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

Takashi Izumo

TL;DR

This paper introduces Coarse-Grained Games (CGGs) to model bounded perception in strategic interactions by partitioning payoff spaces with coarse sets and applying entropy-maximizing preprocessing. It formalizes a CGG as $(N,S,u,\mathfrak{G},\Phi,\Psi)$, constructs coarse-grained payoff matrices, and distinguishes objective versus subjective payoffs to analyze how Nash equilibria are preserved or altered under coarse-graining. The authors prove existence of mixed-strategy equilibria under entropy-maximizing preprocessing, show pure-equilibrium preservation, and illustrate how mixed equilibria can shift, with a detailed Prisoner’s Dilemma example. They extend the analysis to infinitely repeated games, where subjective discount factors can differ across agents, potentially obstructing cooperation, and apply the framework to social-science scenarios such as minor model changes and adverse selection in lemon markets. The work highlights how perceptual resolution shapes equilibrium selection and market outcomes, offering a structural approach to bounded rationality with concrete formal tools and clear applications, while noting limitations and directions for future research.

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.

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

TL;DR

This paper introduces Coarse-Grained Games (CGGs) to model bounded perception in strategic interactions by partitioning payoff spaces with coarse sets and applying entropy-maximizing preprocessing. It formalizes a CGG as , constructs coarse-grained payoff matrices, and distinguishes objective versus subjective payoffs to analyze how Nash equilibria are preserved or altered under coarse-graining. The authors prove existence of mixed-strategy equilibria under entropy-maximizing preprocessing, show pure-equilibrium preservation, and illustrate how mixed equilibria can shift, with a detailed Prisoner’s Dilemma example. They extend the analysis to infinitely repeated games, where subjective discount factors can differ across agents, potentially obstructing cooperation, and apply the framework to social-science scenarios such as minor model changes and adverse selection in lemon markets. The work highlights how perceptual resolution shapes equilibrium selection and market outcomes, offering a structural approach to bounded rationality with concrete formal tools and clear applications, while noting limitations and directions for future research.

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.

Paper Structure

This paper contains 73 sections, 11 theorems, 144 equations, 5 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Comparison to Previous Research
  4. Structure of This Paper
  5. Coarse-Grained Game
  6. Coarse Set
  7. Coarse-Grained Payoff Matrix
  8. Entropy-Maximizing Preprocessing
  9. Cognitive Resolution
  10. Basic Idea
  11. Coarse-Grained Game
  12. Definition
  13. Example: Coarse-Grained Prisoners Dilemma
  14. The Standard Payoff Matrix $M^{\text{base}}$ (Standard Prisoner's Dilemma)
  15. Coarse-Grained Payoff Matrix for Player 1 ($M^{\mathfrak{G}_1}$)
  16. ...and 58 more sections

Key Result

Proposition 1

Given a collection of coarse-grained partitions $\mathcal{C}(\mathbb{R},\leq) = \{(\mathfrak{G}_\theta,\preccurlyeq_\theta) \mid {\theta \in \Theta} \}$, the finest-resolution payoff matrix $M^{\mathfrak{G}^\text{finest}}$ is defined as follows: The coarse-grained partition of $(\mathbb{R},\leq)$ associated with $M^{\mathfrak{G}^\text{finest}}$ is: The finest-resolution payoff matrix $M^{\mathfr

Theorems & Definitions (47)

  • Definition 1: Element-wise Ordering
  • Remark 1
  • Definition 2: Coarse-Grained Partition
  • Remark 2
  • Definition 3: Collection of Coarse-Grained Partitions
  • Example 1
  • Remark 3
  • Definition 4: Coarse-Grained Payoff Matrix
  • Example 2
  • Definition 5: Entropy-Maximizing Preprocessing
  • ...and 37 more