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International Conference on Decision and Game Theory for Security

GameSec 2016: Decision and Game Theory for Security pp 114–126Cite as

Dynamics on Linear Influence Network Games Under Stochastic Environments

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Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 9996))

Abstract

A linear influence network is a broadly applicable conceptual framework in risk management. It has important applications in computer and network security. Prior work on linear influence networks targeting those risk management applications have been focused on equilibrium analysis in a static, one-shot setting. Furthermore, the underlying network environment is also assumed to be deterministic.

In this paper, we lift those two assumptions and consider a formulation where the network environment is stochastic and time-varying. In particular, we study the stochastic behavior of the well-known best response dynamics. Specifically, we give interpretable and easily verifiable sufficient conditions under which we establish the existence and uniqueness of as well as convergence (with exponential convergence rate) to a stationary distribution of the corresponding Markov chains.

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Notes

  1. 1.

    If \(x < 0\), we set \(V_i(x) = -\infty \), representing the fact that negative effective investment is unacceptable.

  2. 2.

    Note that here we do not impose any bounds on the maximum possible investment by any player. If one makes such an assumption, then \(\mathcal {X}\) will be some compact subset of \(\mathbf {R}^N_+\). All the results discussed in this section will still go through. For space limitation, we will not discuss the bounded investment case. Further, note that the unbounded investment case (i.e. without placing any exogenous bound on investments made by any player) which we focus on here is the hardest case.

  3. 3.

    As we shall soon see, the two norms we will be using are weighted \(l_1\) norm and weighted \(l_{\infty }\) norm.

  4. 4.

    The proof utilizes the powerful framework presented in [19, 20].

  5. 5.

    Thi is the well-known metric that is commonly used to characterize the distance between two probability measures [21]. Further, if \((\varOmega , \rho )\) is a separable metric space, as is the case in our application, then convergence in Prokhorov metric implies weak convergence [21].

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Author information

Authors and Affiliations

  1. Department of Electrical Engineering, Stanford University, Stanford, CA, 94305, USA

    Zhengyuan Zhou, Nicholas Bambos & Peter Glynn

  2. Department of Management Science and Engineering, Stanford University, Stanford, CA, 94305, USA

    Nicholas Bambos & Peter Glynn

Authors
  1. Zhengyuan Zhou
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  2. Nicholas Bambos
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  3. Peter Glynn
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Corresponding author

Correspondence to Zhengyuan Zhou .

Editor information

Editors and Affiliations

  1. New York University , New York, New York, USA

    Quanyan Zhu

  2. The University of Melbourne , Melbourne, Victoria, Australia

    Tansu Alpcan

  3. University of Brighton , Brighton, United Kingdom

    Emmanouil Panaousis

  4. University of Southern California , Los Angeles, California, USA

    Milind Tambe

  5. Carnegie Mellon University , Pittsburgh, New York, USA

    William Casey

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© 2016 Springer International Publishing AG

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Zhou, Z., Bambos, N., Glynn, P. (2016). Dynamics on Linear Influence Network Games Under Stochastic Environments. In: Zhu, Q., Alpcan, T., Panaousis, E., Tambe, M., Casey, W. (eds) Decision and Game Theory for Security. GameSec 2016. Lecture Notes in Computer Science(), vol 9996. Springer, Cham. https://doi.org/10.1007/978-3-319-47413-7_7

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  • DOI: https://doi.org/10.1007/978-3-319-47413-7_7

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-47412-0

  • Online ISBN: 978-3-319-47413-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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