[论文解读] Computing Threshold Budgets in Discrete-Bidding Games

Guy Avni, Suman Sadhukhan|arXiv (Cornell University)|Jan 1, 2022

Auction Theory and Applications被引用 2

一句话总结

本文提出了两种新颖的算法，用于计算离散出价公平博弈中的阈值预算，证明该问题属于NP与coNP。第一种算法通过不动点迭代揭示了阈值的结构，第二种算法仅使用线性内存构建策略，首次解决了离散出价博弈中关于复杂度与内存需求的开放问题。

ABSTRACT

In a two-player zero-sum graph game, the players move a token throughout the 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. We distinguish between continuous- and discrete-bidding games. In the latter, the granularity of the players' bids is restricted, e.g., bids must be given in cents. Continuous-bidding games are well understood, however, from a practical standpoint, discrete-bidding games are more appealing. In this paper we focus on discrete-bidding games. We study the problem of finding threshold budgets; namely, a necessary and sufficient initial budget for winning the game. Previously, the properties of threshold budgets were only studied for reachability games. For parity discrete-bidding games, thresholds were known to exist, but their structure was not understood. We describe two algorithms for finding threshold budgets in parity discrete-bidding games. The first algorithm is a fixed-point algorithm, and it reveals the structure of the threshold budgets in these games. Second, we show that the problem of finding threshold budgets is in NP and coNP for parity discrete-bidding games. Previously, only exponential-time algorithms where known for reachability and parity objectives. A corollary of this proof is a construction of strategies that use polynomial-size memory.

研究动机与目标

• 确定离散出价公平博弈中阈值预算——即确保获胜的最小初始预算——的值。
• 解决离散出价博弈中阈值预算计算的计算复杂度这一长期悬而未决的问题。
• 开发仅需线性内存的策略，克服先前构造中指数级内存需求的限制。
• 将阈值预算的结构性理解从可达性博弈扩展至公平博弈，揭示该情境下的平均性质。

提出的方法

• 引入一种不动点算法，通过在各顶点上强制满足平均性质，迭代计算阈值。
• 定义一种新颖的节俭公平目标，确保在到达目标顶点时施加预算约束。
• 构建一个回合制博弈模拟（𝐺𝑇,𝐺），将出价博弈简化为标准公平博弈以进行分析。
• 使用带有阈值函数 𝑇′ 的对偶变换，通过与玩家2的获胜条件对偶，验证阈值的下界。
• 采用阈值函数的多项式大小表示，并利用无记忆策略在多项式时间内验证其正确性。
• 利用从回合制公平博弈到离散出价博弈的约化，建立复杂度界限。

实验结果

研究问题

• RQ1 离散出价公平博弈中的阈值预算是否满足平均性质，如同在可达性博弈中所见？
• RQ2 当预算以二进制形式给出时，计算离散出价公平博弈中阈值预算的问题是否属于NP与coNP？
• RQ3 是否可以仅使用线性内存构造离散出价公平博弈的获胜策略，而非指数级内存？
• RQ4 在公平离散出价博弈中，除了可达性之外，阈值预算分布的结构性质是什么？

主要发现

• 离散出价公平博弈中的阈值预算满足平均性质，该性质此前仅在可达性博弈中被知晓。
• 即使预算以二进制形式给出，计算离散出价公平博弈中阈值预算的问题也属于NP与coNP。
• 本文构造的获胜策略仅需线性内存，相较于先前的指数级内存构造有显著改进。
• 阈值函数可表示为多项式大小，从而实现对候选解的高效验证。
• 从回合制公平博弈到离散出价博弈的新型约化确认，尽管增加了出价机制的复杂性，其复杂度类仍保持为NP ∩ coNP。
• 不动点算法提供了一种系统化方法，将可达性博弈中的结构性洞见推广至公平博弈，从而支持新的算法方法。