Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-26T19:57:14.404Z Has data issue: false hasContentIssue false
  • English
  • Français

n-point, win-by-k games

Published online by Cambridge University Press:  14 July 2016

Kyle Siegrist
Kyle Siegrist*
Affiliation:
University of Alabama in Huntsville
*
∗ Postal address: Department of Mathematics and Statistics, University of Alabama in Huntsville, Huntsville, AL 35899, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

Consider a sequence of Bernoulli trials between players A and B in which player A wins each trial with probability p∈ [0, 1]. For positive integers n and k with k ≦ n, an (n, k) contest is one in which the first player to win at least n trials and to be ahead of his opponent by at least k trials wins the contest. The (n, 1) contest is the Banach match problem and the (n, n) contest is the gambler's ruin problem. Many real contests (such as the World Series in baseball and the tennis game) have an (n, 1) or an (n, 2) format. The (n, k) contest is formulated in terms of the first-exit time of the graph of a random walk from a certain region of the state-time space. Explicit results are obtained for the probability that player A wins an (n, k) contest and the expected number of trials in an (n, k) contest. Comparisons of (n, k) contests are made in terms of the probability that the stronger player wins and the expected number of trials.

Keywords

BERNOULLI TRIALSBANACH MATCH PROBLEMGAMBLER'S RUINREPLICATIONS OF A PAIRED COMPARISON
Type
Research Papers
Information
Journal of Applied Probability , Volume 26 , Issue 4 , December 1989 , pp. 807 - 814
Copyright
Copyright © Applied Probability Trust 1989 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Decani, J. S. (1971) On the number of replications of a paired comparison. Biometrika 58, 169–175.Google Scholar
[2] Feller, W. (1969) An Introduction to Probability Theory and its Applications, Vol. 1, 3rd edn. Wiley, New York.Google Scholar
[3] Maisel, H. (1966) Best k of 2k – 1 comparisons. J. Amer. Statist. Assoc. 61, 329–341.Google Scholar
[4] Menon, V. V. and Indira, N. K. (1983) On the asymptotic normality of the number of replications of a paired comparison. J. Appl. Prob. 20, 554–562.Google Scholar
[5] Uppuluri, V. R. R. and Blot, W. J. (1974) Asymptotic properties of the number of replications of a paired comparison. J. Appl. Prob. 11, 43–52.Google Scholar