Mathematical models of conflict situations using chess. Mathematical models of game theory The mathematical description of a conflict situation is called

14.08.2021

5.7. Brief Remarks on the Issue of Selective Arms Control
We have already said that the main purpose of control is to check whether the other side is complying with the arms control agreement. Control may be exercised by monitoring the production and storage of military materials, the movement of vehicles carrying military materials, the quantity of weapons in certain strategic areas, or the presence or absence of hidden military installations. In nuclear or any other tests prohibited by the treaty, the observer must look for certain evidence that can help him interpret suspicious signals.
It is absurd and impossible to examine all suspicious events to determine whether an agreement is being followed. It has long been established in industry that to control product quality it is not at all necessary to control all products; it is enough to check randomly selected samples. The cost of sampling can be quite high, even if reliable quality control methods are used.
Selective methods applied to arms control problems can vary in complexity. In general, the ideas and methods that are so useful in studying population characteristics are applicable and useful for research.
We do not need to go into detail about the various types of sampling methods, such as random, stratified, group, sequential, etc. Nor do we need to talk about the various methods for obtaining statistical inferences, which use correlation and regression, estimates and test hypotheses. The basic concepts and applications of the methods mentioned can be read in widely available books on statistics and its applications. Here we attempt to outline a typical situation in which sampling techniques can be effectively used to verify an adversary's compliance with an arms control treaty.
The problem of sampling consists of two big questions. The first is to determine the sample size and type of sampling procedure most appropriate in a particular situation. The second is to draw statistical conclusions about the entire population based on sampling data. Both of these issues must be resolved so that the conditions imposed
Disarmament Treaty, and also that they be consistent with other conditions beyond the control of the observer group. The results of the sampling must then be presented in a form that is convenient for decision makers. An area in which sampling methods can be useful for arms control, for example, is the analysis of systems of records that contain information about the transport and production of strategic materials. However, using such records for control is expensive. In addition, it may not be possible to obtain access to these records through negotiation. However, if such records become available to the parties as a result of an agreement, provision must be made for their use. Accountability control aims to create and operate a system of records and reports, recording receipts and departures, in order to prevent the dispersion and loss of materials due to negligence or, if loss has occurred, to ensure the recovery of lost materials and the prevention of similar cases in the future.
Sampling intangibles such as records poses many unusual challenges. One of them is the correspondence of the records to the actual state of affairs. Another is the consistency of the records.
If the existing level of activity in the areas of activity covered by the treaty is indicated in the documents of the parties concerned, then the monitoring team has a basis for finding activities in which the level of activity is not indicated. On the other hand, it is much more difficult to find out whether the level of activity in some area is exceeding activities established by contract
rum, since the flow of materials cannot be divided into black and white, it includes all shades of gray. Therefore, the observation team is required to be attentive and able to untangle complex issues. Naturally, small violations cannot give great advantages to the violator; the production of weapons for the preparation of large military operations requires a wide range of violations.
We believe that these should be approximately the methods applicable in the final stages of disarmament. They will serve as a tool used in the day-to-day activities of implementing the arms control treaty. But long before this stage, the ideas presented in the first five chapters of this book will play an important role in creating measures for real arms reduction.
Short description The problems that arise with selective arms control will be discussed below. Sampling procedures are little used when estimating properties that are relatively rare in population elements. If only a few elements have this property, for example 1 in 10 thousand, then the estimate will be very approximate, provided that the sample is not extremely large (high costs). For example, if the desired property is found in a small sample, then the estimate for the entire population will be greatly overestimated. No change in the sampling procedure can avoid this shortcoming, and care must be taken in selecting sample elements. The same can be said about looking for violations in the production of products for a small number of weapons. It's like looking for a needle in a haystack.
Let's assume that we need to check a factory that produces parts for agricultural machines, but which can also produce a certain number of parts for military equipment. Let us also assume that the number of machines used for peaceful purposes is unknown and, therefore, it cannot be said how many parts of a given type are intended for this purpose. How can it be established that an excess quantity of parts is being produced?
We can set standards for the service life of these parts and the service life of the machines that use these parts. It is also necessary to determine the number of cars produced based on an inspection of the factories where they are produced. Using random samples from a population of machines, we can estimate the size of the population and the need for these parts. We now have an estimate of the number of parts needed to build a new machine and to replace worn parts in old machines. By observing the production rate of these parts and estimating the maximum production volume, we can confirm or refute suspicions that these parts are secretly used in military products.
Statistics serve as a tool for measuring the effectiveness of actions taken in the policy process. These measures or indices serve as criteria for assessing how accurately agreements are being implemented. For example, average levels are often used to show how many activities have been completed. We may sometimes use visual inspection to assess the extent to which requirements have been met. However, if a large number of inspections need to be carried out to cover many areas, statistical methods are needed to obtain a single criterion for fulfilling the requirements. The effectiveness of an action can be judged by the extent to which it corresponds to the goals pursued by the policy. Therefore, in addition to developing viable goals and sustainable courses of action, actions must be taken (as an expression of policy) that ensure the effective implementation of these requirements.
Sometimes it happens that there are no effective actions that can be used to implement a certain policy. This is, for example, the case when two countries block each other's actions. If the state cannot act in accordance with its goals, then the country will experience unrest. In ch. 6 will be considered general concepts disorder, aggression and factors influencing conflict resolution.

Part IV
INTERMEDIATE AND LONG-TERM ISSUES IN ARMS CONTROL - CONFLICT GROWING ANALYSIS, IDEAS AND PERSPECTIVES

CHAPTER 6
CONFLICT STUDY

6.1. Introduction
This chapter will outline some issues regarding the causes of conflicts. We first describe some research on escaping
lations using examples of laboratory-type conflicts and find out what factors determine the growth of conflicts. Then some qualitative considerations will be given regarding war and peace in human history.
“Conflict arises as a result of discontent, and discontent arises as a result of insufficient satisfaction of needs,” say supporters of one of the ideological schools. War and Peace is briefly described as a chain of disorders and recoveries.
Other schools (some of which are briefly mentioned) believe that wars are generated by aggressive instincts, hatred, boredom, mutual misunderstanding, differences in the level of culture, the desire to unite a divided country based on hatred of a common enemy, new scientific discoveries, the desire to stimulate economic growth by creating “artificial” demand, the desire to capture new markets, the struggle for survival, the expansion of a dynamic civilization, the desire for the dominance of the elite of the military-industrial complex, etc. However, be that as it may, the theory set out in section 2.4, makes it possible to rationally resolve the issue of being drawn into a conflict.
The current situation does not look very reliable. Therefore, an attempt is made to paint a picture of the future and show the real possibilities of establishing a lasting peace, provided that we manage to survive the present moment. The final section describes some areas of research and recommended action at this time (and in the near future) that may help resolve conflicts peacefully.

6.2. Experiences with conflict escalation
We sometimes mistakenly believe that if nations understand the dangers of nuclear weapons, then they will strive to intelligently resolve conflicts that arise, in the worst case, using conventional weapons. However, quite naturally, the losing side may resort to the threat of using nuclear weapon to avoid defeat and even regain lost ground. This could end in disaster. In addition, some peoples have a different concept of rationality from ours, especially if they have nothing to lose materially. Until the processes of escalation and how to manage them are fully understood, it is unlikely that a conventional war will be kept under control. Understanding escalation processes and how to manage them will greatly increase hopes of limiting damage should conflict occur. This theory should find its application to a war waged by conventional means, if there are indications in which direction the conflict will develop in the event of certain actions. Such actions are sometimes aimed at de-escalation by suppressing the enemy, but in reality they only intensify the conflict.
Over the past several years, the Disarmament and Arms Control Agency, in collaboration with the Center for Operations Research at the University of Pennsylvania, has been conducting research on the conditions under which conflicts escalate and de-escalate to explore the possibility of influencing the rate of escalation or de-escalation by manipulating the conditions that determine the interaction of parties - participants in the conflict. The research involved: a) analyzing some historical conflicts and studying relevant literature, b) conducting experiments to determine the effect of interactions between various variables, and c) developing a theory based on experimental data and generalizing it to real-world problems.
As a result of the literature review, several hypotheses about escalation and de-escalation were proposed, and then a) their generality and b) the identification of critical variables were tested in experimental situations. Examples of hypotheses: a) in the absence of communication, the likelihood of escalation increases, b) the greater the role played by ideological issues, the more likely escalation, c) escalation depends on economic development, d) escalation is more likely if the conflict develops gradually, e) escalation is more likely in the presence of a multilateral command.
A relatively complex experimental situation was constructed, the so-called “artificial reality” (or “rich game”), which nevertheless was the simplest game that met the following conditions:
1. It is “rich” enough so that many hypotheses expressed about the phenomena under study can be tested, in in this case we are talking about the dynamics of large social conflicts. (Obviously, such experiments cannot confirm a hypothesis about a particular real phenomenon, but they can determine the limits of the validity of the hypothesis or show in what direction it can or should be generalized.) The purpose of the conditions is to create an experimental situation that is realistic enough to most of the properties of real conflict were applicable to her.
2. There must be precise descriptions of the variables and units for their measurement, in addition, simplifications must be indicated (for example, some variable is assumed to be equal to a constant). This enables us to consistently construct ever richer experimental situations by introducing complications.
3. The appropriate behavior in the experimental situation must be expressed quantitatively.
4. The situation should be decomposed into a number of simpler experimental situations and, if possible, these simple situations should already be studied or close to those already studied.
An experimental situation that satisfies these conditions is not a model of reality, but rather can be considered a first step towards creating quantitative models of a real situation; that's why we call it "artificial reality." It is used to accumulate experimental data for the interpretation of which the first theory is built. Experience is gained through the rich play of an experiment designed to systematically test hypotheses about real conflicts that are described in operational and quantitative terms so that they can be used in theoretical constructions.

Notes on the construction of artificial reality
Artificial reality consists of two symmetrical games in which moves are made simultaneously. One is a positive-sum game - the "prisoner's dilemma" - which to some extent depicts an international (two-country) economy. The other is a negative-sum game called roosters, which resembles a confrontation between two countries where they are on a collision course in the hope that the other will make concessions.
END OF PARAGMEHTA BOOKS

The Game Theory section is represented by three online calculators:

Solving a matrix game. In such problems, a payment matrix is specified. It is required to find pure or mixed strategies of players and, game price. To solve, you must specify the dimension of the matrix and the solution method.
Bimatrix game. Usually in such a game two matrices of the same size of payoffs of the first and second players are specified. The rows of these matrices correspond to the strategies of the first player, and the columns of the matrices correspond to the strategies of the second player. In this case, the first matrix represents the winnings of the first player, and the second matrix represents the winnings of the second.
Games with nature. It is used when it is necessary to select a management decision according to the criteria of Maximax, Bayes, Laplace, Wald, Savage, Hurwitz.

In practice, we often encounter problems in which it is necessary to make decisions under conditions of uncertainty, i.e. situations arise in which two parties pursue different goals and the results of the actions of each party depend on the activities of the enemy (or partner).

A situation in which the effectiveness of a decision made by one party depends on the actions of the other party is called conflict. Conflict is always associated with some kind of disagreement (this is not necessarily an antagonistic contradiction).

The conflict situation is called antagonistic, if an increase in the winnings of one of the parties by a certain amount leads to a decrease in the winnings of the other side by the same amount, and vice versa.

In economics, conflict situations occur very often and are of a diverse nature. For example, the relationship between supplier and consumer, buyer and seller, bank and client. Each of them has their own interests and strives to make optimal decisions that help achieve their goals to the greatest extent. At the same time, everyone has to take into account not only their own goals, but also the goals of their partner and take into account the decisions that these partners will make (they may be unknown in advance). In order to make optimal decisions in conflict situations, a mathematical theory of conflict situations has been created, which is called game theory . The emergence of this theory dates back to 1944, when J. von Neumann’s monograph “Game Theory and Economic Behavior” was published.

The game is a mathematical model of a real conflict situation. The parties involved in the conflict are called players. The outcome of the conflict is called a win. The rules of the game are a system of conditions that determine the players’ options for action; the amount of information each player has about the behavior of their partners; the payoff that each set of actions leads to.

The game is called steam room, if it involves two players, and multiple, if the number of players is more than two. We will only consider doubles games. Players are designated A And B.

The game is called antagonistic (zero sum), if the gain of one of the players is equal to the loss of the other.

The choice and implementation of one of the options provided for by the rules is called progress player. Moves can be personal and random.
Personal move- this is a conscious choice by a player of one of the options for action (for example, in chess).
Random move is a randomly selected action (for example, throwing a dice). We will only consider personal moves.

Player strategy is a set of rules that determine the player’s behavior during each personal move. Usually during the game at each stage the player chooses a move depending on the specific situation. It is also possible that all decisions were made by the player in advance (i.e. the player chose a certain strategy).

The game is called ultimate, if each player has a finite number of strategies, and endless- otherwise.

Purpose of Game Theory– develop methods to determine the optimal strategy for each player.

The player's strategy is called optimal, if it provides this player with multiple repetitions of the game the maximum possible average win (or the minimum possible average loss regardless of the opponent’s behavior).

Example 1. Each of the players A or B, can write down, independently of the other, the numbers 1, 2 and 3. If the difference between the numbers written down by the players is positive, then A the number of points equal to the difference between the numbers wins. If the difference is less than 0, he wins B. If the difference is 0, it’s a draw.
Player A has three strategies (action options): A 1 = 1 (write 1), A 2 = 2, A 3 = 3, the player also has three strategies: B 1, B 2, B 3.

B A	B 1 =1	B2=2	B 3 =3
A 1 = 1	0	-1	-2
A 2 = 2	1	0	-1
A 3 = 3	2	1	0

Player A's task is to maximize his winnings. Player B’s task is to minimize his loss, i.e. minimize the gain A. This zero-sum doubles game.

Keywords

CONFLICT / FORMAL LOGIC/ ELEMENTS / LOGICAL OPERATIONS/ LAWS OF LOGIC / STATEMENT / TWO-VALUED LOGIC / MULTI-VALUED LOGIC/ CONFLICT / FORMAL LOGIC ELEMENTS / LOGIC OPERATIONS / LAWS OF LOGIC / STATEMENT / TWO-VALUED LOGIC / MANY-VALUED LOGIC

annotation scientific article on mathematics, author of the scientific work - Vitaly Ilyich Levin, Elena Anatolyevna Nemkova

Relevance. The article discusses current problem adequate mathematical modeling of the behavior of conflicting systems, in relation to systems in which conflicts are not necessarily associated with antagonistic contradictions between system participants. A formal formulation of the problem of logical-mathematical modeling of the process of interaction between conflicting system participants is given. This task consists of constructing algebras of two-valued and multivalued logic, modeling different types of thinking, the difference of which is the source of conflict. Purpose of the article. The purpose of the article is to present and detailed analysis two-digit and many-valued logician, with an emphasis on clarifying the fundamental differences in the laws of these logics, entailing significant differences in the thinking of individuals based on these logics, and the resulting conflicts between the bearers of different logics of thinking. Method. To solve the problem, the traditional method of constructing logical systems is used, based on the introduction of basic constant elements, basic operations on them and identifying the laws to which these operations are subject. In this case, the main attention is paid to the differences between the elements of operations on them and the laws of operations between two-valued and many-valued logicians. Novelty. The position is formulated according to which there are systems in which conflicts between participants are caused not by antagonistic contradictions of their interests, but by the difference in their logic of thinking, the consequence of which is misunderstanding, provoking suspicion, and then aggression. These are so-called imaginary conflicts, the fight against which requires special approaches. Result. A procedure has been developed for constructing an algebra of logic of various meanings that adequately models thinking processes. Two-digit and multivalued logic thinking and their laws. Fundamental differences between two-digit and many-valued logician. An example of an analysis of conflict caused by differences in logics of thinking is given.

Text of scientific work on the topic “Logical-mathematical modeling of conflicts”

Logical-mathematical modeling of conflicts

Levin V. I., Nemkova E. A.

Relevance. The article examines the current problem of adequate mathematical modeling of the behavior of conflicting systems, in relation to systems in which conflicts are not necessarily associated with an antagonistic contradiction between system participants. A formal formulation of the problem of logical and mathematical modeling of the process of interaction between conflicting system participants is given. This task consists of constructing algebras of two-valued and many-valued logic that model different types of thinking, the difference of which is the source of conflict. Purpose of the article. The purpose of the article is the presentation and detailed analysis of two-valued and multivalued logics, with an emphasis on clarifying the fundamental differences in the laws of these logics, entailing significant differences in the thinking of individuals based on these logics, and the resulting conflicts between the bearers of different logics of thinking. Method. To solve the problem, the traditional method of constructing logical systems is used, based on the introduction of basic constant elements, basic operations on them and identification of the laws to which these operations are subject. In this case, the main attention is paid to the differences between the elements of operations on them and the laws of operations between two-valued and multi-valued logic. Novelty. The position is formulated according to which there are systems in which conflicts between participants are caused not by antagonistic contradictions of their interests, but by the difference in their logic of thinking, the consequence of which is misunderstanding, provoking suspicion, and then aggression. These are so-called imaginary conflicts, the fight against which requires special approaches. Result. A procedure has been developed for constructing an algebra of logic of various meanings that adequately models thinking processes. Two-valued and multi-valued logics of thinking and their laws are described. The fundamental differences between two-valued and multi-valued logics are established. An example of an analysis of a conflict caused by a difference in the logic of thinking is given.

Key words: conflict, formal logic, elements, logical operations, laws of logic, statement, two-valued logic, many-valued logic.

Introduction

There is no doubt the importance of the general theory of conflict - a science that deals with the calculation, analysis, synthesis and resolution of general models of conflict situations. At the same time, it is clear that the construction of productive models of conflict should be based on reference to the most important specific classes of conflicting systems. And the greatest interest among these systems is, of course, human society.

Conflicts in human society, with the aim of their practical resolution, are currently dealt with by the humanities - conflictology, which is part of sociology. However, this science does not seek to reveal the internal nature of conflict situations, and without this it is impossible to build appropriate good mathematical models that allow such situations to be studied in detail.

It is generally believed that the source of human conflict is the contradiction between goals that various people put between each other. However, it is no secret that the majority (and perhaps the overwhelming majority) of humanity are people who do not set any special goals for themselves.

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But at the same time, they often conflict with other people - both those who exist aimlessly, like them, and with completely purposeful people. This fact prompts us to assume that the basis of conflicts between people is also some other feature of the human personality, not directly related to a person’s activities and his goals, but inherent to him at the genetic level. This article puts forward and substantiates a hypothesis according to which a person’s feature, which strongly and sometimes decisively influences the occurrence (or absence) of his conflicts with others, is the type, or more precisely, the logic of his thinking. For this purpose, two essentially different types of logic are considered - two-valued and many-valued, and then it is shown that the variants of human thinking based on them are largely incompatible. This incompatibility leads to mutual misunderstanding between adherents of the two indicated types of thinking and, ultimately, to conflicts between them.

1. Two-valued formal logic

Two-valued formal (otherwise known as mathematical, symbolic) logic of statements, also called classical, underlies ordinary human thinking. This logic is built using two constant elements: TRUE (designation I) and false (designation L); variables whose values are the truth values of various statements, and logical operations that can be performed on constant elements. A statement is a statement that can be either true (T) or false (F). Therefore, logical operations can be performed on statements. Logical operations over constant elements or statements P,Q are as follows: negation of P (otherwise “NOT P”), disjunction P V Q (otherwise “P OR Q”), conjunction P l Q (otherwise “P AND Q”), disjunctive disjunction P 0 Q (otherwise “EITHER P OR Q” ), equivalence P « Q (otherwise “ P IS EQUIVALENT TO Q”), implication P ® Q (otherwise “IF P, THEN Q”). These operations are defined in truth tables 1 and 2. In addition to statements that have variable truth values (I or A), there are two statements with constant truth values: an identically true statement or tautology (designation T) and an identically false statement or contradiction (designation P) .

Table 1 - Negation operation

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Table 2 - Operations of disjunction, conjunction, dividing disjunction, equivalence and implication

P Q P V Q P Ù Q P ® Q P « Q P ® Q

L L L L L I I

I L I L I L L

L I I L I L I

I AND I I L I I

In the introduced logic, the following laws are valid:

Commutative law for disjunction and conjunction

Р V Q = Q V Р, Р l Q = Q l Р; (1)

Combination law for disjunction and conjunction

(P V Q) V I = P V (£ V I), (P l Q) l I = P l (£ l I). (2)

Distributive law for conjunction relative to disjunction

(P V Q) l I = (P l I) V (d l I); (3)

Distributive law for disjunction relative to conjunction

(P l Q) V I = (P V I) l (d V I); (4)

De Morgan's Law

P V Q = P l Q, P l Q = P V Q; (5)

law of tautology

Р V Р = Р, Р l Р = Р, (6)

Law of Absorption

P l (P V Q) = P, P V (P l Q) = P; (7)

Law of action over statements with constant truth values

P V P = P, P V T = ^ P l T = P, P l P = P, (8)

Law of double negation

Law of the excluded middle

P V P = T; (10)

Law of contradiction

R l R = P; (eleven)

Law of Implication Transformation

(P ® Q) = PV Q (12)

To prove the laws of two-valued logic, truth tables of both parts are constructed, similar to the table. 1, 2. If it turns out that the tables for both parts coincide, then the law is valid. Logical laws allow you to replace expressions of propositional logic with equivalent, but simpler (or more convenient in some sense) expressions.

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The constructed propositional logic allows us to formally describe the process of human thinking using the formal construction

A1 l A2 l... l Ap ® B. (13)

Here A1,...,An are the original statements (premises), B is the new one

statement (conclusion). A complex statement (13) is called a logical inference. A logical conclusion can be true or false. If it is true for any truth values of the premises and conclusion (i.e., identically true), it is considered true. In other cases, the logical conclusion is considered incorrect. To check the correctness of a logical conclusion, you can build its truth table and make sure that it is identically true, or transform expression (13) of logical conclusion using suitable logical laws and bring it to an identically true statement.

Let us present another logical law - the transitivity of implication, which is important for logical inference

(P ® 0l(0 ® I) ® (P ® I). (14)

Law (14) shows that the operation of implication ® is transitive, which allows logical inference to be carried out as a multi-stage (chain) process.

Two-valued formal logic and automata that implement it are widely used for mathematical modeling of many classes of systems. In particular, conflicting systems.

2. Many-valued formal logic

All the main features of many-valued logic appear starting from the value k = 3. Therefore, we will limit ourselves to three-valued formal propositional logic. This logic underlies human thinking, which is more complex than ordinary thinking. It is built using the same constant elements as two-valued logic: I and L, with the addition of the constant element UNCERTAINTY (designated N). The new element is uncertain in the sense that it is neither true nor false. As in two-valued logic, the truth of various statements is used as variable values. These values can now be I, L or N. Logical operations can be performed on the constant elements I, L and N and on variables (statements) that take the same values I, L and N. In three-valued logic there are the same operations as in double digits. However, the number of possible options for each operation is much greater. In table 3-5 define the three most common variants of the negation operation. In table 6 defines the operations of disjunction Р V 0, conjunction Р l 0, separating disjunction Р Ф 0, equivalence Р « 0, implication Р ® 0 (one option for each operation). In addition to statements with variable truth values (I, L or N), there are three statements with constant truth values: I (called the tautology T), L (called the contradiction P) and N (called the indeterminacy of N).

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The first two coincide with the corresponding ones in two-valued logic, the third is a new statement with a constant truth value.

Table 3 - Mirror negation

Table 4 - Left cyclic negation

Table 5 - Right cyclic negation

Table 6 - Operations of disjunction, conjunction, dividing disjunction, equivalence and implication

P Q P v Q P A Q P ® Q P « Q P ® Q

L L L L L I I

L N N L N N I

L I I L I L I

N L N L N N N

N N N N N N N

N I I N N N I

I L I L I L L

I N I N N N N

I AND I I L I I

In the introduced three-valued logic, the laws of two-valued logic that do not contain negation operations remain valid. These are the laws of commutative, associative and distributive (1)-(4), tautologies, absorption and actions with constants (6)-(8), transitivity (14). However, new laws of actions over statements with a constant truth value H appear

N V L = N, N V I = I, N l L = L, N l I = N. (15)

The main difference between three-valued logic and two-valued logic is a significant change in the laws containing the operation of negation. The specific form of these laws depends on the chosen version of the negation operation. If this is a mirror negation operation (Table 3), then

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De Morgan's laws, double negation and transformation of implication (5), (9), and (12) of two-valued logic are fair, but the law of the excluded middle (10) goes into the next law of the “partially excluded middle”

P V P = T"(P), where T"(P) = (I, with P = I or L; (16)

[I, with P = N; at 7

and the law of contradiction (11) - into the next law of “partial contradiction”

R l R = P"(P), where P"(P) = (L, with P = I or L; (17)

[I, with P = I. y 7

For the operations of left and right cyclic negation (Tables 4 and 5), all laws of two-valued logic containing negation are transformed into the corresponding new, more complex laws of three-valued logic. Thus, the laws of double negation (9), excluded third (10) and contradiction (11) are transformed into the corresponding laws - the law of triple negation

law of the excluded fourth

P V P V P = T (19)

and the law of complete contradiction

R l R l R = P, (20)

and de Morgan's laws (5) and the transformation of implication (12) - into the corresponding more complex laws, the form of which already depends on which cyclic negation is used - left or right. In connection with the discussed problem of the logic of thinking, the specification of law (18) in the form

R f R, "R; (21)

law (19) in the form of the law of “partially excluded third”

GI, with P = I or L, P V P = Tl(P), where Tl(P) = ( " p

[I, with P = I,

P p GI, with P = I or I, P V P = Tp(P), where Tp (P) = ( " p

[And, with P = L,

for right cyclic negation; and law (20) in the form of the law of “partial contradiction”

- „ GL, with P = L or I, R l P = Pl (P), where Pl (P) = ( " p _ tya

[I, with P = I,

for left cyclic negation;

P p GL, with P = L or I, P l P = Pp (P), where Pp (P) = ( " p

[I, with P = I,

for right cyclic negation.

As can be seen from (21), in three-valued logic with the operation of cyclic negation the law of double negation does not apply. Further, from (22) it follows that in this logic the law of the excluded middle does not apply - it is transformed

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into the law of the “partially excluded third”, the specific form of which depends on the version of the cyclic negation operation (right or left). Similarly, from (23) it follows that in this logic the law of contradiction does not apply - it is transformed into the law of “partial contradiction”, the specific form of which also depends on the version of the cyclic negation operation.

3. Logic and conflicts

Every thinking individual in his mental activity always uses consciously or intuitively one or another version of logic. We saw above that there are significant differences between two-valued and many-valued logic. Therefore, all individuals, according to the predominant version of logic used in their thinking, can be divided into two-valued and multi-valued thinkers. Their main differences are that for a two-valued thinker any statement can have only two truth values: true and false, and the negation of one gives the other, while for a polysemantic thinker any statement has at least three truth values: true, false and uncertain. Moreover, the negation operation can be defined in different ways, so that the negation of any truth value can in general give any other truth value.

In view of these profound differences between ambiguous and ambiguous thinkers, a complex problem of their relationship arises. The essence of this problem is that within the framework of two-valued thinking it is difficult to understand the clearly multi-valued nature of the world (from the point of view modern science). This constant misunderstanding leads to suspicion and fear. As a result, the ambiguous thinker begins to conflict with the ambiguous, leaning toward a forceful solution.

Let's consider the simplest typical example. At a banquet, during a feast, the artist, already fairly tipsy, turns to the scientist: “Aren’t you drinking?” - He answers: “I can’t!” The artist continues to insist: “Drink!” The scientist objects: “I won’t!” Then the artist loudly declares: “So you are going to write a denunciation against us!” Our artist, of course, is a typical two-valued thinker, for whom there are only two options: to drink and therefore be unable to denounce, and not to drink and therefore be able to write a denunciation. It does not occur to him that there are other options that are obvious to a scientist - a polysemantic thinker. For example, drinking to the point of unconsciousness, and then reporting something that did not happen, or not drinking at all and not reporting it for moral reasons.

The real version of this semi-fantastic story took place in 1938 at the government dacha in Kuntsevo, near Moscow, when during the next banquet hosted by I.V. Stalin, he failed to force the USSR People's Commissar of Cinematography Boris Shumyatsky to drink. After which, on the orders of the ambiguous thinker Stalin, the suspicious ambiguous thinker Shumyatsky was shot.

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The considerations presented in this section can be used as the basis for a new multi-valued logical approach to modeling conflicts, different from the two-valued logical approach based on the mathematical apparatus discussed in the work. This new approach opens up new perspectives for conflict modeling. In particular, it will increase the number of gradations of interaction between conflicting systems and thereby make the analysis of this interaction more subtle. A detailed presentation of this approach is expected in a separate article.

Conclusion

The article shows that two-valued and multi-valued logics are subject to significantly different laws, due to which they can be used to model various types of thinking. It has been revealed that the source of human conflicts can be not only the contradiction between the goals that different people set for themselves, but also human mutual misunderstanding caused by differences in types of thinking. The advantage of the described approach to the study of conflicts lies in the possibility of a more subtle insight into the essence of the development of conflict situations.

Literature

1. Dmitriev A.V. Conflictology. - M.: IIFRA-M, 2009. - 336 p.

2. Sysoev V.V. Conflict. Cooperation. Idependence: system interaction in a structural-parametric representation. - Moscow: MAEiP, 1999. - 151 p.

3. Svetlov V. A. Conflict Analysis. - St. Petersburg: Rostock, 2001. - 512 p.

4. Levin V.I. Mathematical modeling of systems using dynamic automata // Information Technology. 1997. No. 9. P. 15-24.

5. Levin V.I. Mathematical modeling using automata // Bulletin of Tambov University. Series: Natural and technical sciences. 1997. T. 2. No. 2. P. 67-72.

6. Levin V.I. Automatic model for determining the possible time of holding collective events // Izvestia RAI. Theory and control systems. 1997. No. 3. P. 85-96.

7. Levin V.I. Mathematical modeling of the Bible. Characteristic automaton approach // Bulletin of Tambov University. Series: Natural and technical sciences. 1999. T. 4. No. 3. P. 353-363.

8. Levin V.I. Automatic modeling of collective events // Automation and telemechanics. 1999. No. 12. P. 78-89.

9. Levin V.I. Mathematical modeling of the biblical legend of the Babylonian Pandemonium // Bulletin of Tambov University. Series: Natural and technical sciences. 2001. T. 6. No. 2. P. 123-138.

10. Levin V.I. Automatic modeling of historical processes using the example of wars // Radioelectronics. Computer science. Control. 2002. No. 12. P. 93-101.

11. Levin V.I. Automatic modeling of the processes of emergence and disintegration of a team // Cybernetics and system analysis. 2003. No. 3. P. 92-101.

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12. Levin V.I. Logical-algebraic approach to conflict modeling // Control, communication and security systems. 2015. No. 4. P. 69-87. URL: http://sccs.intelgr.com/archive/2015-04/03-Levin.pdf (accessed 08/01/2016).

1. Dmitriev A.V. Konfliktologia. Moscow, INFRA-M Publ., 2009. 336 p. (in Russian).

2. Sysoev V. V. Konflikt. Sotrudnichestvo. Nezavisimost": sistemnoe vzaimodeistvie v strukturno-parametricheskom predstavlenii. Moscow, MAEP Publ., 1999. - 151 p. (in Russian).

3. Svetlov V. A. Analitika konflikta. Saint-Petersburg, Burgeon Publ., 2001. 512 p. (in Russian).

4. Levin V. I. Mathematical modeling of systems with dynamic machines. Information technologies, 1997, no. 9, pp. 15-24 (in Russian).

5. Levin V. I. Mathematical modeling using automata. Bulletin of the University of Tambov. Series: Natural and Technical Sciences, 1997, vol. 2, no. 2, pp. 67-72. (in Russian).

6. Levin V. I. Automaton model determine the possible time of the collective actions. Izvestiya RAS. Theory and control systems, 1997, no. 3, pp. 85-96. (in Russian).

7. Levin V. I. Mathematical modeling of the Bible. Characteristic automata approach. Bulletin of the University of Tambov. Series: Natural and Technical Sciences, 1999, vol. 4, no. 3, pp. 353-363 (in Russian).

8. Levin V. I. Automatic modeling of collective actions. Automation and Remote Control, 1999, no. 12, pp. 78-89 (in Russian).

9. Levin V. I. Mathematical modeling of the biblical legend of the Tower of Babel. Bulletin of the University of Tambov. Series: Natural and Technical Sciences, 2001, vol. 6, no 2, pp. 123-138 (in Russian).

10. Levin V. I. Automatic modeling of historical processes on the example of the wars. Electronics. Computer science. Control, 2002, no. 12, pp. 93-101 (in Russian).

11. Levin V. I. Automatic modeling of processes of emergence and collapse of collective // Cybernetics and Systems Analysis, 2003, no. 3, pp. 92-101 (in Russian).

12. Levin V. I. Logical-Algebraic Approach to Conflicts Modeling. Systems of Control, Communication and Security, 2015, no. 4, pp. 69-87. Available at: http://sccs.intelgr.com/archive/2015-04/03-Levin.pdf (accessed 01 Aug 2016) (in Russian).

Levin Vitaly Ilyich - Doctor of Technical Sciences, Professor, PhD, Full Professor. Honored Scientist of the Russian Federation. Penza State Technological University. Area of scientific interests: logic;

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mathematical modeling in technology, economics, sociology, history; making decisions; optimization; automata theory; reliability theory; recognition; history of science; problems of education. Email: [email protected]

Nemkova Elena Anatolyevna - Candidate of Technical Sciences, Associate Professor of the Department of Mathematics. Penza State Technological University. Area of scientific interests: logic; mathematical modeling in technology and economics. Email: [email protected]

Address: 440039, Russia, Penza, Baidukova Ave./st. Gagarina, 1 a/11.

Logical-Mathematical Modeling of Conflicts

V. I. Levin, E. A. Nemkova

Keywords: conflict, formal logic elements, logic operations, the laws of logic, statement, the two-valued logic, many-valued logic.

Information about Authors

Vitaly Ilyich Levin - the Doctor of Engineering Sciences, Professor, PhD, Full Professor. Honored worker of science Russian Federation. Penza State Technological University. Field of Research: logic; mathematical modeling in technics, economics, sociology, history; decision-making; optimization; automata theory; theory of reliability; history of science; problems of education. Email: [email protected]

Elena Anatolyevna Nemkova - Ph.D. of Engineering Sciences, Associate Professor at the Department of "Mathematics". Penza State Technological University. Field of Research: logic; mathematical modeling in technics, economics. E-mail:: elenem5 8 @mail. ru

Address: 440039, Russia, Penza, pr. Baydukova / Gagarin st., 1a/11.

A move in a game is the choice and implementation by one player of one of the actions provided for by the rules of the game. The result of one move, as a rule, is not the result of the game, but only a change in the situation. Strategy is the sequence of all moves until the end of the game. Let us denote the payoff of player Pj by vj.

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Teacher: Platonova Tatyana Evgenievna

Lecture 15. Game models of conflict situations

Game theory

Basic concepts of game theory

A game is a mathematical model of a conflict situation. Unlike real conflict situations, in a mathematical model the game is played according to pre-fixed rules and conditions.

Game progress is the choice and implementation by one player of one of the actions provided for by the rules of the game. In a two-person game, moves strictly alternate. The result of one move, as a rule, is not the result of the game, but only a change in the situation.

Strategy is the sequence of all moves until the end of the game. Term the consignment associated with the partial possible implementation of the rules.

Let them take part in the game n partners. Let us denote the player's payoff Pj through v j . In this case, a positive value v j means win, negative means loss, and zero means draw.

The goal of the game is to maximize winnings at the expense of others.

Let us briefly consider the classification of games.

Depending on the number of players, games are paired (n = 2) and multiple (n > 2).
Depending on the number of strategies, games are divided into final , if players have a finite number of strategies, and endless , otherwise.
There are games zero sumif some benefit at the expense of others.
Zero-sum paired games are calledantagonistic.
Ultimate zero-sum games are called matrix
Depending on the relationship between the players, games are divided into cooperative (in which coalitions are predetermined), coalition (players can enter into agreements) andnon-coalition(players are not allowed to enter into agreements).

The players' moves are divided into personal , if the move is chosen consciously, and random , if the move is selected using a random selection mechanism.

There are strategies optimal , which provide the player with the greatest success-winning, and suboptimal

Matrix games

In general, a matrix game is given by a rectangular matrix of dimension mxn:

One player has m possible strategies ( A 1 , A 2 ,…, A m ), and the other player - n possible strategies ( B 1, B 2,…, B n ). A payoff element that the second player pays to the first if the first player chooses the strategy A i , and the second player is the strategy B j . In this case, the winning value may be less than zero.

Let us represent the matrix game in tabular form, calledpayment matrix:


a 11	a 12	a 1n
a 21	a 22	a 2n

a m1	a m2	a mn

Let's formulate basic principle of matrix game: the first player strives to win as much as possible, and the second player strives to lose as little as possible. Based on this principle, both players are conscious, and the game matrix is composed in terms of the first player's payoff; thus, the gain of the first player is simultaneously the loss of the second.

Let's look at the game from the position of the first player. Let the first player consider applying his first strategy (the first row of the matrix). Then his payoff in the worst case will not be less than the minimum element of the first row, i.e. . Similarly, his payoff when applying an arbitrary strategy A i will be a value no less than. Thus, among all his strategies, he can choose the strategy that is the best in the sense of the largest possible minimum payoff. This value of the guaranteed win under the worst conditions of opposition from the second player is calledlower net price of the game maximin):

Now consider the point of view of the second player. When he uses his first strategy, which is represented by the first column of the payment matrix, his maximum loss will be the value for the most unfavorable actions of the first player. Similarly, his loss when applying an arbitrary strategy In j will be a value not greater than. This value of the guaranteed loss under the worst conditions of opposition from the first player is calledtop net price of the game, and it is equal to the following expression ( minimax):

Therefore, the strategies of the first player are called maximin, and the second minimax.

Example 1 . Find the lower and upper net prices of a matrix game with a matrix:

The lower net price of the game is equal, the upper net price of the game is equal. Thus, in this case. The element is called saddle element of the game matrix (it is both minimal in its row and maximal in its column), and the game itselfgame with a saddle point.In this case, the lower and upper net prices of the matrix game coincide, and they are equal to the net price of the game. The optimal strategies of the players are, and it is unprofitable for any of the players to deviate from them.

Example 2 . Let's solve a similar problem for a game with a matrix:

Here we have. Net price of the game. Thus, there is no saddle point in the game. Solving such a game is difficult. Let's clarify this idea. The strategy guarantees the first player to win at least 4 units in the worst case when the second player chooses the strategy. Similarly, the strategy guarantees that the second player will lose no more than 7 units in the worst case when the first player chooses the strategy. The first player can choose a strategy to win 9 units, but the second player will choose a strategy.

A situation is created where the partners are rushing about over strategies. This means that in this case the approach to the game itself needs to be changed.

Pure and mixed player strategies

Pure Player Strategyis a possible move of the player, chosen by him with probability equal to 1.

Let us represent the pure strategies of the players from Example 1 in the form of unit vectors: the strategy of the first player, the strategy of the second player. In general, for a pair of strategies, pure strategies can be written in the form, and in the first vector the unit is on i- th position, and in the second vector on jth position.

Mixed strategythe first (second) player is called the vector:

Here are the probability values of using the corresponding strategies of the first and second players.

The game is called active if.

Based on the considered definitions, the following conclusions can be drawn:

The game becomes random.
The amount of winning (loss) becomes random.
average value winning (mathematical expectation of winning) is a function of mixed strategies: and is calledpayment function of the game.

The strategies are called optimal , if the condition is met for arbitrary strategies.

The value of the payment function for the optimal strategies of the players determines the price of the game, i.e. .

Game solution is called the set of optimal strategies and game prices.

Theorem (the main theorem of matrix game theory is von Neumann's theorem). Any matrix game has at least one solution in mixed strategies two optimal strategies and their corresponding price: .

Methods for solving matrix games

All methods for solving matrix games discussed in our course are based on the theorem about active strategies.

Theorem (about active strategies). If one player adheres to his optimal mixed strategy, then the payoff remains unchanged and equal to the cost of the game if the other player does not go beyond the limits of his active strategies (i.e., uses any of them in its pure form or mixes them in any proportions).

Now let's look at some special cases of solvable matrix games.

A game that has a saddle element in the payoff matrix (saddle point game)

In this case, the first player implements his maximin strategy, and the second player implements his minimax strategy, the lower net price of the game is equal to the upper net price of the game. Then they say thatthe game is decided in pure strategies,deviating from which is not beneficial for anyone (see example 1).

A game with a 2 by 2 payoff matrix that does not have a saddle element.

There is no optimal solution in pure strategies, so a solution is sought in mixed strategies. To find them, we use the theorem about active strategies. If the first player sticks to his optimal mixed strategy, then his average payoff will be equal to the cost of the game, no matter what active strategy the second player uses.

Let the payment matrix be given

(mixed strategies of the players are written around the matrix). Let us write two equations for the first player: the first for the case of the second player using only his first strategy, and then only the elements of the first column of the matrix are used, the second for the case of the second player using only his second strategy, and then only the elements of the second column of the matrix are used. The left sides of these equations calculate the mathematical expectation of the first player's winnings, which is equal to the cost of the game. These two equations contain three unknowns at once - , and the equations themselves are homogeneous, therefore, for the unique solvability of the system, a third equation with a free term is necessary. This additional and very important equation is the normalization condition, according to which the sum of the probabilities of all events must be equal to one. Thus, the final system of equations for the first player looks like this:

This system can be solved very simply for the reason that it is possible to express one unknown quantity through another from the third equation. The solution of this system gives the values of the optimal mixed strategy of the first player and the corresponding game price.

To completely solve the game, it remains to find the optimal mixed strategy of the second player. Here the players seem to change places. The construction of the system of equations is similar to the previous case. The difference is that not the columns of the matrix, but the rows are taken as the coefficients of the system, since it is the rows that correspond to the pure strategies of the first player. So the system looks like this:

Example 3. Find mixed player strategies for the matrix.

Let’s create systems of equations for the first player and the second:

The solution of which gives

Thus, we write the solution of the game in the form:

Graphic solution for a two-on-two game.

Let's consider example 3 again. Let's plot a segment of unit length on the abscissa axis. At the ends of this segment we draw vertical axes I - I and II - II. Let's put it on the axis I - I winning values first player when using first strategies. On axis II - II let's put aside the winnings first player when using second strategies. Let's connect the points with straight line segments. Broken B 1 KB 2 - lower limit of winnings. The player's minimum payoff lies on this boundary A with any mixed strategy. Dot TO , in which this gain reaches a maximum, determines the decision and price of the game. For the mixed strategy of the second player we can also write:

The strategy of the second player can be found directly if the players are swapped on the chart, and instead of the maximum of the lower limit of winning, consider the minimum of the upper limit of loss. Either way, point TO is both a maximin and a minimax point.

Graphic solution of the game.

The construction is similar to the two by two case. Here n enemy strategies will be depicted as segments n straight Next, we consider the lower boundary, which is a broken line. The maximum of the broken line is reached at one of the vertices, where two enemy strategies intersect, which are active

In game theory, it is proven that any finite game has a solution in which the number of active strategies of each side does not exceed the smallest of the numbers or. Therefore the gamehas a solution in which no more than two active strategies are involved on each side. (The game can be solved in the same way). One has only to find these strategies and the game turns into a game.

Example 4 . Solve the game with the following payoff matrix:

This game has 2 strategies on the part of the first player and three strategies on the part of the second. Therefore, we will graphically determine one of the strategies of the second player, which is inactive. Let's build a graph regarding the strategies of the first player.

The graph shows that for the second player, the first strategy, which is inactive, is clearly unprofitable. Thus, we exclude the first column from the game matrix, corresponding to the first strategy of the second player, and arrive at a two-by-two matrix of the following form:

For this matrix, we write a system of equations - for the first player, and a system: - for the second player.Solving these systems gives the following result:

Payoff Matrix Game mx2

As noted above, the game is pre-solved graphically from the point of view of the second player. In this case, the active strategies of the second player are determined. The minimax strategy is applied on the chart and the minimum of the upper limit of the loss is considered. Let's look at an example.

Example . Solve a matrix game with the following matrix:

Let's construct a graph where on the left we plot the values of the losses of the second player when he uses the first strategy, and on the right - the values of the losses of the second player when he uses the second strategy.

The graph shows that the second strategy is unprofitable for the first player, since when it is applied, the first player’s gain (and, accordingly, the second player’s loss) will be less. Thus, the active strategies of the first player will be the first and third. Accordingly, we write down systems of equations for mixed player strategies:

System solution: For the first player the system has the form (strategy A 2 we do not consider it as unpromising):

The solution to the system will be the values. Thus, the solution to the game looks like this: .

Games with dominant and redundant strategies.

Consider two strategies of the first player i yu and k Yu. In this case, let the following conditions be satisfied for all elements of the corresponding rows of the matrix: . In this case they say that i I am the first player's strategy dominates his j th strategy. If each inequality holds as strict, then one strategy is said to bestrictly dominatesover the other. In any case, of the two strategies, the first player will prefer the dominant one, since using the dominated strategy at least his payoff will not increase. In this case, you can accept it.

Similarly, consider the two strategies of the second player - j - yu and l yu, and at the same time the following conditions are met for the elements of the corresponding columns of the matrix: . For the second player, as is known, a strategy that gives a smaller loss is more profitable, so they say that j - i strategy dominates l - y. If pairwise inequalities are strict, then one strategy is said to bestrictly dominatesover the other. At the same time, of course.

If any of the players has two strategies that have only matching elements in the matrix, then these strategies are called duplicating . It does not matter which of them the player chooses to solve the game.

As a result, in the presence of dominant and overlapping strategies, some strategies may not be considered, which in some cases will lead to a significant simplification of the payment matrix.

Equivalent transformation of the payment matrix.

This transformation is applied to simplify calculations without changing the optimal mixed strategies of the players.

Theorem . Optimal mixed strategies of the 1st and 2nd players, respectively, in a matrix game with a price v will also be optimal in a matrix game with price, where.

Example . In a matrix game with a payoff matrix we take b =10, C =-6 . Let's apply the transformation bA+c , then we get a game with the same optimal strategies, but with a different equivalent matrix: .

Equivalence of a matrix game to a pair of dual ZLPs.

Consider a matrix game of size. Let us reduce it to a linear programming problem in general form. We have:

Let's assume that. This can always be done using the theorem on the equivalent transformation of the payment matrix, therefore, we can consider the price of the game to be a positive number, v >0.

For the first player we have a system of inequalities (taking into account the fact that the first player strives to win as much as possible, the cost of the game for him will exceed v):

Let's introduce new variables by dividing by the price of the game: then we get the ZLP:

When constructing the objective function, we take into account that the cost of the game for the first player is maximized.

Similarly, for the second player we have a system of inequalities:

Dividing by the price of the game and introducing new variables, we obtain the ZLP for the second player:

Here the objective function is set to the maximum, because the cost of the game for the second player is minimized.

As a result, we obtained a pair of symmetric dual ZLPs. According to the first duality theorem, therefore, the price of the game v has the same meaning for both players.

The concept of playing with nature (statistical games)

Here one of the participants is a person or a group of people with a common goal, the so-called. statistician (player A), other participant nature (player P), or the whole complex of external conditions under which the statistician has to make a decision. Nature is indifferent to winning and does not seek to turn the mistakes of statistics to its advantage.

The statistician has m strategies; nature can realize n various states. In this case, the probabilities of realizing states of nature can be known. If a statistician can evaluate the use of each of his strategies in any state of nature, then the game can be specified by the payoff matrix:

P 1	P 2	P n
a 11	a 12	a 1n
a 21	a 22	a 2n

a m1	a m2	a mn

When simplifying the payment matrix, one cannot discard certain states of nature, because nature can realize any of its states, regardless of whether it is beneficial to the statistician or not. Nature can even help the player A .

When choosing the optimal strategy, statisticians use various criteria. At the same time, they rely on both the payment matrix and the risk matrix.

Risk statistics. The risk matrix has the same dimension as the payment matrix:

Conversion from the payment matrix to the risk matrix is carried out column by column: in each column of the payment matrix, the largest element is selected, which is replaced by zero in the risk matrix, and the remaining elements of the risk matrix column are obtained by subtracting the corresponding elements from this largest element.

If the probabilities of states of nature are known, use Bayes criterion : the strategy that provides the maximum average winning is selected. Statistics:

When the probabilities of states of nature are unknown, Laplace's principle of insufficient reason is applied, when all states are considered equally probable:

Then the average payoff for each strategy is calculated as the arithmetic mean of the payoffs over all possible states of nature:

An equivalent approach would be to select a strategy that provides the lowest average risk:

with known probabilities of states of nature and

in case these probabilities are unknown. With this approach, the result will be exactly the same as when analyzing the largest average winnings.

If the probabilities of states of nature are unknown, then the Wald, Savage and Hurwitz criteria are more widely used.

The optimal strategy according to the Wald criterion is A i , which provides the smallest payoff of all highest value. In this case, the smallest element is selected from the payoff matrix (i.e., the payment matrix) in each row, and then the largest among these elements is selected:

According to the Savage criterion, the optimal strategy is the one that minimizes the maximum risk, i.e. From each row of the risk matrix, the maximum element is selected, and then, among these elements, the row containing the minimum element is selected:

According to the Hurwitz criterion, the strategy found from the condition is considered optimal:

where is the “pessimism coefficient”. When χ=1 we have the Wald criterion, or the criterion of extreme pessimism, and when χ=0 we have the criterion of “extreme optimism”. It is recommended to choose χ between zero and one, for subjective reasons.

As a result of applying several criteria, they are compared with each other, and the statistical strategy that appears most often as the best is selected as the best.

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A group of scientists led by an employee of the Nizhny Novgorod University. N.I. Lobachevsky Alexandra Petukhova identified the parameters that are needed to manage the system that describes social conflicts. If these characteristics are fully controlled, scientists will be able to create conditions for the occurrence or prevention of such conflict. The results are published in the journal Simulation.

In mathematical modeling of social and political processes it is necessary to take into account that they cannot be strictly specified, since they are subject to constant changes. A social process is often compared to a Brownian particle. Such particles move along a trajectory that, on the one hand, is quite definite, but upon closer examination turns out to be very tortuous, with many small kinks. These small changes (fluctuations) are explained by the chaotic movement of other molecules. In social processes, fluctuations can be interpreted as manifestations of the free will of its individual participants, as well as random manifestations external environment.

In physics, such processes are usually described by Langevin's stochastic diffusion equation, which is relatively often used to model some social processes. An approach based on such equations makes it possible to take into account the manifestations of the free will of its individual participants and random manifestations of the external environment for the social system. In addition, thanks to this approach, it is possible to calculate the behavior of a social system both for a single whole and for individual particles; it also allows us to identify characteristic stable modes of operation of systems depending on various initial conditions. Finally, from the point of view of numerical modeling, diffusion equations have been sufficiently tested and studied.

The new model is based on the idea that individuals interact in society through the field of communication. It is created by each individual in society, modeling information interaction between individuals. However, it should be borne in mind that here we are talking about society, which differs from the objects of classical physics. According to research director Alexander Petukhov, from the point of view of transferring information from individual to individual, space in society combines both classical spatial coordinates and additional specific features. This is due to the fact that in modern world To transmit information you do not need to be near the object of influence.

“Thus, society is a multidimensional, socio-physical space, reflecting the ability of one individual to “reach” another with his communication field, that is, to influence him, his parameters and the ability to move in a given space,” notes Alexander Petukhov. The close proximity of individuals in this model indicates that they regularly exchange information. For this formulation of the problem, a conflict should be considered a variant of interaction between individuals or groups of individuals, as a result of which the distance in this multidimensional space between them increases sharply.

Based on this approach and the developed model, scientists found the following patterns: they were able to establish specific boundary conditions for the emergence of social conflict and its aggravation; discovered a characteristic area of stability for a social system in which a fairly small social distance is maintained between objects; identified dependencies that correspond to some modern ethnosocial conflicts, which makes it possible to use this model as a tool in predicting their dynamics and forming resolution scenarios.

Also, as part of these studies, scientists proved that the transition from a stable state to an unstable state for a multicomponent cognitive system of a distributed type is a threshold effect. According to Alexander Petukhov, the experiments performed revealed the specific parameters necessary to control such a system: they determine the transition from a stable state to an unstable one, which makes it possible, with their full control, to create conditions for the emergence of social conflict, or, on the contrary, to prevent it. “By developing this approach in the future, we will be able to create on its basis a tool for fully predicting social conflicts,” sums up Alexander Petukhov.

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