Heterogeneity as a Natural Source of Randomness

We propose a new framework for studying the evolution of heterogeneous beliefs in a dynamic feedback setting. Beliefs distributions are defined on a beliefs space representing a continuum of possible strategies agents can choose from. Agents base their choices on past performances, re-evaluating strategies as new information becomes available. The distribution of beliefs among agents is updated using a continuous choice model. This leads to price dynamics in which the beliefs distribution evolves together with realized prices. By considering individual choices as random variables, which is natural in a random utility framework, heterogeneity can be seen to act as a 'natural source of randomness'. Allowing for modeling the dynamics explicitly, our framework gives rise to a random dynamical system (RDS), the stochastic properties of which are directly related to the time varying beliefs distribution. We consider some asset pricing examples and discuss several conditions (dependence among agents, unequal market impact) under which the randomness persists even as the number of agents tends to infinity.