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.
| Published in | Science Journal of Applied Mathematics and Statistics (Volume 8, Issue 1) |
| DOI | 10.11648/j.sjams.20200801.11 |
| 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), 2020. Published by Science Publishing Group |
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
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APA Style
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
ACS 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
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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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://article.sciencepublishinggroup.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}
}
TY - JOUR T1 - Symmetrization of the Classical “Attack-defense” Model AU - Pavel Yuryevich Kabankov AU - Alexander Gennadevich Perevozchikov AU - Valery Yuryevich Reshetov AU - Igor Evgenievich Yanochkin Y1 - 2020/01/07 PY - 2020 N1 - https://doi.org/10.11648/j.sjams.20200801.11 DO - 10.11648/j.sjams.20200801.11 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 - 1 EP - 10 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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