=Paper=
{{Paper
|id=Vol-2923/paper35
|storemode=property
|title=Approach of the Attack Analysis to Reduce Omissions in the Risk Management
|pdfUrl=https://ceur-ws.org/Vol-2923/paper35.pdf
|volume=Vol-2923
|authors=Serhii Zybin,Volodymyr Khoroshko,Yuliia Khokhlachova,Valerii Kozachok
|dblpUrl=https://dblp.org/rec/conf/cpits/ZybinKKK21
}}
==Approach of the Attack Analysis to Reduce Omissions in the Risk Management==
https://ceur-ws.org/Vol-2923/paper35.pdf
Approach of the Attack Analysis to Reduce Omissions
in the Risk Management
Serhii Zybina, Volodymyr Khoroshkoa, Yuliia Khokhlachovaa, and Valerii Kozachokb
a
National Aviation University, 1 Liubomyra Huzara ave., Kyiv, 03058, Ukraine
b
Borys Grinchenko Kyiv University, 18/2 Bulvarno-Kudriavska str., Kyiv, 04053, Ukraine
Abstract
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.
Keywords 1
Cyberspace, risk management, sustainable strategy, cyberwar, hybrid war, game
theory, payoff function, counteraction, neutralization, attack on information,
conflict management.
1. Introduction
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 The classic forms of confrontation have been replaced by hybrid methods.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.
Cybersecurity Providing in Information and Telecommunication Systems, January 28, 2021, Kyiv, Ukraine
EMAIL: serhii.zybin@npp.nau.edu.ua (A.1); professor_va@ukr.net (A.2); hohlachova@gmail.com (A.3); v.kozachok@kubg.edu.ua (B.4)
ORCID: 0000-0002-2670-2823 (A.1); 0000-0001-6213-7086 (A.2); 0000-0002-1883-8704 (A.3); 0000-0003-0072-2567 (B.4)
©️ 2021 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
318
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.
2. Literature Survey
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].
3. Purpose and Objectives
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.
4. Solutions of Games with a Choice of Time
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]
𝐾(𝑥, 𝑦) 𝑓𝑜𝑟 𝑥 < 𝑦,
𝑀(𝑥, 𝑦) = { 𝐼(𝑥) 𝑓𝑜𝑟 𝑥 = 𝑦, (1)
𝐿(𝑥, 𝑦) 𝑓𝑜𝑟𝑥 > 𝑦
319
here various restrictions can be imposed on the functions 𝐾, 𝐼, and 𝐿. 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 (1).
The corresponding mutually exclusive classification of all types of games are given in Karlin’s
monograph [27]. Before starting the results, we introduce some notation. We denote the distribution
function P(x), which has a jump in 𝛼 at zero and a jump in 𝛽 at unity, by 𝑃(𝑥) = (𝛼𝐼0 , 𝑃𝑎𝑏 (𝑥), 𝛽𝐼1 )
where the distribution density 𝑃𝑎𝑏 (𝑥) is a continuous function in the entire interval [𝑎, 𝑏] ⊂ [0,1].
Therefore, the following theorem is true.
Theorem 1 [27]. Let the payoff function of a continuous game has the following form:
𝐾(𝑥, 𝑦) 𝑓𝑜𝑟 𝑥 < 𝑦,
𝑀(𝑥, 𝑦) = { 𝐿(𝑥, 𝑦) 𝑓𝑜𝑟 𝑥 > 𝑦, (2)
𝐾(𝑥, 𝑦) = 𝐿(𝑥, 𝑥)
The functions 𝐾 and 𝐿 satisfy the following conditions:
1. The functions 𝐾(𝑥, 𝑦) and 𝐿(𝑥, 𝑦) have continuous third partial derivatives in their domains of
definition.
2. The derivatives 𝐾𝑥𝑥 (𝑥, 𝑦) and 𝐾𝑦𝑦 (𝑥, 𝑦) are strictly negative for 𝑥 ≤ 𝑦, and the derivatives
𝐿𝑥𝑥 (𝑥, 𝑦) and 𝐿𝑦𝑦 (𝑥, 𝑦) are strictly negative for 𝑥 ≥ 𝑦.
3. The function 𝐾(𝑥, 𝑦) strictly increases in y and strictly decreases in x, and the function 𝐿(𝑥, 𝑦)
strictly increases in x and strictly decreases in y.
Then both sides have unique optimal mixed strategies of the following form
𝐹(𝑥) = (∝ 𝐼0 , 𝑓(𝑥), 𝛽𝐼1 ), (3)
𝐻(𝑦) = (𝛾𝐼0 , ℎ(𝑦), 𝛿𝐼1 ). (4)
Here, the function 𝑓(𝑥) and ℎ(𝑦) are continuous in the entire interval [0,1] and are obtained as the
only solutions of a pair of integral equations:
∝ 𝑝1 + 𝛽𝑝2 = 𝑓 + 𝑇𝑓 , (5)
𝛾𝑝1 + 𝛿𝑝2 = ℎ + 𝑅ℎ (6)
𝑦 1
𝐾𝑦𝑦 (𝑥, 𝑦) 𝐿𝑦𝑦 (𝑥, 𝑦)
𝑇𝑓 = ∫ 𝑓(𝑥)𝑑𝑥 + ∫ 𝑓(𝑥)𝑑𝑥 (7)
𝐾𝑦 (𝑦, 𝑦) − 𝐿𝑦 (𝑦, 𝑦) 𝐾𝑦 (𝑦, 𝑦) − 𝐿𝑦 (𝑦, 𝑦)
0 𝑦
𝑥 1
𝐿𝑥𝑥 (𝑥, 𝑦) 𝐾𝑥𝑥 (𝑥, 𝑦)
𝑅ℎ = ∫ ℎ(𝑦)𝑑𝑦 + ∫ ℎ(𝑦)𝑑𝑦 (8)
(𝑥,
𝐿𝑥 𝑥) − 𝐾𝑥 𝑥)(𝑥, 𝐿𝑥 𝑥) − 𝐾𝑥 (𝑥, 𝑥)
(𝑥,
0 𝑥
𝐾𝑦𝑦 (0, 𝑦)
𝑝1 = −
𝐾𝑦 (𝑦, 𝑦) − 𝐿𝑦 (𝑦, 𝑦)
(9)
𝐿𝑦𝑦 (1, 𝑦)
𝑝2 = −
𝐾𝑦 (𝑦, 𝑦) − 𝐿𝑦 (𝑦, 𝑦)
𝐿𝑥𝑥 (𝑥, 0)
𝑞1 = −
𝐿𝑥 (𝑥, 𝑥) − 𝐾𝑥 (𝑥, 𝑥) (10)
320
𝐾𝑥𝑥 (𝑥, 1)
𝑞2 = −
𝐿𝑥 (𝑥, 𝑥) − 𝐾𝑥 (𝑥, 𝑥)
The constants ∝, 𝛽, 𝛾, 𝛿 are determined from the following conditions:
1
∫ 𝑓(𝑥)𝑑𝑥 = 1−∝ −𝛽, (0 ≤∝, 𝛽 ≤ 1) (11)
0
1
∫ ℎ(𝑦)𝑑𝑦 = 1 − 𝛾 − 𝛿, (0 ≤ 𝛾, 𝛿 ≤ 1) (12)
0
Thus, the solution of the game under consideration is reduced to the solution of integral equations.
This solution is a simple task. These equations are classic integral equations. In particular, we use the
expansion of unknown functions 𝑓 and ℎ in a Neumann series in order to find analytical solutions.
There are general results that can be formulated as the following theorem [27].
Theorem 2. Let the payoff function of a continuous game has the following form:
𝐾(𝑥, 𝑦) 𝑓𝑜𝑟𝑥 < 𝑦,
𝑀(𝑥, 𝑦) = { 𝑙(𝑥) 𝑓𝑜𝑟 𝑥 = 𝑦, (13)
𝐿(𝑥, 𝑦) 𝑓𝑜𝑟 𝑥 > 𝑦
The functions 𝐾, 𝑙, 𝐿 satisfy the following conditions:
1. The functions 𝐾(𝑥, 𝑦) and 𝐿(𝑥, 𝑦) are defined and have continuous second partial derivatives on
closed triangles 0 ≤ 𝑥 ≤ 𝑦 ≤ 1 and 0 ≤ 𝑦 ≤ 𝑥 ≤ 1, respectively.
2. The 𝑙(1) value lies between 𝐾(1,1) and 𝐿(1,1); the 𝑙(0) value lies between 𝐾(0,0) and 𝐿(0,0).
3. 𝐾𝑥 (𝑥, 𝑦) > 0 and 𝐿𝑥 (𝑥, 𝑦) > 0 are located in the corresponding closed triangles with the possible
exception of 𝐿𝑥 (1,1) = 0; 𝐾𝑦 (𝑥, 𝑦) < 0 and 𝐿𝑦 (𝑥, 𝑦) < 0 in the corresponding closed triangles with
the possible exception of 𝐾𝑦 (1,1) = 0.
Then, both sides have optimal strategies of the following form
𝐹(𝑥) = (∝ 𝐼0 , 𝑓∝1 , 𝛽𝐼1 ),
𝐻(𝑦) = (𝛾𝐼0 , ℎ∝1 , 𝛿𝐼1 ),
The distribution densities 𝑓∝1 and ℎ∝1 are determined as solutions of the following integral
equations:
1
𝑓𝑎1 (𝑡) − ∫ 𝑇𝑎1 (𝑥, 𝑡)𝑓∝1 (𝑥)𝑑𝑥 =∝ 𝑝1 (𝑡) + 𝛽𝑝2 (𝑡); (14)
𝑎
1
ℎ𝑎1 (𝑢) − ∫ 𝑈𝑎1 (𝑢, 𝑦)ℎ∝1 (𝑦)𝑑𝑦 = 𝛾𝑞1 (𝑢) + 𝛿𝑞2 (𝑢); (15)
𝑎
−𝐾𝑦 (𝑥, 𝑡)
; 𝑎 ≤ 𝑥 < 𝑡 ≤ 1; (16)
𝐾(𝑡, 𝑡) − 𝐿(𝑡, 𝑡)
𝑇𝑎1 (𝑥, 𝑡) =
−𝐿𝑦 (𝑥, 𝑡)
; a ≤ 𝑡 ≤ 𝑥 ≤ 1;
{ 𝐾(𝑡, 𝑡) − 𝐿(𝑡, 𝑡)
321
𝐿𝑥 (𝑢, 𝑦)
; 𝑎 ≤ 𝑦 < 𝑢 ≤ 1; (17)
𝐾(𝑢, 𝑢) − 𝐿(𝑢, 𝑢)
𝑈𝑎1 (𝑢, 𝑦) =
𝐾𝑥 (𝑢, 𝑦)
; 𝑎 ≤ 𝑢 ≤ 𝑦 ≤ 1;
{ 𝐾(𝑢, 𝑢) − 𝐿(𝑢, 𝑢)
−𝐾𝑦 (0, 𝑡)
𝑝1 (𝑡) =
𝐾(𝑡, 𝑡) − 𝐿(𝑡, 𝑡)
(18)
−𝐿𝑦 (1, 𝑡)
𝑝2 (𝑡) =
𝐾(𝑡, 𝑡) − 𝐿(𝑡, 𝑡)
𝐿𝑥 (𝑢, 0)
𝑞1 (𝑢) =
𝐾(𝑢, 𝑢) − 𝐿(𝑢, 𝑢)
(19)
𝐾𝑥 (𝑢, 1)
𝑞2 (𝑢) =
𝑈(𝑢, 𝑢) − 𝐿(𝑢, 𝑢)
The constants ∝, 𝛽, 𝛾, 𝛿 and 𝑎 are determined from the following conditions
1
∫ 𝑓𝑎1 (𝑥)𝑑𝑥 = 1−∝ −𝛽, (0 ≤∝, 𝛽 ≤ 1) (20)
𝑎
1
∫ ℎ𝑎1 (𝑦)𝑑𝑦 = 1 − 𝛾 − 𝛿, (0 ≤ 𝛾, 𝛿 ≤ 1) (21)
𝑎
Remark 1. It follows from the equation (13) that if 𝐾(1,1) < 𝐿(1,1), then the point 𝑥 = 1 and 𝑦 =
1 is a saddle point for 𝑀(𝑥, 𝑦). This follows from condition (2) of Theorem 1.
Corollary 1. For the case of 𝑙(𝑥) = 0 and −𝐾(𝑥, 𝑦) = 𝐿(𝑥, 𝑦), the game is called symmetric.
The symmetric game is investigated for the case when the function 𝑀(𝑥, 𝑦) in the region 0≤ (𝑥 ≤
𝑦 ≤ 1) is continuous in both variables and has continuous first-order partial derivatives 𝑀𝑥 (𝑥, 𝑦) ≥
0, 𝑀𝑦 (𝑥, 𝑦) ≤ 0 for 𝑥 ≤ 𝑦 and the set of points for which 𝑀𝑥 (𝑥, 𝑦) = 0 or 𝑀𝑦 (𝑥, 𝑦) = 0 does not
contain any interval of the form 𝑥 = 𝑐𝑜𝑛𝑠𝑡, 𝛽1 < 𝑦 < 𝛽2 or the form 𝑦 = 𝑐𝑜𝑛𝑠𝑡, ∝1 < 𝑥 <∝2.
The optimal strategy is unique and has the following form for 𝐾(1,1, ) ≤ 0
0 𝑓𝑜𝑟 0 ≤ 𝑥 < 1,
𝐹(𝑥) = 𝐼1 = { (22)
1 𝑓𝑜𝑟 𝑥 = 1.
There is an optimal strategy of the following form for 𝐾(0,1, ) > 0:
0 𝑓𝑜𝑟 𝑥 = 0,
𝐹(𝑥) = 𝐼0 = { (23)
1 𝑓𝑜𝑟 0 < 𝑥 ≤ 1.
In the case 𝐾(0,1, ) < 0 < 𝐾(1,1), we can assume without loss of generality 𝐾(𝑥, 𝑥, ) > 0 for 0 <
𝑥 ≤ 1. Then there is a uniquely defined interval of the form [𝑎, 1], 0 ≤ 𝑎 ≤ 1, such that the optimal
strategy is as follows:
322
0 𝑓𝑜𝑟 𝑥 = 0,
∝ 𝑓𝑜𝑟 0 < 𝑥 ≤ 𝑎, (24)
𝑥
𝐹(𝑥) =
∝ − ∫ 𝑓𝑎1 (𝑧)𝑑𝑧 𝑓𝑜𝑟 𝑎 < 𝑥 ≤ 1
{ 𝑎
The function 𝑓𝑎1 (𝑥) is a continuous, positive function. The parameter ∝ is the jump of 𝐹(𝑥) at zero
and is determined from the normalization equation:
1
∫ 𝑓𝑎1 (𝑧)𝑑𝑧 = 1−∝ (25)
𝑎
From Theorem 1 it follows that the optimal strategy 𝐹(𝑥) for a symmetric game in the case under
consideration exists only if it is possible to find numbers 𝑎, ∝, that satisfy the conditions 0 ≤ 𝑎, ∝< 1
and such a continuous non-negative function 𝑓𝑎1 (𝑥) for 𝑎 < 𝑥 < 1 such that
𝑦 1
𝑎𝐾(0, 𝑦) + ∫ 𝐾(𝑥, 𝑦)𝑓𝑎1 (𝑥)𝑑𝑥 − ∫ 𝐾(𝑦, 𝑥)𝑓𝑎1 (𝑥)𝑑𝑥 = 0, (𝑎 < 𝑦 < 1) (26)
𝑎 𝑦
Remark 2. The case of the function 𝑀(𝑥, 𝑦), which increases in 𝑦 and decreases in 𝑥, using the
substitution 𝑧 = 1 − 𝑥, 𝜂 = 1 − 𝑦 reduces to the case of increasing in 𝑥 and decreasing in 𝑦, which was
considered in the Theorem 1.
Remark 3.
If in the Theorem 1, instead of the condition (1), we assume that (𝐾𝑦 (𝑦, 𝑦) − 𝐿𝑦 (𝑦, 𝑦)) > 0 and
(𝐾𝑥 (𝑥, 𝑥) − 𝐿𝑥 (𝑥, 𝑥)) > 0, then one can verify [27, 28] that the optimal strategies of both parties have
the form of the distribution function 𝐹(𝑥) = (∝ 𝐼𝑎 , 𝑓𝑎𝑏 (𝑥), 𝛽𝐼𝑏 ) and 𝐻(𝑦) = (𝛾𝐼𝑎 , ℎ𝑎𝑏 (𝑦), 𝛿𝐼𝑏 ), where
∝, 𝛽, 𝛾, 𝛿 ≥ 0, and the function 𝑓𝑎𝑏 (𝑥) and ℎ𝑎𝑏 (𝑦) are obtained in the form of Neumann series in the
eigenfunctions of the conjugate integral equations
𝑏
𝑓𝑎𝑏 (𝑡) − ∫ 𝑇𝑎𝑏 (𝑥, 𝑡)𝑓𝑎𝑏 (𝑥)𝑑𝑥 =∝ 𝑝1 (𝑡) + 𝛽𝑝2 (𝑡) (27)
𝑎
𝑏
ℎ𝑎𝑏 (𝑡) − ∫ 𝑈𝑎𝑏 (𝑢, 𝑦)ℎ𝑎𝑏 (𝑦)𝑑𝑦 = 𝛾𝑞1 (𝑢) + 𝛿𝑞2 (𝑢) (28)
𝑎
Further, consider a special class of symmetric games for which 𝑀(𝑥, 𝑦) is not necessarily continuous
in the set of variables at the points (0,0) and (1,1), and it is only required that the following limits exist
𝐾(0,0) = lim 𝐾(0, 𝑦) ; 𝐾(1,1) = lim 𝐾(𝑥, 1). (29)
𝑦→0 𝑥→0
We will assume that
𝑥
𝐾(𝑥, 𝑦) = 𝑘 ( ),
𝑦 (30)
323
The function 𝑘(𝑢) is continuously differentiable in the interval 0 ≤ 𝑢 ≤ 1, and its derivative 𝑘′(𝑢)
does not change the sign on this interval. Moreover, the set of points 𝑢 for which 𝑘 ′ (𝑢) = 0 does not
contain any interval.
It is easy to see that for the equation 𝑘 ′ (𝑢) ≥ 0, the negative strategy is 𝐹(𝑥) = 𝐼1, for the equation
𝑘(1) ≤ 0 and 𝐹(𝑥) = 𝐼0 , for 𝑘(1) ≥ 0. The proof of this fact is based on the idea of finding sustainable
strategies. For this, we write the equality
𝐶1 (𝐹, +0) = 𝐶1 (𝐹, 0)+∝ 𝐾(0,0) = 𝐶1 (𝐹, 0)+∝ 𝑘(0). (31)
The validity of this equality is established using (29). Indeed, for the equation 𝛿 > 0 we have the
following expression
𝛿−0 1
𝐶1 (𝐹, 𝛿) = ∫ 𝐾(𝑥, 𝛿)𝑑𝐹(𝑥) − ∫ 𝐾(𝛿, 𝑥)𝑑𝐹(𝑥), (32)
0 𝛿
1
𝐶1 (𝐹, 0) = − ∫ 𝐾(0, 𝑥)𝑑𝐹(𝑥). (33)
+𝛿
Thus
𝐶1 (𝐹, 𝛿) − 𝐶1 (𝐹, 0) =
𝛿−0 1 1
=∝ 𝐾(0, 𝛿) + ∫ 𝐾(𝑥, 𝛿)𝑑𝐹(𝑥) − ∫ 𝐾(𝛿, 𝑥)𝑑𝐹(𝑥) + ∫ 𝐾(0, 𝑥)𝑑𝐹(𝑥) (34)
+0 𝛿 +𝛿
The first term on the right-hand side of formula (34) as 𝛿 → 0, taking into account (29), tends to ∝
𝐾(0,0). In order to estimate the integrals in (34), for a given 𝜀 > 0, we choose 𝜂 such that the total
variation of 𝐹(𝑥) in [0, 𝜂] is less than 𝜀/4𝐾0, where 𝐾0 = 𝑠𝑢𝑝|𝐾(𝑥, 𝑦)|. Then the first integral will be
less than 𝜀/4, and the next two can be represented as:
𝜂 𝜂
∫ 𝐾(0, 𝑥)𝑑𝐹(𝑥) − ∫ 𝐾(𝛿, 𝑥)𝑑𝐹(𝑥) +
1+0 𝛿
(35)
1+0
+ ∫ (𝐾(0, 𝑥) − 𝐾(𝛿, 𝑥))𝑑𝐹(𝑥) = 𝐼1 + 𝐼2 + 𝐼3 .
𝜂
It is obvious from (35) that all |𝐼𝑖 | ≤ 𝜀/4, 𝑖 = 1,2,3. Hence, this proves the validity of (31).
Let us first take the value 𝑎 = 0. From 𝐶1 (𝐹, 𝑦) = 0 for 𝑎 < 𝑦 < 1 it follows that 𝐶1 (𝐹, +0) = 0.
For 𝑎 > 0, it should be 𝐶1 (𝐹, 0) = 0. It leads to a contradiction with (31), due to the expression 𝑘(0) <
0. On the other hand, for ∝= 0 we have the following expression
1 1
𝐶1 (𝐹, +0) = − ∫ 𝑘(0)𝑓(𝑥)𝑑𝑥 = −𝑘(0) ∫ 𝑓(𝑥)𝑑𝑥 = −𝑘(0) > 0 (36)
0 0
that it is also impossible. If we take ∝> 0, then from 𝐶1 (𝐹, 𝑎) = 0, and strict decrease of the function
we obtain
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1
𝑦
𝐶1 (𝐹, 𝑦) =∝ 𝑘(0) − ∫ 𝑘 ( ) 𝑓(𝑥)𝑑𝑥 (37)
𝑥
𝑎
on the interval 0 < 𝑦 ≤∝ we get 𝐶1 (𝐹, +0) > 0. If ∝> 0 and 𝐶1 (𝐹, 0) = 0, then from expression (31)
we obtain 𝐶1 (𝐹, +0) =∝ 𝑘(0) < 0. This is a contradiction. Hence ∝> 0 and ∝= 0. In this case,
expression (26) is equivalent to the expression 𝐶1 (𝐹, 𝑦) = 0 on the interval (∝ ,1) under the condition
С1′ (𝐹, 𝑦) = 0. It follows from this expression that, we obtain an integral equation for determining the
density 𝑓(𝑥)
𝑦 1
𝑥 𝑥 𝑓(𝑥) ′ 𝑦
2𝑘(1)𝑓(𝑦) = ∫ 2 𝑘 ′ ( ) 𝑓(𝑥)𝑑𝑥 + ∫ 𝑘 ( ) 𝑑𝑥 , (𝑎 < 𝑦 < 1). (38)
𝑦 𝑦 𝑥 𝑥
𝑎 𝑦
In this case, the normalization condition must be satisfied
1
∫ 𝑓(𝑥)𝑑𝑥 = 1.
𝑎
The Remark 4.
For the case of 𝑘 ′ (𝑢) ≤ 0, it can be shown [28, 29] that optimal strategies are 𝐹(𝑥) =∝ 𝐼0 + 𝛽𝐼1. In
addition, it is easy to check the validity of the following expressions
𝐼1 (𝑥)𝑓𝑜𝑟 𝑘(0) < 0,
𝐹(𝑥) = {∝ 𝐼0 (𝑥) + (1−∝)𝐼1 (𝑥) 𝑓𝑜𝑟 𝑘(0) = 0 (0 ≤∝≤ 1),
𝐼0 (𝑥) 𝑓𝑜𝑟 𝑘(0) > 0.
The solution of the game 𝐺(𝑀, [0,1]) with the payoff function 𝑀(𝑥, 𝑦), (0 ≤ 𝑥, 𝑦 ≤ 1) is called a
pair of distribution functions (strategies) 𝐹1∗ and 𝐹2∗ and a real number 𝜐 (value of the game) that satisfies
the condition
1 1
∫ 𝑀(𝑥, 𝑦)𝑑 𝐹2∗ (𝑦) ≤ 𝜐 ≤ ∫ 𝑀(𝑥, 𝑦)𝑑 𝐹1∗ (𝑥), 0 ≤ 𝑥, 𝑦 ≤ 1.
0 0
It follows from this expression that if the player 𝐺1 uses the strategy 𝐹1∗ , then the average payoff is
calculated by the following formula
1
𝐹(𝐹1∗ , 𝐹2 ) = ∬ 𝑀(𝑥, 𝑦)𝑑𝐹1∗ (𝑥)𝑑𝐹2∗ (𝑦).
0
This payoff cannot be less than the number 𝜐, i.e. the player 𝐺1, as it were, neutralizes the opponent’s
actions. And, conversely, if the player 𝐺2 applies the strategy 𝐹2∗ , then his average loss 𝐹(𝐹1 , 𝐹2∗ ) will
always be greater than the number 𝜐, regardless of the actions of the player 𝐺1. Therefore, it is natural
that each player should strive to choose such distribution functions 𝐹1∗ and 𝐹2∗ , which could neutralize
the opponent’s actions. Indeed, for the player 𝐺1, 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 𝐺2 must choose a strategy that would provide him, within reasonable limits, the smallest
possible loss, regardless of the actions of the player𝐺1. Naturally, if the game has an equilibrium
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position on the space of distribution functions, then only in this case the players can choose optimal
strategies [30].
In general, the player 𝐺1 can guarantee himself a payoff of at least
1
𝜐1 = max min ∫ 𝐶1 (𝐹1 )𝑑 𝐹2 (𝑦) = max min 𝐶1 [𝐹1 (𝑦)]. (39)
𝐹1 𝑦 𝐹1 𝑦
0
Here
1
𝐶1 (𝐹1 ) = ∫ 𝑀(𝑥, 𝑦)𝑑𝐹1 (𝑥).
0
Similarly, the player 𝐺2, by the appropriate choice of the distribution function 𝐹2 (𝑦), can guarantee
himself a loss of no more than
1
𝜐2 = min max ∫ 𝐶2 (𝐹2 )𝑑 𝐹1 (𝑥) = min max 𝐶2 [𝐹2 (𝑥)]. (40)
𝐹2 𝐹1 𝐹2 𝑥
0
Here
1
𝐶2 (𝐹2 ) = ∫ 𝑀(𝑥, 𝑦)𝑑𝐹2 (𝑦).
0
From equations (39) and (40) we obtain
𝜐1 ≥ min 𝐶1 (𝐹1 )
𝑦
(41)
𝜐2 ≤ max 𝐶2 (𝐹2 ),
𝑥
Let the player 𝐺2 choose the distribution function 𝐹20 (𝑦) as his strategy, and let the player 𝐺1 know
this choice. Naturally, assuming such an opportunity, the 𝐺2 player should strive to find a sustainable
strategy. It follows from (41), it becomes clear that if the value 𝐶2 (𝐹2 ) has a maximum, then the player
𝐺1 will always get the best result, choosing a point 𝜒0 that corresponds to this maximum
𝜐0 ≤ 𝐶2 (𝐹2 (𝜒0 )) = max 𝐶2 (𝐹2 (𝜒)).
𝜒
It would be beneficial for the player 𝐺2 to bring the value of 𝐶2 (𝐹2 (𝑥)) to a minimum, but this is
not always possible. The player cannot influence the form of the payoff function and the choice of 𝜒0
by the player 𝐺1. Nevertheless, the player 𝐺2 can, in any case, try to choose the strategy 𝐹20 (𝑦) so that
the value of 𝐶2 (𝐹2 ) does not have a single maximum, that is, so that its “curve” has a flat top.
Similarly, if the player 𝐺2 has learned the strategy of the player 𝐺1, then he will always choose the
point 𝑦0 at which the function 𝐶1 (𝐹1 (𝑦)) will take the minimum value. In this case, the task of the
player 𝐺1 is to choose such a strategy 𝐹10 (𝑥) so that the function 𝐶1 (𝐹1 (𝑦)) does not have a single
minimum.
We denote Ω1 = {𝑥: 𝐶2 (𝐹2 (𝑥)) = 𝜈1 = 𝑐𝑜𝑛𝑠𝑡} and Ω2 = {𝑦: 𝐶1 (𝐹1 (𝑦)) = 𝜈2 = 𝑐𝑜𝑛𝑠𝑡}, where 𝜈1
and 𝜈2 are arbitrary numbers, and 𝜐1 ≤ 𝜈1 ≤ 𝜈2 ≤ 𝜐2 .
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If there is such a pair of real numbers (𝜈1 ≤ 𝜈2 ) and a pair of distribution functions (𝐹1 , 𝐹2 ),
which simultaneously satisfies the following conditions
= 𝜈 𝑓𝑜𝑟 𝑦 ∈ Ω2 ,
𝐶1 (𝐹1 (𝑦)) { 1 (42)
> 𝜈1 𝑓𝑜𝑟 𝑦 ∉ Ω2 .
= 𝜈2 𝑓𝑜𝑟 𝑥 ∈ Ω1 ,
𝐶2 (𝐹2 (𝑥)) { (43)
> 𝜈2 𝑓𝑜𝑟 𝑥 ∉ Ω1 ,
then the functions 𝐹1 and 𝐹2 will be called stable [27, 30] strategies.
The question of the existence of sustainable strategies for the payoff function 𝑀(𝑥, 𝑦) 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.
5. Conclusion
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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