Abstract
This paper identifies necessary and sufficient single-profile conditions for a consistent decision under a supermajority rule. It is demonstrated that a preference profile generates a transitive supermajority rule relation if and only if it is not sufficiently balanced. These conditions link transitivity of a supermajority rule to the cardinal outcomes of plurality and anti-plurality elections.
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Notes
For the case of simple majority rule in particular, existence of a maximal element is guaranteed when preferences are intermediate (Grandmont 1978; Rothstein 1990) and when they satisfy the single-crossing property (Gans and Smart 1996). Barberà and Moreno (2011) suggest the top-monotonicity domain restriction, which includes the previous ones, and extend the result of Austen-Smith and Banks (2000) to these types of preferences.
Consider for example a university department which has an open tenure-track position and an open post-doc position and three candidates for both these positions.
For every supermajority rule one may find preferences profiles such that the resulting supermajority rule relation is intransitive (see Greenberg 1979). Caplin and Nalebuff (1988) show, though, that under certain non-extreme domain restrictions, a \(64\%-\)majority rule never results to electoral cycles.
Moreover, Xefteris (2013) provides single-profile conditions for the transitivity of the Pareto extension order.
In anti-plurality elections each voter votes once for her most disliked alternative and the least voted alternative wins.
Notice that \(\mathcal {P}\subset \mathcal {R}\subset \mathcal {C}.\)
We say that a preference relation satisfies transitivity if it satisfies the following four properties (see Chapter [2] in Arrow 1963):
$$\begin{aligned} xPy,yPz\rightarrow xPz&(PP \mathrm{property}) \\ xPy,yIz\rightarrow xPz&(PI \mathrm{property}) \\ xIy,yPz\rightarrow xPz&(IP \mathrm{property}) \\ xIy,yIz\rightarrow xIz&(II \mathrm{property}) \end{aligned}$$Arrow (1963) notion of transitivity is indeed demanding. For a group of weaker transitivity notions one is referred to Cato (2013).
This is without loss of generality since any supermajority rule can be modeled by an \(a^{*}\) that satisfies these constraints. We explain the necessity of this technical assumption after the presentation of the formal analysis.
To see this, suppose that \(N=\{1,2,3\}\) and assume that \(P_{1}=P_{2}\). Moreover, assume that agents \(1\) and \(3\) are mutually exclusive. Then \(N_{0}\) consists of either \(1\) or \(2\). However, the preferences of the agent in \(N_{0}\) remain the same.
Notice that when \(n\) is an even number there is a possibility that the reduced population set is empty \((N_{0}^{\prime }=N)\). In this case (a) condition (ii) is trivially violated (\(a^{*}>0\) implies \(a>-1/2\) and hence \(\frac{n_{0}-1}{2}-a<0\)) and (b) any supermajority rule gives a social ordering with indifferences only, since for every \(x\) and \(y\) exactly half of the population prefers \(x\) over \(y\) (transitivity is satisfied).
On contrary we would have \(\frac{n_{0}-1}{2}-a<0\). This inequality, together with the fact that every alternative is top-ranked and bottom-ranked for at least zero individuals violates condition (ii).
Notice that the second ranked alternative has to be different in each of the \(Q_{i}\)’s. Otherwise we would have a preference profile with a mutually exclusive pair. This implies that the top alternatives could be different in every linear order \(Q_{i}\) (case b) or that there is one alternative that is top (last) choice in two linear orders (case a).
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Acknowledgments
We would like to thank S. Barberà and B. Moreno for helpful discussions and suggestions about the topic. R. Gjorgjiev acknowledges the researchers Grant ECO/1186/2013 provided by Agència de Gestió d’Ajuts Universitaris i de Recerca. R. Gjorgjiev also acknowledges the financial support from Ministerio de Ciencia e Innovación through grant ECO2013-45395-R and Generalitat de Catalunya through Grant 2014-SGR-1360.
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Gjorgjiev, R., Xefteris, D. Transitive supermajority rule relations. Econ Theory Bull 3, 299–312 (2015). https://doi.org/10.1007/s40505-014-0060-6
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DOI: https://doi.org/10.1007/s40505-014-0060-6