S-adapted oligopoly equilibria and approximations in stochastic variational inequalities

This paper deals with a class of dynamic games that are used for modelling oligopolistic competition in discrete time with random disturbances that can be described as an event tree with exogenously given probabilities. The concepts of S-adapted information structure and S-adapted equilibrium are reviewed and a characterization of the equilibrium as the solution of a variational inequality (VI) is proposed. Conditions for existence and uniqueness of the equilibrium are provided. In order to deal with the large dimension of the VI an approximation method is proposed which is based on the use of random sampling of scenarios in the event tree. A proof of convergence is provided and these results are illustrated numerically on two dynamic oligopoly models.