Simple Majorities with Voice but No Vote
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Simple Majorities with Voice but No Vote José Carlos R. Alcantud1
© Springer Nature B.V. 2020
Abstract Oligarchic majority rules bring the voice but no vote principle into effect. We prove characterizations of the oligarchic majority rules for both fixed and unrestricted societies and a binary agenda. This is a general class of rules that includes the simple majority rule as well as dictatorships. Suitable sets of axioms identify a subsociety whose members have voice but no vote, and valid votes are aggregated by the majority rule. Keywords Majority rule · Voting · Oligarchy · Dictatorship
1 Introduction This paper is about two-candidate elections by relative majorities, an analysis whose first axiomatic studies date back to May (1952). In his formulation of the majority rule there are n agents who vote on an issue (or candidate) which can either be accepted, rejected, or left unresolved. And the agents can either vote for acceptance, rejection, or unresolvedness of the issue. The issue is accepted (resp., rejected) when acceptance (resp., rejection) receives higher support than rejection (resp., acceptance); otherwise the issue is not resolved. Alcantud (2019), Aşan and Sanver (2002), Fishburn (1983), Llamazares (2006), May (1952), Miroiu (2004), Quesada (2010a, b), Woeginger (2003), Woeginger (2005) among others also axiomatized the majority rule, be it with a fixed or a variable population. Refinements of the simple majority appear e.g., in García-Lapresta and Llamazares (2010) or Quesada (2013). The first of these two refinements considers preference intensities in order to avoid the criticism that an alternative can beat the others with very small support, whereas the second introduces a tie-breaking rule with the help of an exogenous ranking of the agents. * José Carlos R. Alcantud [email protected] http://diarium.usal.es/jcr 1
BORDA Research Unit and Multidisciplinary Institute of Enterprise (IME), University of Salamanca, 37007 Salamanca, Spain
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Axiomatizations are useful for specifying the scope of applicability of a model. Suppose for example that a voting rule F produces the following aggregate outcomes when the electorate is formed by 3 agents, and −1, 0 and 1 respectively mean rejection, unresolvedness and acceptance:
F(1, 0, −1) = 1, F(1, −1, 0) = 0, F(0, 0, 1) = 0, F(0, 0, −1) = 0. Since the simple majority is anonymous, a comparison of the first and second aggregations of the opinions discards that F is the simple majority rule. And the third and fourth aggregation results happen to contradict another axiom in May’s characterization, namely, positive responsiveness. However these observations of the properties of a voting rule are not fully incompatible with the spirit of relative majoritarianism. Although the third or the fourth aggregations hint that the opinion of the third agent seems to be to some extent superfluous, they are still compatible with the existence of a 2 threshold of excess support for acceptance. And in fact these four particular agg
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