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

TL;DR

This paper investigates optimal control strategies in generalized rock-paper-scissors games with an odd number of strategies, deriving explicit control dynamics and analyzing the limit as strategies increase.

Contribution

It introduces a novel control framework for odd circulant evolutionary games and derives explicit solutions and limiting behavior for large strategy sets.

Findings

Optimal controls satisfy a specific second order differential equation.

Explicit control solutions are derived for the limit of large strategies.

Necessary conditions for the analytic approach are established.

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.