A primary or secondary game for a gaming device. In one embodiment, the player is provided up to three picks, i.e., picks three contestants. The first two contestants both form potential award offers. The game then provides the greater of the two potential award offers to the player as an initial offer. This initial sequence alternatives includes more than two picks, contestants and potential offers. The offer acceptance feature arises after the initial offer. The player must decide whether to keep the initial offer or forego that offer for a final offer. If the player keeps the offer the game provides that initial offer to the player and ends if the player rejects that initial offer, the game proceeds to build a third and final offer, which is then provided to the player automatically.