Robust Value Maximization in Challenge the Champ Tournaments with Probabilistic Outcomes

TL;DR

This paper investigates robust strategies for maximizing total value in Challenge the Champ tournaments with probabilistic match outcomes, highlighting computational challenges and proposing adaptive algorithms for better approximations.

Contribution

It introduces the concept of robust value maximization under probabilistic outcomes and analyzes the complexity, proposing adaptive algorithms for improved approximation.

Findings

Optimal robust value is hard to approximate in simple binary settings.

Adaptive algorithms can achieve better approximations to the optimal value.

Restricting probabilistic matches simplifies the problem.

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}.