The question of how people form their opinion has fascinated economists and
sociologists for quite some time. In many of the models, a group of people in a
social network, each holding a numerical opinion, arrive at a shared opinion
through repeated averaging with their neighbors in the network. Motivated by
the observation that consensus is rarely reached in real opinion dynamics, we
study a related sociological model in which individuals' intrinsic beliefs
counterbalance the averaging process and yield a diversity of opinions.
By interpreting the repeated averaging as best-response dynamics in an
underlying game with natural payoffs, and the limit of the process as an
equilibrium, we are able to study the cost of disagreement in these models
relative to a social optimum. We provide a tight bound on the cost at
equilibrium relative to the optimum; our analysis draws a connection between
these agreement models and extremal problems that lead to generalized
eigenvalues. We also consider a natural network design problem in this setting:
which links can we add to the underlying network to reduce the cost of
disagreement at equilibrium?

通过模拟社会网络中人们的意见动态，研究了一种具有内生信念的模型，发现在这种情况下很少能达成共识，并提供了达成共识的最低成本与社会最优状态的紧密界限。同时，讨论了如何通过调整网络结构来减少达成共识的成本。