Hardness of Learning in Rich Environments and Some Consequences for Financial Markets

This paper examines the computational feasibility of the standard model of learning in economic theory. It is shown that the information update technique at the heart of this model is impossible to compute in all but the simplest scenarios. Specifically, using tools from theoretical machine learning, the paper first demonstrates that there is no polynomial implementation of the model unless the independence structure of variables in the data is publicly known. Next, it is shown that there cannot exist a polynomial algorithm to infer the independence structure; consequently, the overall learning problem does not have a polynomial implementation. Using the learning model when it is computationally infeasible carries risks, and some of these are explored in the latter part of the paper in the context of financial markets. Especially in rich, high-frequency environments, it implies discarding a lot of useful information, and this can lead to paradoxical outcomes in interactive game-theoretic situations. This is illustrated in a trading example where market prices can never reflect an informed trader's information, no matter how many rounds of trade. The paper provides new theoretical motivation for the use of bounded rationality models in the study of financial asset pricing-the bound on rationality arising from the computational hardness in learning.