In a two-player zero-sum graph game, the players move a token throughout a graph to produce an infinite play, which determines the winner of the game. \emph{Bidding games} are graph games in which in each turn, an auction (bidding) determines which player moves the token: the players have budgets, and in each turn, both players simultaneously submit bids that do not exceed their available budgets, the higher bidder moves the token, and pays the bid to the lower bidder (called {\em Richman} bidding). We focus on {\em discrete}-bidding games, in which, motivated by practical applications, the granularity of the players' bids is restricted, e.g., bids must be given in cents. A central quantity in bidding games is are {\em threshold budgets}: a necessary and sufficient initial budget for winning the game. Previously, thresholds were shown to exist in parity games, but their structure was only understood for reachability games. Moreover, the previously-known algorithms have a worst-case exponential running time for both reachability and parity objectives, and output strategies that use exponential memory. We describe two algorithms for finding threshold budgets in parity discrete-bidding games. The first is a fixed-point algorithm. It reveals, for the first time, the structure of threshold budgets in parity discrete-bidding games. Based on this structure, we develop a second algorithm that shows that the problem of finding threshold budgets is in \NP and co\NP for both reachability and parity objectives. Moreover, our algorithm constructs strategies that use only linear memory.

翻译：在双人零和图博弈中，玩家通过移动图上的标记生成无限博弈过程，从而决定博弈的胜负。"竞拍博弈"是一类图博弈，其中每轮通过拍卖（竞拍）决定哪一方移动标记：玩家拥有预算，每轮双方同时提交不超过各自可用预算的出价，出价高者移动标记，并将其出价支付给低出价方（称为"里奇曼竞拍"）。我们聚焦于"离散"竞拍博弈——受实际应用驱动，玩家出价的粒度受限（例如必须以美分为单位出价）。竞拍博弈的核心量是"阈值预算"：赢得博弈的充分必要条件。此前，阈值已被证明存在于奇偶博弈中，但其结构仅在可达博弈中得以理解。此外，现有算法在可达性与奇偶性目标下均具有最坏情况指数级运行时间，且输出策略需指数级内存。我们提出两种用于求解奇偶离散竞拍博弈中阈值预算的算法。第一种是定点算法，首次揭示了奇偶离散竞拍博弈中阈值预算的结构。基于此结构，我们开发了第二种算法，证明在可达性及奇偶性目标下，求解阈值预算问题属于NP与coNP。此外，我们的算法可构造仅需线性内存的策略。