I have been "flirting" with game theory on and off since a
few years but only recently have started to grasp some of its
fundamental ideas, and appreciate its importance
Having come from a background in quantum and classical many-interacting systems, right from the begin I couldn't help but wonder about if there are links (or even deep connections) between game theory, essentially a theory of interactive decision making, and theory of many-particle systems.
Taking a classical systems where particles interact through conservative forces as example:
-each particle can be considered as player
the set of actions for each particle (player) are its position and velocity (the action space is continuous)
-the utility of each particle is the sum of its kinetic and potential energy
-players are engaged in a game of sequential moves
-in each move a particle chooses the action that minimizes its energy
I think this rules would result in the conventional molecular dynamics
For quantum systems, one interesting aspect (I think) is that particles action space is their positions {r_1, ... r_N} and the strategies are mixed strategies, with \psi* \psi, being the joint probability distribution that particle 1 selects action r_1, particle 2 selects action r_2, particle N selects action r_N.
One interesting observation is that in "physical" systems the laws of motion, i.e. how a particle selects its next move, are derived by minimizing a global function, i.e the systems Lagrangian, i.e. somehow it is hypnotized that all particle would move in away that would result in minimization of the global function. On the other hand, in "non-cooperative game theory each player is attempting to maximize its own utility, and generally there is not a notion of a global function. Only in especial cases, like potential games (I believe) utility optimization by selfish players result in optimization of a global function.
Perhaps interestingly one might pose the question, why in the case of physical systems the general rule seems to be that the rules obeyed by individual particles should be derived by hypothesizing global minimization.
Other interesting avenues to explore are if notions such as mean-field solutions, or even Walter Kohn's Density Functional Theory have analogies in many-player game theories.
Having come from a background in quantum and classical many-interacting systems, right from the begin I couldn't help but wonder about if there are links (or even deep connections) between game theory, essentially a theory of interactive decision making, and theory of many-particle systems.
Taking a classical systems where particles interact through conservative forces as example:
-each particle can be considered as player
the set of actions for each particle (player) are its position and velocity (the action space is continuous)
-the utility of each particle is the sum of its kinetic and potential energy
-players are engaged in a game of sequential moves
-in each move a particle chooses the action that minimizes its energy
I think this rules would result in the conventional molecular dynamics
For quantum systems, one interesting aspect (I think) is that particles action space is their positions {r_1, ... r_N} and the strategies are mixed strategies, with \psi* \psi, being the joint probability distribution that particle 1 selects action r_1, particle 2 selects action r_2, particle N selects action r_N.
One interesting observation is that in "physical" systems the laws of motion, i.e. how a particle selects its next move, are derived by minimizing a global function, i.e the systems Lagrangian, i.e. somehow it is hypnotized that all particle would move in away that would result in minimization of the global function. On the other hand, in "non-cooperative game theory each player is attempting to maximize its own utility, and generally there is not a notion of a global function. Only in especial cases, like potential games (I believe) utility optimization by selfish players result in optimization of a global function.
Perhaps interestingly one might pose the question, why in the case of physical systems the general rule seems to be that the rules obeyed by individual particles should be derived by hypothesizing global minimization.
Other interesting avenues to explore are if notions such as mean-field solutions, or even Walter Kohn's Density Functional Theory have analogies in many-player game theories.
