We study zero-sum (combinatorial) games, within the framework of so-called Richman auctions (Lazarus et al. 1996) namely, we modify the alternating play scoring ruleset Cumulative Subtraction (CS) (Cohensius et al. 2019), to a discrete bidding scheme (similar to Develin and Payne 2010). Players bid to move and the player with the highest bid wins the move, and hands over the winning bidding amount to the other player. The new game is dubbed Bidding Cumulative Subtraction (BCS). In so-called unitary games, players remove exactly one item out of a single heap of identical items, until the heap is empty, and their actions contribute to a common score, which increases or decreases by one unit depending on whether the maximizing player won the turn or not. We show that there is a unique bidding equilibrium for a much larger class of games, that generalize standard scoring play in the literature. We prove that for all sufficiently large heap sizes, the equilibrium outcomes of unitary BCS are eventually periodic, with period 2, and we show that the periodicity appears at the latest for heaps of sizes quadratic in the total budget.