Reference-Point Dependence in Multiplicative Kantian Optimization

This note shows that multiplicative Kantian optimization is reference-point dependent. In the standard formulation, the Multiplicative Kantian Equilibrium (MKE) test is applied to absolute contributions, and Nash optimizers may free-ride on Kantians. We compare this with a formulation in which Kantians apply the same multiplicative test to contributions above the individually rational Nash level. With the reference point at the Nash contribution, the Nasher's action is zero in the auxiliary scale. Since zero is a fixed point of every multiplicative transformation, the Kantian test makes the Nash outcome stable in mixed encounters. In a public-good game with a dominant Nash strategy, this neutralizes the free-riding advantage of Nash optimizers, while two Kantians attain the efficient symmetric MKE under the efficient-equilibrium selection convention. The induced type-payoff matrix makes the Kantian type weakly dominant and evolutionarily stable. Thus, the stability of Kantian behavior depends not only on the multiplicative Kantian criterion, but also on the reference point used to formulate it.