A gaming device having an offer/acceptance type bonus game. One embodiment provides a plurality player selectable selections wherein each selection has an associated offer, anti-terminator or terminator. The gaming device enables the player to pick the selections. If a terminator is associated with the selection, the game ends unless the player has previously picked an anti-terminator associated with a selection. When the player picks an anti-terminator, the processor retains or accumulates the anti-terminator until a terminator is selected. The anti-terminator nullifies the effect of the terminator and allows the game to continue. The gaming device thus includes the combination of an offer acceptance scheme having at least one terminator and at least one anti-terminator.