离散竞标博弈中的阈值预算计算
Computing Threshold Budgets in Discrete-Bidding Games
AI总结：
本文研究离散竞标博弈中阈值预算的计算问题，提出定点算法揭示奇偶博弈阈值预算结构，并证明该问题属于NP与coNP，且策略仅需线性内存。
摘要：
在两人零和图上博弈中，玩家在图上移动一个令牌以产生无限博弈过程，该过程决定博弈的胜者。竞标博弈是一类图博弈，其中每一回合由一次拍卖（竞标）决定哪位玩家移动令牌：玩家拥有预算，每一回合双方同时提交不超过其可用预算的竞标，出价较高者移动令牌，并向出价较低者支付其竞标金额（称为Richman竞标）。我们专注于离散竞标博弈，受实际应用驱动，玩家竞标的粒度受到限制，例如竞标必须以分为单位给出。竞标博弈中的一个核心量是阈值预算：赢得博弈所需的充分必要初始预算。此前，阈值已被证明在奇偶博弈中存在，但其结构仅在可达性博弈中被理解。此外，先前已知的算法对于可达性和奇偶目标在最坏情况下具有指数运行时间，并且输出的策略使用指数级内存。我们描述了两种在奇偶离散竞标博弈中寻找阈值预算的算法。第一种是定点算法。它首次揭示了奇偶离散竞标博弈中阈值预算的结构。基于这一结构，我们开发了第二种算法，该算法表明对于可达性和奇偶目标，寻找阈值预算的问题属于NP和coNP。此外，我们的算法构造的策略仅使用线性内存。本文是原论文（arXiv:2210.02773v4）的修正版本，原论文最初于2025年1月22日发布。
英文摘要：
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. 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 Richman bidding). We focus on 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 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 coNP for both reachability and parity objectives. Moreover, our algorithm constructs strategies that use only linear memory. This is a corrected version of the paper (arXiv:2210.02773v4) published originally on Jan 22, 2025.
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