Symmetrization of the Classical “Attack-defense” Model

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.