Bounds on the Price of Anarchy for a More General Class of Directed Graphs in Opinion Formation Games

TL;DR

This paper extends bounds on the price of anarchy in opinion formation games to a broader class of directed graphs, showing that certain influence conditions ensure bounded inefficiency, while slight violations lead to unbounded outcomes.

Contribution

It introduces new influence-based conditions for directed graphs that guarantee bounded price of anarchy in opinion formation games.

Findings

Bounded price of anarchy for a wider class of directed graphs.

Existence of graphs with unbounded price of anarchy when conditions are slightly violated.

Dependence of bounds on influence ratio differences.

Abstract

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.