Abstract
Security games often assume a fixed pattern in which players become active, like leader-follower alternation in Stackelberg games or simultaneous moves in Nash games. Stackelberg games are of particular popularity as models for security since they well describe adversaries that adapt to the defender’s actions. Games in extensive or normal form herein induce a fixed sequence of when players become active. But why would a player in a security game wait for the opponent’s move and not just take further actions to cause more damage or gain more? This work studies generalized interaction patterns motivated from the security context, in which each player can take actions as often as it likes, and receives a payoff from the game upon every activity. The practical scenario motivating this study is an adversary who does not wait for the defender to take action, but rather makes the most of the periods during which the defender is idle. This can mean to learn as possible about the victim system while the defender is not present, or to cause as much damage as possible before the defender can strike back.
We show how to convert the situation of arbitrary, in particular non-synchronized, activity schedules back into the classical setting of games in which players take actions in fixed orders. To this end, we introduce conditions under which Nash- and Stackelberg equilibria are invariant to different speeds of playing, and introduce the separate concept of a synchronized equilibrium, in which each player adapts its activity level optimally to those of its opponents, based on an underlying (Nash) equilibrium. We give constructive results about the existence and computation of a synchronized equilibrium, up to its reachability by online learning.
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Notes
- 1.
We remark that the inclusion of speed is always possible by letting every action be played with every possible speed \(\lambda \in \varLambda \). In our setting, allowing a player to take a different action in each of the \(\lambda \in \varLambda \) repetitions, this would expand the action count from \(\left| AS\right| \) to \(\left| AS\right| ^{\left| \varLambda \right| }\). We want to avoid this combinatorial blow-up.
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Acknowledgment
We thank the anonymous reviewers for invaluable comments on the manuscript, which helped to improve content and readability in our view. This publication resulted from research supported by the Karl-Popper Doctoral Kolleg “Responsible Safe and Secure Robotic Systems Engineering (SEEROSE)”, which is funded and supported by the Alpen-Adria-Universität Klagenfurt.
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Rass, S., König, S. (2023). Synchronization in Security Games. In: Fang, F., Xu, H., Hayel, Y. (eds) Decision and Game Theory for Security. GameSec 2022. Lecture Notes in Computer Science, vol 13727. Springer, Cham. https://doi.org/10.1007/978-3-031-26369-9_7
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