A competitive game method which can be executed via a network is achieved irrespective of the type of the game. Terminal devices receive personal data from a center server, and respectively determine whether it will become a master terminal device in an initial game progress unit based on master determination rules. After the initial master terminal device is determined, a game is started. The master terminal device determines the next master terminal device when the initial game progress unit is completed. The master terminal device transmits a switching notification designating the next master terminal device by, for example, describing an IP address to the other terminal devices. The master terminal device further transmits game data which is necessary for continuing the game to the next master terminal device.