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Nash equilibria in multi-agent motor interactions.

Braun DA, Ortega PA, Wolpert DM - PLoS Comput. Biol. (2009)

Bottom Line: 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.

View Article: PubMed Central - PubMed

Affiliation: Computational and Biological Learning Laboratory, Department of Engineering, University of Cambridge, Cambridge, UK. dab54@cam.ac.uk

ABSTRACT
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.

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The prisoner's dilemma motor game.(A) Pay-off matrix for the classical prisoner's dilemma for twoplayers (players denoted by red and blue). Depending on the choice of eachplayer there are four different outcomes in terms of years that each playerwill serve in prison. (B) The motor version of the prisoner'sdilemma. Each player controls a cursor and moves from a starting bar to atarget bar and experiences a force that resists forward motion. The forcearises 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 theplayers). The stiffness of the springs (K1 &K2) can vary online and each depends on the x-positions of bothplayers' cursors (x1 & x2). (C)Continuous cost landscape for the motor prisoner's dilemma game.Each pair of x-positions (x1, x2) corresponds to aspring constant for each player. The corners of the plane correspond to theclassical prisoner's dilemma matrix (A) and intermediate springconstants are obtained by linear interpolation. The current spring constantsexperienced by the players in B are shown by the points on the surface. Thegame was played by eight pairs of players and by eight individual playersbimanually.
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pcbi-1000468-g001: The prisoner's dilemma motor game.(A) Pay-off matrix for the classical prisoner's dilemma for twoplayers (players denoted by red and blue). Depending on the choice of eachplayer there are four different outcomes in terms of years that each playerwill serve in prison. (B) The motor version of the prisoner'sdilemma. Each player controls a cursor and moves from a starting bar to atarget bar and experiences a force that resists forward motion. The forcearises 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 theplayers). The stiffness of the springs (K1 &K2) can vary online and each depends on the x-positions of bothplayers' cursors (x1 & x2). (C)Continuous cost landscape for the motor prisoner's dilemma game.Each pair of x-positions (x1, x2) corresponds to aspring constant for each player. The corners of the plane correspond to theclassical prisoner's dilemma matrix (A) and intermediate springconstants are obtained by linear interpolation. The current spring constantsexperienced by the players in B are shown by the points on the surface. Thegame was played by eight pairs of players and by eight individual playersbimanually.

Mentions: Here, we develop continuous sensorimotor versions of the prisoner's dilemmaand the rope-pulling game. In the classical prisoner's dilemma [16],two players (prisoners) have a choice (Fig. 1A) between cooperation (claiming the other player is innocent) anddefection (claiming the other player is guilty). If both cooperate, they eachreceive a short sentence (3 years) whereas if both defect they each receive amoderate sentence (7 years). But if one cooperates while the other defects, thedefector is freed and the cooperator receives a lengthy sentence (10 years). Theglobally optimal solution in which the players benefit the most is for both playersto cooperate. However, if one of the players decides to defect, the defector reducestheir sentence at the expense of the other player. In such a non-cooperative settingthe stable Nash solution is for both players to defect. This Nash solutionguarantees in this case that a player minimizes their maximum expected punishment(in this case 7 years) and the player does not have to rely on a particular actionbeing chosen by the other player. The dilemma arises because the Nash solution isnot identical to the globally optimal solution which is cooperative. The samedilemma occurs also in the rope-pulling game (given as a conceptual example in [15]) whereeach of two players is attached by a rope to a mass that they have to pull together.One player is rewarded according to how far he pulls the mass along one directionand the other player is reward according to how far he pulls the mass in anorthogonal direction. Thus, the globally optimal solution is to cooperate and pullthe mass along the diagonal. However, if one of the players defects and pulls intohis own direction he gains even more payoff at the expense of the other player.Therefore, the stable Nash solution in this case is for each player to pull alonghis own direction. In the following we address the question whether human motorinteractions in such motor games can be quantified using a game theoretic framework.


Nash equilibria in multi-agent motor interactions.

Braun DA, Ortega PA, Wolpert DM - PLoS Comput. Biol. (2009)

The prisoner's dilemma motor game.(A) Pay-off matrix for the classical prisoner's dilemma for twoplayers (players denoted by red and blue). Depending on the choice of eachplayer there are four different outcomes in terms of years that each playerwill serve in prison. (B) The motor version of the prisoner'sdilemma. Each player controls a cursor and moves from a starting bar to atarget bar and experiences a force that resists forward motion. The forcearises 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 theplayers). The stiffness of the springs (K1 &K2) can vary online and each depends on the x-positions of bothplayers' cursors (x1 & x2). (C)Continuous cost landscape for the motor prisoner's dilemma game.Each pair of x-positions (x1, x2) corresponds to aspring constant for each player. The corners of the plane correspond to theclassical prisoner's dilemma matrix (A) and intermediate springconstants are obtained by linear interpolation. The current spring constantsexperienced by the players in B are shown by the points on the surface. Thegame was played by eight pairs of players and by eight individual playersbimanually.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC2714462&req=5

pcbi-1000468-g001: The prisoner's dilemma motor game.(A) Pay-off matrix for the classical prisoner's dilemma for twoplayers (players denoted by red and blue). Depending on the choice of eachplayer there are four different outcomes in terms of years that each playerwill serve in prison. (B) The motor version of the prisoner'sdilemma. Each player controls a cursor and moves from a starting bar to atarget bar and experiences a force that resists forward motion. The forcearises 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 theplayers). The stiffness of the springs (K1 &K2) can vary online and each depends on the x-positions of bothplayers' cursors (x1 & x2). (C)Continuous cost landscape for the motor prisoner's dilemma game.Each pair of x-positions (x1, x2) corresponds to aspring constant for each player. The corners of the plane correspond to theclassical prisoner's dilemma matrix (A) and intermediate springconstants are obtained by linear interpolation. The current spring constantsexperienced by the players in B are shown by the points on the surface. Thegame was played by eight pairs of players and by eight individual playersbimanually.
Mentions: Here, we develop continuous sensorimotor versions of the prisoner's dilemmaand the rope-pulling game. In the classical prisoner's dilemma [16],two players (prisoners) have a choice (Fig. 1A) between cooperation (claiming the other player is innocent) anddefection (claiming the other player is guilty). If both cooperate, they eachreceive a short sentence (3 years) whereas if both defect they each receive amoderate sentence (7 years). But if one cooperates while the other defects, thedefector is freed and the cooperator receives a lengthy sentence (10 years). Theglobally optimal solution in which the players benefit the most is for both playersto cooperate. However, if one of the players decides to defect, the defector reducestheir sentence at the expense of the other player. In such a non-cooperative settingthe stable Nash solution is for both players to defect. This Nash solutionguarantees in this case that a player minimizes their maximum expected punishment(in this case 7 years) and the player does not have to rely on a particular actionbeing chosen by the other player. The dilemma arises because the Nash solution isnot identical to the globally optimal solution which is cooperative. The samedilemma occurs also in the rope-pulling game (given as a conceptual example in [15]) whereeach of two players is attached by a rope to a mass that they have to pull together.One player is rewarded according to how far he pulls the mass along one directionand the other player is reward according to how far he pulls the mass in anorthogonal direction. Thus, the globally optimal solution is to cooperate and pullthe mass along the diagonal. However, if one of the players defects and pulls intohis own direction he gains even more payoff at the expense of the other player.Therefore, the stable Nash solution in this case is for each player to pull alonghis own direction. In the following we address the question whether human motorinteractions in such motor games can be quantified using a game theoretic framework.

Bottom Line: 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.

View Article: PubMed Central - PubMed

Affiliation: Computational and Biological Learning Laboratory, Department of Engineering, University of Cambridge, Cambridge, UK. dab54@cam.ac.uk

ABSTRACT
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.

Show MeSH
Related in: MedlinePlus