The Big Bronze Medal is an award that can be obtained by Ryo Hazuki. Once this medal is obtained, Ryo has access to the street fights in Pine Game Arcade.
The most simple way is to grab it from the Vacant Lot in the Green Market Quarter. Another way is to give the waitress at the Lai Lai Eatery in Aberdeen some money after causing trouble while chasing Wong. The last way is to win the Roll It On Top stand in the Lucky Charm Quarter five times in a row.