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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 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 (K1 &                        K2) can vary online and each depends on the x-positions of both                        players' cursors (x1 & x2). (C)                        Continuous cost landscape for the motor prisoner's dilemma game.                        Each pair of x-positions (x1, x2) 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.
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pcbi-1000468-g001: The prisoner's dilemma motor game.(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 (K1 & K2) can vary online and each depends on the x-positions of both players' cursors (x1 & x2). (C) Continuous cost landscape for the motor prisoner's dilemma game. Each pair of x-positions (x1, x2) 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.

Mentions: Here, we develop continuous sensorimotor versions of the prisoner's dilemma and 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) and defection (claiming the other player is guilty). If both cooperate, they each receive a short sentence (3 years) whereas if both defect they each receive a moderate sentence (7 years). But if one cooperates while the other defects, the defector is freed and the cooperator receives a lengthy sentence (10 years). The globally optimal solution in which the players benefit the most is for both players to cooperate. However, if one of the players decides to defect, the defector reduces their sentence at the expense of the other player. In such a non-cooperative setting the stable Nash solution is for both players to defect. This Nash solution guarantees 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 action being chosen by the other player. The dilemma arises because the Nash solution is not identical to the globally optimal solution which is cooperative. The same dilemma occurs also in the rope-pulling game (given as a conceptual example in [15]) where each 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 direction and the other player is reward according to how far he pulls the mass in an orthogonal direction. Thus, the globally optimal solution is to cooperate and pull the mass along the diagonal. However, if one of the players defects and pulls into his 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 along his own direction. In the following we address the question whether human motor interactions 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 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 (K1 &                        K2) can vary online and each depends on the x-positions of both                        players' cursors (x1 & x2). (C)                        Continuous cost landscape for the motor prisoner's dilemma game.                        Each pair of x-positions (x1, x2) 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.
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Related In: Results  -  Collection

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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 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 (K1 & K2) can vary online and each depends on the x-positions of both players' cursors (x1 & x2). (C) Continuous cost landscape for the motor prisoner's dilemma game. Each pair of x-positions (x1, x2) 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.
Mentions: Here, we develop continuous sensorimotor versions of the prisoner's dilemma and 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) and defection (claiming the other player is guilty). If both cooperate, they each receive a short sentence (3 years) whereas if both defect they each receive a moderate sentence (7 years). But if one cooperates while the other defects, the defector is freed and the cooperator receives a lengthy sentence (10 years). The globally optimal solution in which the players benefit the most is for both players to cooperate. However, if one of the players decides to defect, the defector reduces their sentence at the expense of the other player. In such a non-cooperative setting the stable Nash solution is for both players to defect. This Nash solution guarantees 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 action being chosen by the other player. The dilemma arises because the Nash solution is not identical to the globally optimal solution which is cooperative. The same dilemma occurs also in the rope-pulling game (given as a conceptual example in [15]) where each 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 direction and the other player is reward according to how far he pulls the mass in an orthogonal direction. Thus, the globally optimal solution is to cooperate and pull the mass along the diagonal. However, if one of the players defects and pulls into his 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 along his own direction. In the following we address the question whether human motor interactions 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