A gaming system including an offer and acceptance game including a series of offer component accumulations. One or more accumulated offer components are associated with one or more award values which form different offers for acceptance or rejection. One or more of such offer component accumulations are also associated with an elimination of one or more of any previously accumulated offer components. Such a configuration provides that when deciding whether to accept an offer (including the currently accumulated, non-eliminated offer components) or reject the offer and accumulate additional offer components, a player weighs the risks of eliminating one or more previously accumulated offer components.