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Conceived and designed the experiments: DAB PAO DMW. Performed the experiments: DAB. Analyzed the data: DAB PAO DMW. Wrote the paper: DAB DMW. Designed the figures: PAO.

The authors have declared that no competing interests exist.

Social interactions in classic cognitive games like the ultimatum game or the prisoner's dilemma typically lead to Nash equilibria when multiple competitive decision makers with perfect knowledge select optimal strategies. However, in evolutionary game theory it has been shown that Nash equilibria can also arise as attractors in dynamical systems that can describe, for example, the population dynamics of microorganisms. Similar to such evolutionary dynamics, we find that Nash equilibria arise naturally in motor interactions in which players vie for control and try to minimize effort. When confronted with sensorimotor interaction tasks that correspond to the classical prisoner's dilemma and the rope-pulling game, two-player motor interactions led predominantly to Nash solutions. In contrast, when a single player took both roles, playing the sensorimotor game bimanually, cooperative solutions were found. Our methodology opens up a new avenue for the study of human motor interactions within a game theoretic framework, suggesting that the coupling of motor systems can lead to game theoretic solutions.

Human motor interactions range from adversarial activities like judo and arm wrestling to more cooperative activities like tandem riding and tango dancing. In this study, we design a new methodology to study human sensorimotor interactions quantitatively based on game theory. We develop two motor tasks based on the prisoner's dilemma and the rope-pulling game in which we introduce an intrinsic cost related to effort rather than the typical monetary outcome used in cognitive game theory. We find that continuous motor interactions converged to game theoretic outcomes similar to the interaction dynamics reported for other dynamical systems in biology ranging in scale from microorganisms to population dynamics.

Riding a tandem, tango dancing, arm wrestling and judo are diverse but familiar examples of two-player motor interactions. The characteristic feature of such interactions is that the two players influence each others behavior through coupled sensorimotor control with continuous action spaces over repeated trials or continuously in time. In contrast, two-player interactions considered in classical game theory are typically thought to involve cognition in games with discrete actions and discrete time steps for decision-making such as tic-tac-toe, the ultimatum game or the prisoner's dilemma

Here, we develop continuous sensorimotor versions of the prisoner's dilemma and the rope-pulling game. In the classical prisoner's dilemma

(A) Pay-off matrix for the classical prisoner's dilemma for two players (players denoted by red and blue). Depending on the choice of each player there are four different outcomes in terms of years that each player will serve in prison. (B) The motor version of the prisoner's dilemma. Each player controls a cursor and moves from a starting bar to a target bar and experiences a force that resists forward motion. The force arises from a virtual spring that attaches the handle to the starting bar (the springs are only shown on the schematic and are not visible to the players). The stiffness of the springs (K_{1} & K_{2}) can vary online and each depends on the x-positions of both players' cursors (x_{1} & x_{2}). (C) Continuous cost landscape for the motor prisoner's dilemma game. Each pair of x-positions (x_{1}, x_{2}) corresponds to a spring constant for each player. The corners of the plane correspond to the classical prisoner's dilemma matrix (A) and intermediate spring constants are obtained by linear interpolation. The current spring constants experienced by the players in B are shown by the points on the surface. The game was played by eight pairs of players and by eight individual players bimanually.

In our continuous sensorimotor version of prisoner's dilemma (see

Thus, our motor version of the prisoner's dilemma differs from the classic discrete version of the game in at least three different aspects. First, actions are continuous such that there is a continuous coupling between the two players. Second, reward in terms of money or years is replaced by an implicit cost, that is effort. Third, subjects have to learn their optimal strategy since they are unaware of the structure of the coupling, i.e. they have incomplete information about the payoffs. We found a clear distinction between the strategies used at the end of a set for the one-player and the two-player conditions. In

(A) Endpoint distribution of handle positions in the four quadrants corresponding to the cooperate defect (lateral movement) plane with the cooperative solution (top left quadrant), the Nash solution (bottom right quadrant) and the two exploitative solutions (top right or bottom left quadrant). Each plot shows one of the eight games in the two-player version of the prisoner's dilemma. The data is shown for the last 20 trials in each set. (B) Histogram over the four quadrants. C corresponds to cooperation and D to defection. All eight participant-pairs show a strong tendency towards the Nash solution.

(A) Endpoint distribution of handle positions in the four quadrants corresponding to the cooperate defect (lateral movement) plane with the cooperative solution (top left quadrant), the Nash solution (bottom right quadrant) and the two exploitative solutions (top right or bottom left quadrant). Each plot shows one of the eight participants. The data is shown for the last 20 trials in each set. (B) Histogram over the four quadrants. C corresponds to cooperation and D to defection. All eight participants had a strong preference for the cooperative solution.

To investigate the temporal evolution of learning we analyzed the trial-by-trial behavior of the players averaged across all sets. Initially, in both the one-player and the two-player conditions, players acted at chance level in their strategy (

(A,B) The evolution of the probability of cooperative (blue) and Nash (red) solutions across a 40 trial set for the one-player and two-player conditions. In the one-player condition the cooperative solution gains most probability, where as in the two-player condition the Nash solution is predominant. (C,D) The evolution of the probability of the exploitative solutions across the same set of trials for the one-player and two-player conditions. The shading is one standard error of the mean across the participants. As there are four possible behaviors chance level is shown at a probability of 0.25.

In our sensorimotor version of the prisoner's dilemma the cooperative and Nash solutions are two extremes of the one-dimensional control variable (lateral position at the target bar). Therefore, we designed a motor task based on another game, the ‘rope-pulling-game’, which has three additional features. First, the control variable is two-dimensional and the Nash and cooperative solutions are no longer at the boundaries of the control space. Second, unlike the prisoner's dilemma, where each player can achieve their task (reaching the bar) without paying attention to the strategy of the other player, in the new task, coordination is required between the players to jointly achieve the task. Third, the rope-pulling game can be translated into a linear dynamical system allowing for analytical solutions in terms of feedback policies (see

(A) The rope-pulling game in which a mass (circle) is pulled by two players. The arrows show the direction of force for two players for the Nash and cooperative solutions. Red and blue colors represent right and left handles throughout. (B) The motor version of the rope-pulling game. The position of a virtual mass is the sum of the displacements of the two handle positions from their origin (blue and red displacement vectors). However, the visual feedback is only a one-dimensional cursor location that is the y and x values of the mass position for players 1 and 2 respectively. Each player is required to reach a visual target with their cursor. Each robot was used to simulate the forces that would arise from a spring (with constant stiffness) attached between the handle and its origin. The arrow vectors and springs are only shown on the schematic and are not visible to the participants (grayed area not visible). For the one player game, a single participant controls both handles. The game was played by 4 pairs of participants and by 4 different participants individually. (C) Mean end points for each pair of left and right players for the last 40 trials in each set. The ellipses are centered at the average end points across all participants and indicate one standard error. (D) Smoothed frequency histograms (Gaussian kernel sd 20°) of pulling angles for the two-player condition (left: Nash equilibrium shown by vertical lines) and for the one player condition (right: cooperative solution shown by vertical line).

In our version of the rope-pulling game, player 1 and 2's task was to move a virtual mass to a fixed target that was equidistant to both players' origin. Again, each participant grasped the handle of the robotic interface and the location of the virtual mass was the sum of the (possibly rotated—see

We analyzed the distribution of pulling directions after learning (

In our study we have assessed human sensorimotor interactions based on game theoretic predictions with an implicit cost, that is effort. Effort, as a proxy for energy consumption, has been shown to be a fundamental determinant underlying how humans control their own movements

Our results contrast with those obtained in cognitive discrete games in interesting ways. For example, in the classical prisoner's dilemma, contrary to game-theoretic predictions, cooperation plays a significant role: players have been reported to cooperate almost half the time

In our motor version of the prisoner's dilemma the participants showed very little inclination towards cooperative solutions. This could have several reasons. Our participants knew, for example, that the experiment was going to last for 800 trials, i.e. assuming the participants had full knowledge of the game structure their defection is optimal – however, knowing the number of trials does not stop players in discrete cognitive games from cooperating. In our study the action space is continuous. A recent theoretical study has found, for example, that cooperative solutions are less stable in continuous environments where agents can make gradual distinctions of cooperativeness ranging from full cooperation to total defection

In our second motor game, the rope-pulling game, over all players we still observed that the Nash solution was the predominant solution for two-player interactions, but this time the inter-subject differences were quite considerable both in the two-player condition and in the bimanual case. One reason for this could be that the task was substantially more complex than the prisoner's dilemma task, especially in the bimanual case where two two-dimensional movements had to be performed simultaneously. Thus, incomplete learning might have played a crucial role. To model such states of incomplete information a special theory of Bayesian games has been devised dealing with so-called Bayes-Nash solutions

In both our motor games we compared performance of two-players with the performance of a single player. The underlying hypothesis was that the single player condition could be regarded as an instance of a cooperative game where the two motor hemispheres interact to achieve the task. If the two hemispheres were unable to cooperate, for example as might be expected in patients who have undergone commisurectomy

Forty-eight naïve participants provided written informed consent and took part in one of two motor games. The experiments were conducted using two planar robotic interfaces (vBOTs). Participants held the handle of the vBOT that constrained hand movements to the horizontal plane. The vBOT allowed us to record the position of the handle and to generate forces on the hand with a 1 kHz update rate. Using a projection system we overlaid virtual visual feedback into the plane of the movement

All experimental procedures were approved by the Psychology Research Ethics Committee of the University of Cambridge.

Each of the robot handles controlled the position of a cursor in one half of the horizontal workspace (

The position of each robot handle was expressed as a two-dimensional vector position where

Each game was played by eight pairs of participants and by eight different participants individually. All participants were instructed to achieve the task as easily as possible. Participants were also told the number of trials in each set, and the sets were separated during the experiment by short breaks. For the two-player game a divider was used to prevent the participants seeing the cursor or arm of the other player. In the single-player condition subjects saw the same screen that would be displayed to two players in the game condition.

Supplementary Materials

(0.15 MB PDF)

We thank Aldo Faisal, Ian Howard, James Ingram and Luc Selen for their assistance.