3 years ago

# Control Problems with Vanishing Lie Bracket Arising from Complete Odd Circulant Evolutionary Games.

James Fan, Christopher Griffin

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.

Publisher URL: http://arxiv.org/abs/1710.09000

DOI: arXiv:1710.09000v1

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