  In this paper, we study the following problem. Consider a setting where a
proposal is offered to the vertices of a given network $G$, and the vertices
must conduct a vote and decide whether to accept the proposal or reject it.
Each vertex $v$ has its own valuation of the proposal; we say that $v$ is
``happy'' if its valuation is positive (i.e., it expects to gain from adopting
the proposal) and ``sad'' if its valuation is negative. However, vertices do
not base their vote merely on their own valuation. Rather, a vertex $v$ is a
\emph{proponent} of the proposal if the majority of its neighbors are happy
with it and an \emph{opponent} in the opposite case. At the end of the vote,
the network collectively accepts the proposal whenever the majority of its
vertices are proponents. We study this problem for regular graphs with loops.
Specifically, we consider the class $\mathcal{G}_{n|d|h}$ of $d$-regular graphs
of odd order $n$ with all $n$ loops and $h$ happy vertices. We are interested
in establishing necessary and sufficient conditions for the class
$\mathcal{G}_{n|d|h}$ to contain a labeled graph accepting the proposal, as
well as conditions to contain a graph rejecting the proposal. We also discuss
connections to the existing literature, including that on majority domination,
and investigate the properties of the obtained conditions.
