On the purification of Nash equilibria of large games

We consider Salim Rashidʹs asymptotic version of David Schmeidlerʹs theorem on the purification of Nash equilibria. We show that, in contrast to what is stated, playersʹ payoff functions have to be selected from an equicontinuous family in order for Rashidʹs theorem to hold. That is, a bound on the diversity of payoffs is needed in order for such asymptotic results to be valid.