I find this expression strange because it's clearly widely used, but seems sort of "unofficial", the "official" version, meaning the one described in dictionaries and grammar books, being playing games with, not on.
Both expressions seem to be current and valid, according to Google Books. I'm just not entirely clear on the context and the usage.
My understanding is that playing games with implies some sort of tricky, mischievous behavior, which may or may not be intentionally malicious, like office politics or standing someone up on a date ("Jane didn't show up...again. Is she playing games with me?"), while playing games on implies a form of studied deception. ("The ad said that the comic book was in mint condition but there are barely noticeable tears... are they trying to play a game on me?")