Mapping Power Relations: A Geometric Framework for Game-Theoretic Analysis

TL;DR

This paper presents a geometric framework for analyzing power relations in games, providing a utility-independent, structural approach that unifies various game-theoretic concepts and reveals underlying similarities across different strategic models.

Contribution

It introduces a novel geometric model that captures power relations without relying on utility functions, enabling a unified analysis of diverse game types.

Findings

Reveals structural similarities across canonical and economic games

Provides quantitative measures of hierarchy and reciprocity

Bridges cooperative and non-cooperative game theory

Abstract

This paper introduces a geometric framework for analyzing power relations in games, independent of their strategic form. We define a canonical preference space where each player's relational stance is a normalized vector. This model eliminates the arbitrariness of selecting utility functions, a limitation of recent approaches. We show how classical concepts-bargaining power, dependence, reciprocity-are recovered and generalized within this space. The analysis proceeds in two steps: projecting a game's payoffs and outcomes onto the space, and then reducing the resulting landscape using key metrics. These include a Center of Mass (CoM) and structural indices for Hierarchy (H) and Reciprocity (R). Applications to canonical games (Prisoner's Dilemma, Battle of the Sexes) and economic models (Cournot duopoly) demonstrate that the framework reveals underlying structural similarities across different strategic settings and provides a quantitative characterization of relational dynamics. It thus bridges cooperative and non-cooperative game theory by conceptualizing power as a structural property of the mapping from preferences to equilibria.