Moral Geometry: Endogenous Scaling in Nash-Kantian Games

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.