## The Annals of Statistics

### On Most Effective Tournament Plans With Fewer Games than Competitors

Willi Maurer

#### 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

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