At least, he plays one in a new movie Fading Gigolo, written and directed by John Turturro. In a believable/not believable at all turn of events, Turturro (a "middle-aged neophyte hustler") is recruited by Allen (a "bookseller-turned-pimp") to get involved with women ranging from Sharon Stone to Sofia Vergara in order to make some quick cash. The phrase ménage à trois is used more than once.
The movie also stars Liev Schreiber, Vanessa Paradis and Jill Scott, among others. It looks like Woody Allen at his most Woody Allen and yes, that is a compliment this one time.