Minimizing the probability of a tie for first place
type: Item
instance of: scholarly article
title: Minimizing the probability of a tie for first place
zbMATH Open document ID: 0854.60031
author: Bennett Eisenberg
author: G. Stengle
published in: Journal of Mathematical Analysis and Applications
publication date: 1996-09-22T00:00:00Z
review text: 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)].
reviewed by: Ludwig Paditz
Mathematics Subject Classification ID: 60F99
Mathematics Subject Classification ID: 91A60
zbMATH DE Number: 902511
zbMATH Keywords: optimal distribution
zbMATH Keywords: asymptotically optimal distribution
zbMATH Keywords: limit law
zbMATH Keywords: game-theoretic model
zbMATH Keywords: probability of a tie
MaRDI profile type: Publication
full work available at URL: jmaa.1996.0092
OpenAlex ID: W2005389302
DOI: 10.1006/JMAA.1996.0092
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description: scientific article; zbMATH DE number 902511