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A categorical representation of games

Fernando Tohmé, Ignacio Viglizzo

Abstract

Strategic games admit a multi-graph representation, in which two kinds of relations, accessibility, and preferences, are used to describe how the players compare the possible outcomes. A category of games with a fixed set of players $\mathbf{Gam}_I$ is built from this representation, and a more general category $\mathbf{Gam}$ is defined with games having different sets of players, both being complete and cocomplete. The notion of Nash equilibrium can be generalized in this context. We then introduce two subcategories of $\mathbf{Gam}$, $\mathbf{NE}$ and $\mathbf{Gam}^{NE}$ in which the morphisms are equilibria-preserving. We illustrate the expressivity and usefulness of this framework with some examples.

A categorical representation of games

Abstract

Strategic games admit a multi-graph representation, in which two kinds of relations, accessibility, and preferences, are used to describe how the players compare the possible outcomes. A category of games with a fixed set of players is built from this representation, and a more general category is defined with games having different sets of players, both being complete and cocomplete. The notion of Nash equilibrium can be generalized in this context. We then introduce two subcategories of , and in which the morphisms are equilibria-preserving. We illustrate the expressivity and usefulness of this framework with some examples.
Paper Structure (7 sections, 15 theorems, 20 equations, 4 figures)

This paper contains 7 sections, 15 theorems, 20 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Multi-Graph Representation of Strategic Games
  3. Categories of Multi-Graph Representation of Games
  4. Games with a fixed set of players
  5. Games with different sets of players
  6. Subgames and equilibria
  7. Categories with equilibria-preserving morphisms

Key Result

Theorem 3.3

Gam$_I$ is a complete and cocomplete category.

Figures (4)

  • Figure 1: Multi-graph representing the Prisoner's Dilemma in Example \ref{['PD']}. Blue and red lines correspond to players 1 and 2, respectively. Full (undirected) lines correspond to accessibility relations while dashed ones (directed) represent preferences.
  • Figure 2: Multi-graph representation of $G_1$ from Example \ref{['off']}. Blue and red lines correspond to players 1 and 2, respectively. Full (undirected) lines correspond to accessibility relations while dashed ones (directed) represent preferences.
  • Figure 3: Multi-graph representing the game $G_1$ with set of outcomes $O_1=\{o,DL,TR\}$ from Example \ref{['off']}, in which the accessibility relation is not transitive. Blue and red lines correspond to players 1 and 2, respectively. Full lines correspond to accessibility relations and dashed ones to preferences.
  • Figure 4: Multi-graph of $G_{BoS}$. Blue and red lines correspond to players 1 and 2, respectively. Full lines correspond to accessibility relations and dashed ones to preferences.

Theorems & Definitions (27)

  • Theorem 3.3
  • Proposition 3.4
  • proof
  • Theorem 3.5
  • Proposition 3.6
  • proof
  • Proposition 3.7
  • proof
  • Proposition 3.11
  • proof
  • ...and 17 more