  A systematic framework for analyzing dynamical attributes of games has not
been well-studied except for the special class of potential or near-potential
games. In particular, the existing results have shortcomings in determining the
asymptotic behavior of a given dynamic in a designated game. Although there is
a large body literature on developing convergent dynamics to the Nash
equilibrium (NE) of a game, in general, the asymptotic behavior of an
underlying dynamic may not be even close to a NE. In this paper, we initiate a
new direction towards game dynamics by studying the fundamental properties of
the map of dynamics in games. To this aim, we first decompose the map of a
given dynamic into contractive and non-contractive parts and then explore the
asymptotic behavior of those dynamics using the proximity of such decomposition
to contraction mappings. In particular, we analyze the non-contractive behavior
for better/best response dynamics in discrete-action space sequential/repeated
games and show that the non-contractive part of those dynamics is well-behaved
in a certain sense. That allows us to estimate the asymptotic behavior of such
dynamics using a neighborhood around the fixed point of their contractive part
proxy. Finally, we demonstrate the practicality of our framework via an example
from duopoly Cournot games.
