A group of linked gaming machines generates one or more progressive jackpots, common to all the linked machines, where the amount of each jackpot is increased by a percentage of the wagers made by the players. In accordance with the invention, in order to provide incentive for the player to keep playing the machines and to add additional excitement, non-jackpot awards are paid out to players of the linked gaming machines based on triggering thresholds met by each jackpot. One type of triggering threshold may be a threshold value reached by a jackpot. Upon a jackpot reaching the threshold, one or more players will win an award. After an award win, another threshold level will then be set for the jackpot. Since the frequency of the awards being granted is generally higher than the frequency of the full jackpots being won, the players will anticipate awards being granted and will continue to play the machines. The linked gaming machines can be provided with the additional game with little or no changes to the game programs in the gaming machines, since only the existing jackpot meters are used in the game. The game may be carried out by a central processor or a master processor used for administering the jackpot.