Predictions of alternative models of lapses. (a) Effort-dependent disengagement model: In this model, there is an additional cost or mental effort to being engaged in the task which could vary with condition, and an additional random guessing action. If the net payoff of engagement is not greater than the average value of a guess, then it guesses randomly. Such a model does not produce lapses if the effort is fixed across trials (left), but could produce lapses if the effort fluctuates from trial to trial (center). (b) Proportion of trials on which the animal withdrew prematurely does not vary between matched and neutral trials, suggesting that rats are not disengaging preferentially on neutral trials. (c) Predictions of the effort-dependent disengagement model. The model accurately predicts increased lapses on unisensory trials (left panel, green/blue traces) and neutral multisensory trials (middle left panel, orange trace). However, for asymmetric reward manipulations (middle right – reward magnitude, right – reward probability), the model fails to predict our behavioral observation (Figure 4d) that only lapses on the manipulated side are affected. (d) Temporal inattention model: in this model, temporal weighting of evidence differs between matched and neutral trials. To test this, we compared psychophysical kernels on matched and neutral trials. The temporal dynamics of attention are unchanged between the two kinds of trials, arguing against the temporal inattention model. (e) Variable precision model: in this model, the sensory noise (or its inverse, precision) fluctuates from trial to trial, producing heavy tailed performance curves with apparent 'lapses’. The model accurately predicts increased apparent lapses on unisensory trials (left panel, green/blue traces) and neutral multisensory trials (middle left panel, orange trace). However, for asymmetric reward manipulations (middle right, right), the model fails to predict our behavioral observation (Figure 4d) that lapses only on the manipulated side are affected. Like other models of inattention, it predicts that manipulating reward on one side should affect both lapses. (f) Motivation + salience-dependent inattention: in this model, inattention is determined not just by salience, but also motivation, which in turn depends on average reward. This model’s predictions on unisensory, multisensory (left) and neutral (middle left) trials are identical to the inattention model, but on asymmetric reward manipulations, it predicts that total lapse rate should change as a function of total reward. As a result, when reward magnitude on one side is increased or decreased (middle right), total lapse rate also increases or decreases, in addition to the vertical shifts predicted by inattention. However, on the reward probability manipulation (right), it predicts a *decrease* in total lapse rate owing to the overall higher average reward, in addition to a downward shift predicted by inattention, unlike the rat data (Figure 4e) where overall lapse rate *increases* as a consequence of high-rate lapses selectively *increasing*.