Majority decisions when abstention is possible

by Larson and Matteo and Shelah. [LrMaSh:965]
Discrete Math, 2012
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 [Sh:816], but here we allow each voter to abstain. There are four cases.

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