In game theory and statistical decision theory, a random (i.e. mixed) decision strategy often outperforms a deterministic strategy in minimax expected loss. As experimental design can be viewed as a game pitting the Statistician against Nature, the use of a random strategy to choose a design will often be beneficial. However, the topic of minimax-efficient random strategies for design selection is mostly unexplored, with consideration limited to Fisherian randomization of the allocation of a predetermined set of treatments to experimental units. Here, for the first time, novel and more flexible random design strategies are shown to have better properties than their deterministic counterparts in linear model estimation and prediction, including stronger bounds on both the expectation and survivor function of the loss distribution. Design strategies are considered for three important statistical problems: (i) parameter estimation in linear potential outcomes models, (ii) point prediction from a correct linear model, and (iii) global prediction from a linear model taking into account an L 2-class of possible model discrepancy functions. The new random design strategies proposed for (iii) give a finite bound on the expected loss, a dramatic improvement compared to existing deterministic exact designs for which the expected loss is unbounded.