In this paper, we consider variational inequalities over tensor spaces and discuss some basic properties. In particular, we introduce a type of product between two tensors, and with this tensor product, we define a class of affine variational inequalities over tensor spaces. Then, we discuss some properties of the solution set of the affine variational inequality, including existence and uniqueness of the solution and boundedness of the solution set. Finally, we investigate a class of oligopolistic market games and transform it into an affine variational inequality over a tensor space.