The Annals of Statistics

On Most Effective Tournament Plans With Fewer Games than Competitors

Willi Maurer

Full-text: Open access

Abstract

Let $\Omega_n$ denote a set of $n$ players, $p_{ij}$ the probability that player $i$ defeats player $j$ and $\Gamma$ the class of preference matrices $(p_{ij})$ with $p_{1j} \geqq \frac{1}{2}, j > 2$. Under the assumption that the outcomes of games are independent and distributed according to $(p_{ij}) \in \Gamma$, the effectiveness (relative to $(p_{ij})$) of a tournament plan, together with a rule to select a winner, is measured by the probability that player 1 (the "best" player) wins the tournament. A k.o. plan is a tournament plan in which a player is eliminated from the tournament if he loses one game. It is shown that there are no plans on $\Omega_n$ with $n - 1$ games that are more effective than k.o. plans relative to all matrices contained in certain reasonable subclasses of $\Gamma$. Among the k.o. plans for $2^m + k, 0 \leqq k < 2^m$, players, those which consist of a preliminary round of $k$ games followed by a "symmetric" k.o. tournament on the remaining $2^m$ players are more effective than all other plans relative to the preference matrices contained in two large subclasses of $\Gamma$. In order to prove these assertions, the tournament plans are interpreted as mappings with directed digraphs as domain and range.

Article information

Source
Ann. Statist., Volume 3, Number 3 (1975), 717-727.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176343135

Digital Object Identifier
doi:10.1214/aos/1176343135

Mathematical Reviews number (MathSciNet)
MR371712

Zentralblatt MATH identifier
0349.90138

JSTOR
links.jstor.org

Subjects
Primary: 62K05: Optimal designs
Secondary: 05B30: Other designs, configurations [See also 51E30] 05C20: Directed graphs (digraphs), tournaments

Keywords
Tournament plan knockout plan winner rule dominance structure preference matrix complete digraph

Citation

Maurer, Willi. On Most Effective Tournament Plans With Fewer Games than Competitors. Ann. Statist. 3 (1975), no. 3, 717--727. doi:10.1214/aos/1176343135. https://projecteuclid.org/euclid.aos/1176343135


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