Strategy Rescaling and the Stability of Kantian Optimization

Abstract: This study investigates the properties and stability of the Multiplicative Kantian Equilibrium (MKE) in symmetric games. We first demonstrate that MKE lacks strategic equivalence: the Kantian best-response function is not invariant under monotonic strategy rescaling. This strategic non-equivalence implies that the choice of measurement scale - a subjective interpretation of the game - materially impacts equilibrium outcomes. Exploiting this non-equivalence, in a game where players may be Kantian or Nasher, we propose an efficient strategy rescaling that allows Kantians to neutralize the free-rider advantage of Nashers, while preserving Pareto-efficient outcomes among themselves. In a dynamic framework, we show that the subgame-perfect Nash equilibrium with endogenous choice of optimization type leads all players to prefer Kantian optimization over Nash optimization. In an evolutionary setup, we show that Kantian optimization is an evolutionarily stable strategy (ESS). Our results suggest that the inherent strategic non-equivalence of Kantian optimization provides a robust pathway to stable cooperation.

• The paper demonstrates that strategy rescaling affects Kantian best-response functions, altering equilibrium outcomes compared to Nash invariance.
• It establishes an efficient rescaling technique that neutralizes Nash free-riding, ensuring Pareto efficiency in mixed strategy interactions.
• The study confirms that, under optimal rescaling, Kantian optimization is evolutionarily stable and promotes robust cooperative norms.

Strategy Rescaling and Evolutionary Stability in Kantian Optimization

Introduction

This paper analyzes the technical and evolutionary properties of the Multiplicative Kantian Equilibrium (MKE), extending prior work in Kantian game theory by systematically investigating the effect of strategy rescaling. The authors prove that, contrary to Nash equilibrium, the Kantian best-response function is not invariant under monotonic transformations of the strategy space. This lack of strategic equivalence fundamentally alters equilibrium selection, has implications for cooperative stability, and enables design interventions that can neutralize Nash free-riding. The work thus resolves important open questions regarding the rationality and robustness of Kantian optimization in both strategic and evolutionary environments.

Strategic Non-Equivalence and Kantian Optimization

The main theoretical advance presented is the formal demonstration that under MKE, any monotonic rescaling of strategy variables affects the Kantian best-response function and can alter equilibrium outcomes. While Nash equilibrium is invariant under one-to-one transformations of the action space, Kantian responses depend crucially on the measurement scale. The paper delivers an operational characterization: rescalings $x = s(z)$ where $s'(z) z$ is not proportional to $s(z)$ will change the Kantian best-response structure. This finding is essential, as it delegitimizes the assumption that Kantian equilibrium is “objective”—equilibrium selection is contingent on subjective or social choices about strategy labels.

Despite this non-equivalence, the authors show that symmetric, Pareto-optimal MKEs persist under arbitrary monotonic rescaling, thus maintaining the previously established efficiency advantage over Nash equilibria when all agents are Kantians. However, asymmetric interactions between Nash and Kantian optimizers reveal free-rider vulnerabilities: Nashers can exploit Kantian behavior unless an appropriate strategy rescaling is selected. The authors’ formalism extends Roemer’s original definition of MKE to accommodate negative auxiliary variables, allowing for strategy spaces defined relative to a reference state, such as deviation from the Nash equilibrium.

Efficient Rescaling and Nash Neutralization

A key technical contribution is the identification of an “efficient” strategy rescaling—measuring strategies as deviations from the Nash equilibrium—that eliminates Nashers’ free-riding advantage. The transformation $z = x - x^\text{Nash}$ ensures that when Kantians and Nashers interact, both receive the Nash payoff, while Kantian–Kantian pairings maintain full Pareto efficiency. The paper rigorously establishes that, in this rescaled environment, Nashers lose any strategic benefit over Kantians, and the Nash equilibrium becomes a symmetric fixed point even in mixed interactions.

This construction aligns with the indirect evolutionary approach, where the observability of behavioral rules (here, the choice of rescaling) shapes the incentives and responses of opponents. Under perfect information, the Kantian-induced game effectively reconfigures the payoff matrix to eliminate exploitation.

Dynamic and Evolutionary Game-Theoretic Implications

The authors provide two applications—dynamic and evolutionary—that substantiate the robustness of the rescaled Kantian approach:

1. Dynamic Game with Type Selection: Agents first endogenously choose Kantian or Nash types before playing a sequence of pairwise games. Under the efficient rescaling, all agents’ best response is to select Kantian optimization, as Nash types are strictly dominated given the payoffs in all possible matches. The resulting equilibrium is Pareto-efficient, coalition-proof, and robust to individual deviation—a substantial strengthening over earlier models that relied on ad hoc behavior switching.
2. Evolutionary Stability: In a population with observable types, Kantian optimization—under the prescribed rescaling—is shown to be an evolutionarily stable strategy (ESS). Any Nash mutants receive strictly lower payoffs than the Kantian incumbents, while the converse does not hold, thereby guaranteeing the internal stability and persistence of Kantian behavior.

These results generalize Roemer’s paradigm, demonstrating that Kantian optimization can be both rational and evolutionarily favored when actors are permitted to design their own measurement frameworks.

Theoretical and Practical Implications

The findings have several implications for game theory and applied economics:

• Subjectivity in Equilibrium Selection: The choice of strategy rescaling is now recognized as endogenous and impactful. Analysts and designers must specify and justify the measurement framework when applying Kantian solution concepts.
• Norm Emergence and Stability: Institutions or societies that coordinate on efficient rescalings can establish robust norms of cooperation that are resistant to selfish deviations, providing a mechanism for stable collective action beyond the scope of standard Nash reasoning.
• Application to Social Dilemmas: The formal apparatus provides tools for constructing mechanisms or incentive structures in environmental economics, industrial organization, and public goods provision that endogenously enforce cooperative outcomes.
• Role of Observability: The results reinforce that the strategic effect of Kantian optimization is contingent on opponents observing the rule-choosing process, corresponding to broader debates on institution design and the evolution of preferences.

Future Directions

The approach opens new lines of inquiry into the meta-game of equilibrium selection: how groups select among possible Kantian visions, under what information structures rescaling remains effective, and the behavioral micro-foundations of such endogenous selection processes. Incorporating unobserved types, stochastic strategies, or multi-level interaction could further refine models of robust cooperation.

Conclusion

This work demonstrates that the inherent strategic non-equivalence of Kantian optimization, manifested through the rescaling of the strategy space, can be leveraged to achieve both static and evolutionary stability of cooperative behavior. Efficient rescaling enables Kantians to eliminate free-rider exploitation while maintaining Pareto efficiency, leading all individuals to prefer Kantian strategies in dynamic and evolutionary settings. These results deepen the theoretical understanding of Kantian equilibrium, challenge the conventional invariance of solution concepts, and offer principled design levers for fostering stable cooperation in strategic environments.

Reference: "Strategy Rescaling and the Stability of Kantian Optimization" (2605.00692)