A Fixed Point Convergence Theorem in Euclidean Spaces and its Application to Non-Bayesian Learning in Social Networks

Abstract

I characterize the fixed points of continuous and (neighbor-)constricting functions f:ℝ^{v}→ℝ^{v} and show that each recursively defined sequence x^{k 1}=f(x^{k}), k=0,1,2,... converges to a fixed point of f. The results are applied to generalize the existing results on convergence of beliefs of non-Bayesian agents in social networks of DeGroot (1974), DeMarzo, Vayanos and Zwiebel (2003), and Golub and Jackson (2010).