A dedicated gaming device or an on-line game is described wherein the player first plays a primary game. For certain outcomes of the primary game, the player accumulates a bonus point. The player accumulates the bonus points until a certain number of bonus points are accumulated. The player then wins the bonus associated with those bonus points. Accordingly, the player has an incentive to keep playing the on-line game or dedicated gaming machine to ma ke use of the accumulated bonus points that lead to a bonus award. In a further embodiment, a player logs onto the on-line gaming site or begins play of a dedicated gaming machine, and the accumulated bonus points from the player's last game before cashing out is displayed for the current session. Accordingly, the player resumes play with all the accumulated bonus points from the previous session. This provides an incentive for the player to retu rn to the same game whether in a casino or on-line.
ACCUMULATION OF BONUS POINTS IN A GAMBLING GAME