Many models in economics and operations research can be formulated as Variational Inequalities (VI) problems. In this paper, we introduce an equivalence relation called ordinal equivalence transformation (OET) on VI, which has the property of preserving the solution set of VI. We revisit many classical network games in the economics literature, which include games with uni-dimensional or multi-dimensional strategies, games with strategic complementary or substitutes, games with linear or nonlinear best-reply functions, etc. For each of these games, by identifying certain ordinal equivalence transformations, we are able to transform the original VI problem into a much simpler one. The new VI problem (and not the original one) satisfies an integrability condition, which enables us to reformulate this problem as a minimization programming. As a by-product, we explicitly construct a best-response potential function of the original game, from which various properties of Nash equilibrium, such us existence, uniqueness and stability, can be derived. Finally, using this new technique, we also study several new classes of network games with multiple activities and multiple heterogeneous network structures.
(joint work with Yves Zenou from Monash University)