Experimental Design.

A) In both experiments, each participant in the pair grasped the handle of a robotic manipulandum and made reaching movements in the horizontal plane. An LCD projected images (start position, targets, cursors) onto a semi-silvered mirror. Each trial began with each participant’s hand (dark grey circle) within their respective start position (white circle). After a short and random time delay, the self target appeared as a filled dark grey rectangle and the partner target appeared as an unfilled light grey rectangle. Simultaneously, the center cursor (green circle) and partner cursor (light grey circle) also appeared on the screen. After a constant time delay of 500 ms, participants heard a tone that cued them to begin their reach. Participants were instructed to move the center cursor into their own target. Each participant received independent feedback once the center cursor was stabilized within their own target. B) Experimental Conditions. We manipulated the width of both the self and partner targets to be either narrow (task-relevant) or wide (task-irrelevant). The narrow target is task-relevant since participants would need to correct for lateral deviations to successfully complete their task. The wide target is taskirrelevant since participants do not need to correct for lateral deviations to successfully complete their task. Human pairs performed four blocked experimental conditions: i) partner-irrelevant/self-irrelevant ii) partner-relevant/self-irrelevant iii) partner-irrelevant/self-relevant iv) partner-relevant/self-relevant. C-D) Perturbation Trials. On a subset of trials, the center cursor in (C) Experiment 1 or both targets in (D) Experiment 2 jumped 3 cm laterally to the left or right. E-F) Visuomotor Probe Trials. On a subset of trials, the center cursor in Experiment 1 (E) or both targets in Experiment 2 (F) jumped 3 cm laterally for 225 ms, then jumped 3 cm back to the original lateral position. During these probe trials, the hand of both participants in the pair was constrained to a force channel. Here we measured each participant’s visuomotor feedback responses as the force (N) they applied to the wall of the stiff force channel.

Control Model Framework and Hypotheses.

A) Control Model. Human pairs were modelled as controllers within a dynamic game theory framework. Here we depict the feedback control loop from the perspective of one participant (i.e. the self). The self and partner control policy each generate a motor command to produce jointly controlled movement. An efference copy of the motor command passes through an internal model (representation of dynamics) to generate predicted states. Each controller also receives noisy and delayed sensory feedback on the states (e.g., position of the self and partner hand, center cursor, and self and partner targets). Both the self and partner controllers have a state estimator that combines the predicted state and sensory feedback in a statistically optimal manner to produce estimated states. The estimated states are used by the control policy to generate motor commands on each time step. B) Hypotheses. The dynamic game theory framework allowed us to test four distinct hypotheses. The hypotheses test whether the control policy: i) has a representation of a partner, and ii) considers only a self cost or joint (self + partner) cost of accuracy and energy. No Partner Representation & Self Cost Hypothesis: The sensorimotor system has a control policy that does not use a representation of a partner, and only considers a self cost. Partner Representation & Self Cost Hypothesis: The sensorimotor system has a control policy that uses a representation of a partner, but only considers a self cost. Partner Representation & Equal Joint Cost Hypothesis: The sensorimotor system has a control policy that uses a representation of a partner, and equally considers both a self cost and partner cost (i.e., equal joint cost). Partner Representation & Weighted Joint Cost Hypothesis: The sensorimotor system has a control policy that uses a representation of a partner, and that weights the self cost greater than the partner cost (i.e., weighted joint cost). Each of the four hypotheses generate unique predictions of human hand movement (Fig. 3A-P) and visuomotor feedback responses (Fig. 4).

Experiment 1 Hand and Center Cursor Trajectories.

Collectively, the self cursor in models with only a self cost do not laterally deviate to correct for the cursor jump in the partner-relevant/self-irrelevant condition. In contrast, the self cursor in models that consider a self and partner cost laterally deviate to correct for the cursor jump in the partner-relevant/self-irrelevant condition. A-D) Individual hand and center cursor positions of an exemplar pair for each condition in Experiment 1. Thin traces represent each trial. Thick traces represent the average across trials for the human pair. E-H) Group average hand and center cursor positions in Experiment 1. Traces represent the mean and shaded regions reflect ±1 standard error of the mean.

Model Visuomotor Feedback Responses.

Model predictions of visuomotor feedback responses (y-axis) over the time from probe onset (x-axis) for each condition considering the (A) No Partner Representation & Self Cost, (B) Partner Representation & Self Cost, (C) Partner Representation & Equal Joint Cost, and (D) Partner Representation & Weighted Joint Cost models. Solid lines reflect the average visuomotor feedback response to probe trials and shaded error bars reflect ±1 standard deviation of the mean. The inset axis shows the mean visuomotor feedback response between (180 - 230 ms), which aligns with the involuntary time epoch.17 Across the different models, a greater visuomotor feedback response in the partner-relevant/self-irrelevant condition compared to partner-irrelevant/self-irrelevant condition implies that there is a partner representation and a consideration of the partner’s cost. Likewise, a lower feedback response in the partner-relevant/self-relevant condition relative to the partner-irrelevant/self-relevant condition would indicate a partner representation, as well as a higher weighting of the self cost compared to the partner cost.

Visuomotor Feedback Responses in Experiment 1.

A) Visuomotor feedback response (yaxis) over time (x-axis), where 0 ms corresponds to the initial cursor jump. Solid lines represent the group average visuomotor feedback response for each condition. Shaded regions represent ±1 standard error. Vertical grey lines separate involuntary (180 - 230 ms), semi-involuntary (230 - 300 ms), and voluntary (300-400 ms) visuomotor feedback responses. Average B) involuntary, C) semi-involuntary, and D) and voluntary visuomotor feedback response for each condition. Box and whisker plots show 25%, 50%, and 75% quartiles. B) We see significant differences in involuntary visuomotor feedback responses between each condition, matching the predictions of the Partner Representation & Weighted Joint Cost model (see Fig. 4D). Crucially, a greater involuntary visuomotor feedback response in the partner-relevant/self-irrelevant condition compared to the partner-irrelevant/self-irrelevant condition (p < 0.001) suggests a partner representation and some consideration of the partner’s cost. Further, a smaller involuntary visuomotor feedback response in the partner-relevant/self-relevant condition compared to the partner-irrelevant/self-relevant condition (p = 0.002) suggests a higher weighting of the self cost compared to the partner cost. Taken together, our results support the idea that involuntary visuomotor feedback responses express a representation of a partner, while using a joint cost that more heavily weights the self cost over the partner cost.

Visuomotor Feedback Responses in Experiment 2.

A) Visuomotor feedback response (yaxis) over time (x-axis), where 0 ms corresponds to the initial target jump. Solid lines represent the group average visuomotor feedback response for each condition. Shaded regions represent ±1 standard error. Vertical grey lines separate involuntary (180 - 230 ms), semi-involuntary (230 - 300 ms), and voluntary (300-400 ms) visuomotor feedback responses. Average B) involuntary, C) semi-involuntary, and D) and voluntary visuomotor feedback response for each condition. Box and whisker plots show 25%, 50%, and 75% quartiles. B) Critically, a greater involuntary visuomotor feedback response in the partner-relevant/selfirrelevant condition compared to the partner-irrelevant/self-irrelevant condition (p < 0.001) suggests a partner representation and some consideration of the partner’s cost.

Model Hand and Center Cursor Trajectories.

Predicted hand and center cursor positions during left cursor jumps for each condition and model: A-D) No Partner Representation & Self Cost, E-H) Partner Representation & Self Cost, I-L) Partner Representation & Equal Joint Cost, and M-P) Partner Representation & Weighted Joint Cost. Collectively, the self cursor in models with only a self cost do not laterally deviate to correct for the cursor jump in the partner-relevant/self-irrelevant condition. In contrast, the self cursor in models that consider a self and partner cost laterally deviate to correct for the cursor jump in the partner-relevant/self-irrelevant condition.

Experiment 2 Trajectories.

A-D) Individual hand and center cursor positions of an exemplar pair for each condition in Experiment 2. Thin traces represent each trial. Thick traces represent the average across trials for the human pair. E-F) Group average hand and center cursor positions in Experiment 2. Traces represent the mean and shaded regions reflect ±1 standard error of the mean. The group average behaviour in Experiment 2 closely aligns with the Partner Representation & Weighted Joint Cost model (Supplementary Fig. S1), suggesting that voluntary behaviour reflects a partner representation and consideration of a partner’s cost.

Model Final Hand Lateral Deviation.

Final lateral hand deviation for A) No Partner Representation & Self Cost B) Partner Representation & Self Cost C) Partner Representation & Equal Joint Cost D) Partner Representation & Weighted Joint Cost. The final lateral hand deviation was calculated as the mean of the absolute value of the difference between the final hand position on non-perturbation trials and the final hand position on perturbation trials.

Experimental Final Hand Lateral Deviation.

Final lateral hand deviation for A) Experiment 1 and B) Experiment 2. The final lateral hand deviation was calculated as the mean of the absolute value of the difference between the final hand position on non-perturbation trials and the final hand position on perturbation trials. We saw significant differences between each of our statistical comparisons for both Experiment 1 and Experiment 2. The results for both experiments closely match the Partner Representation & Weighted Joint Cost model (Supplementary Fig. S3D), showing that voluntary behaviour considers a partner representation and a weighted joint cost.

Model Parameters