Universal Decision Models

Abstract

Humans are universal decision makers: we reason causally to understand the world; we act competitively to gain advantage in commerce, games, and war; and we are able to learn to make better decisions through trial and error. Whilst these individual modalities of decision making have been studied for decades in various subfields of AI and ML, there has been commensurately less effort in developing formalisms that unify these various modalities into common framework. In this paper, we propose Universal Decision Model (UDM), a mathematical formalism based on category theory, to address this challenge. Decision objects in a UDM correspond to instances of decision tasks, ranging from causal models and dynamical systems such as Markov decision processes and predictive state representations, to network multiplayer games and Witsenhausen's intrinsic models, which generalizes all these previous formalisms. A UDM is a category of objects, which include decision objects, observation objects, and solution objects. Bisimulation morphisms map between decision objects that capture structure-preserving abstractions. We formulate universal properties of UDMs, including information integration, decision solvability, and hierarchical abstraction. Information integration consolidates data from heterogeneous sources by forming products or limits in the UDM category. Abstraction simulates complex decision processes by simpler processes through bisimulation morphisms by forming quotients, co-products and co-limits in the UDM category. Finally, solvability of a UDM decision object is defined by a fixed point equation, and it corresponds to an isotonic order-preserving morphism across the topology induced by UDM objects. We describe universal functorial representations of UDMs, and propose an algorithm for computing the minimal object in a UDM using algebraic topology. We sketch out an application of UDMs to causal inference in network economics, using a complex multiplayer producer-consumer two-sided marketplace.