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

arXiv: https://arxiv.org/abs/1710.09000 · HTML: https://mlsanity.com/paper/1710.09000/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Oct 2017.

## Built on by (most cited first, 1 of 1)

- [Fourier Learning Machines: Nonharmonic Fourier-Based Neural Networks for Scientific Machine Learning](https://mlsanity.com/paper/2509.08759/) (2025)

## Abstract (arXiv)

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 γ. 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.