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On the waiting time in a janken game

Part of: Limit theorems Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Hiroshi Maehara  and
Sumie Ueda
Hiroshi Maehara*
Affiliation:
Ryukyu University
Sumie Ueda*
Affiliation:
Institute of Statistical Mathematics, Tokyo
*
∗Postal address: College of Education, Ryukyu University, Nishihara, Okinawa, 903-0213 Japan. Email address: hmaehara@edu.u-ryukyu.ac.jp
∗∗Postal address: The Institute of Statistical Mathematics, 4–6–7 Minami-Azabu Minato-ku Tokyo, 106-8569 Japan. Email address: ueda@ism.ac.jp
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Abstract

Consider a janken game (scissors-paper-rock game) started by n players such that (1) the first round is played by n players, (2) the losers of each round (if any) retire from the rest of the game, and (3) the game ends when only one player (winner) is left. Let Wn be the number of rounds played through the game. Among other things, it is proved that (2/3)nWn is asymptotically (as n → ∞) distributed according to the exponential distribution with mean ⅓, provided that each player chooses one of the three strategies (scissors, paper, rock) with equal probability and independently from other players in any round.

Keywords

Jankenscissors-paper-rockwaiting timeasymptotic distribution

MSC classification

Primary: 60E99: None of the above, but in this section 60F99: None of the above, but in this section
Type
Short Communications
Information
Journal of Applied Probability , Volume 37 , Issue 2 , June 2000 , pp. 601 - 605
Copyright
Copyright © by the Applied Probability Trust 2000 

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References

Kendall, M. G., and Stuart, A. S. (1969). The Advanced Theory of Statistics, Vol. I. Griffin, London.Google Scholar
Ohbayashi, T., Kishino, U., Sougawa, T. and Yamashita, S. (Eds) (1998). Encyclopedia of Ethnic Play And Games (Japanese). Taishuukan-shoten, Tokyo.Google Scholar