Minimizing the probability of a tie for first place
Sometimes it happens that a tie for the first place in a competition is undesirable, inappropriate or costly. A simple model is investigated for which it makes sense to consider minimizing the probability of such an outcome. Suppose that the scores of \(k\) players (in an ordered set of cardinality \(n)\) are independent integer-valued r.v.'s with the probabilities \((p_j)_{j = 1(1)n}\), where \(\sum^n_{i = 1} p_i = 1\), and that the highest score wins. Let \(S_k\) be the event that there is a simple winner among the \(k\) players and let \(F(j) : = \sum^j_{i = 1} p_i\). Then \[ P(S_k) = \sum^n_{j = 2} k \bigl( F(j) - F(j - 1) \bigr) \bigl( F(j - 1) \bigr)^{k - 1}. \] The basic problem is to maximize the function \(F(j)\), \(j \in \{1, 2, \dots, n\}\), and to characterize the optimal distribution for fixed \(k\) if \(n\) and \(j\) become large. Finally the result of the distributions which minimize the probability of a tie for the highest score is a limit law: \[ \lim_{n \to \infty} G_{n,k} (x) : = \lim_{n \to \infty} \sum_{j \leq nx} p_j = x^{2/k}, \quad 0 \leq x \leq 1. \] [See also a previous paper by the authors and \textit{G. Strang}, Ann. Appl. Probab. 3, No. 3, 731-745 (1993; Zbl 0787.60029)].
scientific article; zbMATH DE number 902511