A variational formulation of network games with random utility functions

We consider a class of games played on networks in which the utility 4 functions consist of both deterministic and random terms. In order to find the Nash 5 equilibrium of the game we formulate the problem as a variational inequality in a 6 probabilisticLebesguespacewhichissolvednumericallytoprovideapproximations 7 for the mean value of the random equilibrium. We also numerically compare the 8 solution thus obtained, with the solution computed by solving the deterministic 9 variational inequality derived by taking the expectation of the pseudo-gradient of 10 the game with respect to the random parameters.