A category theoretic approach to approximate game theory

This paper uses category theory to guide the development of an entirely new approach to approximate game theory. Game theory is the study of how different agents within a multi-agent system take decisions. At its core, game theory asks what an optimal decision is in a given scenario. Thus approximate game theory asks what is an approximately optimal decision in a given scenario. This is important in practice as—just like in much of computing—exact answers maybe too difficult (oreven impossible) to compute given inherent uncertainty in input. We consider first Selection Functions which are a simple model of compositional game theory. We develop i) a simple yet robust model of approximate equilibria; ii) the algebraic properties of approximation with respect to selection functions; and iii) relate approximation to the compositional structure of selection functions. We then repeat this process for Open Games — a more advanced model of game theory featuring the key operation of sequential composition of games. Finally, we use approximation to develop new metrics on game theory and show these yield core theorems in what one might term”Metric Game Theory”.