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Group Decision and Negotiation

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05-01-2016

Probabilities of Consistent Election Outcomes with Majorities Based on Difference in Support

Authors: Mostapha Diss, Patrizia Pérez-Asurmendi

Published in: Group Decision and Negotiation | Issue 5/2016

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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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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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Title: Probabilities of Consistent Election Outcomes with Majorities Based on Difference in Support
Authors: Mostapha Diss
Patrizia Pérez-Asurmendi
Publication date: 05-01-2016
Publisher: Springer Netherlands
Published in: Group Decision and Negotiation / Issue 5/2016
Print ISSN: 0926-2644
Electronic ISSN: 1572-9907
DOI: https://doi.org/10.1007/s10726-015-9467-1

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