A gaming system and methods provide a plurality of gaming devices configured to operate a multiple-player bonus or secondary game. Each gaming device of the gaming system generates targets upon occurrences of target-generation events. Each gaming device also generates collectors upon occurrences of collector-generating events. The collectors may be used in conjunction with the aforementioned targets to obtain awards in a bonus sequence as described. Upon the occurrence of a redemption event at one of the plurality of gaming devices, the gaming device provides all players playing at one of the gaming devices in the gaming system the opportunity to redeem all accumulated collectors and targets in a bonus sequence. In the bonus sequence, the gaming system determines any awards associated with the redemption of accumulated collectors and targets and provides them to the players.