Bounds on the Price of Anarchy for a More General Class of Directed   Graphs in Opinion Formation Games
  In opinion formation games with directed graphs, a bounded price of anarchy is only known for weighted Eulerian graphs. Thus, we 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, where the bounds depend on such difference (in a ratio). We also show that there exists an example just slightly violating the conditions with an unbounded price of anarchy. 
Po-An Chen, Yi-Le Chen, Chi-Jen Lu
cs.GT cs.SI