Robust Value Maximization in Challenge the Champ Tournaments with Probabilistic Outcomes

2026-02-16

AI Summary

This study addresses the problem of maximizing Value-not-at-Risk (VnaR) in challenge-based tournament settings where match outcomes are probabilistic. It extends robust value maximization to stochastic scenarios by introducing the VnaR metric and proves that the problem is inapproximable under non-adaptive strategies. However, when adaptive scheduling of participants is permitted or reasonable structural assumptions are imposed on win probabilities, the paper develops efficient approximation algorithms. The technical approach integrates combinatorial optimization, approximation algorithm analysis, adaptive decision-making, and probabilistic graphical models, thereby elucidating the critical roles of adaptivity and probabilistic structure in determining the problem’s tractability.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Game Theory and Economic Paradigms: Adversarial Learning
• Search and Optimization: Adversarial Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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 \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}.

Problem

• Robust Value Maximization
• Challenge the Champ Tournaments
• Probabilistic Outcomes
• Value Not at Risk
• Tournament Seeding

Innovation

• robust value maximization
• Challenge the Champ tournament
• probabilistic outcomes
• adaptive algorithms
• VnaR