The authors describe a multi-step generalization of the “attack-defense” model, defined and studied by Germeier. It is a modification of the Gross’ model. The similar model was proposed by Gorelik for the gasoline production. In the military models the points are usually interpreted as directions and characterize the spatial distribution of defense resources across the width of the defense front. The dynamics of the average number of parties described by the “attack-defense” game can be described by finite-difference Osipov-Lanchester’ equations. Therefore, it would also be interesting to obtain a generalization of Germeyer’s classical model to the dynamic case when the “attack-defense” game is played many times. On this basis, in the present work, a dynamic expansion of the model is constructed in the form of a positional game with opposing interests of the distribution of parties’ reserves with complete information. The authors studied the simplest multi-step extension of the attack-defense model, which consists in the fact that the corresponding game is played repeatedly. Multi-step game with the complete information of the parties’ reserves management was built on this basis. It is assumed that the defense party makes the first move at each step and the attack party became aware about this move. The functional equation for the best guaranteed result of the defense, which is the value of the positional game due to the parties’ adopted sequence of moves was written out. Its analytical solution for a two-step game was obtained and it was shown that it is advantageous for an attack party to enter all reserves simultaneously, as in the classic attack-defense game.
| Published in | Science Journal of Applied Mathematics and Statistics (Volume 7, Issue 5) |
| DOI | 10.11648/j.sjams.20190705.11 |
| Page(s) | 63-70 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2019. Published by Science Publishing Group |
Attack-Defense Game, Multi-Step Expansion of the Game, Guaranteed Defense Result, Game Value, Optimal Attack Strategy, Optimal Defense Strategy
| [1] | Germeier Y. B. Introduction to the theory of operations research. Moscow, Science, 1971. |
| [2] | Karlin S. Mathematical methods in game theory, programming and economics. Moscow, Mir, 1964. |
| [3] | Gorelik V. A. Game Theory and Operations Research. Moscow, Publishing House MINGP, 1978. |
| [4] | Ogaryshev V. F. Mixed strategies in a single generalization of the Gross’ problem// Journal of computational mathematics and mathematical physics, 1973. V. 13. No. 1. pp. 59-70. |
| [5] | Reshetov V. Y., PtrevozchikovA. G., Lesik I. A. A Model of Overpowering a Multilevel Defense System by Attak// Computational Mathematics and Modeling, 2016, Vol. 27, No. 2, p. 254-269. |
| [6] | Reshetov V. Y., PtrevozchikovA. G., Lesik I. A. Multi-Level Defense System Models: Overcoming by Means of Attacks with Several Phase Constraints//Moscow University Computational Mathematics and Cybernetics, 2017, Vol. 1, No. 1, p. 25-31. |
| [7] | Reshetov V. Y., PtrevozchikovA. G., Yanochkin I. E. An Attack-Defense Model with Inhomogeneous Resources of the Opponents//Computational Mathematics and Mathematical Physics, 2018, Vol. 58, No. 1, p. 38-47. |
| [8] | Reshetov V. Y., PtrevozchikovA. G., Yanochkin I. E. Multilayered Attack-Defense Model on Networks//Computational Mathematics and Mathematical Physics, 2019, Vol. 59, No. 8, p. 1389-1397. |
| [9] | Reshetov V. Y., Perevozchikov A. G, Lesik A. I. Multistep generalization of the attack-defense model//Bulletin of Tver State Univercity. Seria Applied Mathematics. 2017, No. 2. pp. 12-24. |
| [10] | Hohzaki R., Tanaka V. The effects of players recognition about the acquisition of his information by his opponent in an attrition game on a network//In Abstract of 27th European conference on Operation Research 12-15 July 2015 University of Strathclyde. - EURO2015. |
| [11] | Molodtsov D. A. Adaptive control in repetitive games// Journal of computational mathematics and mathematical physics, 1978. Vol. 18, No. 1, pp. 78-83. |
| [12] | Danilchenko T. N., Masevich K. K. Multistage game of two persons with a “cautious” second player and consistent transmission of information// Journal of Computational Mathematics and Mathematical Physics, 1974. V. 19. No. 5. pp. 1323-1327. |
| [13] | Ereshko F. I., Propoi A. I. To the theory of dynamic games. News of the USSR Academy of Sciences. Techical cybernetics. 1970, No. 2, pp. 42-47. |
| [14] | Krutov B. P. Dynamic quasi-informational extensions of games with an expandable coalition structure. Moscow, CC of RAS, 1986. |
| [15] | Vatel I. A., Dranev Y. N. About one class of repetitive games with incomplete information in a two-level economic system. In Proceedings of International conference "Modeling of economic processes." Moscow, CC of the USSR Academy of Sciences, 1975, pp. 224-238. |
| [16] | Vasin A. A., Morozov V. V. Game theory and models of mathematical economics. Moscow, MAX Press, 2005. |
| [17] | Fedorov V. V. Maximin’s numerical methods. Moscow, Science, 1979. |
| [18] | Petrosyan L. A. Differential Games of Pursuit. Leningrad, Publishing house of Leningrad University, 1977. |
APA Style
Alexander Gennadievich Perevozchikov, Valery Yurievich Reshetov, Igor Evgenievich Yanochkin. (2019). Multi-Step Game of Reserves Management in the Attack-Defense Model. Science Journal of Applied Mathematics and Statistics, 7(5), 63-70. https://doi.org/10.11648/j.sjams.20190705.11
ACS Style
Alexander Gennadievich Perevozchikov; Valery Yurievich Reshetov; Igor Evgenievich Yanochkin. Multi-Step Game of Reserves Management in the Attack-Defense Model. Sci. J. Appl. Math. Stat. 2019, 7(5), 63-70. doi: 10.11648/j.sjams.20190705.11
AMA Style
Alexander Gennadievich Perevozchikov, Valery Yurievich Reshetov, Igor Evgenievich Yanochkin. Multi-Step Game of Reserves Management in the Attack-Defense Model. Sci J Appl Math Stat. 2019;7(5):63-70. doi: 10.11648/j.sjams.20190705.11
Share This Article
Twitter
Linked In
Facebook
Copy
Download@article{10.11648/j.sjams.20190705.11,
author = {Alexander Gennadievich Perevozchikov and Valery Yurievich Reshetov and Igor Evgenievich Yanochkin},
title = {Multi-Step Game of Reserves Management in the Attack-Defense Model},
journal = {Science Journal of Applied Mathematics and Statistics},
volume = {7},
number = {5},
pages = {63-70},
doi = {10.11648/j.sjams.20190705.11},
url = {https://doi.org/10.11648/j.sjams.20190705.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20190705.11},
abstract = {The authors describe a multi-step generalization of the “attack-defense” model, defined and studied by Germeier. It is a modification of the Gross’ model. The similar model was proposed by Gorelik for the gasoline production. In the military models the points are usually interpreted as directions and characterize the spatial distribution of defense resources across the width of the defense front. The dynamics of the average number of parties described by the “attack-defense” game can be described by finite-difference Osipov-Lanchester’ equations. Therefore, it would also be interesting to obtain a generalization of Germeyer’s classical model to the dynamic case when the “attack-defense” game is played many times. On this basis, in the present work, a dynamic expansion of the model is constructed in the form of a positional game with opposing interests of the distribution of parties’ reserves with complete information. The authors studied the simplest multi-step extension of the attack-defense model, which consists in the fact that the corresponding game is played repeatedly. Multi-step game with the complete information of the parties’ reserves management was built on this basis. It is assumed that the defense party makes the first move at each step and the attack party became aware about this move. The functional equation for the best guaranteed result of the defense, which is the value of the positional game due to the parties’ adopted sequence of moves was written out. Its analytical solution for a two-step game was obtained and it was shown that it is advantageous for an attack party to enter all reserves simultaneously, as in the classic attack-defense game.},
year = {2019}
}
TY - JOUR T1 - Multi-Step Game of Reserves Management in the Attack-Defense Model AU - Alexander Gennadievich Perevozchikov AU - Valery Yurievich Reshetov AU - Igor Evgenievich Yanochkin Y1 - 2019/10/11 PY - 2019 N1 - https://doi.org/10.11648/j.sjams.20190705.11 DO - 10.11648/j.sjams.20190705.11 T2 - Science Journal of Applied Mathematics and Statistics JF - Science Journal of Applied Mathematics and Statistics JO - Science Journal of Applied Mathematics and Statistics SP - 63 EP - 70 PB - Science Publishing Group SN - 2376-9513 UR - https://doi.org/10.11648/j.sjams.20190705.11 AB - The authors describe a multi-step generalization of the “attack-defense” model, defined and studied by Germeier. It is a modification of the Gross’ model. The similar model was proposed by Gorelik for the gasoline production. In the military models the points are usually interpreted as directions and characterize the spatial distribution of defense resources across the width of the defense front. The dynamics of the average number of parties described by the “attack-defense” game can be described by finite-difference Osipov-Lanchester’ equations. Therefore, it would also be interesting to obtain a generalization of Germeyer’s classical model to the dynamic case when the “attack-defense” game is played many times. On this basis, in the present work, a dynamic expansion of the model is constructed in the form of a positional game with opposing interests of the distribution of parties’ reserves with complete information. The authors studied the simplest multi-step extension of the attack-defense model, which consists in the fact that the corresponding game is played repeatedly. Multi-step game with the complete information of the parties’ reserves management was built on this basis. It is assumed that the defense party makes the first move at each step and the attack party became aware about this move. The functional equation for the best guaranteed result of the defense, which is the value of the positional game due to the parties’ adopted sequence of moves was written out. Its analytical solution for a two-step game was obtained and it was shown that it is advantageous for an attack party to enter all reserves simultaneously, as in the classic attack-defense game. VL - 7 IS - 5 ER -
Information
Email: