Upper and Lower Values in Zero-Sum Stochastic Games with Asymmetric Information
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Upper and Lower Values in Zero-Sum Stochastic Games with Asymmetric Information Dhruva Kartik1
· Ashutosh Nayyar1
© Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract A general model for zero-sum stochastic games with asymmetric information is considered. In this model, each player’s information at each time can be divided into a common information part and a private information part. Under certain conditions on the evolution of the common and private information, a dynamic programming characterization of the value of the game (if it exists) is presented. If the value of the zero-sum game does not exist, then the dynamic program provides bounds on the upper and lower values of the game. Keywords Dynamic games · Asymmetric information · Upper and lower values
1 Introduction Zero-sum games have been widely used as a model of strategic decision making in the presence of adversaries. Such decision-making scenarios arise in a range of domains including (i) security of cyber-physical and infrastructure systems such as the power grid and water networks in the presence of cyber or physical attacks [2,3,38,41,42,44], (ii) cyber-security of networked computing and communication systems [1,41], (iii) designing anti-poaching measures [7–9], (iv) military operations in the presence of hostile agents [15] and (v) competitive markets and geopolitical interactions [4,24]. In many cases, the adversarial interactions occur over time in a dynamic and uncertain environment. Zero-sum stochastic games provide a useful model for these situations. In these games, two players may jointly control the evolution of the state of a stochastic dynamic system with one player trying to minimize the total cost while the other trying to maximize it. In stochastic games with symmetric information, all players have the same information about the state and action histories. Such games have been
Preliminary version of this paper appears in the proceedings of the 58th Conference on Decision and Control (CDC), 2019 [17].
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Dhruva Kartik [email protected] Ashutosh Nayyar [email protected]
1
Ming Hsieh Department of Electrical and Computer Engineering, University of Southern California, Los Angeles, CA 90007, USA
Dynamic Games and Applications
extensively studied in the literature in both zero-sum and nonzero-sum settings [6,10,11]. In many situations of interest, however, the players may have different information about the state and action histories. A potential attacker of a cyber-physical system, for example, may not have the same information as the defender; adversaries in a battlefield may have different information about the surroundings and about each other. The focus of this paper is on such asymmetric information settings. We adopt a model of asymmetric information that was originally developed for decentralized stochastic control [28]. This model partitions each player’s information at each time into a common information part and a private information part. The common information at time t is known to all
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