AM  Vol.13 No.2 , February 2022
A Statistical Analysis of Games with No Certain Nash Equilibrium Make Many Results Doubtful
Abstract: Some games may have a Nash equilibrium if the parameters (e.g. probabilities for success) take certain values but no equilibrium for other values. So there is a transition from Nash equilibrium to no Nash equilibrium in parameter space. However, in real games in business and economics, the input parameters are not given. They are typically observed in several similar occasions of the past. Therefore they have a distribution and the average is used. Even if the averages are in an area of Nash equilibrium, some values may be outside making the entire result meaningless. As the averages are sometimes just guessed, the distribution cannot be known. The main focus of this article is to show this effect in an example, and to explain the surprising result by topological explanations. We give an example of two players having three strategies each (e.g. player and keeper in penalty shooting) where we demonstrate the effect explicitly. As the transition of Nash equilibrium to no equilibrium is sharp, there may be a special form of chaos which we suggest to call topological chaos.
Cite this paper: Klinkova, G. and Grabinski, M. (2022) A Statistical Analysis of Games with No Certain Nash Equilibrium Make Many Results Doubtful. Applied Mathematics, 13, 120-130. doi: 10.4236/am.2022.132010.

[1]   Palacios-Huerta, I. (2014) Beautiful Game Theory: How Soccer Can Help Economics. Princeton University Press, USA.

[2]   The Economist (2018) The Lucky 12 Yards. The Economist.

[3]   Schuster, H.G. (1984) Deterministic Chaos. Physik Verlag, Weinheim, Germany.

[4]   Grabinski, M., Klinkova, G. (2019) Wrong Use of Average Implies Wrong Results from Many Heuristic Models. Applied Mathematics, 10, 605-618.

[5]   Battulga, G., Altangerel, L. and Battur, G. (2018) An Extension of One-Period Nash Equilibrium Model in Non-Life Insurance Markets. Applied Mathematics, 9, 1339-1350.

[6]   Cojocaru, M. and Jaber, A. (2018) Optimal Control of a Vaccinating Game toward Increasing Overall Coverage. Journal of Applied Mathematics and Physics, 6, 754-769.

[7]   von Mangoldt, H. and Knopp, K. (1975) Einführung in die Höhere Mathematik. Vol. 4, 2nd Edition, Hirzel Verlag, Stuttgart, Germany.

[8]   Gamelin, T.W. and Greene, R.E. (1999) Introduction to Topology. 2nd Edition, Dover Publications, New York, USA.

[9]   Munkres, J.R. (2000) Topology. 2nd Edition, Prentice Hall, Upper Saddle River, USA.

[10]   Bronshtein, I.N., Semendyayev, K.A., Musiol, G. and Muehlig, H. (2007) Handbook of Mathematics. 5th English Edition, Springer, Berlin Heidelberg, Germany.

[11]   Letellier, C. and Gilmore, R (2013) Topology and Dynamics of Chaos. World Scientific Publishing, Singapore.

[12]   Klinkova, G. (2018) The Effects of Chaos on Business Operations. Ph.D. Thesis, Neu-Ulm University, Neu-Ulm, Germany.

[13]   Appel, D. and Grabinski, M. (2011) The Origin of Financial Crisis: A Wrong Definition of Value. Portuguese Journal of Quantitative Methods, 2, 33-51.

[14]   Klinkova, G. and Grabinski, M. (2017) Due to Instability Gambling Is the Best Model for Most Financial Products. Archives of Business Research, 5, 255-261.

[15]   Appel, D., Dziergwa, K. and Grabinski, M. (2012) Momentum and Reversal: An Alternative Explanation by Non-Conserved Quantities. International Journal of Latest Trends in Finance & Economic Sciences, 2, 8-16.