Settable Systems: An Extension of Pearl's Causal Model with Optimization, Equilibrium, and Learning

Judea Pearl's Causal Model is a rich framework that provides deep insight into the nature of causal relations. As yet, however, the Pearl Causal Model (PCM) has had a lesser impact on economics or econometrics than on other disciplines. This may be due in part to the fact that the PCM is not as well suited to analyzing structures that exhibit features of central interest to economists and econometricians: optimization, equilibrium, and learning. We offer the settable systems framework as an extension of the PCM that permits causal discourse in systems embodying optimization, equilibrium, and learning. Because these are common features of physical, natural, or social systems, our framework may prove generally useful for machine learning. Important features distinguishing the settable system framework from the PCM are its countable dimensionality and the use of partitioning and partition-specific response functions to accommodate the behavior of optimizing and interacting agents and to eliminate the requirement of a unique fixed point for the system. Refinements of the PCM include the settable systems treatment of attributes, the causal role of exogenous variables, and the dual role of variables as causes and responses. A series of closely related machine learning examples and examples from game theory and machine learning with feedback demonstrates some limitations of the PCM and motivates the distinguishing features of settable systems.