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 \textsc{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 \textsc{VnaR}.

翻译：挑战冠军是一种简单的锦标赛形式，其中需要确定选手的排序——称为种子排序。排序中的首位选手为初始冠军，并与下一位选手对决。每场比赛的结果决定当前冠军，该冠军将迎战排序中的下一位选手。每位选手还具有受欢迎度，每场比赛的价值等于获胜者的受欢迎度。先前已有研究在每场比赛结果确定的情况下锦标赛中的价值最大化问题。然而，比赛结果通常具有概率性而非确定性。本研究探讨当比赛胜者具有概率性时，挑战冠军锦标赛中的稳健价值最大化问题。即我们寻求最大化获得的总价值，且该最大化策略不受概率性比赛结果的影响。我们证明即使在简单的二元设定中，对于非自适应算法，最优稳健价值——我们称之为\textsc{VnaR}（无风险价值）——也难以近似求解。然而，若允许采用根据先前比赛结果动态调整挑战者顺序的自适应算法，或限制具有概率性结果的比赛场次，我们能够获得对最优\textsc{VnaR}的良好近似解。