Abstract
The main criticism to the aggregation of individual preferences under majority rules refers to the possibility of reaching inconsistent collective decisions from the election process. In these cases, the collective preference includes cycles and even could prevent the election of any alternative as the collective choice. The likelihood of consistent outcomes under a class of majority rules constitutes the aim of this paper. Specifically, we focus on majority rules that require certain consensus in individual preferences to declare an alternative as the winner. Under majorities based on difference of votes, the requirement asks to the winner alternative to obtain a difference in votes with respect to the loser alternative taking into account that individuals are endowed with weak preference orderings. Same requirement is asked to the restriction of these rules to individual linear preferences.
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Notes
To calculate the probabilities presented here, \(m\) takes the following values: 3, 4, 5, 10, 100, 1000 and 100,000.
Given \(\tau > 0.5\), supermajority rules are defined by \(x_i P^{\tau } x_j \Leftrightarrow \frac{\# \{p \mid r_{ij}^p = 1\}}{m} \ge \tau \).
The free software to calculate the integer points under the Parameterized Barvinok’s algorithm can be found in http://freecode.com/projects/barvinok.
Such simplification is done with Maple software.
Notice that in this case, the four original validity domains from Proposition 3 reduce to two, given that, for the three last ones, the probabilities are the same when considering large electorates.
Notice that, in Lepelley and Martin (2001), the probability 0.042 is obtained as an estimate value through computer simulations.
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Acknowledgments
The authors are grateful to the editor, two anonymous referees, Ahmad Fliti, José Luis García-Lapresta, Bonifacio Llamazares, Ana Pérez Espartero, and Rachid Seghir for their valuable suggestions and comments. This work is partially supported by the Spanish Ministry of Economy and Competitiveness (Projects ECO2012-32178 and ECO2012-34202).
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Diss, M., Pérez-Asurmendi, P. Consistent collective decisions under majorities based on difference of votes. Theory Decis 80, 473–494 (2016). https://doi.org/10.1007/s11238-015-9501-4
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DOI: https://doi.org/10.1007/s11238-015-9501-4