inertial dynamics for games Nash-Kuiper theorem nash equilirium

http://www.atzjg.net/admin/do/listqsbyctgry.php?cid=120

http://arxiv.org/pdf/1305.0967.pdf

在经典黎曼几何中, 我们从光滑流形 X  开始, 然后研究丛 S 2 T ∗ X  的光滑、正定截面 g  . 为了引进协变导数和曲率的基本概念(cf.[Grl-Kl-Mey] or [Milnor] MT   , Ch.2), 仅运用了 g  的可微性而并未用到它的正定性, 正如在广义相对论中使用洛伦兹几何(Lorentzian geometry)阐述的那样. 与之对比, X  中曲线长度的概念以及相应于度量 g  的测地距离概念依赖于下面的事实: g  在 X  的切空间 T x X  上给出了一族连续范数. 我们将研究长度(length)和距离(distance)的相关概念.

 

NS TO CONSTRAINED OPTIMIZATION

 

 

RIDA LARAKI AND PANAYOTIS MERTIKOPOULOS

Abstract. We derive a class of inertial dynamics for games and constrained optimization

 

 

problems over simplices by building on the well-known “heavy ball

with friction” method. In the single-agent case, the dynamics are generated by

endowing the problem’s configuration space with a Hessian–Riemannian structure

and then deriving the equations of motion for a particle moving under the

influence of the problem’s objective (viewed as a potential field); for normal form

 

games, the procedure is similar, but players are instead driven by the unilateral

 

 

gradient of their payoff functions. By specifying an explicit Nash–Kuiper embedding

of the simplex, we show that global solutions exist if and only if the interior

of the simplex is mapped isometrically to a closed hypersurface of some ambient

Euclidean space, and we characterize those Hessian–Riemannian structures

which have this property. For such structures, low-energy solutions are attracted

to isolated minimizers of the potential, showing in particular that pure Nash

equilibria of potential games are attracting; more generally, when the game is not

a potential one, we establish an inertial variant of the folk theorem of evolutionary

game theory,

 

 

 

 

 

等效原理与引力的规范理论

郝刘祥

 ( 中国科学院自然科学史研究所，北京100190)