Majority Digraphs

Abstract

A majority digraph is a finite simple digraph $G=(V,\to)$ such that there exist finite sets $A_v$ for the vertices $v\in V$ with the following property: $u\to v$ if and only if "more than half of the $A_u$ are $A_v$". That is, $u\to v$ if and only if $ |A_u \cap A_v | > \frac{1}{2} \cdot |A_u|$. We characterize the majority digraphs as the digraphs with the property that every directed cycle has a reversal. If we change $\frac{1}{2}$ to any real number $\alpha\in (0,1)$, we obtain the same class of digraphs. We apply the characterization result to obtain a result on the logic of assertions "most $X$ are $Y$" and the standard connectives of propositional logic.