In a race game comprising a preliminary round and a main race, a player can start the game without waiting for other players, the boredom of waiting time is eliminated, and skillful players can be satisfied with the game. An image processing device for generating game images, and a plurality of game units each provided respectively with communications control devices connected to this image processing device, which control communications with other devices, are provided, and the communications control devices are connected to each other. When any of the players inserts a coin, the corresponding communications control device immediately starts a preliminary round, and when this preliminary round is completed, it displays a demonstration screen during the waiting period before the start of the main race. In the main race, it determines the start position of the player according to the results from the preliminary round, suspends the game for any player who runs out of time, and when a main race is finished, it displays a demonstration screen until the start of the next main race.