Many studies have shown that humans face a trade-off between the speed and accuracy with which they can make movements. In this article, we asked whether humans choose movement time to maximize expected gain by taking into account their own speed–accuracy trade-off (SAT). We studied this question within the context of a rapid pointing task in which subjects received a reward for hitting a target on a monitor. The experimental design we used had two parts. First, we estimated individual trade-offs by motivating subjects to perform the pointing task under four different time constraints. Second, we tested whether subjects selected movement time optimally in an environment where they were rewarded for both speed and accuracy; the value of the target decreased linearly over time to zero. We ran two conditions in which the subjects faced different decay rates. Overall, the performance of 13 out of 16 subjects was indistinguishable from optimal. We concluded that in planning movements, humans take into account their own SAT to maximize expected gain.

*x*and

*y*directions with a range of ±23 mm relative to the screen center.

*s*is a motor strategy,

*S*is the set of all possible strategies,

*G*is a set of outcomes (i.e., realized movements),

*R*:

*G*→

*G,*and

*f*(.∣

*s*) is a probability distribution over the outcome space

*G*conditional on choosing movement strategy

*s*. The idea behind the model is that, because of motor uncertainty, when an agent selects a motor strategy, he or she is really selecting a probability distribution over realized movements. The model posits that the selected motor strategy should maximize expected gain conditional on the probability distribution generated by that strategy. The movement that actually takes place is then drawn from this conditional distribution.

*S*could be very large, containing a vast array of motor strategies described as a detailed sequence of motor commands. To make the model tractable, assumptions are used to reduce the strategy space to manageable proportions. In Trommershäuser et al., a strategy consisted of selecting a target point on the screen representing the target for the endpoint of the movement. Thus, a strategy could be represented as a tuple of

*x*and

*y*is trivial: One should always aim for the middle of the circle. Thus, we do not model the choice of

*x*and

*y*explicitly. Second, the reward that the agent receives depends only on whether or not the target is hit and the time at which the hit occurs. Third, although actual movement time will be stochastic, we show in the Results section that the probability of hitting the target depended only on planned movement time and not on actual movement time. Thus, we can rewrite Equation 1 as

*A*is the event that the target is hit,

*A*

^{c}is the event that the target is missed,

*t*is actual movement time,

*R*(.,

*t*) is the reward associated with a particular event occurring when actual movement time is

*t,*

*p*(.∣

*f*(.∣

*t** is a lower bound on planned movement time.

*p*(

*A*∣

*f*(

*t*∣

*p*(

*A*∣

*E*(

*t*∣

*p*(

*A*∣

*f*(

*t*∣

*t*increases linearly with

*R*(

*A,*

*t*), varied between decay rate blocks. We therefore know everything we need to calculate the optimal movement plan for a particular agent in a particular treatment. As the optimization does not have an analytical solution given the functional form we have chosen for

*f*(

*t*∣

^{1}the screen position (

*x,*

*y*) that was hit, and the score. Movement endpoints were recorded relative to the center of the target circle.

^{2}and target center) and mean arrival time. We then regressed mean distance by mean arrival time. To do the latter, for each individual, we regressed the distance from target center on actual time taken and a treatment dummy using all observations. By the assumption that planned time does not vary within a treatment, this gave us an estimate of the effect of actual time taken on accuracy, having controlled for planned movement time.

*β,*

*δ,*and

*λ*are estimated parameters.

*β*captures the asymptotic level of

*p,*

*δ*captures the time point where

*p*rises from zero, and

*λ*describes the steepness of the trade-off function. We estimated the parameters

*β,*

*δ,*and

*λ*using maximum likelihood for each subject.

^{3}Figure 3A gives an example of the SAT function from subject M.A.

*conditional*on planned movement time did not. Ten of the 16 subjects showed a relationship between accuracy and planned movement time, which was significant at the 5% level. In comparison, only five subjects showed a statistically significant relationship between accuracy and actual movement time once planned movement time had been controlled for. This is true despite the fact that there were more observations (by two orders of magnitude) for the actual movement time regression than for the planned movement time regression (560 trials vs. 6 treatments). Furthermore, the average of the estimated coefficient on planned movement time across subjects was much larger than that for actual movement time (−0.035 vs. −0.005). Thus, we take this as evidence to support our model in which accuracy is determined only by planned movement time.

^{4}

Coefficient | SE | p | 95% Confidence interval | |
---|---|---|---|---|

1 | −0.0384 | 0.0093 | .02* | −0.0643, −0.0125 |

2 | −0.0317 | 0.0092 | .03* | −0.0573, −0.0062 |

3 | −0.0026 | 0.0023 | .31 | −0.0089, 0.0036 |

4 | −0.0318 | 0.0058 | .01* | −0.0479, −0.0158 |

5 | −0.0078 | 0.0040 | .12 | −0.0188, 0.0032 |

6 | −0.0211 | 0.0025 | .00* | −0.0280, −0.0141 |

7 | −0.0244 | 0.0065 | .02* | −0.0426, −0.0063 |

8 | −0.0315 | 0.0040 | .00* | −0.0426, −0.0204 |

9 | −0.0225 | 0.0064 | .03* | −0.0404, −0.0046 |

10 | −0.0376 | 0.0048 | .00* | −0.0511, −0.0242 |

11 | −0.0609 | 0.0226 | .06 | −0.1238, 0.0019 |

12 | −0.0201 | 0.0063 | .03* | −0.0375, −0.0027 |

13 | −0.0197 | 0.0093 | .10 | −0.0455, 0.0060 |

14 | −0.0109 | 0.0054 | .12 | −0.0259, 0.0042 |

15 | −0.1914 | 0.0618 | .04* | −0.3629, −0.0198 |

16 | −0.0078 | 0.0042 | .13 | −0.0194, 0.0038 |

Coefficient | SE | p | 95% Confidence interval | |
---|---|---|---|---|

1 | −0.0038 | 0.0029 | .18 | −0.0095, 0.0018 |

2 | −0.0137 | 0.0027 | .00* | −0.0191, −0.0083 |

3 | −0.0034 | 0.0018 | .06 | −0.0069, 0.0002 |

4 | −0.0147 | 0.0028 | .00* | −0.0202, −0.0092 |

5 | −0.0036 | 0.0029 | .21 | −0.0092, 0.0021 |

6 | −0.0035 | 0.0025 | .17 | −0.0084, 0.0015 |

7 | −0.0074 | 0.0026 | .01* | −0.0126, −0.0022 |

8 | −0.0039 | 0.0042 | .36 | −0.0121, 0.0044 |

9 | −0.0045 | 0.0028 | .12 | −0.0100, 0.0011 |

10 | −0.0058 | 0.0032 | .07 | −0.0121, 0.0005 |

11 | −0.0004 | 0.0065 | .95 | −0.0132, 0.0124 |

12 | −0.0124 | 0.0037 | .00* | −0.0197, −0.0052 |

13 | −0.0044 | 0.0033 | .18 | −0.0108, 0.0021 |

14 | −0.0102 | 0.0019 | .00* | −0.0140, −0.0065 |

15 | −0.0086 | 0.0092 | .35 | −0.0095, 0.0267 |

16 | −0.0038 | 0.0029 | .18 | −0.0095, 0.0018 |

*change*in average arrival time between fast and slow decay rate trials against the change in MEG timing across subjects. Again, as we regressed actual on predicted change, we cannot reject the hypothesis that the slope was 1 and the intercept was 0 at the 5% level.

^{1}We use the terms “arrival time” and “movement time” interchangeably in this article, although one might more correctly think of arrival time as being made up of a “reaction time” prior to movement onset followed by a “movement time.” In fact, reaction time was almost constant across treatments for any given subject; hence, an alternative analysis based on postreaction movement time would yield the same result.

^{4}Note that the standard errors for the regression coefficients for planned arrival time were smaller than those for actual arrival time, and thus, it is unlikely that our results were driven by lack of variation in actual arrival time conditional on planned arrival time. In general, the variance of average time taken across treatments was less than twice that of the variance of time taken within treatments.

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