🤖 AI Summary
Game-theoretic power analysis traditionally relies on specific strategic forms and suffers from arbitrariness in utility function specification, undermining objectivity and cross-game comparability.
Method: This paper proposes a game-form-agnostic geometric framework that maps players’ preferences into a standardized vector space. It eliminates utility-based subjectivity via canonical preference-space modeling and vectorized representation of relational postures. Power structure is then characterized through vector projection, centroid computation, and two structural metrics—hierarchicality (H) and reciprocity (R)—enabling dimensionality-reduced, quantitative, and comparative analysis.
Results: Empirical application to the Prisoner’s Dilemma, Battle of the Sexes, and Cournot model demonstrates consistent cross-context representation of bargaining power, dependence, and reciprocity. The framework achieves, for the first time, geometric, quantitatively comparable, and dynamically tractable analysis of power structures in strategic interactions.
📝 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.