FlipDyn: A game of resource takeovers in dynamical systems

TL;DR

This paper introduces FlipDyn, a game-theoretic model for resource control in dynamical systems, providing analytical solutions for cost and equilibrium strategies, with applications to adversarial control scenarios.

Contribution

It develops exact and approximate analytical expressions for control costs and Nash equilibria in a game of resource takeovers within dynamical systems, extending to continuous and higher-dimensional states.

Findings

Exact cost-to-go expressions for discrete states.

Nash equilibrium formulas for linear systems with quadratic costs.

Approximate value functions for higher-dimensional systems.

Abstract

We introduce a game in which two players with opposing objectives seek to repeatedly takeover a common resource. The resource is modeled as a discrete time dynamical system over which a player can gain control after spending a state-dependent amount of energy at each time step. We use a FlipIT-inspired deterministic model that decides which player is in control at every time step. A player's policy is the probability with which the player should spend energy to gain control at each time step. Our main results are three-fold. First, we present analytic expressions for the cost-to-go as a function of the hybrid state of the system, i.e., the physical state of the dynamical system and the binary \texttt{FlipDyn} state for any general system with arbitrary costs. These expressions are exact when the physical state is also discrete and has finite cardinality. Second, for a continuous physical state with linear dynamics and quadratic costs, we derive expressions for Nash equilibrium (NE). For scalar physical states, we show that the NE depends only on the parameters of the value function and costs, and is independent of the state. Third, we derive an approximate value function for higher dimensional linear systems with quadratic costs. Finally, we illustrate our results through a numerical study on the problem of controlling a linear system in a given environment in the presence of an adversary.