We consider a class of decomposition methods for variational inequalities, which is related to the classical Dantzig–Wolfe decomposition of linear programs. Our approach is rather general, in that it can be used with set-valued or nonmonotone operators, as well as various kinds of approximations in the subproblems of the functions and derivatives in the single-valued case. Also, subproblems can be solved approximately. Convergence is established under reasonable assumptions. We also report numerical experiments for computing variational equilibria of the game-theoretical models of electricity markets. Our numerical results illustrate that the decomposition approach allows to solve large-scale problem instances otherwise untractable if the popular PATH solver is applied directly, without decomposition.