In this paper we formulate and investigate experimentally a model of how individuals choose between time sequences of monetary outcomes. The theoretical model assumes that a decision-maker uses, sequentially, two criteria to screen options. Each criterion only permits a decision between some pairs of options, while the other options are incomparable according to that criterion. When the first criterion is not decisive, the decision maker resorts to the second criterion to select an alternative. This type of decision procedures has encountered the favour of several psychologists, though it is quite under-explored in the economics domain. In the experiment we find that: 1) traditional economic models based on discounting alone cannot explain a significant (almost 30%) proportion of the data no matter how much variability in the discount functions is allowed; 2) our model, despite considering only a specific (exponential) form of discounting, can explain the data much better solely thanks to the use of the secondary criterion; 3) our model explains certain specific patterns in the choices of the ‘irrational’ people. We can safely reject the hypothesis that anomalous behaviour is due simply to random ‘mistakes’ around the basic predictions of discounting theories: the deviations are not random and there are clear systematic patterns of association between ‘irrational’ choices.