Majority of Majorities on Regular Graphs with Loops

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 proponent of the proposal if a majority of its neighbors are happy with it and an opponent in the opposite case. At the end of the vote, the network collectively accepts the proposal whenever a majority of its vertices are proponents. We study this problem on 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 and investigate some properties of the obtained conditions.