Enhanced amusement game tournament play is achieved by awarding prizes to one or more players maintaining a position on a threshold game score achieving, bump board listing, a prize qualifying cumulative threshold game score total, and/or a prize qualifying cumulative actual game score total wherein individual game scores are required to meet a threshold game score prior to being added to the cumulative total. These methods for awarding prizes may also include the awarding of prizes to one or more players achieving a prize qualifying individual game score and/or a prize qualifying cumulative game score total.