Recent work has derived the optimal policy for two-alternative value-baseddecisions, in which decision-makers compare the subjective expected reward oftwo alternatives. Under specific task assumptions — such as linearutility, linear cost of time and constant processing noise — the optimalpolicy is implemented by a diffusion process in which parallel decisionthresholds collapse over time as a function of prior knowledge about averagereward across trials. This policy predicts that the decision dynamics of eachtrial are dominated by the difference in value between alternatives and areinsensitive to the magnitude of the alternatives (i.e., their summed values).This prediction clashes with empirical evidence showing magnitude-sensitivityeven in the case of equal alternatives, and with ecologically plausible accountsof decision making. Previous work has shown that relaxing assumptions aboutlinear utility or linear time cost can give rise to optimal magnitude-sensitivepolicies. Here we question the assumption of constant processing noise, infavour of input-dependent noise. The neurally plausible assumption ofinput-dependent noise during evidence accumulation has received strong supportfrom previous experimental and modelling work. We show that includinginput-dependent noise in the evidence accumulation process results in amagnitude-sensitive optimal policy for value-based decision-making, even in thecase of a linear utility function and a linear cost of time, for both single(i.e., isolated) choices and sequences of choices in which decision-makersmaximise reward rate. Compared to explanations that rely on non-linear utilityfunctions and/or non-linear cost of time, our proposed account ofmagnitude-sensitive optimal decision-making provides a parsimonious explanationthat bridges the gap between various task assumptions and between various typesof decision making.