Discrete Richman-bidding scoring games

We study zero-sum combinatorial games within the framework of so-called Richman auctions (Lazarus et al., in Games No Chance 29:439–449, 1996). We modify the alternating play scoring ruleset Cumulative Subtraction (Cohensius et al., in Electron J Combin 26(P4):52, 2019), to a discrete bidding scheme, similar to Develin and Payne (Electron J Combin 17(1):85, 2010). Players bid to move, and the player with the highest bid wins the move and hands over the winning bid amount to the other player. The new game is dubbed Bidding Cumulative Subtraction. 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 wins 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. We prove that for all sufficiently large heap sizes, the equilibrium outcomes of unitary games are eventually periodic, with period 2. We show that the periodicity appears at the latest for heaps of sizes quadratic in the total budget.