L.sub.C] contains all lotteries defined on agenda C.
Candidates are equipped with preferences over L [union] {[empty set]}. These preferences are complete, reflexive, and transitive binary relations that are consistent with the postulates of Expected Utility Theory. Additionally, candidates always prefer every [lambda] [member of] L to the empty set. Hence, for each pair [lambda], [lambda]' [member of] L, a candidate a [member of] C equipped with preferences over candidates [P.sub.a] [member of] [P.sup.a] and utility function [u.sub.a] representing [P.sub.a], prefers [lambda] to [lambda]', if and only if
[summation over (b[member of]C)][lambda](b)[u.sub.a](b) > [summation over (b'[member of]C)][lambda]'(b')[u.sub.a](b').
Having defined the strict component of candidates' preferences over lotteries, the weak component is defined in the usual way. For each pair [lambda], [lambda]' [member of] L, a candidate a is indifferent between them if neither she prefers [lambda] to [lambda]' nor she prefers [lambda]' to [lambda].
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 : [2.sup.C]
{[empty set]}...
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