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We study general quadratic games with multidimensional actions, stochastic payoff interactions, and rich information structures. We first consider games with arbitrary finite information structures. In such games, we show that there generically exists a unique equilibrium. We then extend the result to games with infinite information structures, under an additional assumption of linearity of certain conditional expectations. In that case, there generically exists a unique linear equilibrium. In both cases, the equilibria can be explicitly characterized in compact closed form. We illustrate our results by studying information aggregation in large asymmetric Cournot markets and the effects of stochastic payoff interactions in beauty contests. Our results apply to general games with linear best responses, and also allow us to characterize the effects of small perturbations in arbitrary Bayesian games with finite information structures and smooth payoffs.