Mathematical models of conflict situations. Mathematical modeling of conflicts What is the name of the mathematical model of a conflict situation

Recently, the method of mathematical modeling has been increasingly used to study intergroup and interstate conflicts. Its significance stems from the fact that experimental studies of such conflicts are rather laborious and complex. The presence of model descriptions allows us to study the possible development of the situation in order to select the optimal option for their regulation.

Mathematical modeling with the involvement of modern computer technology allows you to go from simple accumulation and analysis of facts to forecasting and assessing events in real time of their development. If the methods of observation and analysis of intergroup conflict make it possible to obtain a single solution of a conflict event, then mathematical modeling of conflict phenomena using a computer makes it possible to calculate various options for their development with the prediction of a probable outcome and influence on the result.

Mathematical modeling of intergroup conflicts allows you to replace the direct analysis of conflicts by analyzing the properties and characteristics of their mathematical models. The mathematical model of the conflict is a system of formalized relationships between the characteristics of the conflict, divided into parameters and variables. The parameters of the model reflect the external conditions and slightly changing characteristics of the conflict, the variable components are the main characteristics for this study. Changing these conflict values is the main goal of modeling. Substantial and operational explainability of the variables and parameters used is a necessary condition for the effectiveness of modeling.

The use of mathematical modeling of conflicts began in the middle of the 20th century, which was facilitated by the emergence of electronic computers and a large number of applied research on conflict. It is difficult to give a clear classification of the mathematical models used in conflict management. The used mathematical apparatus (differential equations, probability distributions, mathematical programming, etc.) and objects of modeling (interpersonal conflicts, interstate conflicts, conflicts in the animal world, etc.) can be put at the basis of the classification of models. It is possible to highlight the typical mathematical models used in conflict management.
Probability distributions are the simplest way to describe variables by indicating the proportion of elements in the population with a given value of the variable.
Statistical studies of addictions are a class of models widely used to study social phenomena. These are primarily regression models that represent the relationship between dependent and independent variables in the form of functional relationships.
Markov chains describe such mechanisms of distribution dynamics, where the future state is determined not by the entire prehistory of the conflict, but only by the “present”. The main parameter of the finite Markov chain is the probability of the transition of a statistical individual (in our case, the component) from one state to another in a fixed period of time. Each action brings a private gain (loss); the resulting gain (loss) is made up of them.

Purposeful behavior models represent the use of purposeful functions for the analysis, forecasting and planning of social processes. These models usually take the form of a mathematical programming problem with a given objective function and constraints. Currently, this direction is focused on modeling the processes of interaction of purposeful social objects, including determining the likelihood of a conflict between them.

Theoretical models are intended for the logical analysis of certain meaningful concepts when it is difficult to measure the main parameters and variables (possible interstate conflicts, etc.). Simulation models are a class of models implemented in the form of algorithms and computer programs and reflecting complex dependencies that do not lend themselves to analytical analysis. Simulation models are a means of machine experimentation. It can be used for both theoretical and practical purposes. This method of modeling is used to study the development of ongoing conflicts.

A group of scientists led by an employee of the N.N. N.I. Lobachevsky Alexandra Petukhova identified the parameters that are needed to manage a system that describes social conflicts. With full control over these characteristics, scientists will be able to create conditions for the occurrence or prevention of such a conflict. The results are reported in the Simulation journal.

In the mathematical modeling of social and political processes, one must take into account the fact that they cannot be strictly specified, since they are subject to constant changes. The 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 close 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 of the external environment.

In physics, such processes are usually described by the Langevin stochastic diffusion equation, which is relatively often used to model some social processes. An approach based on such equations allows one to take into account the manifestations of the free will of its individual participants and the 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 individual particles; it also allows one to identify characteristic stable modes of systems functioning depending on various initial conditions. Finally, from the point of view of numerical modeling, the diffusion equations have been sufficiently tested and studied.

The new model is based on the idea that individuals interact in society through the communication field. 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 a society that differs from the objects of classical physics. According to the head of research, 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 the modern world one does not need to be near the object of influence to transmit information.

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

Based on this approach and the developed model, scientists have found the following patterns: they were able to establish specific boundary conditions for the emergence of social conflict and its aggravation; found 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 ethno-social conflicts, which makes it possible to use this model as a tool in predicting their dynamics and forming settlement scenarios.

Also, within the framework of these studies, scientists have proven that the transition from a steady state to an unstable one 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 for managing 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. “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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5.7. Brief remarks on selective arms control
We have already said that the main purpose of control is to check whether the other party is complying with an arms control agreement. Control can be exercised by observing the production and storage of military materials, the movement of vehicles with military materials, the number of weapons in certain strategic areas, or the presence or absence of covert military installations. In nuclear or any other tests prohibited by the treaty, the observer must look for certain evidence that can assist him in interpreting suspicious signals.
It is absurd and impossible to study all suspicious events to find out if the agreement is being respected. It has long been established in the industry that to control the quality of products it is not at all necessary to control all products, it is enough to check the selected samples at random. The cost of sampling inspection can be quite high even if reliable quality control methods are used.
The sampling techniques applied to arms control issues can vary in complexity. In general, the ideas and methods that are so useful in studying the characteristics of the population are applicable and useful for research.
We do not need to go into the details of different types of sampling methods, such as random, layer-by-layer, group, sequential, etc. We also do not need to talk about different methods of obtaining statistical inference that use correlation and regression, estimates and hypotheses about testing. You can read about the basic concepts and applications of the methods mentioned in the widely used books on statistics and its applications. Here we will attempt to outline a typical situation in which sampling techniques can be effectively used to verify adversary compliance with an arms control treaty.
The sampling problem consists of two big questions. The first is to determine the sample size and the type of sampling procedure that is most appropriate for a particular situation. The second is to obtain statistical conclusions about the entire population based on sampling data.Both of these issues should be resolved so that the conditions imposed by
A disarmament treaty, and also that they be agreed upon with other conditions beyond the control of the observer group. The results of the sampling should then be presented in a form suitable for decision-makers. An area in which sampling techniques can be useful for arms control, for example, is the analysis of a system of records that contain information on the transport and production of strategic materials. However, the use of such records for control is expensive. In addition, it may turn out that it is impossible to gain access to these records through negotiations. However, if such records become available to the parties as a result of an agreement, provision should be made for their use. Reporting control is aimed at creating and operating a system of reports and reports, registration of arrivals and departures, in order to prevent the dispersion and loss of materials due to negligence or, if the loss has occurred, to ensure that the lost is found and to prevent similar cases in the future.
Selective inspection of intangible things such as records presents 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 agreement is indicated in the documents of the interested parties, then the group of observers has a basis for finding activities in which the level of activity is not specified.On the other hand, it is much more difficult to find out whether the level of activity in a certain area activities established by the contract
rum, since the flow of materials cannot be divided into black and white, it includes all shades of gray. Therefore, a group of observers is required to be attentive and able to unravel complex issues. Naturally, small violations cannot give great advantages to the violator, and the production of weapons for the preparation of large military operations presupposes a wide range of violations.
We believe that this should be approximately the same as the methods applicable in the last stages of disarmament. They will serve as a tool used in the day to day implementation of the arms control treaty. But long before this stage, the ideas outlined in the first five chapters of this book will play an important role in shaping the actual arms reduction measures.
A brief description of the challenges posed by selective arms control will be given below. Sampling procedures are little used in assessing properties that are relatively rare in the elements of a population. If only a few items 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 avoids this drawback, and care should be taken in the selection of sampling elements. The same can be said about the search for irregularities in the production of products for a small number of weapons. It's like looking for a needle in a haystack.
Suppose we have to check a ZavbD, which produces parts for agricultural machines, but on which a certain number of parts for military equipment can be produced. Let us also assume that the number of machines used for peaceful purposes is unknown and, therefore, it is impossible to say how many parts of a given type are intended for this purpose. How can it be established that an excessive amount of parts is being produced?
We can set standards for the life of these parts and the life of the machines that use these parts. It is also necessary to determine the number of machines produced based on the inspection of the factories in which 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 required to build a new machine and to replace worn parts in old machines. By observing the speed of manufacture of these parts and evaluating the maximum volume of production, we can confirm or deny 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 well the agreements are being implemented. For example, average levels are often used to show how many transactions have been completed. Sometimes we can use visual control to assess the degree of fulfillment of requirements. However, if a large number of checks are to be carried out to survey many areas, statistical methods are needed to obtain a single criterion for meeting the requirements. The effectiveness of an action can be judged by the extent to which it corresponds to the goals pursued by this policy. Therefore, in addition to developing consistent goals and stable behaviors, actions must be taken (as an expression of policy) to ensure that these requirements are effectively met.
Sometimes it happens that there are no effective actions that could be used to implement some 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 unrest breaks out in the country. In ch. 6 will consider the general concepts of disorder, aggression and factors influencing the resolution of conflicts.

Part IV
INTERMEDIATE AND LONG-TERM PROBLEMS OF ARMS CONTROL - ANALYSIS OF INCREASING CONFLICTS, IDEAS AND PROSPECTS

CHAPTER 6
CONFLICT RESEARCH

6.1. Introduction
This chapter will outline some of the issues related to the causes of conflicts. First, we describe some of the studies of esca-
We will discuss the examples of laboratory-type conflicts and find out what factors determine the growth of conflicts. Then some qualitative reasoning about war and peace in the history of mankind will be presented.
"Conflict arises as a result of dissatisfaction, and dissatisfaction as a result of insufficient satisfaction of needs," supporters of one of the ideological schools argue. War and peace are briefly described as a chain of frustration and recovery.
Other schools (some of them are briefly mentioned) believe that wars are engendered 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 seize new markets, the struggle for survival, the expansion of a dynamic civilization, the desire for the domination of the elite of the military-industrial complex, etc. However, be that as it may, the theory set forth in Sec. 2.4, makes it possible to rationally resolve the issue of being drawn into the 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 can survive the present moment. The last section describes some areas of research and recommended actions during this period (and in the near future) that can help to resolve conflicts peacefully.

6.2. Experiences with escalating conflicts
We sometimes mistakenly believe that if peoples understand the full danger of nuclear weapons, then they seek to reasonably resolve emerging conflicts, at worst using conventional weapons. However, quite naturally, the losing side may resort to the threat of using nuclear weapons to avoid defeat and even regain lost ground. It could end in disaster. In addition, for some peoples the concept of rationality differs from ours, especially if they have nothing to lose materially. Until the processes of escalation and their management are fully understood, it is unlikely that conventional warfare will be kept under control. An awareness of escalation processes and how to manage them will greatly increase the hopes of limiting damage in the event of a conflict. This theory should also find its application to the war, which is 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 exacerbate the conflict.
Over the past several years, the Agency for Disarmament and Arms Control, in conjunction with the Center for Operations Research at the University of Pennsylvania, has conducted research on the conditions under which conflicts escalate or de-escalate, in order to investigate the possibility of influencing the speed of escalation or de-escalation by managing the conditions that determine the interaction of the parties - participants in the conflict. The study included: a) analyzing some historical conflicts and studying the 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 problems.
As a result of the analysis of the literature, several hypotheses about escalation and de-escalation were proposed, and then in experimental situations the following were tested: a) their generality and b) identification of critical variables. Examples of hypotheses: a) in the absence of communications, the likelihood of escalation increases, b) the more important ideological issues play, the more likely escalation, c) escalation depends on economic development, d) escalation is more possible 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 was nevertheless the simplest game that met the following conditions:
1. It is “rich enough” to test many of the hypotheses expressed about the phenomena under study, in this case we are talking about the dynamics of major social conflicts. (Obviously, such experiments cannot confirm the hypothesis about this or that real phenomenon, but they can determine the limits of the hypothesis or show in which 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 variables and units for their measurement, in addition, simplifications must be indicated (for example, a certain variable is assumed to be a constant). This enables us to consistently construct increasingly rich experimental situations by introducing complications.
3. The appropriate behavior in the experimental situation should be quantified.
4. The situation should be decomposed into a number of simpler experimental situations and, if possible, these simple situations should be already 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 the first step towards creating quantitative models of a real situation; therefore we call it “artificial reality”. It is used in order to accumulate experimental data, for the interpretation of which the first theory is built. Experience is accumulated through rich play in an experiment designed to systematically test hypotheses about real conflicts, which are described in operational and quantitative terms so that they can be used in theoretical constructions.

Remarks on Artificial Reality Construction
Artificial reality consists of two symmetrical games in which the moves are made simultaneously. One of them is the positive-sum game - the prisoner's dilemma, which to some extent depicts the international (two countries) economy. The other is a negative-sum game called "roosters," which is reminiscent of a confrontation between two countries when they head on a collision course in the hope that the enemy will make concessions.
KOHETS FRAGMEHTA BOOKS

Game theory is a collection of mathematical tools for building models, and in socio-economic applications it is an inexhaustible source of flexible concepts.

The game is a mathematical model of collective behavior that reflects the interaction of participants-players in an effort to achieve a better outcome, and their interests may be different. Mismatch, antagonism of interests give rise to conflict, and coincidence of interests leads to cooperation. Often, interests in socio-economic situations are neither strictly antagonistic nor exactly the same. The seller and the buyer agree that it is in their common interest to negotiate a sale, provided, of course, that the deal is beneficial to both. They bargain vigorously on a mutually beneficial price within the constraints. Game theory allows you to develop optimal rules of behavior in conflicts.

The possibility of conflict is inherent in the very essence of human life. The causes of conflicts are rooted in the anomalies of social life and the imperfection of the person himself. Among the reasons giving rise to conflicts, it is necessary to name, first of all, socio-economic, political and moral reasons. They are a breeding ground for the emergence of various kinds of conflicts. The emergence of conflicts is influenced by the psychophysical and biological characteristics of people.

In all spheres of human activity, when solving a wide variety of tasks in everyday life, at work or rest, one has to observe conflicts that are different in their content and strength. Newspapers write about this every day, broadcast on radio, and broadcast on television. They occupy a significant place in the life of every person, and the consequences of some conflicts are too tangible even over many years of life. They can consume the life energy of one person or group of people for days, weeks, months, or even years. It happens, however, unfortunately, rarely that the resolution of some conflicts is very correct and professional, competently, while others, which happens much more often - unprofessional, illiterate, with bad outcomes sometimes for all participants in the conflict, where there are no winners, but only defeated. Obviously, recommendations are needed on a rational course of action in conflict situations.

Moreover, more often than not, some of the conflicts are far-fetched, artificially inflated, created to cover up professional incompetence by some persons and are harmful in commercial activities.

Other conflicts, being an inevitable companion in the life of any team, can be very useful and serve as an impetus for the development of commercial activities for the better.

Conflicts are currently a key problem in the life of both individuals and entire groups.

The actions of literary characters, heroes are inevitably accompanied by the manifestation, development of some kind of life conflict, which in one way or another is resolved sometimes peacefully, sometimes dramatically or tragically, for example, in a duel. The best sources of our knowledge of human conflict are classic tragedies, serious and deep novels, their adaptation or theatrical production.

Human activities can be opposed in a conflict by the interests of other people or the elemental forces of nature. In some conflicts, the opposite side is a consciously and purposefully acting active adversary who is interested in our defeat, deliberately hinders success, tries to do everything in his power to achieve his victory by any means, for example, with the help of a killer.

In other conflicts, there is no such a conscious enemy, but only "blind forces of nature" operate: weather conditions, the state of commercial equipment at the enterprise, illness of employees, etc. In such cases, nature is not malicious and acts passively, sometimes to the detriment of man, and sometimes to his benefit, but her condition and manifestation can significantly affect the result of commercial activity.

The driving force in the conflict is a person's curiosity, the desire to win, maintain or improve their position, for example, security, stability in a team, or the hope of success in achieving an explicit or implicit goal.

How to act in a given situation is often unclear. A characteristic feature of any conflict is that none of the parties involved knows in advance exactly and completely all their possible solutions, as well as the other parties, their future behavior, and, therefore, everyone is forced to act in conditions of uncertainty.

The uncertainty of the outcome can be due to both the conscious actions of active opponents and unconscious, passive manifestations, for example, the elemental forces of nature: rain, sun, wind, avalanche, etc. In such cases, the possibility of an accurate prediction of the outcome is excluded.

The commonality of all conflicts, regardless of their nature, consists in a clash of interests, aspirations, goals, ways to achieve goals, in the absence of consent of two or more parties to the conflict. The complexity of conflicts is due to the reasonable and calculating actions of individuals or groups with different interests.

Uncertainty about the outcome of the conflict, curiosity, interest and desire for victory induce people to consciously enter the conflict, which attracts both participants and observers to conflicts.

Mathematical game theory provides scientifically based recommendations for behavior in conflict situations, showing how to play so as not to lose. To apply this theory, it is necessary to be able to represent conflicts in the form of games.

The basis of any conflict is the presence of a contradiction, which takes the form of a disagreement. A conflict can be defined as a lack of agreement between two or more parties - individuals or groups, manifested when trying to resolve a contradiction, and often against the background of acute negative emotional experiences, although it is known, according to V. Hugo's definition, that “of two quarreling, the one who is smarter is to blame. ".

It should be noted that the involvement of a large number of people in the conflict can dramatically increase the number of alternatives and outcomes, which is an important positive function of the conflict, associated with expanding horizons, increasing the number of alternatives and, accordingly, possible outcomes.

In the process of commercial negotiations, one has to look for an area of mutual interests (Fig. 3.4), in which a compromise solution is found. By making large concessions on aspects that are less significant for the firm, but more significant for the opponent, the merchant gets more on other positions that are more significant and beneficial for the firm. These concessions have minimum and maximum boundaries of interest. This condition is called Pareto principle named after the Italian scientist V. Pareto.

Current conditions of market relations are characterized by situations similar to cooperative games with two players seeking a successful agreement, for example, when buying and selling an apartment, a car, etc. In such cases, the outcomes of the interaction of the participants can be represented as a set of decisions S on the plane (see Fig. 3.4) among the total payoffs X and U. This set is convex, closed, bounded above, and optimal solutions are located on the right upper northeastern boundary. At this border, it stands out between R and Р 2 set optimal Pareto solutions(P), in which an increase in the partner's payoff is possible only by decreasing the payoff of the other partner. Threat point T (x t, y t) determines the amount of gains that players can get without entering into a coalition with each other. On the set (P), F x and P 2, negotiation set F, within which-

Rice. PER

it makes sense to negotiate where the point stands out N, corresponding to the Nash equilibrium, - Nash point, in it the maximum of the product max (x L. - x m) (h y - y t), in which the factors represent the excess of the winnings of each player over the payments that can be received without an operation. The Nash point is the most attractive reference point in finding the optimal solution.

One of the typical socio-psychological interpersonal conflicts is unbalanced role interaction. The theoretical basis for the analysis of interpersonal conflicts was proposed by the American psychologist E. Byrne, who presented a description of the role interaction of partners (Fig. 3.5, a - no conflict b - possible conflict) in the form of network models.

Rice. 35

Each person in the process of interacting with others is forced to play more than a dozen roles, and not always successfully. In the proposed model, each partner can imitate the role of C - senior, P - equal, or M - junior. If the role interaction is balanced, then communication can develop without conflict, otherwise, if the roles are imbalanced, a conflict is possible.

In long-term conflicts, the share of business content often decreases over time and the personal sphere begins to dominate, which is shown in Fig. 3.6.

A conflict is a process that develops in time (Fig. 3.7), which can be divided into several periods, i.e. presented in the form of dynamic models of the development of the conflict. These, for example, can be the pre-conflict period (/ „), conflict interaction (? / E) and the post-conflict period ( t c).

Tensions over time in the pre-conflict period (? 0 ~ t) gradually (1) or avalanche (2) para-

Rice. 3.6

melts and then reaches its highest value at the moment of climax? 2 and then falls off. It should be noted that often conflict interaction has a duration (? 3 - 1 1) only about 1 minute, and the post-conflict period can be 600-2000 and more times greater than it. Moreover, the indicators of the outcome of the conflict for both parties may not contain winning indicators at all, i.e. only damages.

The assessment of the partner's state in interaction can be interpreted graphically as a combination of the degree of his activity A and the level of mood (Fig. 3.8).

These indicators can be measured from an average, neutral (0) level. Then the state point is determined by a vector with the corresponding coordinates, for example M (x,1 ) 2 ). State defined by another vector N (pci, y [) y less active at= (z / 2 - Have) The state of the partner determined by the vector Oh 3, y / 2), differs in a more nasty mood than the state determined by the vector B (x 2 , y 2).

Rice. 3.7

Rice. 3.8

In fig. 3.9 shows a model of the interaction of partners, the states of which are fixed by vectors A and V, which can be used to construct the resulting conflict vector E. This zone of preparedness for conflict is the most unfavorable of all quadrants. Using such graphical models for assessing the state of partners, one can prepare in advance for the possible outcomes of their interaction.

The game model of the conflict can be represented as a combination of the display (Fig. 3.10) of possible positive and negative alternatives (moves) of the participants-players K and P and variants of outcomes for each pair of moves K, P in the form of a payment matrix B =|| And, the element of which can be determined by the formula

Rice. 3.9

Rice. 3.10

where Boogie M * - accordingly estimated nka characteristics of the outcome of the conflict in points and its weight, k = 1 NS.

In fig. 3.10 shows that the actions of both sides with negative alternatives (- / -) indicate that it is impossible to understand each other with the help of "wars". Positive action on both sides leads to a peaceful outcome. Variants of alternatives (- / +) or (+/-) can lead to a peaceful variant of consent, which is determined by a chain of causal alternatives in a multi-way interaction.

Example 3.14. Let's consider an example of solving a conflict situation.

The woman paid at the market for 2 kg of tomatoes, and the control scales showed an underweight of 200 g. She asked the seller to take the tomatoes and return the money. The seller refused and insulted the customer.

Customer alternatives: IIi - call the administration, P 2 - contact the law enforcement agencies, P 3 - insult the seller and demand to return the money.

Seller alternatives: TO - return the money, K 2 - insult the customer and do not return the money, K 3 - do not return the money.

Let us choose the following characteristics for assessing the outcome of the conflict.

E - the power of emotional excitement, dB (0.19)

tk - conflict interaction time, min (0.17)

t - duration of negative emotions, min (0.15)

About s - the number of offensive, rude words, pcs. (0.13)

L k - the number of participants in the conflict, people (0.11)

t cn - post-conflict period, min (0.09);

T - total time spent, min (0.07);

З m - material costs, rubles. (0.05);

t n- pre-conflict period, min (0.03);

t + - duration of positive

The characteristics are arranged by rank, their weight is indicated in parentheses. M/ 0 found by the method of paired comparisons (p. 1.3).

Let's introduce a 10-point assessment of the characteristics of the conflict on a scale worse (B /, = 1) - better (B * = 10) and form a matrix of their possible values (Table 3.22).

and neutral emotions, min (0.01).

Table 3.22

Now it is necessary for each pair of alternatives (P „K,) to establish the actual values of the characteristics of the conflict RU, determine the score for the characteristics of B / CL)) * and then calculate the values of the outcomes by according to the formula

where T - the number of characteristics of the conflict; M - the weight k- characteristics of the conflict; B b (Ru) - point value k-th characteristics of the conflict outcome of a pair of alternatives II /, K, -.

For example, for a pair of alternatives Пj, TO and the conditional values of the characteristics, we find the value of the outcome B n

Similarly, we calculate the outcomes by for the remaining pairs of alternatives and thus construct a game model of the conflict situation in the form of a payment matrix

Using the minimax principle, we find the lower and upper prices of the game, which are equal to a = P = 3.23, then the pair of alternatives 11 (, K] determines the saddle point of the game. Consequently, the minimax strategies of the parties to the conflict П [, Kj are optimal.

In fact, the buyer did just that: she called the administrator, who confiscated the weights from the seller, banned the trade, and the seller took the tomatoes back and returned the money.

It should be noted that for other values of the conflict indices, a matrix can be constructed that does not contain a saddle point, then you can use the Wald, Savage, Hurwitz criteria, and also use the simplex linear programming method to solve the game in mixed strategies.

Keywords

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

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

Relevance. The article deals with the actual problem of adequate mathematical modeling of the behavior of conflicting systems, in relation to systems, conflicts in which are not necessarily associated with an antagonistic contradiction between the participants in the system. A formal statement of the problem of logical and mathematical modeling of the process of interaction of conflicting participants in the system is given. This problem consists in constructing two-valued algebras and multi-valued logic modeling different types of thinking, the difference of which is the source of the conflict. Purpose of the article. The purpose of the article is to present and analyze in detail two-digit and multivalued logics, with an emphasis on clarifying the fundamental differences between the laws of these logics, entailing significant differences in the thinking of individuals, based on these logics, and the conflicts arising from this difference between carriers of different logics of thinking. Method. To solve this problem, the traditional method of constructing logical systems is used, based on the introduction of basic constant elements, the main operations on them and the identification of the laws that govern these operations. 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-digit and multivalued logics... Novelty. A position is formulated according to which there are systems, conflicts between whose 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. This is the so-called imaginary conflicts, the fight against which requires special approaches. Result. A procedure has been developed for constructing the algebra of logic of various meanings, which adequately simulates the processes of thinking. Described two-digit and multivalued logic thinking and their laws. Found fundamental differences between two-digit and multivalued logics... An example of the analysis of the conflict caused by the difference in the logic of thinking is given.

The text of the scientific work on the topic "Logical and mathematical modeling of conflicts"

Logical and mathematical modeling of conflicts

Levin V.I., Nemkova E.A.

Relevance. The article deals with the actual problem of adequate mathematical modeling of the behavior of conflicting systems, in relation to systems, conflicts in which are not necessarily associated with an antagonistic contradiction between the participants in the system. A formal statement of the problem of logical and mathematical modeling of the process of interaction of conflicting participants in the system is given. This task consists in constructing algebras of two-valued and multi-valued logic that model various 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 of two-valued and multi-valued logics, with an emphasis on clarifying the fundamental differences between the laws of these logics, which entail significant differences in the thinking of individuals based on these logics, and the conflicts arising from this difference between carriers of different logics of thinking. Method. To solve this problem, the traditional method of constructing logical systems is used, based on the introduction of basic constant elements, the main operations on them and the identification of the laws that govern these operations. 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 multivalued logics. Novelty. A position is formulated according to which there are systems, conflicts between whose 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. This is the so-called imaginary conflicts, the fight against which requires special approaches. Result. A procedure has been developed for constructing the algebra of logic of various meanings, which adequately simulates the processes of thinking. The two-valued and multi-valued logic of thinking and their laws are described. The fundamental differences between two-valued and multi-valued logics are established. An example of the analysis of the conflict caused by the difference in the logic of thinking is given.

Key words: conflict, formal logic, elements, logical operations, laws of logic, statement, two-valued logic, multivalued 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 binding 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 inner nature of conflict situations, and without this, it is impossible to build appropriate good mathematical models that allow a detailed study of such situations.

It is generally believed that the source of human conflict is the contradiction between the goals that different people set among themselves. However, it is no secret that a large (and possibly overwhelming) part of humanity is people who do not set themselves any special goals.

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But at the same time, they often conflict with other people - both aimlessly existing, like them, and with completely purposeful people. This fact prompts the assumption that the basis of conflicts between people is also some other feature of the human personality, not directly related to human activities and his goals, but inherent in him at the genetic level. In this article, a hypothesis is put forward and substantiated, according to which a feature of a person, which strongly, and sometimes decisively, affects the occurrence (or absence) of his conflicts with others, is the type, or rather, the logic of his thinking. For this purpose, two essentially different types of logic are considered - two-valued and multi-valued, and then it is shown that the variants of human thinking based on them are largely incompatible. This incompatibility leads to misunderstanding between the adherents of the two types of thinking and, ultimately, to conflicts between them.

1. Two-valued formal logic

Two-valued formal (otherwise - mathematical, symbolic) logic of statements, also called classical, lies at the basis of ordinary human thinking. This logic is built using two constant elements: TRUE (designation AND) and false (designation A); variables, the values of which 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 (L). Therefore, logical operations can be performed on statements. Logical operations on constant elements or statements P, Q are as follows: negation P (otherwise "NOT P"), disjunction P VQ (otherwise "P OR Q"), conjunction P l Q (otherwise "P and Q"), separating disjunction P 0 Q (otherwise "EITHER P, OR Q"), equivalence P "Q" (otherwise "P IS EQUAL TO Q"), implication P ® Q (otherwise "IF P, THEN Q"). These operations are defined in truth tables 1 and 2. In addition to statements with variable truth values (I or A), there are two statements with constant truth values: identically true statement or tautology (notation T) and identically false statement or contradiction (notation P) ...

Table 1 - Operation of negation

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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 L I L I L L

L I I L I L I

AND AND AND AND AND L AND AND

In the introduced logic, the following laws are valid:

The displacement law for disjunction and conjunction

P V Q = Q V P, P l Q = Q l P; (1)

Combination law for disjunction and conjunction

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

Distribution law for conjunction with respect to disjunction

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

Distribution law for disjunction with respect to conjunction

(R l Q) V I = (R 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)

Absorption law

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

The law of action on statements with constant truth values

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

The law of double negation

The excluded third law

P V P = T; (ten)

The law of contradiction

R l R = P; (eleven)

Implication transformation law

(Р ® Q) = PV Q (12)

To prove the laws of two-valued logic, truth tables of both parts are built, similar to 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 logic of statements allows you to formally describe the process of human thinking using a formal construction

A1 l A2 l ... l Ap ® V. (13)

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

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

Let us give one more logical law - the transitivity of the implication, which is important for logical inference

(R ® 0l (0 ® Y) ® (R ® Y). (14)

Law (14) shows that the operation of the implication ® is transitive, which makes it possible to carry out logical inference as a multistage (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. Multivalued formal logic

All the main features of multi-valued logic are manifested, starting with the value k = 3. Therefore, we restrict ourselves to the three-valued formal logic of statements. This logic underlies human thinking, which is more complex than usual. It is constructed using the same constant elements as the two-valued logic: AND and L, with the addition of the constant element UNCERTAINTY (notation H). The new element is uncertainty 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 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 two-digit. However, the number of possible options for each operation is much larger. Table 3-5, the three most common variants of the negation operation are identified. 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 variant for each operation). In addition to statements with variable truth values (I, L, or H), there are three statements with constant truth values: I (called the tautology T), L (called the contradiction P), and H (called the uncertainty H).

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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 circular negation

Table 5 - Right circular negation

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

P Q P v Q P A Q P ® Q P «Q P ® Q

L L L L L 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

AND AND AND AND AND L AND AND

In the introduced three-valued logic, the laws of two-valued logic remain valid, which do not contain the operation of negation. These are the laws of transposition, combination and distribution (1) - (4), tautology, absorption and actions with constants (6) - (8), transitivity (14). However, new laws of actions on statements with a constant truth value H

H V L = H, H V I = I, H l L = L, H 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 variant of the negation operation. If this is an operation of mirror negation (Table 3), then there remain

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the laws of de Morgan, double negation and transformation of implication (5), (9), and (12) of two-valued logic are valid, however, the law of the excluded third (10) turns into the following law of the "partially excluded third"

Р V Р = Т "(Р), where Т" (Р) = (И, at Р = И or Л; (16)

[And, at P = H; 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)

[And, with P = I. y 7

For the operations of left and right cyclical 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. So, the laws of double negation (9), the excluded third (10) and contradiction (11) are transformed into the corresponding laws - the law of triple negation

excluded fourth law

Р V Р V Р = Т (19)

and the law of complete contradiction

R l R l R = P, (20)

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

R f R, "R; (21)

law (19) in the form of a "partially excluded third" law

ГИ, at Р = И or Л, Р V Р = Тл (Р), where Тl (Р) = ("р

[And, at P = And,

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

[And, at P = L,

for right cyclical negation; and law (20) in the form of the law of "partial contradiction"

- „GL, with R = L or I, R l R = Pl (R), where Pl (R) = (“ p _ ty

[And, at P = And,

for left circular negation;

P p G L, for P = L or I, P l P = Pn (P), where Pn (P) = ("p

[And, at P = And,

for right cyclical 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 the law of the excluded middle does not apply in this logic - it is transformed

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

3. Logic and conflicts

Every thinking individual in his thinking activity always uses, consciously or intuitively, one or another version of logic. We saw above that there are significant differences between two-valued and multi-valued logics. Therefore, all individuals, according to the preferred version of logic used in their thinking, can be divided into two-valued and polysemantic 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 multi-valued thinker, any statement has at least three truth values: true, false and vague. In this case, the negation operation can be defined in different ways, so that the negation of any truth value in the general case can give any other truth value.

In view of these profound differences between double-valued and polysemantic thinkers, a complex problem arises of their relationship. 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 of modern science). This constant misunderstanding leads to suspicion and fear. As a result, the two-valued thinker begins to conflict with the ambiguous, leaning towards a forceful solution.

Let's consider the simplest typical example. At a banquet, during a feast, the artist, already pretty tipsy, turns to the scientist: "Why aren't you drinking?" - He replies: "I can not!". The artist continues to insist: "Drink!" The scientist objects: "I will not!" Then the artist announces loudly: "So you are going to write a denunciation on us!" Our artist is, of course, a typical double-valued thinker, for whom there are only two options: to drink and therefore be unable to convey and not 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 multi-valued thinker. For example, to get drunk to unconsciousness, and then convey about what was not, or not drink at all and at the same time not inform for moral reasons.

The real version of this semi-fantastic story took place in 1938 at a government dacha in Kuntsevo, near Moscow, when, during a regular banquet hosted by I.V. Stalin, he failed to force the USSR People's Commissar for Cinematography Boris Shumyatsky to drink. After that, by order of the double-digit thinker Stalin, the suspicious polysemantic thinker Shumyatsky was shot.

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

Conclusion

The article shows that two-valued and multi-valued logics obey significantly different laws, due to which they can be used to model various types of thinking. It was 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 misunderstanding caused by the difference in types of thinking. The advantage of the described approach to the study of conflicts lies in the possibility of a more subtle penetration into the essence of the development of conflict situations.

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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. Research interests: logic;

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mathematical modeling in engineering, economics, sociology, history; making decisions; optimization; theory of automata; 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. Research interests: logic; mathematical modeling in engineering and economics. Email: [email protected]

Address: 440039, Russia, Penza, Baidukova Avenue / st. Gagarin, 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 of the Russian Federation. Penza State Technological University. Field of Research: logic; mathematical modeling in technics, economy, 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, economy. E-mail :: elenem5 8 @mail. ru

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

Mathematical models of conflict situations. Mathematical modeling of conflicts What is the name of the mathematical model of a conflict situation

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

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The text of the scientific work on the topic "Logical and mathematical modeling of conflicts"