Robust Value Maximization in Challenge the Champ Tournaments with Probabilistic Outcomes

TL;DR

It is shown that even in simple binary settings, for non-adaptive algorithms, the optimal robust value -- which is the \textsc{VnaR}, or the value not at risk -- is hard to approximate, but if adaptive algorithms that determine the order of challengers based on the outcomes of previous matches, or restrict the matches with probabilistic outcomes, can obtain good approximations to the optimal robust value.

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}.

Figures (4)

• Figure 1: Illustration of a strength graph, and a spanning arborescence. The instance contains two popular players, $a$ and $b$, and eight unpopular players $x_1, \ldots, x_4$ and $y_1, \ldots, y_4$. Solid(dashed) edges are deterministic(uncertain). Edges between unpopular players are uncertain and omitted for clarity. The backbone of the arborescence is formed by the vertices $\{b, y_2, y_1, a\}$. The VnaR is 7, and is witnessed by the seeding $\langle a, x_1, x_2, x_3, x_4, y_1, y_2, b, y_3, y_4 \rangle$.
• Figure 2: The backbone witnessed by the seeding $\langle v^{n_p}_p, v^{n_p}_u, \dots, v^1_p, v^1_u \rangle$
• Figure 3: Illustration of a tagged block due to partners $(v^1_p, v^2_p)$. Here, the bold circles represent the popular backbone players, and the normal circles are unpopular players. The solid(dashed) arrows represent the deterministic (uncertain) edges between players.
• Figure 4: A figure illustrating the key ideas used thus far in the proof of \ref{['thm:multiplicative-approximation-NP-hardness']}. Here we represent $A_\sigma$ using the same conventions as \ref{['fig:tagged-pair-example']}. The green and orange ovals denote clean blocks and high-scoring tagged blocks, respectively. The first low-scoring tagged block starts with $v^{k,1}_p$. The red area is the removed portion because of a low-scoring tagged block.

Theorems & Definitions (53)

• proposition 1
• proof
• theorem 1.1
• theorem 1.2
• theorem 1.3
• theorem 1.4
• theorem 1.5
• proof
• theorem 1.6
• lemma 1