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

Five natural axioms — multiplicative form, permutation equivariance, first-order shift invariance, boundary preservation, and first-order regularity — characterize the imitative class of discrete selection kernels on the probability simplex. 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 invariance from first-order to exact forces the kernel to take the exponential weights form. The Price equation (in both absolute-fitness and relative-fitness presentations), the continuous-time replicator dynamics, and the characterization of Nash equilibria as stationary measures of the kernel all follow as direct corollaries of the first-order theorem. The framework also locates the discrete-vs-continuous asymmetry in evolutionary game theory — the phenomenon, documented by Cabrales–Sobel through Falniowski–Mertikopoulos, that distinct discrete kernels with shared continuous-time limits can exhibit divergent finite-step behavior including Li–Yorke chaos — as a structural consequence of the gap between first-order and exact gauge invariance, together with the orbit-equivalence ambiguity identified by Akin in 1979. The continuous limit erases information that the discrete kernel preserves; the theorems characterize what is being projected out.