Selection on the Simplex from Natural Gauge Invariance: Price Equation, Replicator Dynamics, and Nash Equilibria as Corollaries

We give an axiomatic characterization of discrete selection kernels on the probability simplex. Five natural axioms — multiplicative form, permutation equivariance, first-order shift equivariance, boundary preservation, and first-order regularity — characterize the imitative class of discrete update rules. Any kernel satisfying these axioms agrees with the Darwin kernel up to second order in the time step, with continuous-time limit the standard replicator dynamics rescaled by a permutation-symmetric positive rate function. Strengthening shift equivariance to hold exactly in the time step produces a sharper conclusion: any kernel in the imitative class with exact gauge invariance must take the exponential weights form, with possibly state-dependent time parameter. The Price equation, the continuous-time replicator dynamics, and the Nash equilibria as stationary measures of the kernel emerge as direct corollaries of the characterization. The axioms further separate the two fitness conventions of the literature: the absolute-fitness Price equation is what a gauge-invariant discrete kernel produces, while the relative-fitness form of population genetics is a reparametrization that no such kernel produces, since dividing by mean fitness breaks shift invariance at first order in the time step. The gap between first-order and exact gauge invariance is the structural origin of the discrete-vs-continuous asymmetry observed in the literature: kernels with shared continuous limits exhibit substantially different finite-step behaviors, including phenomena ranging from global convergence to Li–Yorke chaos. The framework places selection dynamics within the Lie-algebraic and Hamiltonian formalism developed by Raju and Krishnaprasad. Submitted May 2026 to International Journal of Game Theory