Abstract
Computer simulations are used to evaluate the likelihood of consistent outcomes under the class of majorities based on difference in support. These majorities require certain consensus in collective preferences to declare an alternative as the winner. More precisely, individuals show preference intensities in the unit interval among each pair of alternatives and it is required that the winner alternative obtains a difference in the sum of the intensities with respect to the loser alternative. This difference is a real number located between 0 and the total number of voters. We introduce the values of the required threshold for which majorities based on difference in support lead to transitive and triple-acyclic collective decisions with a probability of 1. Our results improve the previous theoretical ones since they require softer thresholds to reach consistent collective decisions.
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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.
Assuming a proportion of consistent outcomes P on the population of a 50 %, the proportion p in a random sample of size \(n \ge 30\) for a confidence level of 99 %, diverges from the one of the population in an error of less than \(\epsilon \):
$$\begin{aligned} Prob(|P-p|\le \epsilon ) \ge 0.99. \end{aligned}$$Taken into account that the sample proportion p is distributed as \(N\left( P, \sqrt{P(1 - P)/n} \right) \), the sampling error \(\epsilon \) is as follows:
$$\begin{aligned} \epsilon = z_{\alpha /2}\sqrt{P(1 - P)/n}. \end{aligned}$$In our case, \(n=\)100,000 and the corresponding percentile of the normal distribution for a confidence level of 99 % is \(z_{\alpha /2}= 2.57\). Thus, \(\epsilon \le 0.00407\)
Notice that \(m/3 = \lfloor (2\times 100)/3\rfloor /2\) when m is multiple of three.
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Acknowledgments
The authors are grateful to two anonymous reviewers, Ahmad Fliti, José Luis García-Lapresta, Bonifacio Llamazares, and Ana Pérez Espartero for their valuable suggestions and comments. We also thank participants at the 2014 Social Choice and Welfare Conference (Boston) and the 2015 European Conference on Operational Research (Glasgow) for helpful comments. This work is partially supported by the Spanish Ministry of Economy and Competitiveness (Projects ECO2012-32178 and ECO2012-34202). Financial support by the National Agency for Research (ANR)—research program “Dynamic Matching and Interactions: Theory and Experiments” (DynaMITE)—is gratefully acknowledged.
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Appendix
Appendix
See Tables 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28 and 29.
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Diss, M., Pérez-Asurmendi, P. Probabilities of Consistent Election Outcomes with Majorities Based on Difference in Support. Group Decis Negot 25, 967–994 (2016). https://doi.org/10.1007/s10726-015-9467-1
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DOI: https://doi.org/10.1007/s10726-015-9467-1