Game models of conflict situations. Mathematical models of game theory Models of conflict situations in game theory

Funk Maxim

The relevance of this work lies in the ability to expand one's own ideas about the application of mathematics, to show its possibilities in the field of social sciences, which by their nature describe the behavior of both individuals and groups. The mathematical study of conflicts makes it possible not only to consider the actions of a person in a given situation, but also to determine their consequences, especially when they depend on a combination of strategies used by the participants in this situation. The paper shows how mathematics and chess come to the aid of each other in different situations.

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Mathematical models of conflict situations using chess Completed by: Funk Maxim, student of grade 5 A, MBOU "Secondary School No. 71" Supervisor: Senatorova LG, teacher of mathematics. Novokuznetsk, 2017

That's what chess is all about. Today you give your opponent a lesson, and tomorrow he will teach you. Robert Fischer, 11th World Chess Champion

The game is understood as a process in which two or more parties participate, fighting for the realization of their interests.

The relevance of this study: * expand your own ideas about the application of mathematics and chess knowledge; * to consider by mathematical study of conflicts not only the possible actions of a person, but also to determine their consequences.

The object of the study is mathematical models of conflict situations. The purpose of the study is to consider the basic concepts of game theory and their application in specific situations. Hypothesis - mathematical models using chess help resolve conflict situations.

Game Senet Game Kings of Ur

The formation of game theory began in the 17th century and continued until the middle of the 20th century.

John von Neumann (1903–1957) Hungarian-American Jewish mathematician who made important contributions to quantum physics, quantum logic, functional analysis, set theory, computer science, economics and other branches of science

Legend of the Four Diamonds

Coordinates. From latitude and longitude to abscissa and ordinate

Waking up in the morning, ask yourself: "What should I do?" In the evening, before falling asleep: "What have I done?" Pythagoras

Winning and losing on the chessboard White winning. Checkmate White loses. Mat

Let's Play!

No one will regret the time devoted to chess, because it will help in any profession... Tigran Petrosyan, 9th world chess champion Who has been involved in mathematics since childhood develops attention, trains his brain, his will, cultivates perseverance and perseverance in achieving the goal. A. Markushevich, mathematician

Internet resources: https:// ru.wikipedia.org http:// chessmaestro.ru http:// life-prog.ru http:// www.magichess.uz http:// stuki-druki.com http:/ / home.onego.ru https://www.google.ru

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Introduction 3

1. History of the emergence and development of game theory 5

2. Basic concepts of game theory 7

3. Chess and mathematics 8

4. Coordinate system 11

5. The Pythagorean theorem on the chessboard 13

6. Conclusion 15

7. References 16

Introduction

I chose this topic because I have been playing chess since the age of four, and mathematics is one of my favorite school subjects. Moreover, mathematics and chess have a lot in common. The eminent mathematician Godfrey Hardy, drawing a parallel between these two types of human activity, once remarked that “the solution of the problems of a chess game is nothing but a mathematical exercise, and chess itself is the whistling of mathematical melodies.” There is even the concept of chess mathematics.

After a little thought, I realized that this connection can help in mastering both chess and mathematical knowledge. In mathematics, there are problems that can be solved by creating a mathematical model, and when playing chess, conflict situations constantly arise that can be resolved by creating a model.

I worked on this plan:

1. Study game theory.

2. Understand how with the help of chess knowledge you can solve difficult situations in mathematics.

3. Consider examples.

4. Make a conclusion.

Game theory A branch of mathematics that deals primarily with decision making. Game theory is applicable in many situations where there is conflict, when the parties must make the best decision based on their own interests, without knowing anything about the decision of opponents. Under game is understood as a process in which two or more parties participate, fighting for the realization of their interests. Each side has its own goal and uses some strategy, which can lead to a win or a loss - depending on the behavior of other players. Game theory helps to choose the best strategies, taking into account ideas about other participants, their resources and possible actions.

Relevance of this studylies in the ability to expand one's own ideas about the application of mathematics, to show its possibilities in the field of social sciences, which by their nature describe the behavior of both individuals and groups. The mathematical study of conflicts makes it possible not only to consider the actions of a person in a given situation, but also to determine their consequences, especially when they depend on a combination of strategies used by the participants in this situation.

So the objectthis study -mathematical models of conflict situations.

Purpose of the study– consider the basic concepts of game theory and their application in specific situations.

To achieve the goal, the following tasks:

study game theory and its basic concepts;
to study the algorithm for constructing a mathematical model of conflict situations using the example of a chess game;
consider the method of constructing a chess game.

Hypothesis - mathematical models with the use of chess help to resolve conflict situations.

The following were used during the work methods :

search method; modeling; analysis method.

1. The history of the emergence and development of game theory

Since ancient times, the history of mathematics is full of references to games and entertaining problems. From the inception of games to the 19th century serious and entertaining mathematics cannot be separated from each other, since they are closely intertwined. Already in the two great civilizations of antiquity, Babylonian and Egyptian, where mathematics was only of a practical nature, board games and entertaining tasks are found: the game "Senet", the board game of the Ur kings.

Serious and entertainingmathematics have coexisted side by side since ancient times, but at the beginning of the 17th century a special direction appeared, devoted to the analysis of games. In 1612 the first book devoted only to entertaining mathematics. Its author is Claude Gaspard Bacher de Meziriac. This book contains descriptions of problems about the wolf, goat and cabbage, magic squares, problems about weighing.

From this point on, a lot of similar books appear. And in the 17th century, Christian G. Eugens (1629-1695) and Gottfried W. Leibniz (1646-1716) proposed to create a discipline that would use scientific methods to study human conflicts and interactions through games. Throughout the 18th century, almost no work on game analysis was written that had such a goal. In the 19th century, many economists created simple mathematical models to analyze the simplest competitive situations. Among them, the work of the French economist Antoine Auguste Cournot "Investigation of the mathematical principles of the theory of wealth" (1838) stands out. Nevertheless, game theory as a fundamental mathematical theory appeared only in the first half of the 20th century.

At the beginning of the 20th century, the theoretical basis of modern game theory began to take shape, finally taking shape in the middle of the century. The authorship of the first theorem belongs to the logician Ernst Zermelo (1871–1956). He formulated and proved it in 1912. This theorem confirms that any finite game with complete information (such as checkers or chess) has an optimal solution in pure strategies, that is, in the absence of an element of uncertainty. But this theorem does not describe how such strategies can be found.

Around 1920, the great mathematician Émile Borel became interested in the burgeoning theory and introduced the idea of a mixed strategy (in which there is an element of chance). Soon John von Neumann began to work on this topic.

John von Neumann, known for his work in many fields, is one of the most important mathematicians of the 20th century. He made significant contributions to many areas of science. One of his most important achievements, related to applied mathematics in economics, is the creation of the first book with a systematic presentation of game theory and an approach to the analysis of economic problems called "Game Theory and Economic Behavior". In 1943, Neumann wrote it together with Oscar Morgenstern. This work is considered fundamental in game theory. It marked the creation of game theory, which a few years later, starting in the 1950s, began to find application in the analysis of many real situations.

The main issues dealt with by game theorists in the 1950s and 60s were related, among other things, to foreign policy, in particular nuclear deterrence and an arms race.

In Russia, mathematicians are mainly engaged in game theory - Olga Bondareva, Elena Yanovskaya, Sergey Pechersky, Victoria Kreps, Victor Domansky, Levon Petrosyan in St. Petersburg, Victor Vasiliev in Novosibirsk, Nikolai Kukushkin and Vladimir Danilov in Moscow.

2. Basic concepts of game theory

Situations in which the interests of two parties collide and the result of any operation carried out by one of the parties depends on the actions of the other party are called conflict .

Conflict situation taken from real life is usually quite complex. In addition, its study is hampered by the presence of various circumstances, some of which do not have a significant impact on either the development of the conflict or its outcome. Therefore, in order for the analysis of the conflict situation to be possible, I need to abstract from these secondary factors. I will talk about the conflict situation from the conventional point of view, where the formalized conflict model is called game (checkers, chess, cards, etc.). The game differs from a real conflict situation in that in the game the opponents act according to strictly defined rules.

Hence the terminology of game theory: the conflicting parties are called players , one exercise of the game - party, the outcome of the game - win or lose.

A typical conflict is characterized by three main components:

interested parties
possible actions of these parties,
the interests of the parties.

The actions that players perform are called strategies . When the optimal strategy contains an element of uncertainty and must be kept secret, such a strategy is called mixed . If the optimal strategy does not contain an element of chance, then it is called clean.

Games can be classified in various ways depending on the selected criteria: place to play, number of participants, game length, difficulty level, etc. With regard to mathematics, games can be divided into two large groups depending on whether random events are present in them or not. Random events can appear both in the initial conditions of the game and when making moves. For example, in most card games, cards are dealt randomly by players. The same goes for dominoes.

Strategy games are games in which random events never occur. Everything is determined only by the decision of the players. Due to the lack of randomness, games of this type can be analyzed and a way to win (chess) can be found.

3. Chess and mathematics

Chess is a game that is closely related to mathematics and conflict resolution. Therefore, I suggest you consider the chessboard.

Fig.1

A chessboard is not just 64 squares. It has coordinates, symmetry, and geometry (Fig. 1).In mathematical problems and puzzles on a chessboard, the matter, as a rule, is not complete without the participation of pieces. However, the board itself is also a rather interesting mathematical object. The clarity and correctness of the lines reminds us that the resolution of the conflict must be carried out correctly, reasonably, in compliance with the rules that will not harm opponents. Consider situations that can be resolved with the help of chess.

I would like to remind you of one old legend about the origin of chess, connected with arithmetic calculation on the board.

When the Indian king first got acquainted with chess, he was delighted with their originality and abundance of beautiful combinations. Having learned that the sage who invented the game was his subject, the king called him to personally reward him for his ingenious invention. The sovereign promised to fulfill any request of the sage, and was surprised by his modesty when he wished to receive wheat grains as a reward. On the first field of the chessboard - one grain, on the second - two, and so on, for each subsequent field there are twice as many grains as for the previous one. The king ordered that the inventor of chess be given his insignificant reward as soon as possible. However, the next day, the court mathematicians informed their master that they were unable to fulfill the wish of the cunning sage. It turned out that there was not enough wheat for this, stored not only in the barns of the whole kingdom, but in all the barns of the world. The sage modestly demanded

1+2+2 2 + … +2 63 =2 64 − 1

grains. This number is written in twenty digits and is fantastically large. The calculation shows that a barn for storing the necessary grain with a base area of 80 m 2 must extend from the earth to the sun.

This amount of grain is about 1800 times the world's wheat harvest in a year, that is, it exceeds the entire wheat harvest harvested in the entire history of mankind.

S = 18446744073709551615

Eighteen quintillion four hundred and forty-six quadrillion seven hundred forty-four trillion seventy-three billion seven hundred nine million five hundred and fifty-one thousand six hundred and fifteen.

Of course, the connection with mathematics here is somewhat arbitrary, but the unexpected outcome of the story clearly illustrates the grandiose mathematical possibilities hidden in the game of chess.

It is appropriate to give one hypothesis that uses some of the mathematical properties of the board. According to this hypothesis, chess originated from the so-called magic squares.

The magic square of order n is a square tableau n× n filled with integers from 1 to n 2 and having the following property: the sum of the numbers of each row, each column, as well as the two main diagonals is the same. For magic squares of order 8, it is equal to 260 (Fig. 2).

Rice. 2. Almujannah 1 and magic square

The regularity of the arrangement of numbers in magic squares gives them the magical power of art. No wonder the outstanding German artist A. Dürer was so fascinated by these mathematical objects that he reproduced the magic square in his famous engraving “Melancholia”.

Similar examples (their number can be increased) allow us to make a hypothesis about the connection between magic squares and chess. And the disappearance of traces of this connection can be explained by the fact that in the distant era of superstition and mysticism, the ancient Hindus and Arabs attributed mysterious properties to the numerical combinations of magic squares, and these squares were carefully hidden. Maybe that's why the legend about the sage who invented chess was invented.

Among the mathematical problems and puzzles about the chessboard, the most popular problems are cutting the board. The first of them is also connected with the legend.

Almujannah 1 - old opening tabia (initial arrangement of pieces)

Rice. 3. The legend of the four diamonds

One eastern ruler was such a skillful player that he suffered only four defeats in his entire life. In honor of his winners, the four wise men, he ordered four diamonds to be inserted into his chessboard - on the squares on which his king was mated (see Fig. 3, where horses are depicted instead of diamonds).

After the death of the ruler, his son, a weak player and a cruel despot, decided to take revenge on the wise men who had beaten his father. He ordered them to divide the chessboard with diamonds into four parts of the same shape so that each contained one diamond. Although the sages complied with the demand of the new ruler, he still took their lives, and, as the legend says, for the execution of each sage he used his part of the board with a diamond.

This board cutting problem is often found in entertaining literature.

Cut the board into four identical parts (coinciding when superimposed) so that each of them has one knight. It is assumed that the cuts pass only along the boundaries between the verticals and horizontals of the board.

One of the solutions to the problem is shown in Fig. 3. By placing four knights on different squares of the board, we get a lot of cutting problems. Of interest in them is not only finding one necessary cut, but also counting the number of all ways to cut the board into four identical parts containing one knight each. It has been established that the largest number of solutions - 800 - with the location of the knights in the corners of the board.

As we can see, wise men come out of these chess situations with dignity; people who have knowledge and believe in it. In communicating with each other, situations arise that require coordination of actions and the manifestation of a benevolent attitude towards rivals, the ability to give up personal desires in order to achieve common goals, and sometimes the truth. Unfortunately, not everyone and not always, even at the chessboard, are able to adequately get out of the current situation. It's hard, everyday work. And chess teaches that.

In our school, there are 78 students in the 5th grade parallel, 25 of them (21%) are engaged in chess and study at "4" and "5".

It's easy to draw a conclusion. Chess is not just a game, but a sport that trains and develops mental processes. The link between learning and play is undeniable.

4. Coordinate system

More than 100 years BC. the Greek scientist Hipparchus proposed to encircle on the map Earth parallels and meridians and enter now well-known geographic coordinates: latitude and longitude - and designate them with numbers.

In the fourteenth century the French mathematician N. Oresme introduced, by analogy with geographical coordinates, on a plane. He proposed to cover the plane with a rectangular grid and call latitude and longitude what we now call the abscissa and ordinate.

This innovation proved to be extremely productive. On its basis, the method of coordinates arose, which connected geometry with algebra. The main merit in creating the coordinate method belongs to the French mathematician R. Descartes.

Cartesian coordinate system on the planeis given by mutually perpendicular coordinate lines with a common origin at the point ABOUT and the same scale. Point O is called the origin of coordinates.The horizontal line is called x-axis or x-axis , vertical - y-axis or y-axis. The coordinate plane is ho.

Let the point P lies on the plane ho. Let us drop the perpendiculars from this point onto the coordinate axes; denote the base of the perpendiculars R x and R y . Abscissa point R called the coordinate x point P x on the x-axis , ordinate - coordinate at the point P y on the Oy axis.

Fig.4

Distance between two points R 1 (x 1; y 1) and R 2 (x 2; y 2) on the plane is determined using the Pythagorean theorem. I will talk about this further.

Rice. five

In the pictures we see tickets to the circus and theater. Each of them gives a description of where the place of the owner is located. this ticket: row number and seat number in that row.

Description of where this or that object (object, place) is located, they call it coordinates . So on a ticket to the circus, the row number and the seat number in the row are the coordinates of this place.

The chessboard also has coordinates. In a professional game, they usually keep records (the designation of the pieces and the coordinates of these pieces).

In figure 6, we see some algorithm for determining the coordinates of the black king.

(Cr. c2)

Fig.6

The coordinate system is used not only in chess, but also in other games (Naval battle, board games, Biathlon, drawing by dots, graphic dictations etc.)

I think that if most people played such games (in the family, with friends), then a huge number of domestic conflicts could be avoided. Because play is one way to overcome differences. And the ability to resolve small conflicts through compromise will be improved, which means that more serious problems can also be solved.

5. The Pythagorean theorem on a chessboard.

We all know the famous Pythagorean theorem."In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs".

Fig.7

Let ABC - a given right triangle with a right angle FROM . Draw height CD from the top of a right angle FROM . AC 2 + BC 2 \u003d AB 2.

This theorem has been studied by schoolchildren for several hundred years. With its help solve problems, it is used by engineers, architects, designers, fashion designers. The Pythagorean theorem is widely used in everyday life.

Consider the proof of this theorem on a chessboard.

Fig.8 Fig.9

Let's divide the board into a square and four identical right-angled triangles (Fig. 8). Figure 9 shows the same four triangles and two squares. Triangles in both cases occupy the same area, and, consequently, the same area is occupied by the remaining parts of the board without triangles (in Fig. 8 there is one square, and in Fig. 9 there are two). Since the large square is built on the hypotenuse of a right triangle, and the small squares are built on its legs, the famous Pythagorean theorem is proved!

One can prove the theorem as follows:

Fig.10

Draw in the center of the chessboard triangle ABC(Fig. 10). Construct squares on the legs and hypotenuse of this triangle, and the square built on the hypotenuse consists of the squares included in the partitions of the squares built on the legs.

Squares 1 and 2 consist of eight small squares, in total we get the number of squares that make up square 3 built on the hypotenuse.

If you look closely at this picture, you will see a beautiful house. These are usually drawn by us - children. There are definitely no conflicts in such a house, because everything is calculated and built with the help of the oldest game - chess and one of the oldest sciences - mathematics. This home is cozy and comfortable.

6. Conclusion

At the very beginning of my work, I set a goal - to consider the resolution of conflict situations in mathematics with the help of chess, and I think that I completed the task. Using examples, I analyzed the use of chess for solving mathematical problems.

Output: mathematics helps chess players to play and win. And chess, in turn, helps us solve both the simplest and the most complex math problems, help develop logic, attention and know mathematics perfectly, build logical chains, even resolve conflicts.

The spirit of competition in the game, in solving problems helps to develop, think, find the right solutions, and in case of a loss, do not give up, but seek and win.

My coach, giving me a book about chess, wrote: “The goal in life is not the main thing. The main thing is how you achieved it!

I am sure that by learning to play chess and mastering mathematics, I will be able to find the right solutions in conflict situations. In the future, I plan to continue playing chess and try to figure out what remains a mystery to me.

7. References

Gardner, M. Mathematical miracles and secrets / M. Gardner. - Moscow: Nauka, 1978. - 127 p.
Gik, E. Ya. Mathematics on a chessboard / E. Ya. Gik. - Moscow: World of Encyclopedias Avanta +, Astrel, 2009. - 317s; ill. – (Avanta+ Library).
Gik, E. Ya. Chess and mathematics / E. Ya. Gik. - Moscow: Nauka, 1983. - 173 p.
Gik, E. Ya. Entertaining mathematical games / E. Ya. Gik. - Moscow: Knowledge, 1982. - 143 p.
Gusev, V. A. Extracurricular work in mathematics in grades 6-8: a manual / V. A. Gusev, A. I. Orlov, A. L. Rozental. - Moscow: Education, 1984.
Gusev, V.A. Mathematics - reference materials / V.A. Gusev, A.G. Mordkovich. - Moscow: Education, 1986.- 271p.
Ignatiev, E. I. In the realm of ingenuity / E. I. Ignatiev. - Moscow: Nauka, 1984. - 189 p.
Loyd, S. Mathematical mosaic / S. Loyd. - Moscow: Mir, 1984. - 311 p.
Saaty, T. L. Mathematical models of conflict situations / T. L. Saaty. - Moscow: Soviet Radio, 1977. - 300 p.
Savin, A. P. encyclopedic Dictionary young mathematician / A. P. Savin. - Moscow: Pedagogy, 1989.- 349 p.
Seirawan, Ya. Diamond games: a chess textbook / Yasser Seirawan; per. from English by A. N. Elkova. - Moscow: Astrel, 2007. - 259 p.: ill. - (Win-lose chess).

Recently, to study intergroup and interstate conflicts, the method of mathematical modeling. Its significance stems from the fact that experimental studies of such conflicts are quite time-consuming and complex. The presence of model descriptions makes it possible to study the possible development of the situation in order to select the optimal variant of their regulation.

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

Mathematical modeling of intergroup conflicts makes it possible to replace the direct analysis of conflicts with an analysis of 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 external conditions and slightly changing characteristics of the conflict, variable components are the main characteristics for this study. Changing these conflict values represents the main goal of the simulation. The meaningful 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 conflict research. It is still difficult to give a clear classification of mathematical models used in conflictology. The classification of models can be based on the mathematical apparatus used ( differential equations, probability distributions, mathematical programming, etc.) and modeling objects (interpersonal conflicts, interstate conflicts, conflicts in the animal world, etc.). It is possible to single out typical mathematical models used in conflictology.
Probability distributions are the simplest way to describe variables by specifying the proportion of population elements with a given value of the variable.
Statistical dependency studies are a class of models widely used to study social phenomena. This is first of all regression models, representing the connection of dependent and independent variables in the form of functional relations.
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 opponent) from one state to another in a fixed period of time. Each action brings a private gain (loss); they add up to the resulting gain (loss).

Models of purposeful behavior represent the use of objective functions for the analysis, forecasting and planning of social processes. These models usually take the form of a mathematical programming problem with 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 are not amenable to analytical analysis. Simulation models are a means of machine experiment. It can be used for both theoretical and practical purposes. This modeling method is used to study the development of ongoing conflicts.

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

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

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

In the economy, conflict situations are very common and have a diverse character. For example, the relationship between the supplier and the consumer, the buyer and the seller, the bank and the client. Each of them has its own interests and strives to make optimal decisions that help to achieve the goals set to the greatest extent. At the same time, everyone has to reckon not only with their own goals, but also with the goals of the partner and take into account the decisions that these partners will make (they may not be known 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 the monograph "Game Theory and Economic Behavior" by J. von Neumann was published.

A 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 winning. The rules of the game are a system of conditions that determine the options for the players to act on; the amount of information each player has about the behavior of partners; the payoff that each set of actions leads to.

The game is called steam room, if two players participate in it, and multiple if the number of players is more than two. We will consider only paired 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 for action provided for by the rules is called move player. Moves can be personal and random.

personal move- this is a conscious choice by the player of one of the options for action (for example, in chess).

Random move- this is a randomly selected action (for example, throwing a dice). We will consider only personal moves.

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

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

The Purpose of Game Theory- develop methods for determining the optimal strategy for each player.

The player's strategy is called optimal, if it provides this player with the maximum possible average gain (or the minimum possible average loss regardless of the behavior of the opponent) when the game is repeated many times.

The Game Theory section is represented by three online calculators:

1. Matrix game solution. In such problems, a payoff matrix is given. It is required to find pure or mixed strategies of the players and, game price. To solve, you must specify the dimension of the matrix and the solution method.
2. Bimatrix game. Usually in such a game, two matrices of the same size of the payoffs of the first and second players are set. 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 payoffs of the first player, and the second matrix shows the payoffs of the second.
3. Games with nature. It is used when it is necessary to choose a management decision according to the criteria of Maximax, Bayes, Laplace, walda, Savage, Hurwitz.

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 wins the number of points equal to the difference between the numbers. If the difference is less than 0, wins B. If the difference is 0 - a draw.

Player A three strategies (action options): A1= 1 (write down 1), A2= 2, A3= 3, the player also has three strategies: B1, B2, B3.

B A

Player task A- maximize your winnings. Player task B- minimize your loss, i.e. minimize gain A. This steam room Basic concepts of game theory

In economic practice, conflict situations often take place. Game models are basically simplified mathematical models of conflicts. Unlike a real conflict, the game is played according to clear rules. To simulate conflict situations, a special apparatus has been developed - the mathematical theory of games. The parties involved in the conflict are called players.

Each formalized game (model) is characterized by:

1. the number of subjects - players involved in the conflict;
2. a variant of actions for each of the players, called strategies;
3. functions of winning or losing (paying) the outcome of the conflict;

A game in which two players A and B participate is called a pair game. If the number of players is more than two, then it is a multiple game. We will consider doubles only models.

A game in which the gain of one of the players is exactly equal to the loss of the other is called antagonistic game or a zero-sum game. Let's start with consideration of models of antagonistic games.

To simulate (solve) an antagonistic game means to indicate for each player strategies satisfying the condition optimality, i.e. player A must receive the maximum guaranteed payoff, no matter what strategy player B adheres to, and player B must receive the minimum loss, no matter what strategy player A adheres to. optimal strategy.

Note. There are cooperative and non-cooperative games, with complete information and incomplete. In a game with complete information, before each move, each player knows all possible moves (strategies of behavior) and payoffs. In cooperative games, the possibility of preliminary negotiations between players is allowed. We will consider non-cooperative games with complete information.

Mathematical game theory is a branch of mathematics that studies decision making in conflict situations.

Let us define the basic concepts of game theory.

A game- a simplified formalized model of a conflict situation. Player- one of the parties in the game situation. Depending on the statement of the problem, a collective or even a whole state can act as a party. Each player can have their own strategies. The strategy of the i-th player x2 is one of the possible solutions from the set of feasible solutions of this player.

According to the number of strategies, games are divided into final, in which the number of strategies is limited, and endless, which have infinitely many different strategies.

Each of the n participants in the game can choose their own strategy. The set of strategies x=x1,x2,…,xn chosen by the game participants is called game situation.

It is possible to evaluate the situation x from the point of view of the goals pursued by the decision maker by constructing objective functions (or quality criteria) that associate each situation x with numerical estimates f1(x),f2(x),…,fn(x) (for example, the income of firms in situations x or their costs, etc.).

Then the goal of the i-th decision maker is formalized as follows: choose your own solution xi so that in the situation x=x1,x2,…,xn the number fi(x) is as large (or smaller) as possible. However, the achievement of this goal depends only partially on him, since other participants in the game influence general situation x in order to achieve their own goals (optimize their objective functions). The value of the objective function in a particular game situation can be called player's payoff in this situation.
According to the nature of the payoffs, games can be divided into zero-sum and non-zero-sum games. IN zero sum games the sum of winnings in each game situation is equal to zero. Two-player zero-sum games are called antagonistic. In these games, the gain of one player is equal to the loss of the other.

In games with non-zero sum all participants in the game can win or lose.

According to the type of payoff function, games can be divided into matrix, bimatrix, continuous, separable, etc.

Matrix games zero-sum finite games of two players are called. In this case, the row number of the matrix corresponds to the number of strategy Ai of player 1, and the number of the column corresponds to the number of strategy Bj of player 2.

The elements of the matrix aij are the payoff of player 1 for the situation (implementation of strategies) AiBj. Since we are considering a zero-sum matrix game, player 1's gain is equal to player 2's loss.

It can be shown that any matrix game with a known payoff matrix can be reduced to solving a linear programming problem.

Since situations that reduce to matrix games are not very common in applied problems of economics and management, we will not dwell on solving these problems.

Bimatrix game - it is a finite non-zero-sum game between two players. In this case, for each game situation AiBj, each of the players has its payoff aij for the first player and bij- for the second player. For example, the behavior of producers in markets of imperfect competition is reduced to a bimatrix game. Topic 6 of this tutorial is devoted to the analysis of this problem.

According to the degree of incompleteness of information possessed by decision makers, games are divided into strategic and statistical.

Strategy games These are games in conditions of complete uncertainty.

statistical games are games with partial uncertainty. In a statistical game, there is always one active player who has his own strategies and goals. Another player (passive, not pursuing its goals) is nature. This player implements his strategies (states of nature) randomly, and the probability of implementing a particular state can be estimated using a statistical experiment.

Since the theory of economic decision making is closely related to the theory of statistical games, in what follows we will restrict ourselves to consideration of only this class of games.

Generalization. It consists in the study of the properties, connections and relations of the conflict, which characterize not a single conflict, but a whole class of conflicts that are homogeneous in this respect. When generalizing, it is important to be able to single out the singular, that which is characteristic only of this conflict situation, and the general, which is inherent in a number of conflicts. This method is used in most scientific disciplines that study conflict.

Comparative method. It involves comparing a number of aspects of the conflict and clarifying the similarities or differences in their manifestations in various conflicts. As a result of the comparison, differences in the parameters of the conflict are established, which makes it possible to manage conflict processes in a differentiated way.

Mathematical modeling of conflicts

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 quite time-consuming and complex. The presence of model descriptions makes it possible to study the possible development of the situation in order to select the optimal variant of their regulation.

Mathematical modeling of interpear conflicts makes it possible to replace the direct analysis of conflicts with an analysis of 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 external conditions and slightly changing characteristics of the conflict, the variable components are the main characteristics for this study.

Changing these conflict values represents the main goal of the simulation. The meaningful and operational explainability of the variables and parameters used is a necessary condition for the effectiveness of modeling.

probability distributions represent the simplest way to describe variables by indicating the proportion of elements of the population with a given value of the variable;

statistical studies dependencies - a class of models widely used to study social phenomena. These are, first of all, regression models that represent the relationship of 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 a finite Markov chain is the probability of a transition of a statistical individual (in our case, an opponent) from one state to another in a fixed period of time. Each action brings a private gain (loss); the resultant gain (loss) is formed from them;

purposeful behavior patterns represent the use of objective functions for the analysis, forecasting and planning of social processes. These models usually take the form of a mathematical programming problem with 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 designed 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 represent a class of models implemented in the form of algorithms and computer programs and reflecting complex dependencies that are not amenable to meaningful analysis. Simulation models are a means of machine experiment. It can be used for both theoretical and practical purposes. This modeling method is used to study the development of ongoing conflicts.

Topic 10. Conflict prevention

1. Features of prevention and forecasting of conflicts. Objective and organizational and managerial conditions that contribute to the prevention of destructive conflicts.

2. conflict prevention technology. Change your attitude to the situation and behavior in it. Methods and techniques for influencing the opponent's behavior. Psychology of constructive criticism.

3. Factors preventing the emergence of conflicts.

4. Methods of psycho-correction of conflict behavior: socio-psychological training; individual psychological counseling; autogenic training; intermediary activity of a psychologist (social worker); self-analysis of conflict behavior.

1. Features of prevention and forecasting of conflicts. Objective and organizational and managerial conditions that contribute to the prevention of destructive conflicts.

Forecasting the emergence of conflicts is the main prerequisite for effective action to prevent them. Forecasting and prevention of conflicts are areas of managerial activity to regulate social contradictions.

Features of conflict management are largely determined by their specificity as a complex social phenomenon.

An important principle of conflict management is the principle of competence.

Intervention in the natural development of a conflict situation should be carried out by competent people.

First, people who intervene in the development of a conflict situation must have general knowledge about the nature of the emergence, development and end of conflicts in general.

Secondly, it is necessary to collect the most versatile, detailed meaningful information about a particular situation.

Another principle .

Conflict management requires not blocking, but striving to resolve it in non-conflict ways.

Still, it is better to give people the opportunity to defend their interests, but to ensure that they do this through cooperation, compromise, avoiding confrontation.

Consider the content of such a concept as conflict management.

Conflict management is a conscious activity in relation to it, carried out at all stages of its occurrence, development and completion by the parties to the conflict or a third party.

Conflict management includes: diagnostics, forecasting, prevention, prevention, mitigation, settlement, resolution.

Conflict management is more effective if it is carried out in the early stages of the emergence of social contradictions. Early detection of social contradictions, the development of which can lead to conflicts, is provided by forecasting.

Forecasting conflicts consists in a reasonable assumption about their possible future occurrence or development.

Before predicting conflicts, science must go through two stages in their knowledge.

First, it is necessary development of descriptive models various types of conflicts. It is necessary to determine the essence of conflicts, give their classification, reveal the structure, functions, describe the evolution and dynamics.

Secondly, you must explanatory models conflicts.

Signs of social tension can be detected by routine observation. The following methods of predicting a "ripening" conflict are possible:

1. spontaneous mini-gatherings (conversations of several people);

2. increase in absenteeism;

3. increase in the number of local conflicts;

4. decrease in labor productivity;

5. increased emotional and psychological background;

6. mass dismissal of one's own free will;

7. spreading rumors;

8. spontaneous rallies and strikes;

9. growth of emotional tension.

Identifying the sources of social tension and predicting the conflict at an early stage of its development significantly reduces costs and reduces the possibility of negative consequences. An important way to manage conflicts is to prevent them.

Prevention of conflicts - consists in such an organization of the life of the subjects of social interaction, which eliminates or minimizes the likelihood of conflicts between them. Conflict prevention - this is their warning in the broadest sense of the word. Preventing conflicts is much easier than constructively resolving them. Prevention of conflicts is no less important than the ability to constructively resolve them. It requires less effort, money and time.

Game theory is a set of mathematical tools for building models, and in socio-economic applications 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 the coincidence of interests leads to cooperation. Often interests in socio-economic situations are neither strictly antagonistic nor exactly coinciding. The seller and the buyer agree that it is in their common interest to agree on the sale, of course, provided that the transaction is beneficial to both. They trade vigorously at a win-win price within the limits. Game theory allows you to develop optimal rules of behavior in conflicts.

The possibility of conflict is inherent in the essence of human life itself. The causes of conflicts are rooted in the anomalies of social life and the imperfection of the person himself. Among the reasons that give rise to conflicts, one should first of all name 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 leisure, one has to observe conflicts that are different in content and strength of manifestation. Newspapers write about it every day, broadcast on the radio, and broadcast on television. They occupy a significant place in the life of every person, and the consequences of some conflicts are too felt even over many years of life. They can eat up the life energy of one person or group of people for several days, weeks, months or even years. It happens, however, unfortunately, rarely that the resolution of some conflicts takes place very correctly and professionally, competently, while others, which happens much more often, are 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, most of the conflicts are far-fetched, artificially inflated, created to cover up the professional incompetence of some people and are harmful in commercial activities.

Other conflicts, being an inevitable companion of 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 teams.

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

Human activities can be opposed in 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 enemy, interested in our defeat, deliberately hindering success, trying 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 conscious adversary, and only “blind forces of nature” act: weather conditions, the state of commercial equipment at the enterprise, illnesses 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 its state and manifestation can significantly affect the result of commercial activity.

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

What to do 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 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 conscious actions of active opponents, and unconscious, passive manifestations, for example, of the elemental forces of nature: rain, sun, wind, avalanches, etc. In such cases, the possibility of an accurate prediction of the outcome is excluded.

The commonality of all conflicts, regardless of their nature, lies in the clash of interests, aspirations, goals, ways to achieve goals, the lack of consent of two or more parties - participants in the conflict. The complexity of conflicts is determined by the reasonable and prudent actions of individuals or groups with different interests.

The uncertainty of the outcome of the conflict, curiosity, interest and the desire to win encourage people to consciously enter into a conflict, which attracts both participants and observers to conflicts.

Mathematical game theory gives 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, which manifests itself when trying to resolve a contradiction, and often against the background of acute negative emotional experiences, although it is known, according to the definition of V. Hugo, that “of the two quarreling, the one who is smarter is to blame ".

It should be noted that the involvement in the conflict of a large number of people allows you to 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 interest (Fig. 3.4), in which there is a compromise solution. By making large concessions on less significant aspects 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 limits of interest. This condition is called Pareto principle named after the Italian scientist V. Pareto.

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

Rice. BEHIND

it makes sense to negotiate where the dot stands out N, corresponding to the Nash equilibrium, - Nash point, 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 of the players over the payments that can be received without the operation. The Nash point is the most attractive guide in finding the optimal solution.

One of the typical socio-psychological interpersonal conflicts is an unbalanced role interaction. Theoretical basis analysis of interpersonal conflicts was proposed by the American psychologist E. Burn, who presented a description of the role interaction of partners (Fig. 3.5, but - 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 unbalanced, 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.

The conflict is a process that develops over time (Fig. 3.7), which can be divided into several periods, i.e. present in the form of dynamic models of conflict development. 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-like (2) para-

Rice. 3.6

fades and then reaches the greatest value at the climax? 2 and then falls off. It should be noted that often the conflict interaction has a duration (?3 - 1 1) only about 1 minute, and the post-conflict period can be 600-2000 or more times longer than it. Moreover, indicators of the outcome of the conflict for both sides may not contain winning indicators at all, i.e. one damage.

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

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

Rice. 3.7

Rice. 3.8

On fig. 3.9 shows a model of interaction between partners whose states are fixed by vectors BUT And IN, which can be used to construct the resulting conflict-vector E. This conflict readiness zone is the most unfavorable of all the 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 displaying (Fig. 3.10) possible positive and negative alternatives (moves) of the participants-players K and P and outcome options for each pair of moves K, P in the form of a payoff matrix B =|| And, whose element can be determined by the formula

Rice. 3.9

Rice. 3.10

where boogie m* - respectively nc characteristics of the outcome of the conflict in points and its weight, k = 1 at t.

On 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. Options of alternatives (-/+) or (+/-) can lead to a peaceful option of consent, which is determined by a chain of cause-and-effect alternatives in a multi-way interaction.

Example 3.14. Consider an example of conflict resolution.

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 pick up the tomatoes and return the money. The seller refused and insulted the buyer.

Buyer's alternatives: IIi - call the administration, P 2 - contact law enforcement agencies, P 3 - insult the seller and demand a refund.

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

Let us choose the following as characteristics of assessing the outcomes of the conflict.

E - strength of emotional arousal, dB (0.19)

tk- time of conflict interaction, min (0.17)

t - duration of negative emotions, min (0.15)

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

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

tcn- post-conflict period, min (0.09);

T - total time spent, min (0.07);

З m - material costs, rub. (0.05);

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

t+ - duration of positive

Characteristics are arranged by rank, their weight is indicated in brackets M/ 0 found by the method of paired comparisons (section 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 (П„ К,) to establish the actual values of the characteristics of the conflict RU, determine the scoring of the B/CL characteristics)) * and then calculate the values of the outcomes by according to the formula

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

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

Similarly, we calculate outcomes by for the remaining pairs of alternatives and thus construct a game model of a conflict situation in the form of a payoff 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. Therefore, the minimax strategies of the conflict participants П[, Kj are optimal.

In fact, the buyer did just that: she called the administrator, who seized 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 indicators, a matrix can be constructed that does not contain a saddle point, then you can use the criteria of Wald, Savage, Hurwitz, and also use the simplex linear programming method to solve the game in mixed strategies.