[论文解读] Robust Value Maximization in Challenge the Champ Tournaments with Probabilistic Outcomes

Umang Bhaskar, Juhi Chaudhary+3 more|arXiv (Cornell University)|Feb 16, 2026

Artificial Intelligence in Games被引用 0

一句话总结

论文研究具有概率结果的挑战冠军赛中鲁棒价值最大化，证明非自适应种子分配的困难性，并提供具有可证明近似到最优 VnaR 的自适应策略。

ABSTRACT

Challenge the Champ is a simple tournament format, where an ordering of the players -- called a seeding -- is decided. The first player in this order is the initial champ, and faces the next player. The outcome of each match decides the current champion, who faces the next player in the order. Each player also has a popularity, and the value of each match is the popularity of the winner. Value maximization in tournaments has been previously studied when each match has a deterministic outcome. However, match outcomes are often probabilistic, rather than deterministic. We study robust value maximization in Challenge the Champ tournaments, when the winner of a match may be probabilistic. That is, we seek to maximize the total value that is obtained, irrespective of the outcome of probabilistic matches. We show that even in simple binary settings, for non-adaptive algorithms, the optimal robust value -- which we term the extsc{VnaR}, or the value not at risk -- is hard to approximate. However, if we allow adaptive algorithms that determine the order of challengers based on the outcomes of previous matches, or restrict the matches with probabilistic outcomes, we can obtain good approximations to the optimal extsc{VnaR}.

研究动机与目标

• 在概率比赛结果下通过最大化不在风险中的价值（VnaR）来激励鲁棒锦标赛设计。
• 刻画非自适应种子分配的计算上界并开发具有保证的自适应方法。
• 将 VnaR 与骨干/有向树结构联系起来，并在自适应下给出算法结果。

提出的方法

• 将 VnaR 定义为在所有概率比赛 realizations 中的最小锦标赛值。
• 证明对最优 VnaR 的近似在非自适应种子分配下的困难性。
• 引入依赖于前几场比赛结果的自适应种子分配算法。
• 将 VnaR 的近似与受欢迎球员子图的染色数相关联。
• 给出在确定性或受限不确定性设定下可获得有效近似的条件。
• 建立一个多项式时间的自适应算法，达到 VnaR ≥ (n_p + n_u − 1) / 2。

Figure 1: Illustration of a strength graph, and a spanning arborescence. The instance contains two popular players, $a$ and $b$ , and eight unpopular players $x_{1},\ldots,x_{4}$ and $y_{1},\ldots,y_{4}$ . Solid(dashed) edges are deterministic(uncertain). Edges between unpopular players are uncertai

实验结果

研究问题

• RQ1 在非自适应种子分配下，达到最优 VnaR 的加法或乘法近似的计算困难性是什么？
• RQ2 在概率比赛结果下，自适应种子分配策略是否能实现对最优 VnaR 的有意义近似？
• RQ3 不确定性的结构（例如流向热门玩家之间的边与不热门玩家之间的边）如何影响可实现的保证？
• RQ4 图的染色数和顶点覆盖等图参数在自适应 VnaR 保证中的作用是什么？

主要发现

• 在非自适应多项式时间的种子分配下，除非 P = NP 或 ETH 失败，否则无法实现对最优 VnaR 的良好加法或乘法近似。
• 一个自适应种子分配算法能够实现依赖于热门玩家子图的染色数的 VnaR 加法近似。
• 如果每条不确定边都位于热门集合内或不热门集合内，则可以在没有自适应的情况下获得与自适应方法性能相匹配的保证。
• 存在一个多项式时间的自适应算法，保证 VnaR 至少为 (n_p + n_u - 1) / 2。
• VnaR 的上界为 n_p + n_u − 1，为近似提供了自然基准。

Figure 2: The backbone witnessed by the seeding $\langle v^{n_{p}}_{p},v^{n_{p}}_{u},\dots,v^{1}_{p},v^{1}_{u}\rangle$