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On Most Effective Tournament Plans With Fewer Games than Competitors

Willi Maurer
The Annals of Statistics
Vol. 3, No. 3 (May, 1975), pp. 717-727
Stable URL: http://www.jstor.org/stable/2958441
Page Count: 11
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On Most Effective Tournament Plans With Fewer Games than Competitors
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Abstract

Let Ωn denote a set of n players, pij the probability that player i defeats player j and Γ the class of preference matrices (pij) with $p_{1j} \geqq \frac{1}{2}, j > 2$. Under the assumption that the outcomes of games are independent and distributed according to (pij) ∈ Γ, the effectiveness (relative to (pij)) 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 Ωn with n - 1 games that are more effective than k.o. plans relative to all matrices contained in certain reasonable subclasses of Γ. 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 2m players are more effective than all other plans relative to the preference matrices contained in two large subclasses of Γ. In order to prove these assertions, the tournament plans are interpreted as mappings with directed digraphs as domain and range.

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