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.

Timing, a central aspect of decision-making in security scenarios, is a subject of growing academic interest; frequently in the context of stealthy attacks, or advanced persistent threats (APTs). A key model in this research landscape is FlipIt [1]. However, a limiting simplifying assumption in the FlipIt literature is that costs and gains are not subject to discounting, which contradicts the typical treatment of decision-making over time in most economically relevant contexts. Our recent work [2] introduces an adaptation of the FlipIt model that applies time-based exponential discounting to the value of a protected resource, while allowing players to choose from among the same canonical strategies as in the original game. This paper extends the study of games of timing by introducing two new classes of strategies that are fundamentally motivated by a time-discounted world view. Within our game model, we compute player utilities, best responses and give a partial characterization of the game's Nash equilibria. Our model allows us to re-interpret the APT model using a finite total valuation, and a finite time horizon. By applying time-based discounting to the entire decision-making framework, we increase the level of realism as well as applicability to organizational security management.

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