🤖 AI Summary
This study addresses the limitation of Multiplicative Kantian equilibria in failing to cover the Pareto frontier due to constraints imposed by the parameterization of the strategy space. To overcome this, the authors propose a coordinate translation—specifically, imposing lower bounds on actions—to reconstruct the strategy space. Through geometric analysis leveraging the common tangent property of players’ indifference curves, they constructively demonstrate that for any interior Pareto-efficient outcome, there exists a translation under which it becomes a Multiplicative Kantian equilibrium. This result establishes, for the first time, that such equilibria can span the entire set of interior points on the Pareto frontier, thereby decoupling efficiency from fairness: any normative criterion can be implemented without compromising Pareto optimality. This significantly broadens the theoretical applicability and explanatory power of Kantian equilibria.
📝 Abstract
Multiplicative Kantian equilibrium explains cooperative behavior in social dilemmas without abandoning methodological individualism. However, its outcomes depend critically on the parametrization of the strategy space - the property of strategic non-equivalence. We investigate what fraction of the Pareto frontier can be attained by varying the strategy space. We show that the set of achievable Kantian equilibria is the entire Pareto frontier: for any interior Pareto-efficient point there exists a shift of coordinates - imposing lower bounds on actions - that makes it a Multiplicative Kantian equilibrium. The proof is constructive and relies on a intuitive geometric property: moving the origin to a point on the common tangent to players' indifference curves. This result separates the problem of efficiency from the problem of fairness, allowing any normative criterion to be implemented without loss of Pareto optimality.