Opinion Formation Games with Aggregation and Negative Influence

We study continuous opinion formation games with aggregation aspects. In many domains, expressed opinions of people are not only affected by local interaction and personal beliefs, but also by influences that stem from global properties of the opinions in the society. To capture the interplay of such global and local effects, we propose a model of opinion formation games with aggregation, where we concentrate on the average public opinion as a natural way to represent a global trend in the society. While the average alone does not have good strategic properties as an aggregation rule, we show that with a reasonable influence of the average public opinion, the good properties of opinion formation models are preserved. More formally, we prove that a unique equilibrium exists in average-oriented opinion formation games. Simultaneous best-response dynamics converge to within distance \(\varepsilon \) of equilibrium in \(O(n^2 \ln (n/\varepsilon ))\) rounds, even in a model with outdated information on the average public opinion. For the Price of Anarchy, we show a small bound of \(9/8 + o(1)\), almost matching the tight bound for games without aggregation. Moreover, some of the results apply to a general class of opinion formation games with negative influences, and we extend our results to the case where expressed opinions come from a restricted domain.