Control Problems with Vanishing Lie Bracket Arising from Complete Odd Circulant Evolutionary Games. Griffin, C. & Fan, J. arXiv: Optimization and Control, 2017. MAG ID: 2766566204doi abstract bibtex 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.
@article{griffin_control_2017,
title = {Control {Problems} with {Vanishing} {Lie} {Bracket} {Arising} from {Complete} {Odd} {Circulant} {Evolutionary} {Games}.},
doi = {10.3934/jdg.2022002},
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 \${\textbackslash}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.},
journal = {arXiv: Optimization and Control},
author = {Griffin, Christopher and Fan, James},
year = {2017},
doi = {10.3934/jdg.2022002},
note = {MAG ID: 2766566204},
}
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