Cybersecurity Investments with Nonlinear Budget Constraints: Analysis of the Marginal Expected Utilities

  • Patrizia DanieleEmail author
  • Antonino Maugeri
  • Anna Nagurney
Part of the Springer Optimization and Its Applications book series (SOIA, volume 113)


In this paper, we consider a recently introduced cybersecurity investment supply chain game theory model consisting of retailers and consumers at demand markets with the retailers being faced with nonlinear budget constraints on their cybersecurity investments. We construct a novel reformulation of the derived variational inequality formulation of the governing Nash equilibrium conditions. The reformulation then allows us to exploit and analyze the Lagrange multipliers associated with the bounds on the product transactions and the cybersecurity levels associated with the retailers to gain insights into the economic market forces. We provide an analysis of the marginal expected transaction utilities and of the marginal expected cybersecurity investment utilities. We then establish some stability results for the financial damages associated with a cyberattack faced by the retailers. The theoretical framework is subsequently applied to numerical examples to illustrate its applicability.


Cybersecurity Investments Supply chains Game theory Nash equilibrium Variational inequalities Lagrange multipliers Stability 

MSC 2010:

49K40 65K10 65K15 90C33 90C46. 



The research of the first author was partially supported by Istituto Nazionale di Alta Matematica Francesco Severi (Progetto di Ricerca GNAMPA 2015: Nuove frontiere dei problemi di equlibrio su rete: dallo sviluppo sostenibile alla dinamica dei disastri ambientali ai crimini informatici). The research of the third author was supported, in part, by the National Science Foundation under Grant No. 1551444. This support is gratefully acknowledged.


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Patrizia Daniele
    • 1
    Email author
  • Antonino Maugeri
    • 1
  • Anna Nagurney
    • 2
  1. 1.Department of Mathematics and Computer ScienceUniversity of CataniaCataniaItaly
  2. 2.Isenberg School of ManagementUniversity of MassachusettsAmherstUSA

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