This is a tough question: if the Japanese release date for the arcade Mario Bros. was 7/14/83, then wouldn't I be led to believe that the U.S. release of the same game in arcades was but a few weeks later (think the week ending 8/6/83 or 8/13/83), inching close to the later August release date for the Atari 2600 (VCS) console version?
~Ben