Content: "The only way in which the declaration vˆi
influences the maximization above
is through the term v_i(x (\hat{v}))." This isn't quite true, is it? \hat{v}_i influences the result of x(\hat{v}) and thus also the term \sum_{j \neq i} x(\hat{v}). Nonetheless i has to set v_i = \hat{v}_i to obtain a maximizer for its best response.