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

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 ®. 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.