arxiv: 2605.00692 · v1 · submitted 2026-05-01 · 💰 econ.TH

Recognition: unknown

Strategy Rescaling and the Stability of Kantian Optimization

Gerasimos Lianos, Igor Sloev

Pith reviewed 2026-05-09 14:46 UTC · model grok-4.3

classification 💰 econ.TH

keywords Kantian optimizationNash optimizationstrategy rescalingevolutionary stabilitysymmetric gamesfree-rider problemmultiplicative Kantian equilibrium

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The pith

In symmetric games, Kantian optimizers neutralize Nash free-riding via monotonic strategy rescaling, leading all players to select Kantian optimization in dynamic equilibrium and making it evolutionarily stable.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that Multiplicative Kantian Equilibrium changes when strategies are rescaled monotonically, unlike standard Nash best responses. This scale dependence lets Kantians pick a rescaling that offsets the free-rider gain Nash players enjoy in mixed groups while keeping efficient play among Kantians. In a dynamic game where agents choose their optimization type, the subgame-perfect equilibrium has everyone switch to Kantian optimization. An evolutionary model then confirms that Kantian optimization resists invasion by Nash optimization.

Core claim

Multiplicative Kantian Equilibrium is not invariant under monotonic strategy rescaling, so the subjective scale chosen to measure actions alters equilibrium outcomes. This non-equivalence is used to construct a rescaling under which Kantians neutralize the free-rider advantage of Nash players. In the resulting dynamic setting, the subgame-perfect Nash equilibrium of endogenous type choice selects Kantian optimization for all agents. In an evolutionary game, Kantian optimization is an evolutionarily stable strategy.

What carries the argument

Monotonic rescaling of the strategy space that changes the Kantian best-response function without altering the underlying payoff matrix, allowing Kantians to adjust their scale to offset Nash free-riding.

If this is right

All agents adopt Kantian optimization in the subgame-perfect equilibrium of the dynamic choice game.
Kantian optimization remains uninvadable by Nash optimization under replicator or other standard evolutionary dynamics.
Pareto-efficient outcomes among Kantians are preserved even as they neutralize free-riding by Nash players.
Equilibrium play depends on the chosen measurement scale for strategies, so different subjective scales produce different aggregate outcomes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same rescaling logic could be tested in asymmetric games or games with more than two types of optimizers.
Experimental designs could check whether subjects spontaneously discover and apply rescalings that favor cooperation.
If scale choice is costly or observable, the stability result may weaken, suggesting a role for communication or norms about measurement conventions.

Load-bearing premise

The game must be symmetric and the rescaling must be monotonic while leaving the payoff structure unchanged, with players able to costlessly select their optimization type.

What would settle it

Observe whether, in a laboratory symmetric public-goods game, participants who can choose between Kantian and Nash optimization and apply a monotonic rescaling end up selecting Kantian optimization at higher rates than when no rescaling option is given.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Rescaling exploits Kantian non-invariance to let Kantians offset Nash free-riding in symmetric games, leading to claimed SPNE and ESS selection of Kantian types, but the type-choice stage may hide coordination issues.

read the letter

The paper shows that Multiplicative Kantian Equilibrium is not invariant under monotonic rescaling of strategies, so the choice of scale can be used strategically. Kantians can pick a rescaling that removes the free-rider gain Nash players would otherwise enjoy while keeping Kantian-Kantian play Pareto efficient. In the dynamic game where agents first pick their optimization type, the subgame-perfect equilibrium has everyone choosing Kantian; in the evolutionary version, Kantian is an ESS. That combination of non-equivalence plus stability results is the actual addition beyond earlier MKE papers. The construction is straightforward once the non-invariance is granted, and the symmetry assumption keeps the best-response functions manageable. The claims are internally consistent on their own terms and rest on standard game-theoretic objects rather than fitted parameters. The main soft spot is whether the rescaling parameter is fully inside the strategy space when types are chosen. If Nash players can also propose scales or if the best-response correspondence becomes set-valued under rescaling, the type-selection stage could have multiple equilibria or fail to select Kantian uniquely. The abstract and stress-test note leave this open, so the proofs need to show explicitly that the extended game still delivers the claimed unique selection without extra coordination assumptions. The symmetry restriction is standard but narrows the scope. This is for readers already working on Kantian or other non-Nash solution concepts and on endogenous preference for behavioral rules. It is coherent enough to deserve referee time; the math is checkable and the mechanism is specific. Send it to review and ask for the explicit rescaling map plus verification that no new multiplicity arises in the type game.

Referee Report

2 major / 2 minor

Summary. The paper investigates the Multiplicative Kantian Equilibrium (MKE) in symmetric games. It shows that MKE lacks strategic equivalence because the Kantian best-response function is not invariant under monotonic strategy rescaling. Exploiting this, it constructs an efficient rescaling allowing Kantians to neutralize the free-rider advantage of Nash optimizers while preserving Pareto efficiency among Kantians. In a dynamic game with endogenous choice of optimization type, the subgame-perfect Nash equilibrium selects Kantian optimization for all players. In an evolutionary setup, Kantian optimization is an evolutionarily stable strategy (ESS).

Significance. If the derivations hold, the result offers a mechanism for the endogenous emergence and stability of Kantian cooperation in symmetric games via scale choice, without external enforcement. The observation of strategic non-equivalence under rescaling is a useful contribution to the literature on Kantian optimization and could inform models of endogenous preference or type selection. The dynamic SPNE and ESS results, if rigorously established, would strengthen the case for Kantian optimization as a robust alternative to Nash behavior.

major comments (2)

[dynamic framework section] The dynamic framework claims that the SPNE with endogenous type choice leads all players to prefer Kantian optimization. However, the rescaling construction (described in the section on efficient strategy rescaling for Kantians vs. Nashers) appears to introduce a continuous parameter into the strategy space. This risks turning the type-selection stage into a game with an additional dimension, potentially creating multiple mutual best responses or allowing Nash types to propose counter-rescalings, which would undermine the claimed unique selection of Kantian optimization.
[evolutionary setup section] In the evolutionary setup, Kantian optimization is asserted to be an ESS. The proof sketch relies on the rescaling neutralizing free-rider advantages unilaterally while keeping Kantian-Kantian outcomes efficient. But if best-response correspondences are not guaranteed to remain single-valued and continuous under the chosen rescaling (as assumed in the weakest-assumption note), the invasion-resistance argument may fail when small perturbations in scale choice are allowed.

minor comments (2)

[Abstract] The abstract states the main results but provides no visible derivations or model details; the full manuscript should include explicit best-response functions, the exact form of the monotonic rescaling, and the payoff matrix or functional forms used for the SPNE and ESS arguments.
[rescaling construction] Notation for the rescaling parameter and its entry into the strategy space should be clarified to avoid ambiguity about whether it is chosen jointly with the optimization type or separately.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which help clarify potential modeling issues in our dynamic and evolutionary analyses. We respond point by point below and indicate planned revisions.

read point-by-point responses

Referee: [dynamic framework section] The dynamic framework claims that the SPNE with endogenous type choice leads all players to prefer Kantian optimization. However, the rescaling construction (described in the section on efficient strategy rescaling for Kantians vs. Nashers) appears to introduce a continuous parameter into the strategy space. This risks turning the type-selection stage into a game with an additional dimension, potentially creating multiple mutual best responses or allowing Nash types to propose counter-rescalings, which would undermine the claimed unique selection of Kantian optimization.

Authors: We agree that the continuous rescaling parameter requires explicit treatment to preserve uniqueness of the SPNE. In the manuscript the rescaling is chosen by Kantians to neutralize free-riding while maintaining efficiency among themselves; we implicitly treat it as committed prior to type selection. To address the referee's concern we will revise the dynamic framework section to specify the exact timing (rescaling chosen first, then type), demonstrate that Nash players cannot profitably propose counter-rescalings (any such deviation yields lower payoffs against committed Kantians), and prove that best responses remain unique under this ordering. These additions will confirm the unique selection of Kantian optimization. revision: yes
Referee: [evolutionary setup section] In the evolutionary setup, Kantian optimization is asserted to be an ESS. The proof sketch relies on the rescaling neutralizing free-rider advantages unilaterally while keeping Kantian-Kantian outcomes efficient. But if best-response correspondences are not guaranteed to remain single-valued and continuous under the chosen rescaling (as assumed in the weakest-assumption note), the invasion-resistance argument may fail when small perturbations in scale choice are allowed.

Authors: Our ESS argument is conditioned on the weakest-assumption note that best-response correspondences are single-valued and continuous under the rescaling. We will expand the evolutionary setup section (and add a short appendix verification) to confirm that the specific efficient rescaling we construct preserves these properties even when small perturbations in scale are permitted. With this explicit check the invasion-resistance argument continues to hold, so Kantian optimization remains an ESS. revision: partial

Circularity Check

0 steps flagged

No significant circularity; results follow from equilibrium analysis on rescaling non-equivalence

full rationale

The derivation begins with a demonstration that MKE is not invariant under monotonic rescaling, then constructs a specific rescaling to offset Nash free-riding while preserving Kantian efficiency. This leads to SPNE selection of Kantian types in the dynamic game and ESS in the evolutionary setup. All steps rely on standard definitions of best responses, subgame perfection, and evolutionary stability applied to the augmented type-choice game; none reduce by construction to fitted parameters, self-definitions, or unverified self-citations. The central claims remain independently falsifiable via the stated assumptions on symmetry and monotonicity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on standard domain assumptions of game theory with no new free parameters or invented entities visible in the abstract.

axioms (2)

domain assumption Players have common knowledge of rationality and the game structure.
Standard background assumption invoked for equilibrium analysis in symmetric games.
domain assumption The game is symmetric across players.
Explicitly stated as the setting for studying MKE and rescaling effects.

pith-pipeline@v0.9.0 · 5464 in / 1405 out tokens · 50773 ms · 2026-05-09T14:46:12.287503+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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