  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.
