No Show Paradox and the Golden Number in Generalized Condorcet Voting Methods

Abstract

This paper extends the analysis of the so called no show paradox to the context of generalized $q$-Condorcet voting correspondences, in which the $q$-Condorcet winner is defined as the candidate who wins to any other by a qualified majority $q$ higher than half the number of voters. This paradox occurs when a group of voters is better off by not voting than by voting according to its preferences. We try to progress, for the case of general voting correspondences, in the resolution of an open problem proposed in Holzman (Discrete Appl Math 22:133–141, 1988/1989). He asked for the range of $q$ quota values for which all $q$-Condorcet voting rules are subjected to the paradox. This also means to extend a known general result of Moulin (J Econ Theory 45:53–64, 1988) stating that all conventional Condorcet voting rules ($q=\frac{1}{2}$) suffer the paradox. Interestingly, the well-known mathematical constant $\upvarphi$, called the golden number, appears in the main theorem of the paper. This theorem establishes that a result on voting correspondences of Jimeno et al. (Soc Choice Welf 33:343–359, 2009), similar to Moulin result, does not extend to $q$-Condorcet voting correspondences if $q$ is equal or higher than $\frac{1}{\varphi }$. More specifically, we find, for every $q$, a particular correspondence that suffers from the paradox if $q<\frac{1}{\varphi }$, but is free if $q\ge \frac{1}{\varphi }$.