On Most Effective Tournament Plans With Fewer Games than Competitors

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 p₁ⱼ 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 2ᵐ + k, 0 k < 2ᵐ, 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.