Cybersecurity investments with nonlinear budget constraints and conservation laws: variational equilibrium, marginal expected utilities, and Lagrange multipliers

Abstract

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.

A projected dynamic system associated with a cybersecurity investment model with budget constraints and fixed demands

“…As in [1], the demand at each demand market j, d j , is assumed to be fixed and known and such that it satisfies the following conservation law:…”

“…As in [1], we assume that the demand price functions are continuously differentiable and are given byρ…”

“…Subsequently the cyber criminals shared the information stolen from the collective hacker Digital Revolution, which sent it to some newspapers. In the literature cybersecurity models have been studied both in the case of elastic (see [2] and [12]) and fixed demands (see [1]). Further, an analysis of the dual problem and its associated Lagrange multipliers has been performed, together with the study of the marginal expected transaction utility for each retailer.…”

“…In this paper we start from the model in [1] and characterize the solutions of the deriving variational inequality through the critical points of a suitable projected dynamic system. The paper is organized as follows.…”

Allocating revenues in a Smart TV ecosystem

“…For the reasons given above among others, there is great interest for many types of problems arising in networks, in particular in wireless communications and/or its integration with the Internet, in the Operations Research community (for recent references on the topic, see, e.g., Ribeiro et al., ; Risso et al., ; Morais and Mateus, ; Pagès‐Bernaus et al., ; Ye et al., ). Likewise, game theory has proved to be an interesting tool with which to analyze different problems, such as allocation/sharing problems (see, e.g., Petrosjan and Zaccour, ; Moretti and Patrone, ; Nagarajan and Sošić., ; Ackermann et al., ; Guajardo and Rönnqvist, ; Gutiérrez et al., ) or cooperation (see, e.g., Ahmadi‐Javid and Hoseinpour, ; Basso et al., ; Quintero‐Araujo et al., ), from many different areas of knowledge, in particular in communication network problems (see, e.g., Acemoglu and Ozdaglar, ; Gozalvez et al., ; Zhu and Başar, ; Bahbouni and Moussa, ; Goyal and Kaushal, ; van Hove, ; Taleizadeh et al., ; Wang et al., ; Colajanni et al., ; Geng and Mallik, ; Zeng et al., ). Furthermore, game theory has been also utilized to analyze engineering problems in the broadest sense (Sanchez‐Soriano, ).…”