Positional scoring rules are frequently used for rank aggregation (for example in social choice and in sports). These rules are highly sensitive to the weights associated to positions: depending on the weights, a different winner may be selected. In this paper we explicitly consider the role of weight uncertainty in both the case of monotone decreasing weights and of convex decreasing weights. First we discuss the problem of finding possible winners (candidates that may win for a feasible in-stantiation of the weights) based on previous works that established a connection with the notion of stochastic dominance. Second, we adopt decision-theoretic methods (minimax regret, maximum advantage, expected value) to pick a winner based on the weight uncertainty and we provide a characterization of these methods. Finally, we show some applications of our methodology in real datasets.