How to Make Knockout Tournaments More Popular?

TL;DR

This work studies how to seed knockout tournaments to maximize total profit or popularity when player strength induces a linear order. It formalizes Tournament Value Maximization with a seed-dependent tournament value $V_\sigma= abla_{(i,j,r)\, ext{played}} v(i,j,r)$ and analyzes complexity under various restrictions, establishing NP-hardness and APX-hardness for restricted game-value functions. On the algorithmic side, it delivers a $(1/\,\log n)$-approximation for round-oblivious values, a quasipolynomial DP for win-count oriented values, linear-time and FPT strategies for popularity-based settings, and an FPT algorithm parameterized by the size of an influential set of players. The results offer both theoretical hardness boundaries and practical seeds for maximizing tournament appeal, with open questions around non-round-oblivious values, stronger approximations, and alternative parameterizations. Overall, the paper lays a comprehensive foundation for algorithmic seed design that optimizes profitability and engagement in knockout formats.

Abstract

Given a mapping from a set of players to the leaves of a complete binary tree (called a seeding), a knockout tournament is conducted as follows: every round, every two players with a common parent compete against each other, and the winner is promoted to the common parent; then, the leaves are deleted. When only one player remains, it is declared the winner. This is a popular competition format in sports, elections, and decision-making. Over the past decade, it has been studied intensively from both theoretical and practical points of view. Most frequently, the objective is to seed the tournament in a way that "assists" (or even guarantees) some particular player to win the competition. We introduce a new objective, which is very sensible from the perspective of the directors of the competition: maximize the profit or popularity of the tournament. Specifically, we associate a "score" with every possible match, and aim to seed the tournament to maximize the sum of the scores of the matches that take place. We focus on the case where we assume a total order on the players' strengths, and provide a wide spectrum of results on the computational complexity of the problem.

Figures (7)

• Figure 1: An illustration of the seeding of the players in $T_{\phi}$ in \ref{['lem:1']}. Here, we assume that $x_{1}^{F/T}$ appears as a literal in clauses $c_{1}$, $c_{2}$, and $c_{3}$.
• Figure 2: An illustration of the seeding of the players in $T_{\phi}$ in \ref{['lem:4']}. Here, we assume that $x_{1}^{F/T}$ appears as a literal in clauses $c_{1}$, $c_{2}$, and $c_{3}$.
• Figure 3: Here, we are given an instance of the optimization version of Tournament Value Maximization with $n$ players and round-oblivious game-value function defined as: $v(n-1,j)=1$ for all $j\in[n-2]$, $v(n-1,n)=1+\varepsilon$ (where $\varepsilon>0$ is a constant), and $v(i,j)=0$ for all $i,j\in [n]\setminus \{n-1\}$. Now, note that Algorithm $\mathcal{A}$ (given in \ref{['thm:approx']}) will return a seeding with tournament value $1+\epsilon$ (by making sure that players $n$ and $n-1$ play against each other in the first round). However, the optimal tournament value for this given instance is $\log n+\epsilon$ (one of the possible seedings is $(n-1,1,2,\ldots,n-2,n)$).
• Figure 4: A tournament with $n=8$ players where the winning player $w^*$ is seeded into position one. The red vertices represent the roots of the three subtournaments that open.
• Figure 5: An illustration of the swapping of the players on positions $\sigma(i), \ldots, \sigma(i)+2^{r'}-1$ with players on positions $\sigma(j), \ldots, \sigma(j)+2^{r'}-1.$ Here, note that $r'=2$ and $r=3$. Also, for clarity, only a part of the whole tournament is shown.

Theorems & Definitions (56)

• Definition 1: Tournament Value
• proof
• proof
• Definition 2: Win-Count Oriented Game-Value Function
• Proposition 3
• proof
• Definition 3: Player Popularity-Based Game-Value Function
• Definition 4: $\mathsf{L}$-reduction, papadimitriou1988optimizationausiello2012complexity
• Lemma 6
• proof