  In this paper, we use mean field games (MFGs) to investigate approximations
of $N$-player games with uniformly symmetrically continuous heterogeneous
closed-loop actions. To incorporate agents' risk aversion (beyond the classical
expected total costs), we use an abstract evaluation functional for their
performance criteria. Centered around the notion of regret, we conduct
non-asymptotic analysis on the approximation capability of MFGs from the
perspective of state-action distribution without requiring the uniqueness of
equilibria. Under suitable assumptions, we first show that scenarios in the
$N$-player games with large $N$ and small average regrets can be well
approximated by approximate solutions of MFGs with relatively small regrets. We
then show that $\delta$-mean field equilibria can be used to construct
$\varepsilon$-equilibria in $N$-player games. Furthermore, in this general
setting, we prove the existence of mean field equilibria.
