Hacking Game Theory: A Computational Approach to Finite-Space Mean Field Game Models of Cybersecurity

Abstract

Considerable literature exists on modeling the spread of virus on computer networks. The majority of the current research, however, models cybersecurity as a zero sum game in which an attacker faces off against a network administrator, whose job is to defend the network of defenders. This approach, with only two players to consider, is a good starting point due to its simplicity, and as a result is popular in the mathematical literature. But what if we remove the network administrator, and instead pit the attacker against the network of defenders directly? By removing the intermediary, we can treat each user of a computer in the network as an independent agent rather than a simple pawn that agrees to the policy set forth by the network administrator. In this paper we treat this interaction as a game between an attacker and all of the computers in a network. We begin with a simple model in which the attacker is treated as an exogenous part of the system, and then move on to a more complex version where the attacker is modeled endogenously. We will analyze the behavior of the network through the lens of Mean Field Games, comparing the numerical results to the theoretical predictions. This approach, while requiring a few assumptions, will allow computation in an otherwise computationally impossible problem. We hope that applying numerics to this cybersecurity example will provide inspiration for further computational work on Mean Field Games moving forward.This paper builds primarily on recent developments in Mean Field Game Theory, focusing on the works of Rene Carmona, V.N. Kolokoltsov, and Alain Bensoussan. We use their work as a theoretical compass, but then focus on obtaining numerical results, rather than analytical ones. The R Programming Language will be used for computation.