Exponential discounting in security games of timing

Abstract

Strategic game models of defense against stealthy, targeted attacks that cannot be prevented but only mitigated are the subject of a significant body of recent research, often in the context of advanced persistent threats (APTs). In these game models, the timing of attack and defense moves plays a central role. A common assumption, in this literature, is that players are indifferent between costs and gains now and those in the distant future, which conflicts with the widely accepted treatment of intertemporal choice across economic contexts. This article investigates the significance of this assumption by studying changes in optimal player behavior when introducing time discounting. Specifically, we adapt a popular model in the games of timing literature, the FlipIt model, by allowing for exponential discounting of gains and costs over time. We investigate changes of best responses and the location of Nash equilibria through analysis of two well-known classes of player strategies: those where the time between players’ moves is constant, and a second class where the time between players’ moves is stochastic and exponentially distributed. By introducing time discounting in the framework of games of timing, we increase its level of realism as well as applicability to organizational security management, which is in dire need of sound theoretic work to respond to sophisticated, stealthy attack vectors.

Generalized Hyperbolic Discounting in Security Games of Timing

“…These well-studied functions, introduced by Loewenstein and Prelec in 1992 [16], smoothly interpolate an agent's temporal valuation between an arbitrary rate of exponential discounting and a constant valuation, nesting exponential and hyperbolic discounting as special cases. As we cover the entire space of generalized hyperbolic discounting functions, our conclusions subsume those of earlier research efforts involving persistent resource control-including some that do not involve discounting and work on exponential discounting [13,14].…”

“…The interesting strategic aspects of FlipIt have led to numerous follow-up studies. Most closely related to our work are studies on the impact of exponential discounting in the framework of the FlipIt game [13][14][15]. These have found new predictions of optimal defense investments due largely to the fact that an exponentially discounted investment has a finite cumulative valuation over time.…”

“…This presents some conceptual challenges, as nothing is everlasting, and also results in some questionable predictions, such as every resource being attacked by attackers at non-zero rates. In [13,14], Merlevede et al introduced exponential discounting of future gains and costs, addressing some of these concerns. In [15], these authors also introduce a novel class of "discounted strategies" to the exponentially discounted game in which players move less frequently as the resource loses value.…”

“…While this line of work has explored diverse facets of security decision-making, a limitation in most studies on games of timing in general and on FlipIt-like games in particular is that they do not consider economic discounting in the valuation of the contested resource-so although the environment changes over time, the value of the resource and the costs of attacking or defending it do not. A few more recent efforts have attempted to bridge this deficiency by considering the effect of exponential discounting on the valuation and defense costs over time [13][14][15]. Such studies have found new predictions of optimal defense investments due mainly to the fact that an exponentially discounted investment has a finite cumulative valuation over time.…”

Multi-agent Coordination with Deferred Asynchronous Messaging in a Distributed Coordination Space

“…Of great interest there are also approaches to solving the problem of multi-agent coordination at the intersection of different research areas, for example, combining the principles of game theory and reinforcement learning methods [9]. It is important to note that from the point of view of the general problematics of multi-agent coordination, models and solutions in the field of games of timing [10], including games of timing with many players [11], as well as applied solutions in this area [12] are of particular interest.…”