A dynamical systems approach for network oligopolies and variational inequalities

The variational inequality problem has been used to formulate and study a plethora of competitive equilibrium problems at the equilibrium state. In this paper, we focus on oligopolistic market network equilibrium problems in which firms are spatially located and seek to determine their profit-maximizing production out-puts and shipments, in the presence of transportation costs. In particular, we utilize the equivalence between the set of stationary points of a dynamical system and the set of solutions to the associated variational inequality problem governing the network oligopoly problem to explore the underlying dynamics both qualitatively and numerically. Although the dynamical system is nonstandard in that the right-hand side is discontinuous, recent theoretical results have shown that the important qualitative and quantitative results of ordinary differential equations are applicable under the standard Lipschitz continuity assumptions. The identification between solutions to dynamical systems and associated variational inequality problems unveils a new tool for addressing the behavior of competitive network systems over time.

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