## The Annals of Statistics

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

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

#### 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