Bounds on the price of anarchy for a more general class of directed graphs in opinion formation games

The opinion forming process in a social network could be naturally modeled as an opinion influencing and updating dynamics. This already attracted researchers interest a while ago in mathematical sociology, and recently in theoretical computer science. In so-called opinion formation games, when underlying networks are directed, a bounded price of anarchy is only known for weighted Eulerian graphs, which may not be the most general class of directed graphs that give a bounded price of anarchy. Thus, we aim to bound the price of anarchy for games with directed graphs more general than weighted Eulerian graphs in this paper. We first bound the price of anarchy for a more general class of directed graphs with conditions intuitively meaning that each node does not influence the others more than she is influenced by herself and the others, where the bounds depend on such influence differences. This generalizes the previous results on directed graphs, and recovers and matches the previous bounds in some specific classes of (directed) Eulerian graphs. We then show that there exists an example that just slightly violates the conditions with an unbounded price of anarchy. We further propose more directions along this line of research.