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Cybersecurity investments with nonlinear budget constraints and conservation laws: variational equilibrium, marginal expected utilities, and Lagrange multipliers
Author(s) -
Colajanni Gabriella,
Daniele Patrizia,
Giuffrè Sofia,
Nagurney Anna
Publication year - 2018
Publication title -
international transactions in operational research
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.032
H-Index - 52
eISSN - 1475-3995
pISSN - 0969-6016
DOI - 10.1111/itor.12502
Subject(s) - lagrange multiplier , variational inequality , conservation law , mathematical optimization , nonlinear system , economics , nash equilibrium , mathematical economics , computer science , mathematics , quantum mechanics , mathematical analysis , physics
In this paper, we propose a new cybersecurity investment supply chain game theory model, assuming that the demands for the product are known and fixed and, hence, the conservation law of each demand market is fulfilled. The model is a generalized Nash equilibrium model with nonlinear budget constraints for which we define the variational equilibrium, which provides us with a variational inequality formulation. We construct an equivalent formulation, enabling the analysis of the influence of the conservation laws and the importance of the associated Lagrange multipliers. We find that the marginal expected transaction utility of each retailer depends on this Lagrange multiplier and its sign. Finally, numerical examples with reported equilibrium product flows, cybersecurity investment levels, and Lagrange multipliers, along with individual firm vulnerability and network vulnerability, illustrate the obtained results.