Relationship between saccade duration, endpoint accuracy, and expected value of reward. Relationship between saccade duration, endpoint accuracy, and expected value of reward. A, Top, We computed probability of success (probability that the motor commands will place the target on the fovea) for a 10° saccade as a function of movement duration. Signal-dependent noise magnitude was set to κ = 0.0066. Movement duration that maximizes probability of success is indicated by the vertical line. Bottom, Expected discounted value of the reward attained at the completion of the saccade (blue line) under the assumption that reward is discounted hyperbolically in time (red line). The temporal discount function is F(τ) = 1/(1 + τ). For a 10° saccade, the movement duration that maximizes the probability of success is similar to that which maximizes the expected value of reward. B, Same as A but for a 40° saccade. For this saccade amplitude, the movement duration that maximizes the expected value of reward is much shorter than one that maximizes the probability of success. C, Relationship between saccade amplitude and saccade duration predicted by maximum probability of success hypothesis (green), and maximum expected value of reward hypothesis (red and blue). For the expected value of reward hypothesis, durations that maximize hyperbolically discounted expected rewards are shown in red, and durations that maximize exponentially discounted expected rewards are shown in blue. For hyperbolic discounting, F(τ) = 1/(1 + τ). For exponential discounting, F(τ) = exp(−τ). Also plotted are experimental data from Collewijn et al. (1988) (filled circles; vertical bars indicate ±1 SD). The dashed green line indicates predictions of the maximum probability of success model under a noise model fitted to generate accurate predictions for a 40° saccade (κ = 0.0055). Adrian M. Haith et al. J. Neurosci. 2012;32:11727-11736 ©2012 by Society for Neuroscience