  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.

Computer Science and Game Theory;Combinatorics