Symmetrization of the Classical “Attack-defense” Model

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 Paretonon-dominated part of the boundary and its extreme points is given.


Introduction
The work is based on the results from [1][2] and is a further development of the constructions in [3][4]. Germeyer's classical "attack-defense" model was defined and studied in [5]. It is a modification of the Gross' model [6]. In the military models points are usually interpreted as directions and characterize the spatial distribution of defense resources across the width. However, in reality there is also a spatial distribution of defense resources in depth, characterized by the number of levels of defense lines in this direction.
The simplest model was proposed in [5], taking into account the defense's lines. A game model that generalized the Gross and Germeyer models was studied in [7]. In this model a constructive description of the set of all optimal mixed attack strategies was obtained. The Gross's model with the opposite interests of the parties was studied in [8], and dynamic extensions of the model were studied in [9,10]. a direct generalization of the attack-defense game on networks describing the topology of the paths leading to defended objects was proposed in [11]. The further generalization of the classical "attack-defense" game may consist in its symmetrization, which leads in the general case to problems of finding equilibria with concave criteria, that can be reduced to solving a system of inclusions [12]. 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 forces of the parties and explicit expressions for them are obtained, which can be used in the dynamic expansion of the model according to the scheme [13].

The Simplest Multi-Line Generalization of the Model
The simplest model, taking into account the defense's lines, consists in modifying the classical model [5], in which the function of attack's winning has the form [ (1) and the vectors , x y belong to the sets 1 1 , , where i T is the number of defense lines in the i − th direction 1,..., i n = , i r − the maximum number of actions that one unit of defense can produce, i p − the probability of hitting one enemy attack means with one effect in the i − th direction, which is assumed to be independent from the number 1,..., i t T = of the defense line, 1 i i q p = − − the corresponding probability of non-defeat, X and Y is the total number of attack and defense means, which are considered homogeneous and infinitely divisible, i x and i y -the number of attack and defense means in the i − th direction. In particular, formally, when 1 i p = we obtain the Germeier's classical model [5].
The content of the model is interpreted as follows: the attack party strive to maximize the total amount of means that break through, distributing its means in n directions.
Defense party, on the contrary, strive to minimize the number of means that break through by distributing its means in n directions. In each direction, the attack party must overcome the layered defense of the enemy. The result of a combat collision at one defense line is given by a function max ; that is the result of the Osipov-Lanchester's discrete one-step model of dynamics of average. It is obtained as follows: the number of attack means that have received an impact at a given line is min ; providing that exactly one defense means effect one attack means. The mathematical amount of attack's losses will be i i i m p n = . As a result, the mathematical expectation of the number of means that break through will be max ; The final formula for the mathematical expectation of attack means overcoming all i T defense's lines will be max ; into account the optimization for the defense party of the distribution of its means i y at the lines and derived in [1]. Using the convexity of the function ( , ) f x y , it was proved for this antagonistic game (see [1]) that the best guaranteed result (BGR) of defense will be the value of the game and the minimax defense strategy is optimal. In this case, the optimal attack strategy is a mixed strategy, consisting in concentrating all forces in one direction in accordance with the optimal probability distribution, which can be obtained by the formulas given in [1]. Let's denote for brevity 1 1,2,..., 1

Model's Symmetrization
Let's suppose that strike forces and defense forces take part in a game of two sides , B A . The proportion of strike forces and defense forces is , B A σ σ from the total number of parties' means , B A Y Y . Then the losses of the opposite sides will be according to formula (3), taking into account the accepted notation (4) The payment functions of the parties are continuous and concave; therefore, by virtue of Theorem 8 in [12, p. 90], there are situations of parties' equilibrium that satisfying the inclusions where [0,1] [0,1] ( ) max ( , ), sets of the parties' best answers. In this case, it can be easy to find explicitly.

Parameterization of Equilibria
Parameterization of all Nash's equilibria can be obtained by choosing specific points from segments (9) in inclusion (7),  (12) -(14) are studied depending on the initial ratio of the parties forces and, in particular, the extreme points of the Pareto sets.
For the attack party, which has a noticeable excess of the balance of forces in its favor, it makes sense to choose an equilibrium corresponding to the minimum value of the share allocated to cover.
Let's consider the various cases that arise when the minima are revealed in (12). Depending on whether the minimum in the first and second equations in (12) is equal to the left (L) or right (R) expression, we'll consider the LL, LR, RL, and RR options.

The Main Case LL
Let's suppose that the minima in (12) are reached on the first component. Then (12), (14) are equivalent to the system with conditions λ µ = taking into account the accepted notation, has the form of conditions It is easy to verify that the left inequalities in (19) to be verified, taking into account (18), are collectively equivalent to the right. Therefore, system (19) is equivalent to the conditions It is easy to verify that the left inequalities in (19) that were subjects to verification are collectively equivalent to the right inequalities taking into account (18). Therefore the system (19) is equivalent to the conditions

Case When the Left Sides in (20) Are Nonpositive
We Note that due to inequalities (15), the right side of (22) is larger than the left one.

Case When
Then condition (20) is equivalent to the right inequality which should be solved in conjunction with (21). However, the latter is false due to the fact that 0 B σ < by virtue of (23).

Symmetric Case
Remark 3. The only solution that appears in Theorem 1 is stable provided that the latter adheres to a strategy of reasonable sufficiency of defense, i.e. it strive to minimize the share of its funds allocated for defense when reaching the maximum of its criterion, as the second parties in the Stackelberg equilibrium (see [12], p. 122). Therefore, it can be called the Stackelberg symmetrized equilibrium. Remark 4. It seems unnatural to play along with your opponent in the war game using the strategies of the second players in the Stackelberg equilibria, but this contradiction is easily eliminated by moving to the vector criteria of the parties ( ( , ),1 ) , accordingly, to the lexicographic equilibrium, which is defined similarly to the classical Nash's equilibrium using the concept of the lexicographic maximum of the vector criterion. This equilibrium will also be equivalent to a system of inclusions of parties strategies to point-to-multiple mappings that implement the lexicographic maximum of each party, as multivalued functions of the strategy of the opposite side. After that, each party "plays along" with itself according to the second criterion 1

Researching the Obtained Solution
The general solution of system (16) relatively B σ has the form 1 , ( , )
Inequality (17) is equivalent to the right inequality which should be solved in conjunction with the condition In fact, it follows from (14) and (33) The condition (33) is remains, which, taking into account (32), takes the form The latter is equivalent to the inequality Hence we obtain the condition In the view of (35), we finally obtain the condition (1 )(1 ). ( Let's note that by virtue of condition (36), the fraction on the right side is a non-negative quantity.

Conditions for x , Under
The latter is equivalent to the inequality The equality holds on the Pareto border, whence, taking into account conditions (13), we'll obtain its parametric notation in the form The set Ω and its Pareto boundary ′ Ω are shown in the Figure 1.

The Case When
Then the left inequality in (36) is satisfy, and the right inequality for minimal equilibria on B σ is satisfy as equality This follows from condition (30) of the strict increase of the function B σ on , λ µ , which in this case is satisfied by virtue of (41

LR and RL Options
Lemma 3. Under condition (15) and 1 , [0,1), 1 Proof. In fact, in the LP option we'll get , 1 from which only the right inequality in the right double inequality is nontrivial, which is equivalent to the condition 1 1

RR Option
In this case equilibria do not exist. This can be seen directly from the general parametric representation of equilibria (12) -(14). In fact, in this case the restrictions has the form and obviously incompatible.
Now, by formula (32), we have There is the condition (46) by Lemma 3 under condition (39) 1 , [0,1), 1 Therefore, there is a solution of the necessary sufficiency of defense for the B party (47) , 1 which coincides with (52), (53) taken into account (55). Thus, the LL and LR options provide the same solution.

Dominates the Solution
Under Condition (44), the Solutions Obtained in Lemmas 2 and 3 Are Comparable. By Lemma 2, the Solution Is (45) , 1.
By the formula (49) we'll obtain which dominates the solution (58), (59). Thus, the LR option provides a better solution than LL option.  [13], in which the equations of system's motion will have the form:

Case When There Are no Solutions
where / , 0,1,... Thus, for the side B it is enough to allocate 40% of its forces to defense, and for the side A it is enough to allocate 43% of its forces to defense.

Conclusion
In this work, we proposed the symmetrization of the "attack-defense" model defined and studied by Germeyer. In the military models points are usually interpreted as directions and characterizes the spatial distribution of defense resources across the width. It is also possible to distribute the resources in depth in relation with the separation of the defense. The parties' resources are generally heterogeneous. All these areas of generalization of the classical "attack-defense" model were studied by the authors in previous works. In reality, there is also a symmetry of the conflict, when both sides attack and defend at the same time. Therefore, in the present work, a symmetric extension of the model was proposed, in which the parties simultaneously participate in two games, notably in one game the each side is an attack party and in the other game the each side is a defense party. The solution in the resulting doubled game is defined as the non-dominated Nash's equilibrium by Pareto. A classification of such equilibria is given depending on the balance of parties' forces. The extreme points of the Pareto' sets corresponding to the minimum of the share of the stronger side directed to defense are distinguished. Or the same thing, the extreme points of the Pareto' sets corresponding to the maximum share of the stronger side directed to the attack, which makes sense when planning a defense's breakthrough of the weaker side.