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We examine the existence and behavior of game-theoretic solutions for robust linear filters and predictors. Our basic uncertainty class includes th-order time-varying discrete-time systems with uncertain dynamics, uncertain initial state covariance, and uncertain nonstationary input and observation noise covariance. Our results include recursive (Kalman filter/predictor) realizations for the resulting robust procedures. Our approach is based on saddle-point theory. We emphasize the notion of a least favorable prior distribution for the uncertain parameter values to obtain a worst case design technique. In this paper, we highlight the role such distributions with finite support play in these decision models. In particular, we demonstrate that, in these decision models, the least favorable prior distribution is always discrete.