  This paper proposes and studies a general form of dynamic $N$-player
non-cooperative games called $\alpha$-potential games, where the change of a
player's value function upon her unilateral deviation from her strategy is
equal to the change of an $\alpha$-potential function up to an error $\alpha$.
Analogous to the static potential game (which corresponds to $\alpha=0$), the
$\alpha$-potential game framework is shown to reduce the challenging task of
finding approximate Nash equilibria for a dynamic game to minimizing the
$\alpha$-potential function. Moreover, an analytical characterization of
$\alpha$-potential functions is established, with $\alpha$ represented in terms
of the magnitude of the asymmetry of value functions' second-order derivatives.
For stochastic differential games in which the state dynamic is a controlled
diffusion, $\alpha$ is explicitly identified in terms of the number of players,
the choice of admissible strategies, and the intensity of interactions and the
level of heterogeneity among players. This is achieved by introducing a
suitable linear derivative of the value functions with respect to unilateral
deviations of strategies and via analyzing the sensitivity processes of state
dynamics with respect to controls.
  For games with mean-field type interactions, $\alpha$ is shown to decay to
zero as the number of players goes to infinity, even with heterogeneity in
state dynamics, cost functions, and admissible strategy classes. For
distributed games, if a static potential function can be derived from the cost
functions, then $\alpha=0$. For crowd aversion games, $\alpha$ is capable of
capturing the subtle difference between the choice of admissible strategies.
