Science Journal of Applied Mathematics and Statistics

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Symmetrization of the Classical “Attack-defense” Model

Received: Dec. 07, 2019    Accepted: Dec. 18, 2019    Published: Jan. 07, 2020
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Abstract

The article considers Germeyer’s “doubled” classic “attack-defense” game, which is symmetrical for the participants in the sense that in one game each participant is an “attack” party and in the other game each participant is a “defense” party. This corresponds to the logic of bilateral active-passive operations, when the parties simultaneously conduct defensive-offensive operations against each other. The mathematical expectation of the number of destroyed enemy means is taken as criteria for the effectiveness of the parties, which should be maximized implicitly. Thus, both sides are placed in a “defense” position. Under otherwise equal conditions, the parties strive to minimize shares aimed at defense, guided by a strategy of reasonable sufficiency of defense. The authors study Pareto-dominated equilibria depending on the initial ratio of the parties forces and, in particular, the extreme points of Pareto sets. Formulas are obtained for such equilibria depending on the parties’ balance of forces, which allows us to build a dynamic expansion of the model in the future. The main research method is the parametrization of Nash’s equilibria. The parameterization of the equilibria shows that they fill the two-dimensional subregion of a unit square with a boundary. Therefore, for its narrowing, it makes sense to distinguish from it the Pareto-non-dominated part of the boundary and its extreme points. The latter provide an opportunity to assess the maximum share of the strike means of the parties, which they can afford to allocate without prejudice to the defense. It is shown that these fractions represent piecewise continuous functions of the initial ratio of the parties’ forces and explicit expressions for them are obtained. A numerical example of the construction of the Pareto-non-dominated part of the boundary and its extreme points is given.

DOI 10.11648/j.sjams.20200801.11
Published in Science Journal of Applied Mathematics and Statistics ( Volume 8, Issue 1, February 2020 )
Page(s) 1-10
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), 2024. Published by Science Publishing Group

Keywords

Germeyer’s Classical “Attack-Defense” Game, Multi-Turn Generalization, Best Guaranteed Result of Defense, Game’s “Doubling”, Equilibrium Strategies Parameterization, Pareto-Minimal Set of Equilibria, Pareto-Minimal set Extreme Points

References
[1] Perevozchikov A. G., Lesik I. A., Shapovalov T. G. Multilevel generalization of the “attack-defense” model // Bulletin of TSU. Seria Applied Mathematic. 2017, No. 1, p. 39-51.
[2] Reshetov V. Y., Perevozchikov A. G., Lesik I. A. A Model of Overpowering a Multilevel Defense System by Attack // Computational Mathematics and Modeling, 2016, Vol. 27, No. 2, p. 254-269.
[3] Reshetov V. Y., Ptrevozchikov A. 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.
[4] Reshetov V. Y., Perevozchikov A. G., Yanochkin I. E. An Attack-Defense Model with Inhomogeneous Resources of the Opponents // Journal of Computational Mathematics and Mathematical Physics, 2018, Vol. 58, No. 1, p. 38-47.
[5] Germeier Y. B. Introduction to the theory of operations research. Moscow, Science, 1971.
[6] Karlin S. Mathematical methods in game theory, programming and economics. Moscow, Mir, 1964.
[7] Ogaryshev V. F. Mixed strategies in one generalization of the Gross’ problem // Journal of Computational Mathematics and Mathematical Physics. 1973. Vol. 13. No. 1. p. 59-70.
[8] Molodtsov D. A. Gross’ model in case of conflicting interests // // Journal of Computational Mathematics and Mathematical Physics. 1972. Vol. 12, No. 2, p. 309-320.
[9] 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. Vol. 19. No. 5. pp. 1323-1327.
[10] Krutov B. P. Dynamic quasi-informational extensions of games with an expandable coalition structure. Moscow, CC of RAS, 1986.
[11] 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.
[12] Vasin A. A., Morozov V. V. Game theory and models of mathematical economics. Moscow, MAX Press, 2005.
[13] Lesik A. I., Perevozchikov A. G., Reshetov V. Y. A multi-step generalization of the “attack-defense” model // Bulletin of TSU. Seria Applied Mathematic. 2017, No. 2, p. 99-110.
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    Pavel Yuryevich Kabankov, Alexander Gennadevich Perevozchikov, Valery Yuryevich Reshetov, Igor Evgenievich Yanochkin. (2020). Symmetrization of the Classical “Attack-defense” Model. Science Journal of Applied Mathematics and Statistics, 8(1), 1-10. https://doi.org/10.11648/j.sjams.20200801.11

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    Pavel Yuryevich Kabankov; Alexander Gennadevich Perevozchikov; Valery Yuryevich Reshetov; Igor Evgenievich Yanochkin. Symmetrization of the Classical “Attack-defense” Model. Sci. J. Appl. Math. Stat. 2020, 8(1), 1-10. doi: 10.11648/j.sjams.20200801.11

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    AMA Style

    Pavel Yuryevich Kabankov, Alexander Gennadevich Perevozchikov, Valery Yuryevich Reshetov, Igor Evgenievich Yanochkin. Symmetrization of the Classical “Attack-defense” Model. Sci J Appl Math Stat. 2020;8(1):1-10. doi: 10.11648/j.sjams.20200801.11

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  • @article{10.11648/j.sjams.20200801.11,
      author = {Pavel Yuryevich Kabankov and Alexander Gennadevich Perevozchikov and Valery Yuryevich Reshetov and Igor Evgenievich Yanochkin},
      title = {Symmetrization of the Classical “Attack-defense” Model},
      journal = {Science Journal of Applied Mathematics and Statistics},
      volume = {8},
      number = {1},
      pages = {1-10},
      doi = {10.11648/j.sjams.20200801.11},
      url = {https://doi.org/10.11648/j.sjams.20200801.11},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.sjams.20200801.11},
      abstract = {The article considers Germeyer’s “doubled” classic “attack-defense” game, which is symmetrical for the participants in the sense that in one game each participant is an “attack” party and in the other game each participant is a “defense” party. This corresponds to the logic of bilateral active-passive operations, when the parties simultaneously conduct defensive-offensive operations against each other. The mathematical expectation of the number of destroyed enemy means is taken as criteria for the effectiveness of the parties, which should be maximized implicitly. Thus, both sides are placed in a “defense” position. Under otherwise equal conditions, the parties strive to minimize shares aimed at defense, guided by a strategy of reasonable sufficiency of defense. The authors study Pareto-dominated equilibria depending on the initial ratio of the parties forces and, in particular, the extreme points of Pareto sets. Formulas are obtained for such equilibria depending on the parties’ balance of forces, which allows us to build a dynamic expansion of the model in the future. The main research method is the parametrization of Nash’s equilibria. The parameterization of the equilibria shows that they fill the two-dimensional subregion of a unit square with a boundary. Therefore, for its narrowing, it makes sense to distinguish from it the Pareto-non-dominated part of the boundary and its extreme points. The latter provide an opportunity to assess the maximum share of the strike means of the parties, which they can afford to allocate without prejudice to the defense. It is shown that these fractions represent piecewise continuous functions of the initial ratio of the parties’ forces and explicit expressions for them are obtained. A numerical example of the construction of the Pareto-non-dominated part of the boundary and its extreme points is given.},
     year = {2020}
    }
    

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    T1  - Symmetrization of the Classical “Attack-defense” Model
    AU  - Pavel Yuryevich Kabankov
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    AU  - Igor Evgenievich Yanochkin
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    DO  - 10.11648/j.sjams.20200801.11
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    JF  - Science Journal of Applied Mathematics and Statistics
    JO  - Science Journal of Applied Mathematics and Statistics
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    PB  - Science Publishing Group
    SN  - 2376-9513
    UR  - https://doi.org/10.11648/j.sjams.20200801.11
    AB  - The article considers Germeyer’s “doubled” classic “attack-defense” game, which is symmetrical for the participants in the sense that in one game each participant is an “attack” party and in the other game each participant is a “defense” party. This corresponds to the logic of bilateral active-passive operations, when the parties simultaneously conduct defensive-offensive operations against each other. The mathematical expectation of the number of destroyed enemy means is taken as criteria for the effectiveness of the parties, which should be maximized implicitly. Thus, both sides are placed in a “defense” position. Under otherwise equal conditions, the parties strive to minimize shares aimed at defense, guided by a strategy of reasonable sufficiency of defense. The authors study Pareto-dominated equilibria depending on the initial ratio of the parties forces and, in particular, the extreme points of Pareto sets. Formulas are obtained for such equilibria depending on the parties’ balance of forces, which allows us to build a dynamic expansion of the model in the future. The main research method is the parametrization of Nash’s equilibria. The parameterization of the equilibria shows that they fill the two-dimensional subregion of a unit square with a boundary. Therefore, for its narrowing, it makes sense to distinguish from it the Pareto-non-dominated part of the boundary and its extreme points. The latter provide an opportunity to assess the maximum share of the strike means of the parties, which they can afford to allocate without prejudice to the defense. It is shown that these fractions represent piecewise continuous functions of the initial ratio of the parties’ forces and explicit expressions for them are obtained. A numerical example of the construction of the Pareto-non-dominated part of the boundary and its extreme points is given.
    VL  - 8
    IS  - 1
    ER  - 

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Author Information
  • Department of System Design, JSC NPO RusBITekh-Tver, Tver, Russia

  • Department of System Design, JSC NPO RusBITekh-Tver, Tver, Russia

  • Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Moscow, Russia

  • Department of System Design, JSC NPO RusBITekh-Tver, Tver, Russia

  • Section