New Prospective on Multiple Dice Rolling Game and Its Statistical Implications

Jimbo Henri Claver^{1, 2,} , Jawad Azimi³ and Takeru Suzuki²

¹Department of Applied Mathematics and Statistics, Joint Waseda University, Tokyo, Japan and American University of Afghanistan, Faculty Building 1, D-22, Po.Box 458, Central Post, Kabul, Afghanistan

²Department of Applied Mathematics, Waseda University, Tokyo, Japan

³Japan International Cooperation Agency (JICA), Head Office, Kabul, Afghanistan

Cite this paper:
Jimbo Henri Claver, Jawad Azimi and Takeru Suzuki. New Prospective on Multiple Dice Rolling Game and Its Statistical Implications. American Journal of Applied Mathematics and Statistics. 2017; 5(5):169-174. doi: 10.12691/ajams-5-5-3

Abstract

We present a mathematical formulation of the Multiple Dice Rolling (MDR) game and develop an adaptive computational algorithm to simulate such game over time. We use an extended version of the well-known Chapman-Kolmogorov Equations (CKEs) to model the state transition of the probability mass function of each side of the dice during the game and represent the time-dependent propensity of the game by a simple regression process, which enable to capture the change in the expectation over time. Furthermore, we perform a quantitative analysis on the outcome of the game in a framework of Average Probability Value (APV) of appearance of a side of the dice over trials. The power of our approach is demonstrated. Our results also suggest that in the MDR game, the APV of appearance of a side of a dice can be appropriately predicted independently of the number of sides and trials.

Keywords:
MDR Game Chapman-Kolmogorov Equations simulation propensity statistics expectation and regression

This work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References:

[1]	Ferguson, T. S. (1968). Mathematical statistics - A decision theoretic approach. Academic Press, New York.
[2]	Krasovskij, N. N. (1970), Game theoretic problem of capture. Nauka, Moscow.
[3]	Murray, H. J. R. (2002). A history of chess. Oxford University Press, London, Reprint.
[4]	Alon, N. & Krilievich, M. (2006). Extremal and probabilistic combinatory. Canad. J. Math, 11:34-38.
[5]	Backwell, D. & Girshick, M.A. (1954) Theory of games and statistical decisions. John Wiley & Sons, New York.
[6]	Conway, J. H. (1976). On numbers and games. Academic Press, New York.
[7]	Kolmogorov, A.N. (1988). Foundation of the theory of probability. (3rd eds). Phasis, Moscow.
[8]	Aumann, R.J. (1989). Lecture note on Game Theory, Westview Press Inc., Boulder, Colorado
[9]	Iverson, G. R. & Longcour, W. H. (1971). Bias and runs in dice throwing and recording: A Few million throws. Psychometrika, 36, 01.
[10]	Kreps, D. M. (1990) Game theory and Economic Modeling, Oxford University Press.
[11]	Maitra, A. P. & Sudderth, W. D. (1996). Discrete gambling and stochastic games: Applications of mathematics 32. Springer-Verlag.
[12]	Morris, P. (1994) Introduction to game theory. Springer-Verlag, New York.
[13]	Roth, A. E. & Sotomayor, M. A. O. (1990). Two sides matching- A study in game theoretic modeling and analysis. Cambridge University Press.
[14]	Feller, W. (1972). An introduction to probability theory and its applications. John Wiley & Son, New York, vol. 2.
[15]	Jimbo, H.C. (2004). Distribution characterization in a practical moment problem. Acta Mathematica, no 4, 73, 1.
[16]	Kac. J. L, Logan, L. L. (1976). Fluctuating phenomena. Eds. E.S.W. Montroll & J:L: Lebowitz, North Holland, Amsterdam.
[17]	Knifer R. (1999). Dice game properly explained. Elliot Right Wy Books. ISBN 0-7160-2112-9W.