The not-quite non-atomic game: Normal approximation

Measure games with a large number of players are frequently approximated by nonatomic games. In fact, however, while it is true that values of large measure games will, under certain reasonable circumstances, converge to the value of a non-atomic game, it is also true that this convergence is quite slow. Using the multi-linear extension and the central limit theorem, we obtain an approximation which (because it is based on the normal distribution) we call the normal approximation. We show that, for two examples with several hundred and several thousand players respectively, the normal approximation is much better than the non-atomic approximation.