What matters in making demand-based decisions: Time alone or difficulty too?

Which task is easier, doing arithmetic problems of specified form for some specified duration, or carrying a bucket of specified weight over some specified distance? If it is possible to choose between the "more cognitive" task and the "more physical" task, how are the difficulty levels of the tasks compared? We conducted two experiments in which participants chose the easier of two tasks, one that involved solving addition or multiplication problems (Experiment 1) or addition problems with different numbers of addends (Experiment 2) for varying amounts of time (in both experiments), and one that involved carrying a bucket of different weights over a fixed distance (in both experiments). We found that the probability of choosing to do the bucket task was higher when the bucket was empty than when it was weighted, and increased when the cognitive task was harder and its duration grew. We could account for the choice probabilities by mapping the independent variables onto one abstract variable, Φ. The functional identity of Φ remains to be determined. It could be interpreted as an inferred effort variable, subjective duration, or an abstract, amodal common code for difficulty.