Experimental design.

(A) Organization of the different experiments. Each experiment began with a familiarization block, followed by a baseline block. In the main and second control experiments, participants completed four additional conditions, whereas in the first control experiment, only two additional conditions were tested. The order of conditions (excluding the familiarization and baseline blocks) was randomized across participants. To minimize potential after-effects, the first 5 trials of each block were “wash-out” trials, using the same parameters as the baseline condition. (B) Illustration of the experimental setup. Participants were connected to an HRX-1 robotic exoskeleton (in blue) in front of a computer screen. Two EMG electrodes were placed over the wrist flexor and extensor muscle groups. (C) Example of one trial from the main experiment. The trial began with a reach phase, in which participants moved a red dot with a velocity proportional to the applied torque. Once reaching the patch, a delay elapsed during which the patch gradually filled. This was followed by the harvest phase, during which participants resisted a virtual force field to stay within the patch and collect points. Pressing the space bar ended the trial, resetting the red dot to the starting position and initiating the next trial. (D) Example of one trial from the control experiments. A delay period was introduced before the reach phase, during which a target gradually emptied over time. (E) Relationship between reach duration and the average normalized torque applied for each reach effort condition. (F) The harvest function (in yellow) depicts the number of points collected over time within a patch. This function is subject to maximization. In terms of minimization, the equivalent function is the ‘cost of reward’ (in blue), obtained by negating f and adding the maximum reward per patch, α (see modeling section below). BL: Baseline; HRE: High Reach Effort; LRE: Low Reach Effort; HD: High Delay; LD: Low Delay; HHE: High Harvest Effort; LHE: Low Harvest Effort.

Common-utility and separate-cost models.

(A) Utility as a function of the reach and harvest durations. In this utility-based framework, reach and harvest phases are jointly regulated, with optimal durations selected to maximize a single time-discounted utility function that integrates effort, reward, and time. (B) An alternative model is proposed such that reach (red, left panel) and harvest (blue, right panel) are optimized separately. Here, optimal durations are selected to minimize distinct subjective cost functions (Cr and Ch; thick dot-dashed lines), each comprising at least a cost of time (Gr and Gh; dashed lines) and a cost that decreases over time (Ur and R; solid lines). Note that the cost of time integrates all preceding delays, implying mutual influence of one phase over another depending on their onset order. Here, the general case is considered without any assumption on the phase order, but see supplementary information (Fig. S6 and Supplementary Video) for representation of a typical trial of the experiments illustrating the inter-dependence induced by the offset. Uh: harvest’s cost of effort; or: reach time offset.

Effects of time and reach effort on reach and harvest durations (main experiment).

Reach (A) and harvest durations (B) for each condition of the main experiment. Colored circles represent median durations, and bars represent interquartile ranges (n = 20 for each condition). Individual data points are displayed as small colored dots. Thin grey lines connect compared observations across conditions. The stars represent p-values associated with Wilcoxon signed-rank tests. (C) Left panel: heatmap of Spearman correlation coefficients between all pairs of condition for reach (lower triangle) and harvest (upper triangle) durations. Right panel: heatmap of Spearman correlation coefficients between reach and harvest durations for each pair of conditions. The stars represent p-values associated with the analyses, after correction for multiple comparisons using the Benjamini-Hochberg procedure (Ntests = 45). *** p < 0.001. BL: Baseline (green); HRE: High Reach Effort (blue); LRE: Low Reach Effort (light blue); HD: High Delay (orange); LD: Low Delay (yellow).

Effects of delay’s timing onset on reach and harvest durations (first control experiment).

Reach (A) and harvest durations (B) for each condition of the first control experiment. Colored circles represent median durations, and bars represent interquartile ranges (n = 14 for each condition). Individual data points are displayed as small colored dots. Thin grey lines connect compared observations across conditions. The stars represent p-values associated with Wilcoxon signed-rank tests. (C) Left panel: heatmap of the correlations separately for reach (lower part) and harvest (upper part) durations between each pair of conditions. Right panel: heatmap of the correlations between reach and harvest durations for each pair of conditions. Spearman correlation coefficients (rho) are reported for each correlation. The stars represent p-values associated after correction for multiple comparisons using the Benjamini-Hochberg procedure (Ntests = 15). * p < 0.05 ** p < 0.01; *** p < 0.001. BL: Baseline (green); HD0: High Delay (orange); LD0: Low Delay (yellow).

Effects of time and harvest effort on reach and harvest durations (second control experiment).

Reach (A) and harvest durations (B) for each condition of the first control experiment. Colored circles represent median durations, and bars represent interquartile ranges (n = 10 for each condition). Individual data points are displayed as small colored dots. Thin grey lines connect compared observations across conditions. The stars represent p-values associated with Wilcoxon Signed-rank tests. (C) Left panel: heatmap of the correlations separately for reach (lower part) and harvest (upper part) durations between each pair of conditions. Right panel: heatmap of the correlations between reach and harvest durations for each pair of conditions. Spearman correlation coefficients (rho) are reported for each correlation. The stars represent p-values associated after correction for multiple comparisons using the Benjamini-Hochberg procedure (Ntests = 45). *** p < 0.001. BL: Baseline (green); HHE: High Harvest Effort (dark green); LHE: Low Harvest Effort (light green); HD0: High Delay (orange); LD0: Low Delay (yellow).

Model predictions of reach and harvest durations.

(A, B, D, E) Colored crosses represent the median harvest (x axis) and reach (y axis) durations across all participants in the main (left panel) and second control (right panel) experiments. Predicted values from the common model (A, B) and the separate model (D, E) are shown as colored circles with black borders. Arrows and rectangles illustrate the deviation between model predictions and actual data. Residual sum of squares for each condition in the main experiment and second control experiment from the common (C) and separate (F) models. BL: Baseline (green); HRE: High Reach Effort (blue); LRE: Low Reach Effort (light blue); HHE: High Harvest Effort (dark green); LHE: Low Harvest Effort (light green); HD0: High Delay (orange); LD0: Low Delay (yellow).

Overview of model fitting results.

This table summarizes the best-fitting parameters, the RSS and the AIC for the common-utility and separate-cost models, in the main and second control experiments. The associated functions of each free parameter are also indicated. Uh: harvest effort; Ur: reach effort; G: temporal discount; Gh: harvest’s cost of time; Gr: reach’s cost of time.

Summary of theoretical predictions and experimental observations.

The direction of the predicted effects was determined from the parameters of the models fitted to the experimental data. *Indirect effects were not considered here; further details are provided in the main text. BL: Baseline; HRE: High Reach Effort; LRE: Low Reach Effort; HD: High Delay; LD: Low Delay; HHE: High Harvest Effort; LHE: Low Harvest Effort; HD0: High Delay; LD0: Low Delay.

Experimental deviations from the predicted changes in duration.

Difference in harvest (A) and reach (B) durations between the high- and low-reach effort conditions (HRE-LRE; in blue) and between the high- and low-delay conditions (HD-LD; in orange) in the main experiment. Difference in harvest (C) and reach (D) durations between the high- and low-harvest effort conditions (HHE-LHE; in green) and between the high- and low-delay conditions (HD0-LD0; in orange) in the second control experiment. These observed differences are compared with the predicted differences made by the common-utility model (white circle) and the separate-cost model (black circle). Colored circles represent median differences in duration, and bars represent interquartile ranges (n = 20 and n = 10 for each condition in the main and second control experiments, respectively). Individual data points are displayed as small colored dots. The stars represent p-values associated with the Wilcoxon signed-rank test. ** p < 0.01. HRE: High Reach Effort; LRE: Low Reach Effort; HHE: High Harvest Effort; LHE: Low Harvest Effort; HD0: High Delay; LD0: Low Delay.

Inflection points of the costs of time according to the time offset.

(A) Large colored circles represent the inflection point of each cost of time in relation to the time offset (red for reach phases, blue for harvest phases) for the main experiment (filled circles) and the second control experiment (empty circles). Small colored crosses represent each condition’s time offset with the corresponding inflection point. A power-law function was fitted (black line) such that ti(o) = a1oa2, where o is the average time offset and a1 and a2 are two free parameters. Fit result: R² = 0.98, RSS = 1.18 s²; a1 = 1.26; a2 = 0.99. The dotted line represents theoretical values of the inflection point if it occurs exactly at the start of the phase of interest. (B) Representation of each cost of time of the reach (red) and harvest (blue) phases found for the main (solid lines) and second control experiment (dashed lines). The costs of time were normalized by their parameter a. Dotted lines represents the corresponding average time offsets of the reach (or1, or2 respectively for the main and second control experiment) and harvest phases (oh1, oh2 respectively for the main and second control experiment). Note how the inflection point location (colored circles) and the steepness of the curve increase with the time offset. BL: Baseline (green); HRE: High Reach Effort (blue); LRE: Low Reach Effort (light blue); HD / HD0: High Delay (orange); LD / LD0: Low Delay (yellow); HHE: High Harvest Effort (dark green); LHE: Low Harvest Effort (light green).