What majority decisions are possible with possible abstaining

TL;DR

This paper explores the possible majority decisions in a voting scenario where voters can abstain, extending previous work by providing a complete characterization for these cases.

Contribution

It introduces a framework allowing voters to abstain and characterizes the resulting majority decision functions across four different cases.

Findings

Characterization of majority decision functions with abstentions

Extension of previous models to include abstaining voters

Analysis of four distinct abstention scenarios

Abstract

Suppose we are given a family of choice functions on pairs from a given finite set. The set is considered as a set of alternatives (say candidates for an office) and the functions as potential "voters". The question is, what choice functions agree, on every pair, with the majority of some finite subfamily of the voters? For the problem as stated, a complete characterization was given in \citet{shelah2009mdp}, but here we allow each voter to abstain. There are four cases.