We extend statistical minimax theorems for the average risk by providing general conditions under which maximin priors exist and are saddle points. We show that these conditions apply not only when the parameter space is compact, but also under the weaker condition that the priors have bounded moments. We illustrate the practicality of these conditions in the normal mean problem and the sparse normal mean problem where we readily recover known existence results and derive new ones. We then cast doubt on the possibility of extending Huber’s approach to derive new existence results in minimax games for the average risk without any boundedness conditions on the parameter or the priors. We illustrate this issue in the normal mean problem and the sparse normal mean problem when the parameter space is the whole real line. As a corollary of independent interest, we show that Brown’s identity does not hold for subprobability measures on the reals. We finally show that existence results obtained in Bickel (1983) and Bickel and Collins (1983) when the parameter space is the extended real line [−∞, +∞] is not equivalent to Huber’s approach and impose much stronger bounds on the parameter than may appear at first. As a consequence, we call for caution when working with maximin priors for the average risk in absence of explicit bounds on the parameter space or on the priors’ moments.