A Variational Problem On The Probability Simplex

We investigate a variational problem on the probability simplex with a path cost expressed as a sum of two terms reminiscent of Lagrangian mechanics. This arose first in a 1972 paper of Y. M. Svirezhev on mathematical genetics, where it was demonstrated that solutions to certain equations governing evolutionary processes are extremals of the variational problem. In the present work, we show that this result holds generally for replicator dynamics, a natural class of dynamical systems on the probability simplex, of great interest in evolutionary game theory. The Lagrangian of Svirezhev respects time-translation symmetry and hence has a conserved quantity, the energy. In particular, solutions to replicator dynamics are extremals confined to the zero level set of energy. Solutions of the dual Hamiltonian system are also of interest. Here we investigate their properties in relation to 2 x 2 matrix games, and assert existence of periodic orbits under suitable hypotheses.