Maximizing Value in Challenge the Champ Tournaments

TL;DR

This work addresses the problem of maximizing the total value of matches in Challenge the Champ tournaments under various strength-graph models and valuation families. It develops polynomial-time solutions for DAG strength graphs in several settings (notably player-popularity-based and win-count-based valuations) and establishes NP-hardness for cyclic graphs, including binary and ternary valuations and linear-after-threshold variants. The results are built via dynamic programming on DAGs, backbone/caterpillar representations, and reductions from classic problems such as 3-D-Matching, revealing a nuanced complexity landscape and identifying when Challenge the Champ seeding is optimal among all single-elimination tournaments. The study also provides approximation techniques and highlights structural connections to Hamiltonian paths and caterpillar arborescences, with implications for broader tournament design and value maximization in competitive settings.

Abstract

A tournament is a method to decide the winner in a competition, and describes the overall sequence in which matches between the players are held. While deciding a worthy winner is the primary goal of a tournament, a close second is to maximize the value generated for the matches played, with value for a match measured either in terms of tickets sold, television viewership, advertising revenue, or other means. Tournament organizers often seed the players -- i.e., decide which matches are played -- to increase this value. We study the value maximization objective in a particular tournament format called Challenge the Champ. This is a simple tournament format where an ordering of the players is decided. The first player in this order is the initial champion. The remaining players in order challenge the current champion; if a challenger wins, she replaces the current champion. We model the outcome of a match between two players using a complete directed graph, called a strength graph, with each player represented as a vertex, and the direction of an edge indicating the winner in a match. The value-maximization objective has been recently explored for knockout tournaments when the strength graph is a directed acyclic graph (DAG). We extend the investigation to Challenge the Champ tournaments and general strength graphs. We study different representations of the value of each match, and completely characterize the computational complexity of the problem.

Figures (6)

• Figure 1: A Challenge the Champ tournament, with players seeded $(e_1,e_{2},\ldots,e_{n})$. In each round $i$, player $e_{i+1}$ challenges the current champ --- the winner of the previous round.
• Figure 2: The first figure gives an example with 7 players showing the seeding $(3,4,2,5,7,1,6)$. In the strength graph (not shown), each player $j$ defeats all players $i < j$. Hence player 7 is the strongest and player 1 the weakest. The second figure shows the resulting spanning caterpillar arborescence.
• Figure 3: An example of a (partial) strength graph with cycles where a Challenge the Champ tournament is not optimal. Each leaf $b_i$ beats all the non-leaf nodes except $a_{i}$, creating cycles. The root has popularity $2$, intermediate $a_i$ vertices have popularity $1$, and the leaves have popularity $0$. Here, the maximum value achievable by a single elimination tournament is $9$ whereas the maximum value achievable in a Challenge the Champ tournament is $7$.
• Figure 4: The red paths illustrate the Hamiltonian paths. A directed edge between two boxes represents all directed edges connecting the vertices from one box to those in the other.
• Figure 5: An edge from a vertex to a box denotes all directed edges between that vertex and vertices within the box in the same direction. Edges not depicted are arbitrarily directed. The dotted edges are directed from the vertices in the box to the outside vertex.

Theorems & Definitions (21)

• Proposition 1: redei1934kombinatorischer
• Theorem 2
• proof
• Theorem 3
• proof
• Proposition 3
• proof
• Theorem 4
• proof
• Theorem 5