A Behavioral Perspective on Price Convergence via Perturbed Metric Spaces with An Extended Contraction

In this study, we examine how investors update their price forecasts over time within a "perturbated metric space," which incorporates behavioral influences and market friction. Classical metric structures are inadequate when the measured distance changes with perceived deviations. Therefore, a new structure is proposed in which the measured distance is modified by perceived deviations. In this context, the existence of a fixed point is guaranteed through an extended contraction inequality, and the convergence behavior of the model is analyzed using different examples. Simulations established under different linear and nonlinear update functions demonstrate that the model can reflect both slow and fast market behaviors that reach equilibrium. The proposed approach mathematically demonstrates that investors can reach a common price expectation in the long run, even with heterogeneous psychological responses.