Coarse-Grained Games: A Framework for Bounded Perception in Game Theory

Abstract

In everyday life, we frequently make coarse-grained judgments. When we say that Olivia and Noah excel in mathematics, we disregard the specific differences in their mathematical abilities. Similarly, when we claim that a particular automobile manufacturer produces high-quality cars, we overlook the minor variations among individual vehicles. These coarse-grained assessments are distinct from erroneous or deceptive judgments, such as those resulting from student cheating or false advertising by corporations. Despite the prevalence of such judgments, little attention has been given to their underlying mathematical structure. In this paper, we introduce the concept of coarse-graining into game theory, analyzing games where players may perceive different payoffs as identical while preserving the underlying order structure. We call it a Coarse-Grained Game (CGG). This framework allows us to examine the rational inference processes that arise when players equate distinct micro-level payoffs at a macro level, and to explore how Nash equilibria are preserved or altered as a result. Our key findings suggest that CGGs possess several desirable properties that make them suitable for modeling phenomena in the social sciences. This paper demonstrates two such applications: first, in cases of overly minor product updates, consumers may encounter an equilibrium selection problem, resulting in market behavior that is not driven by objective quality differences; second, the lemon market can be analyzed not only through objective information asymmetry but also through asymmetries in perceptual resolution or recognition ability.