A Variational Formulation of Network Games with Random Utility Functions

We consider a class of games played on networks in which the utility functions consist of both deterministic and random terms. In order to find the Nash equilibrium of the game we formulate the problem as a variational inequality in a probabilistic Lebesgue space which is solved numerically to provide approximations for the mean value of the random equilibrium. We also numerically compare the solution thus obtained, with the solution computed by solving the deterministic variational inequality derived by taking the expectation of the pseudo-gradient of the game with respect to the random parameters.