The article is dedicated the attack analysis and reducing mistakes and miscalculations in risk management. Widespread use of game theory in the analysis of attacks on information resources and countering them can significantly reduce errors and miscalculations that occur in risk management, which in turn minimizes the negative and adverse political, social, and financial consequences for the subjects of information warfare. The solution to the problems of information confrontation is impossible without the development of new theoretical and methodological principles for the analysis of confrontation processes. The authors have offered and studied the scheme of finding sustainable strategies, which ensure the neutralization of the enemy. The scheme for finding sustainable strategies always turns out to be useful in many problems.
In recent years, due to the rapid development of operations research of systems engineering in solving risk management problems, it has become possible to study conflict situations taking into account situations of uncertainty.
The theoretical basis of risk management in conflict situations is game theory. Hybrid war and cyber warfare contributed to the widespread adoption of game theory [1]. New forms and methods of counteraction have appeared. The classic forms of confrontation have been replaced by hybrid methods. They are of a hidden nature and are carried out mainly in the political, economic, informational, and other spheres. Solving the issues of risk management and information protection, countering attacks, and information impacts remain relevant for all of us.
Nowadays, game-theoretic methods [2] are successfully used to solve a wide variety of issues. The application of game theory in solving problems of risk estimating in information wars, information and information-psychological confrontation, information and geopolitical areas gives especially great benefit.
Game theory is a mathematical theory of conflict situations. In these situations, the interests of two or more parties collide, which pursue different, opposite goals. The direct subject of study of the game theory is the mathematical analysis of a formalized model of conflict, which takes into account the peculiarities of a real conflict situation. The technique itself is the formalization of a specific conflict situation does not apply to the mathematical theory of games. It is within the competence of specialists in the field, which is affected by this conflict situation.
Each conflict situation, in terms of risk assessment, is a complex situation. Risk analysis is hampered by many secondary factors. Therefore, in order to make possible a mathematical analysis of the situation, it is necessary to abstract from random factors and builds a simplified formalized model of the process and risk management factors. In this case, the formalization should be such that the possible ways of behavior of the participants and the results are visible, to which all possible combinations of actions of all participants in the conflict lead.
Following modern research trends can be identified in this field: building influence models (information cascades (IC) [3]; linear thresholds (LT) [4], probabilistic models [5]; construction of effective algorithms for maximizing the impact (based on the apparatus of submodular functions (greedy algorithm) and its improvement, CELF [6], CELF ++ [7]); using local properties of the graph (LDAG [8], SimPath [9]); thinning the graph [10]; simulated annealing [11]; network monitoring optimization algorithms [6]; variations of the influence maximization problem and solution algorithms (maximizing influence blocking [12], maximizing influence taking into account time [13], thematic distribution of influence [14]); game-theoretic models of information influence [15, 16].
The analysis of scientific and technical literature [17–21] showed that to date the following issues of game theory application have not been solved within the problem of risk management for information protection: The task of risk management for information protection has not been structured. No areas of risk quantitative estimates have been found. No guaranteed assessments of the risk level of information security were found. Optimal strategies for attacking and protecting information have not been found. The solution of information protection issues described by stochastic models is not fully found; The behavior of information attacks during information confrontations has not been studied.
Modeling of information attack processes involves the risk management and reflection in the developed models of dynamic properties due to the conflicting nature and related ideas about the optimal distribution of information resources of players [22].
Mathematical modeling of physical processes by methods of game theory is based on the following factors that verbally determine the essence of this theory [23]:
The presence of a system of differential equations, which describes the change over time in the parameters of the processes being modeled.
Definition of admissible controls of players, in the form of a class of functions on which the corresponding restrictions are imposed.
Goals of players in the form of functionalities. Information that is available to players at the beginning of the game and in the process.
Thus, the use of game theory in information warfare for the purpose of risk management requires detailed research, which is the purpose of this article.
Tasks related to the timing of actions occur in many problems of information confrontation, which use game theory applications [24]. In such situations, the possible actions of the players are set in advance. During the action, the goal is set by strategic decisions of the players (the attacker and the defending side). In general, the payoff function of such games has the following form [25] ( , ) ( , ) = { ( ) ( , )
< ,
= ,
>
(
( , )are strictly negative for ≥ .
The functions and satisfy the following conditions: 1. The functions ( , ) and ( , ) have continuous third partial derivatives in their domains of 2. The derivatives ( , ) and
( , ) are strictly negative for ≤ , and the derivatives 3. The function ( , ) strictly increases in y and strictly decreases in x, and the function ( , ) strictly increases in x and strictly decreases in y.
Here, the function ( ) and ℎ( )are continuous in the entire interval [0,1] and are obtained as the only solutions of a pair of integral equations: ∝ 1 + 2 = + , 1 + 2 = ℎ+ ℎ here various restrictions can be imposed on the functions , , and . They are determined by the specific conditions of the problem being solved.
Many kinds of research [26, 27] have been devoted to the study of games with payoff function (
( )is a continuous function in the entire interval [ , ] ⊂ [0,1].
0 ( , )− ( , ) ( ) ℎ( ) 1 1 + ∫ ( , )− ( , )
( ) ( , ) ( , ) + ∫ ( , )− ( , )
ℎ( )
(0, )
(1, )
1 = − ( , )− ( , )
2 = − ( , )− ( , )
1 = − ( , )− ( , )
( ,0)
(
≤ ≤ 1, respectively.
2. The (
( ) = (∝ 0, ∝1, 1), ( ) = ( 0, ℎ∝1, 1), equations:
The distribution densities ∝1 and ℎ∝1 are determined as solutions of the following integral 1 1 1( ) − ∫ 1 ( , ) ∝1( )
=∝ 1( ) + 2( ); ℎ 1( ) − ∫ 1( , )ℎ∝1( )
= 1( ) + 2( ); 1( , ) =
− ( , ) ( , ) − ( , )
− ( , )
{ ( , ) − ( , )
; ≤ < ≤ 1;
; a ≤ ≤ ≤ 1;
(
( ) = 1−∝ − ,(0 ≤∝, ≤ 1) ∫ℎ 1( ) = 1− − ,(0 ≤ , ≤ 1)
Remark 1. It follows from the equation (13) that if (
Corollary 1. For the case of ( ) = 0 and − ( , ) = ( , ), the game is called symmetric.
The symmetric game is investigated for the case when the function ( , ) in the region 0≤ ( ≤ ≤ 1) is continuous in both variables and has continuous first-order partial derivatives ( , ) ≥ 0, ( , ) ≤ 0 for ≤ and the set of points for which ( , ) = 0 or ( , ) = 0 does not contain any interval of the form = , 1 < < 2 or the form = , ∝1< <∝2.
There is an optimal strategy of the following form for (
< ≤ 1 1 ∫ 1 ( )
= 1−∝ 1 and is determined from the normalization equation:
The function 1( ) is a continuous, positive function. The parameter ∝ is the jump of ( ) at zero From Theorem 1 it follows that the optimal strategy ( ) for a symmetric game in the case under consideration exists only if it is possible to find numbers , ∝, that satisfy the conditions 0 ≤ , ∝< 1 and such a continuous non-negative function 1( ) for < < 1 such that (24) (25) (26) (27) (28) (29) (30)
Further, consider a special class of symmetric games for which ( , ) is not necessarily continuous
in the set of variables at the points (0,0) and (
→0
→0 ( , ) = ( ),
Remark 2. The case of the function ( , ), which increases in and decreases in , using the substitution = 1 − , = 1 − reduces to the case of increasing in and decreasing in , which was considered in the Theorem 1.
∝, , , ≥ 0, and the function
( ) and ℎ eigenfunctions of the conjugate integral equations
If in the Theorem 1, instead of the condition (
=∝ 1( ) + 2( ) ℎ ( ) − ∫ ( , )ℎ ( )
= 1( ) + 2( ) 1( , ) = ∫ ( , ) ( ) − ∫ ( , ) ( ),
1( , 0) = − ∫ (0, ) ( ).
∫ (0, ) ( ) − ∫ ( , ) ( ) + + ∫ ( (0, ) − ( , )) ( ) = 1 + 2 + 3. following expression (31) (32) (33) (34) (35) (36)
The function ( ) is continuously differentiable in the interval 0 ≤ ≤ 1, and its derivative ′( ) does not change the sign on this interval. Moreover, the set of points for which ′( ) = 0 does not contain any interval. strategies. For this, we write the equality
It is easy to see that for the equation ′( ) ≥ 0, the negative strategy is ( ) = 1, for the equation
(
The validity of this equality is established using (29). Indeed, for the equation > 0 we have the It is obvious from (35) that all | | ≤ /4, = 1,2,3. Hence, this proves the validity of (31).
Let us first take the value = 0. From 1( , ) = 0 for < < 1 it follows that 1( , +0) = 0. For > 0, it should be 1( , 0) = 0. It leads to a contradiction with (31), due to the expression (0) < 0. On the other hand, for ∝= 0 we have the following expression 1( , +0) = − ∫ (0) ( ) = − (0) ∫ ( ) = − (0) > 0 we obtain that it is also impossible. If we take ∝> 0, then from 1( , ) = 0, and strict decrease of the function
1( , ) − 1( , 0) = −0 +0 =∝ (0, ) + ∫ ( , ) ( ) − ∫ ( , ) ( ) + ∫ (0, ) ( )
The first term on the right-hand side of formula (34) as → 0, taking into account (29), tends to ∝
(0,0). In order to estimate the integrals in (34), for a given > 0, we choose such that the total
variation of ( ) in [0, ] is less than /4 0, where 0 =
less than /4, and the next two can be represented as:
| ( , )|. Then the first integral will be
1+0
1+0
1
0
on the interval 0 < ≤∝ we get 1( , +0) > 0. If ∝> 0 and 1( , 0) = 0, then from expression (31)
we obtain 1( , +0) =∝ (0) < 0. This is a contradiction. Hence ∝> 0 and ∝= 0. In this case,
expression (26) is equivalent to the expression 1( , ) = 0 on the interval (∝ ,1) under the condition
С1′( , ) = 0. It follows from this expression that, we obtain an integral equation for determining the
2 (
1 + ∫ ( ) ′ ( ) , ( < < 1).
addition, it is easy to check the validity of the following expressions
For the case of ′( ) ≤ 0, it can be shown [28, 29] that optimal strategies are ( ) =∝ 0 + 1. In ( ) = {∝ 0( ) + (1−∝) 1( )
(0) = 0 (0 ≤∝≤ 1), (0) < 0, (0) > 0. the condition
The solution of the game ( , [0,1]) with the payoff function ( , ), (0 ≤ , ≤ 1) is called a pair of distribution functions (strategies) 1∗ and 2∗ and a real number (value of the game) that satisfies 1 0 ∫ ( , ) 2∗( ) ≤ ≤ ∫ ( , ) 1∗( ), 0 ≤ , ≤ 1.
( 1∗, 2) = ∬ ( , ) 1∗( ) 2∗( ). calculated by the following formula
It follows from this expression that if the player 1 uses the strategy 1∗, then the average payoff is This payoff cannot be less than the number , i.e. the player 1, as it were, neutralizes the opponent’s actions. And, conversely, if the player 2 applies the strategy 2∗, then his average loss ( 1, 2∗) will always be greater than the number , regardless of the actions of the player 1. Therefore, it is natural that each player should strive to choose such distribution functions 1∗ and 2∗, which could neutralize the opponent’s actions. Indeed, for the player 1, the best strategy is a strategy that makes his average winnings as large as possible within reason, regardless of the opponent’s actions. Moreover, conversely, the player 2 must choose a strategy that would provide him, within reasonable limits, the smallest possible loss, regardless of the actions of the player 1. Naturally, if the game has an equilibrium
Here 1 0 1 0 position on the space of distribution functions, then only in this case the players can choose optimal strategies [30].
1 = max min ∫ 1( 1) 2( ) = max min 1[ 1( )].
1( 1) = ∫ ( , ) 1( ). himself a loss of no more than
Similarly, the player 2, by the appropriate choice of the distribution function 2( ), can guarantee 2
1 2 = min max ∫ 2( 2) 1( ) = min max 2[ 2( )]. (39) (40) (41) 1 0 1 0 2( 2) = ∫ ( , ) 2( ).
1 ≥ min 1( 1) 2 ≤ max 2( 2),
Let the player 2 choose the distribution function 20( ) as his strategy, and let the player 1 know this choice. Naturally, assuming such an opportunity, the 2 player should strive to find a sustainable 1 will always get the best result, choosing a point 0 that corresponds to this maximum strategy. It follows from (41), it becomes clear that if the value 2( 2) has a maximum, then the player 0 ≤ 2( 2( 0)) = max 2( 2( )).
minimum.
It would be beneficial for the player 2 to bring the value of 2( 2( )) to a minimum, but this is not always possible. The player cannot influence the form of the payoff function and the choice of 0 by the player 1. Nevertheless, the player 2 can, in any case, try to choose the strategy 20( ) so that the value of 2( 2
) does not have a single maximum, that is, so that its “curve” has a flat top.
Similarly, if the player 2 has learned the strategy of the player 1, then he will always choose the point 0 at which the function 1( 1( )) will take the minimum value. In this case, the task of the player 1 is to choose such a strategy 10( ) so that the function 1( 1( )) does not have a single We denote Ω1 = { : 2( 2( )) = 1 = } and Ω2 = { : 1( 1( )) = 2 = }, where 1 and 2 are arbitrary numbers, and 1 ≤ 1 ≤ 2 ≤ 2.
If there is such a pair of real numbers ( 1 ≤ 2) and a pair of distribution functions ( 1, 2), which simultaneously satisfies the following conditions then the functions 1 and 2 will be called stable [27, 30] strategies.
The question of the existence of sustainable strategies for the payoff function ( , ) in most cases remains unsolved. The scheme itself finding sustainable strategies is always useful in many applications and, in particular, in the game theory with a choice of a moment in time. Such games do not require the definition of strategies that neutralize the enemy. It turns out [27, 30] that instead of them one can be content with partially stable strategies, i.e. strategies that provide the player with a stable position in a certain subinterval of the unit interval.
Widespread use of game theory in the analysis of attacks on information resources and countering them can significantly reduce errors and miscalculations that occur in the risk management of information security, which in turn minimizes the negative and adverse political, social, and financial consequences for the subjects of information warfare.
Systematic studies of the behavior for complex dynamic processes require consideration of a large number of risks, features, and relationships of typical attacks on information and informational influences. The investigated features contradict one another; however, each of them cannot be neglected, since they give us a complete picture of the process that is investigated or simulated.
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