Title: Control Problems with Vanishing Lie Bracket Arising from Complete Odd Circulant Evolutionary Games,
Abstract: We study an optimal control problem arising from a generalization of
rock-paper-scissors in which the number of strategies may be selected from any
positive odd number greater than 1 and in which the payoff to the winner is
controlled by a control variable $\gamma$. Using the replicator dynamics as the
equations of motion, we show that a quasi-linearization of the problem admits a
special optimal control form in which explicit dynamics for the controller can
be identified. We show that all optimal controls must satisfy a specific second
order differential equation parameterized by the number of strategies in the
game. We show that as the number of strategies increases, a limiting case
admits a closed form for the open-loop optimal control. In performing our
analysis we show necessary conditions on an optimal control problem that allow
this analytic approach to function.