Control problems with vanishing Lie Bracket arising from complete odd circulant evolutionary games

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ \gamma $\end{document}</tex-math></inline-formula>. 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.</p>

Approximation of optimal control surfaces for 2 × 2 skew-symmetric evolutionary game dynamics

“…In future work we will generalize the evolutionary game problems we consider to more strategies in an attempt to analyze the control problems found in [6]. In particular, this will require dealing with non-integrable dynamical systems.…”

“…In particular, this will require dealing with non-integrable dynamical systems. The work in [6] shows that the general non-linear controller in cyclic games with an odd number of strategies exhibit oscillations whose properties may be elucidated by this Fourier approximation method. In addition, we will consider control problems for evolutionary games on graphs where chaos can emerge [51].…”

“…More recently [5] studies convergence of best-response strategies on graphs. Fan and Griffin [6] study optimal control of odd circulant games generalizing work in [7] in which control of the Bass model is considered. At the same time, there has been extensive work on reinforcement learning (RL) based methods for control of dynamical systems [8,9] with more recent work in (deep) neural network based methods coming to the fore [10].…”

Spatial dynamics of higher order rock-paper-scissors and generalisations

“…Then the N × N circulant matrix A N defined by A N1 is the payoff matrix for the N strategy generalisation of RPS. Griffin and Fan [87] showed that the replicator dynamics equation ( 1) have a unique interior fixed point at u = 1 N , . .…”