pith. sign in

arxiv: 2606.21967 · v1 · pith:WKTTDGWEnew · submitted 2026-06-20 · 💰 econ.TH

Moral Geometry: Endogenous Scaling in Nash-Kantian Games

Igor Sloev , Gerasimos Lianos This is my paper

Pith reviewed 2026-06-26 11:08 UTC · model grok-4.3

classification 💰 econ.TH
keywords Kantian equilibriumstrategic complementsstrategic substitutesStackelbergcommitmentscalingNash equilibrium
0
0 comments X

The pith

By publicly choosing a scale, a Kantian player can implement the Stackelberg leader outcome under strategic complements and earn more than the Nash payoff.

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

The paper shows that the non-invariance of multiplicative Kantian equilibrium under monotone transformations of the strategy space creates a commitment opportunity. A Kantian player selects a smooth increasing scale before interacting with a Nash player; this scale changes the elasticity weights in the Kantian first-order condition while leaving payoffs and actions fixed. Under strategic complements the chosen scale lets the Kantian stationary-implement the Stackelberg leader outcome and obtain a strictly higher payoff than the Nash benchmark. Under substitutes the same mechanism is only defensive and cannot exceed the Nash level. A reader would care because it turns the choice of how to measure proportional deviations into a strategic instrument that links moral optimization to leadership advantages.

Core claim

The Kantian player can characterize stationary outcomes implementable by a common monotone scale. A sharp dichotomy emerges: under strategic substitutes scaling is defensive and the Kantian approaches but cannot exceed player 2's Nash benchmark; under strategic complements scaling is offensive and the Kantian can stationary-implement the Stackelberg leader outcome and obtain a payoff strictly above the Nash benchmark. In Cournot and differentiated Bertrand examples the scales are explicitly constructed and verified as local transformed Nash-Kantian equilibria.

What carries the argument

The publicly chosen smooth increasing scale, which alters the Kantian first-order condition through endogenous elasticity weights and serves as a commitment device.

If this is right

  • The Kantian player obtains a payoff strictly above the Nash benchmark under strategic complements.
  • Under strategic substitutes the Kantian eliminates the payoff loss from naive behavior but stays at or below the Nash benchmark.
  • Allowing player-specific scales aligns the Kantian condition with the full Stackelberg reaction curve under complements but violates monotonicity under substitutes.
  • Endogenous scaling acts as a commitment mechanism connecting Kantian optimization to strategic leadership.

Where Pith is reading between the lines

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

  • Moral agents in repeated interactions might select representations of actions to gain strategic advantages.
  • The trade-off between universality and strategic flexibility could appear in other equilibrium refinements that depend on how actions are measured.
  • Experimental tests could ask subjects to choose scales in simple games and observe whether they select ones that implement desired outcomes.

Load-bearing premise

There exists a smooth increasing scale whose local elasticity ratios make the Kantian first-order condition coincide with the Stackelberg condition while preserving monotonicity and satisfying second-order conditions.

What would settle it

A counterexample in a strategic complements game where no monotone scale satisfies the required local elasticity ratios for the Kantian first-order condition to match the Stackelberg outcome.

read the original abstract

We study the strategic implications of the non-invariance of multiplicative Kantian equilibrium (MKE) under monotone transformations of the strategy space. Before interacting with a standard Nash player, a Kantian player publicly selects a smooth increasing scale that determines how proportional deviations are evaluated. Material payoffs and feasible actions remain unchanged, but the chosen scale alters the Kantian first-order condition through endogenous elasticity weights. The representation of actions therefore becomes a commitment device. We characterize the stationary outcomes implementable by a common monotone scale. A sharp dichotomy emerges. Under strategic substitutes, the Kantian player can approach the Nash payoff arbitrarily closely but cannot exceed player 2's Nash benchmark; scaling is defensive and eliminates the payoff loss associated with naive Kantian behavior. Under strategic complements, scaling becomes offensive: the Kantian player can stationary-implement the Stackelberg leader outcome and obtain a payoff strictly above the Nash benchmark. In the canonical Cournot and differentiated Bertrand examples, we explicitly construct scales satisfying the required local elasticity ratios and verify the second-order conditions, so the stationary outcomes are local transformed Nash-Kantian equilibria. Allowing player-specific scales would align the Kantian first-order condition with the Stackelberg condition along the entire reaction curve under complements, but would violate monotonicity under substitutes. This reveals a trade-off between universality and strategic flexibility. The results identify endogenous scaling as a commitment mechanism and connect Kantian optimization to strategic leadership and strategic non-equivalence.

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.

Referee Report

0 major / 2 minor

Summary. The manuscript investigates how a Kantian player can use a publicly chosen smooth increasing scale on the strategy space to modify the multiplicative Kantian equilibrium when facing a standard Nash player. The scale acts as a commitment device by altering the Kantian first-order condition through endogenous elasticity weights, without changing material payoffs. The paper characterizes the stationary outcomes that can be implemented with a common monotone scale, identifying a dichotomy based on strategic substitutes versus complements. Under substitutes, defensive scaling allows the Kantian to approach the Nash payoff but not exceed it. Under complements, offensive scaling enables implementation of the Stackelberg leader outcome with a strictly higher payoff. Explicit constructions of such scales are provided for the Cournot and differentiated Bertrand games, with verification of the second-order conditions ensuring local equilibria.

Significance. This paper offers a novel perspective on Kantian optimization by framing endogenous scaling as a strategic commitment tool, linking it to concepts like Stackelberg leadership. The provision of explicit scale constructions in canonical examples and the verification of local equilibrium conditions strengthen the practical applicability of the theoretical results. The analysis of common versus player-specific scales reveals an interesting trade-off between universality and flexibility, contributing to the understanding of non-equivalence in strategic settings.

minor comments (2)
  1. The notation for the scale function, elasticity ratios, and transformed Nash-Kantian equilibrium could be introduced with greater clarity in the introduction and setup sections to aid readers.
  2. In the Cournot and Bertrand examples, a diagram showing the constructed scale function and its local elasticity adjustment would improve intuition for the constructions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, their accurate summary of the results on endogenous scaling as a commitment device, and their recommendation to accept. We are gratified that the dichotomy between strategic substitutes and complements, the explicit constructions in the Cournot and differentiated Bertrand examples, and the discussion of common versus player-specific scales were viewed as strengthening the contribution.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper characterizes which stationary outcomes are implementable when a Kantian player publicly selects a smooth increasing scale function that reweights proportional deviations via local elasticities. The explicit constructions in the Cournot and differentiated Bertrand sections set those elasticities so the transformed Kantian first-order condition coincides with a target (Nash or Stackelberg) point while preserving monotonicity and satisfying second-order conditions. This is a direct demonstration that such functions exist and produce local equilibria; it does not derive an independent prediction that is then shown to equal its own inputs by construction. No self-citation chain, fitted-parameter prediction, or uniqueness theorem imported from prior work is invoked to force the result. The derivation therefore proceeds from the stated primitives (material payoffs unchanged, scale chosen publicly, monotonicity required) without circular reduction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The model rests on standard assumptions of game theory plus the definition of multiplicative Kantian equilibrium; the scale function itself is chosen endogenously rather than fitted to external data.

free parameters (1)
  • monotone scale function
    Chosen by the Kantian player to satisfy local elasticity ratios that align the first-order condition with the target outcome.
axioms (2)
  • domain assumption The scale is smooth and strictly increasing
    Required for the representation of actions and for monotonicity of the transformation.
  • domain assumption Stationary outcomes are local transformed Nash-Kantian equilibria
    Invoked when verifying second-order conditions in the Cournot and Bertrand examples.

pith-pipeline@v0.9.1-grok · 5785 in / 1288 out tokens · 27564 ms · 2026-06-26T11:08:04.807438+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

  1. [1]

    Econometrica, 81(6), pp.2269 -2302

    Homo moralis -preference evolution under incomplete information and assortative matching. Econometrica, 81(6), pp.2269 -2302. https://doi.org/10.3982/ECTA10637 Andreoni, J.,

    work page doi:10.3982/ecta10637

  2. [2]

    The Quarterly Journal of Economics, 110(1), pp.1-21

    Warm -glow versus cold -prickle: the effects of positive and negative framing on cooperation in experiments. The Quarterly Journal of Economics, 110(1), pp.1-21. https://doi.org/10.2307/2118508 Christensen, D.M., Serafeim, G. and Sikochi, A.,

    work page doi:10.2307/2118508

  3. [3]

    The Accounting Review, 97(1), pp.147 -175

    Why is corporate virtue in the eye of the beholder? The case of ESG ratings. The Accounting Review, 97(1), pp.147 -175. https://doi.org/10.2308/TAR-2019-0506 Diewert, W.E.,

    work page doi:10.2308/tar-2019-0506 2019

  4. [4]

    Bulletin of economic research, 44(3)

    The measurement of productivity. Bulletin of economic research, 44(3). https://doi.org/10.1111/j.1467-8586.1992.tb00542.x Fershtman, C. and Judd, K.L.,

    work page doi:10.1111/j.1467-8586.1992.tb00542.x 1992

  5. [5]

    Environmental science & technolog y, 43(16), pp.6414 -6420

    Carbon footprint of nations: a global, trade - linked analysis. Environmental science & technolog y, 43(16), pp.6414 -6420. https://doi.org/10.1021/es803496a Mas-Colell, A., Whinston, M.D. and Green, J.R.,

    work page doi:10.1021/es803496a

  6. [6]

    Scandinavian Journal of Economics, 112(1), pp.1-24

    Kantian equilibrium. Scandinavian Journal of Economics, 112(1), pp.1-24. https://doi.org/10.1111/j.1467-9442.2009.01592.x Roemer, J.E.,

    work page doi:10.1111/j.1467-9442.2009.01592.x 2009

  7. [7]

    Journal of Public Economics, 127, pp.45-57

    Kantian optimization: A microfoundation for cooperation. Journal of Public Economics, 127, pp.45-57. https://doi.org/10.1016/j.jpubeco.2014.03.011 Roemer, J. E. (2019). How we cooperate. Yale University Press. https://doi.org/10.12987/9780300245325 Roemer, J. E. (2020). Response to Braham and van Hees, Sher, Vallentyne, and Laslier. Erasmus Journal for Ph...

    work page doi:10.1016/j.jpubeco.2014.03.011 2014

  8. [8]

    Harvard university press

    The Strategy of Conflict: with a new Preface by the Author. Harvard university press. Sher, I. (2020). Normati ve aspects of Kantian equilibrium. Erasmus Journal for Philosophy and Economics, 13(2), 43-84. https://doi.org/10.23941/ejpe.v13i2.514 Sloev, I ., and Lianos, G . (2026a) Strategy Rescaling and the Sta bility of Kantian Optimization. Available at...

    work page doi:10.23941/ejpe.v13i2.514 2020

  9. [9]

    The Economic Journal, 95 (Supplement), pp.138-147

    Delegation and the theory of the firm. The Economic Journal, 95 (Supplement), pp.138-147. https://doi.org/10.2307/2232877

    work page doi:10.2307/2232877