The Duck Race is a secret mini-game in Shenmue II.
To unlock the Duck Race you need to get the Bronze and Silver Medals. Ryo Hazuki has to defeat both Izumi Takano and Eileen M. Edelweiss by going up to the 3rd floor in the Pine Game Arcade. Once Izumi is defeated, she will ask Ryo to visit her at the Tomato Convenience Store in Golden Qr. Then she will show you something interesting when you press the X button, now from this point on you can go to the Duck Races.