The role and significance of reflexive games. Reflexive business games

Goals:

- exercise children in rolling, throwing and catching the ball;

- the ability to coordinate movement with the word;

- develop attention, dexterity.

Target Audience: students in grades 1-5.

Formed UUD: tocommunicative, regulatory, personal.

Content

Players in a circle roll the ball from one to another, saying:

- An apple rolls into a circle of a round dance,

- Whoever raised it, that governor ...

The child who at this moment has the ball is the governor. He says:

- Today I am a governor.

- I'm running from the round dance.

Runs around the circle, puts the ball on the floor between two players. The children say in chorus:

One, two, do not crow

And run like fire!

Players run in a circle in opposite directions, trying to grab the ball before their partner. The one who runs first and grabs the ball rolls it in a circle. The game continues.

Recommendations

Roll or throw the ball only to a nearby player. You can not interfere with a player running around the circle. The one who first touched the ball won.

The game

"Find and shut up"

The game

"Find

where is hidden"

The game

"Magic Chair"

The game

"Professions"

Game "Find where it's hidden"

Goals:

- learn to navigate in a room or on a site;

- perform actions on a signal.

The target audience:

Formed UUD: communicative, regulatory.

Content

The players stand along the wall. The leader shows them an object and says that he will hide it. The host invites the players to turn to the wall. After making sure that none of the children are looking, he hides the object, after which he says: “It's time!”. Children begin to look for an object.

The game "Find and keep silent"

Goals:

- learn to navigate in space;

- educate endurance, ingenuity.

The target audience: preschool and primary school age.

Formed UUD: communicative, regulatory.

Content

The host shows the player the object, and after they close their eyes, he hides it. Then he offers to search, but not to take it, but to say in your ear where it is hidden. Whoever found the first one is the leader in the next game.

Game "Professions"

Goals :

development of the imagination;

development of observation, empathy, expressiveness of movements.

The target audience: atstudents in grades 1-5.

Formed UUD: communicative, regulatory.

Content

Children, breaking into pairs, show expressive movements to each other, on the instructions of a leading, cheerful and sad artist, dancer, conductor, educator, janitor, builder, imitating the movements characteristic of people of this profession.

Recommendations

One child in a pair shows a sad person, the other a cheerful person, and each time the roles change, the children change according to their emotional presentation.

Magic chair game

Goals :

to help increase the child's self-esteem;

help improve relationships between children.

The target audience: atstudents in grades 1-5.

Formed UUD: tocommunicative, personal.

Content

This game can be played with a group of children for a long time. Previously, an adult must find out the “history” of the name of each child - its origin, what it means. In addition, it is necessary to make a crown and a "Magic Chair" - it must be necessarily high. The adult conducts a short introductory conversation about the origin of the names, and then says that he will talk about the names of all the children in the group (the group should not be more than 5-6 people), and the names of anxious children are best called in the middle of the game. The one whose name is told becomes the king. Throughout the story of his name, he sits on a throne wearing a crown.

Recommendations

ATAt the end of the game, you can invite children to come up with different variants his name (gentle, affectionate). You can also take turns saying something nice about the king.

The game

"Journalists"

The game

"Continue the story"

The game

"The best astronaut"

The game

"Guess who am I?"

The game "Continue the story"

Goals :

contributes to the formation of a culture of behavior, friendly collective relationships;

encourages children to fantasize;

develops speech.

The target audience: primary school age.

Formed UUD: tocommunicative, personal.

Content

We compose a fairy tale, and then we shoot a film - a continuation of the fairy tale. We distribute the roles of screenwriters, director, actors, etc. among the children in advance. In the process of composing a fairy tale, students can offer new acting characters, so the development of the plot in most cases depends directly on the actions taken by the participants in the game.

Everyone is free to choose their own strategy of behavior, which only initially had an installation for certain restrictions that determine the image of the character to be played and the rules of the game.

Goals :

formation of adequate self-esteem;

development of creative activity;

bonding of group members.

The target audience: students in grades 5-11.

Formed UUD: tocommunicative, personal.

Content

We choose the editor-in-chief of the magazine, the class is divided into "departments" in which journalists work on a specific topic.

We call the magazine according to the topic of the lesson or lesson, for example, "The Red Book of Russia".

The task of students - journalists is to pick up interesting material on the topic, discuss it in groups, analyze it, then "present the material in the issue." At the last stage - a discussion of the resulting journal, each conducts a self-assessment and self-analysis of their activities.

Initially, students need to be introduced to the rules necessary for the success of business communication of the members of each group into which the class is divided;

so that each group includes students with different levels training, and everyone found an activity according to their interests.

The game "Guess who am I?"

Goals :

development of reflection and self-awareness;

creativity, empathy and sensitivity.

The target audience: atstudents in grades 1-5.

Formed UUD: tocommunicative, personal.

Content

Children are given the task: to imagine themselves in the role of some fairy tale hero, writer, artist, animal and, having come to class for a lesson, move and speak on his behalf. You can use group roles (for example, Little Red Riding Hood and grey Wolf). The rest of the students must guess who their classmate has become.Whose stories were the most interesting for you? Did you enjoy the task?

Recommendations

It is necessary to agree in advance whether it is possible to use elements of the costume, scenery. After the end of the game, analyzing the results of its conduct, it is necessary to note the successful performance of a particular role, but in no case should direct evaluation comments be made, otherwise the next time the child simply does not want to play and refuses to participate.

The game "The best astronaut"

Target: the game helps children learn to evaluate and appreciate the work of their comrades, supports the child's desire to learn something new.

The target audience: atstudents in grades 1-5.

Formed UUD: educational, tocommunicative, personal.

Content

The teacher/facilitator draws 10 rockets with different numbers on the board. 11 students are called at once. Around the table, where cards with examples are laid out, the children walk, holding hands, and recite:

“Fast rockets are waiting for us to walk around the planets. Whatever we want, we'll fly to such! But there is one secret in the game: there is no place for latecomers.

As soon as the last word is said, the teacher gives each student cards with examples that encode the number of the rocket on which the astronaut will fly. Children solve examples by determining the number of their rocket and write the example under the corresponding rocket number.

Recommendations

Can be used in different lessons and classes by changing the examples to other types of tasks.

To art library of games

The game

"Target Shooting"

The game

"Zoological Domino"

The game

"Who am I?"

The game "Target Shooting"

Target : the game helps children learn to evaluate and appreciate the work of their comrades, supports the child's desire to learn something new, develops accuracy.

The target audience: students in grades 2-6.

Formed UUD: tocommunicative, personal, regulatory.

Content

Work in pairs (mutual assessment). Each child receives a form for work, writes his name on the back and draws three lines for evaluation. The rules of the game are shown by the teacher with one of the students on the blackboard. Child - the shooter takes the chalk and puts it on the "pistol" point. The teacher-commander commands: "Get ready - aim - fire!". At the word "plee" the shooter leads the line to the target. The teacher evaluates the result as follows: if you shoot accurately, then the bullet should fly straight and fast at the target. 2nd attempt. The pairs had to agree on who would shoot first.Recommendations

The game starts and ends with a call. The commander evaluates "shooting" according to the criteria "accuracy", "speed", whether the bullet flew smoothly. The commander puts an assessment on the shooter's leaflet. Whoever wants to thank his commander for justice shakes his hand.

Compilers

Brovkina L.A.,

Grigoryeva E.A., physics teacher, MAOU "Beloyarsk secondary school No. 1"

Ezhova N.A.,

Kryukova E.M., teacher-organizer of MAOU "Beloyarsk secondary school No. 1"

Filimonova M.I., mathematics teacher, MAOU "Beloyarsk secondary school No. 1"

Khamitova A.G., primary school teacher MAOU "Beloyarsk secondary school No. 1"

Shnaider A.V., primary school teacher MAOU "Beloyarsk secondary school No. 1"

Yurina L.N., teacher of history and social studies, MAOU "Beloyarsk secondary school No. 1"

Game "Who am I?"

Target: formation and evaluation of the level of formation of personal reflection, aimed at understanding by adolescents their motives, needs, aspirations, the use of adjectives.

The target audience: atstudents in grades 5-8.

Formed UUD: personal.

Content

1. Setting up for the lesson. Each student receives pictures with a symbolic image of his mood - a joyful little man, neutral, sad. In a pre-prepared box, everyone lowers the little man with the mood that corresponds to him at the beginning of the lesson. The teacher opens the box and reports the mood in which most of the children came. If the mood is sad, find out what happened. Sets in a positive way.2. We give compliments. Each student compliments his neighbor - he looks good, has a nice blouse, etc.3. Introspection. Who am I? Everyone writes (based on a pre-prepared list of words that students can use - funny, brave, timid, witty, snooty, fair, etc.). Each student, without showing his piece of paper, goes to the blackboard, sits on a chair and listens to the opinions of his classmates about himself. How many matches - so many points.

Zoological domino game

Target: strengthening the knowledge of schoolchildren about wild and domestic animals; education of intelligence and attention. The target audience: atstudents in grades 1-5.

Formed UUD: educational, communicative.

Content game rule The first player to put down all their cards is the winner. Game actions combine attentiveness, the ability not to skip a move, and put your card on time.Game progress . The cards depict wild and domestic animals. The subjects were divided into micro-teams (4 people each). The cards are laid out face down. Younger students were asked to count 6 cards. Then the teacher reminds the rules of the game: you can only put the same picture side by side.Recommendations

If the desired picture is not available, then the child skips a turn. If one of the players is left without cards, he is considered the winner of the game. The game is repeated, but the cards are moved and other canvas cards are dealt..

Reflective game

Reflective game- the process of social interaction, during which each of the participants in the game exercises reflexive control (the author of the term is V. A. Lefevre) by other participants, trying to implement his management strategy to form his own version of social reality (implementation of a social innovation project). In the space of a reflexive game, social management technologies are tested for effectiveness, so the gain in such a game is an increase in the level of skill.

Reflective play as a type of social interaction

Reflexive play refers to open type games. Unlike games that proceed according to a script and according to predetermined rules (such as business games or role-playing games), reflexive games are a process of social interaction in which roles, rules, and plot moves are generated by the participants right in the course of the game action. At the same time, the course of the reflexive game can be controlled using the individual personal characteristics of the participants, the configuration of their business and personal interests, preferences, expectations, goals, fears and temptations.

In a reflexive game, the advantage is given to the one who is more equipped with the tools to manage people and social processes, the one who is more sophisticated in analyzing and calculating situations of social interaction. Of all existing options reflexive games, the most famous are ODI (organizational-activity games), which were first used to solve social tasks G. P. Shchedrovitsky. However, there are other varieties of reflexive games developed by students, followers and even ardent opponents of G. P. Shchedrovitsky.

Typology of reflexive games

Every reflective game has task or set of tasks which it must solve by the participants. According to the tasks solved in the course of reflexive games, they can be divided into two types.

To first type include reflexive games, the main task of which is to create conditions for the individual development of participants. Reflexive games of the first type can be used for professional training or retraining of participants who are not elements of any one social system. Reflexive games of the first type can be used to identify and develop cultural and mythological stereotypes and attitudes that are characteristic of the participants. They can also be used to form megamachines (social systems made up of elements and parts of other social systems) - for example, during political campaigns or nationwide (or interstate) projects, the implementation of which involves the capabilities of a large number of different social systems.

Co. second type include reflexive games, the main task of which is to generate social innovation that will change the social system, the elements of which are the subjects-participants. Such games can be used to implement such specific activities for the development of social systems as management consulting.

Reflective Game Project

Reflexive games of the first and second types are conducted under the control of a team of game technicians who ensure such a flow of the game process, during which the tasks assigned to the game are solved. To conduct a reflexive game (of the first or second type), a team of game technicians needs game project. The project is created by a team of game engineers under the guidance of methodologists. (A methodologist is a thinker who is able to develop new approaches to the design of reflexive games, using the apparatus of methodological thinking for this (see SMD methodology)). The process of generating a game project can be called a metagame, which is carried out by methodologists with a team of game technicians on the eve of a reflexive game. In the course of game design, a team of game technicians carefully works out diagnostic information (information about the participants and about the social system (systems), the elements of which are the subjects-participants), detail the topics and meanings around which the discussion will be built within the groups and during the game, determine the best for solving this problem is a way to divide players into groups and calculate the dynamics of intragroup and intergroup gaming interaction.

Organizational structure of reflexive play

Reflexive games of the first and second types can proceed only under the condition that a rigid organizational structure of the game(dividing the participants into groups, each of which is assigned at least one game technician, determining the time regulations for game events, determining the forms of intra-group and general game communication and strict rationing communication). The structure of the game is a meta-norm that is held by the team of game technicians in relation to the players. Players have a lot of freedom, however, this freedom is limited by clearly defined limits, for example: players cannot move from group to group, the group cannot exceed the time allotted for the report during the general game meeting, the group must clearly state the topic of the report and not deviate from it, you can ask questions for understanding to the report, but you can’t mix them with judgments about what you heard in the report, you can’t ask questions about what was not in the report, etc. The norms, rules and elements of the organizational structure of the reflexive game may vary depending on on the preferences and methods used by the gaming team and methodologists conducting the game.

Rationing of players' actions, self-reflection and reflexive state

There are quite a lot of norms that hold the structure of the game, and the team of game engineers can invent them in the course of the game action. Basic normative questions: “What are you doing now?” or “What is the meaning of your action here and now?” are uncomfortable enough to draw the speaker's or the group's attention to their own thinking process, thus putting them in a self-reflexive position. When a player learns, while maintaining an autoreflexive position, to set goals, design and implement actions and analyze the result of the actions taken, he passes into a reflective state and acquires a new degree of intellectual and creative freedom, since from a reflective state he is able to work with the meanings of his own and other people's actions and to carry out reflexive management actions, the effectiveness of which is significantly higher than that of actions carried out not from a reflexive state.

There are several levels of the reflective state (“reflexive levels” or “reflexive layers”). The more reflexive levels a player is able to build on (the higher his reflexive potential), the more opportunities he gains in designing social innovations, in managing people and social processes, and in calculating social situations. A subject with a higher reflexive potential has a significant advantage over a subject whose reflexive potential is lower.

Thus, the reflexive game does not in any way impose topics or options for self-determination on the players, but it creates for each player a unique chance to expand their thinking and increase their reflexive potential.

Literature

• S.A. Kutolin"Reflexive Literature"
• S.A. Kutolin"Literature as illumination by reflection"
• Lefevre V. A. conflicting structures. M.: Soviet radio, 1973;
• Novikov D. A., Chkhartishvili A. G. Reflexive games. - M.: Sinteg, 2003.
• Makarevich V. N. Game practices, methodologists: the invisible community "comes out of the underground.//Sotsis, 1992, N 7;
• Kotlyarevsky Yu. L., Shantser A. S. The art of modeling and the nature of play. M., Progress, 1992. - 104 p.;
• Shchedrovitsky G.P. Selected works. M., Shk. Cult. Polit., 1995.

see also

Links

• Shokhov A. S. Method of live modeling in the study and consulting of organizations

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See what "Reflexive game" is in other dictionaries:

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A business game is a method of simulating decision-making by executives or specialists in various production situations, carried out according to given rules by a group of people or a person with a PC in an interactive mode, in the presence of conflicts ... ... Wikipedia

Gaming, (gaming) community that arose originally in the USSR and now exists in Russia and Belarus large group specialists practicing the use of games, game methods in various areas of public life: in the economy, ... ... Wikipedia

A business game is a method of simulating decision-making by executives or specialists in various production situations, carried out according to given rules by a group of people or a person with a computer in an interactive mode, in the presence of conflicts ... ... Wikipedia

- (English practice firm, fictitious firm) an active form of training schoolchildren, students and trainees of practical activities at the enterprise. A training company is created on the basis of a school, primary vocational, secondary specialized or higher ... ... Wikipedia

Activities aimed at improving the efficiency of firms, companies, organizations. Contents 1 Types of management consulting 1.1 Strategic consulting ... Wikipedia

- "Reflexive processes and management" international scientific and practical interdisciplinary journal. Founders: Institute of Psychology of the Russian Academy of Sciences and Lepsky Vladimir Evgenievich, with the participation of the Institute of Reflective Processes and Control Since 2001, 2 have been published ... ... Wikipedia

If the awareness structure has finite complexity, then we can construct reflexive game graph, which clearly shows the relationship between the actions of agents (both real and phantom) participating in the equilibrium.

The vertices of this directed graph are the actions r e?+ corresponding to pairwise non-identical structures informed™ /., or components of the awareness structure in" or simply the number r of a real or phantom agent, r e Z+.

Arcs are drawn between the vertices according to the following rule: to each vertex x s arcs were drawn from (P- 1) vertices corresponding to structures I mp j e N(/) If two vertices are connected by two oppositely directed arcs, we will depict one edge with two arrows.

We emphasize that the graph of a reflexive game corresponds to the system of equations (2.3.1) (that is, the definition of informational equilibrium), while its solution may not exist.

So the Count g, reflexive game Г (see the definition of a reflexive game in the previous section), whose information structure has finite complexity, is defined as follows:

• - graph vertices G t correspond to real and phantom agents participating in the reflexive game, that is, pairwise non-identical structures of awareness;
• - graph arcs G t reflect the mutual awareness of agents: if there is a path from one agent (real or phantom) to another agent, then the second one is adequately informed about the first one.

If at the vertices of the graph G/ represent the representations of the corresponding agent about the state of nature, then the reflexive game G, with a finite awareness structure / can be given as a tuple Г, = (N,(A)), e N,f(), e ,v, G/), where N- many real agents, x,- the set of admissible actions of the z"-th agent, f(-) 0 x X -> 9?" - its objective function, / "e N, G,- reflexive game graph.

Note that in many cases it is more convenient (and visual) to describe a reflexive game in terms of the graph G/, not the information structure tree.

Consider several examples of finding informational equilibrium.

Examples 2.4.1-2.4.3. These examples involve three agents with objective functions of the following form:

where Xi> 0, / € N= (1, 2, 3}; in e 0 = (1, 2).

For brevity, we will call the agent who believes that demand is low (0= 1), a pessimist, and one who believes that demand is high (0 = 2) is an optimist. Thus, in examples 2.4.1-2.4.3, the situations differ only due to different structures of awareness.

Example 2.4.1. Let the first two agents be optimists, and the third one a pessimist, all three being equally informed. Then, in accordance with Statement 2.2.5, for any a e I the identities /st] = /b/st2 = h, Dz = h-

In accordance with property 2 of the definition of information equilibrium, X*.

It can be seen that any structure of awareness is identical to one of the three that form the basis: (/b/2, D). Therefore, the complexity of this awareness structure is equal to three, and the depth is equal to one. The graph of the reflexive game is shown in Fig. eight.

Rice. eight.

Thus, the actions of agents in a situation of informational equilibrium will be as follows: X! = x 2 =1/2, =0.*

Example 2.4.2. Let the first two agents be optimists, and the third be a pessimist who considers all sin agents to be equally informed pessimists. The first two agents are equally informed, and both of them are adequately informed about the third agent.

We have: I x ~ I 2 , I >h, h > h,1 ~z I 2~z h? The graph of the reflexive game is shown in Fig. 9.

Rice. 9.

These conditions can be written as the following identities, which hold for any ae I (we use the corresponding definitions and statements 2.2.1, 2.2.2, and 2.2.5):

12а = ha, 1а = ha, ha = ha, hla = ha, ha = h, ha2 = hi, hal = h-

Similar relations hold for equilibrium actions X". The left-hand sides of these identities show that any structure 1 p for |c|>2 is identical to some structure fn |r|

Thus, the complexity of this awareness structure is equal to five, and the depth is equal to two.

To find the informational equilibrium, it is necessary to solve the following system of equations (see expression (2.3.1)):

Thus, the actions of real agents in a situation of informational equilibrium will be as follows: Х) \u003d x 2\u003d 9/20, x 3 * \u003d 1/5.

Example 2.4.3. Let all three agents be optimists, the first and second are mutually informed, the second and third are also mutually informed. According to the first agent, the third considers all three to be equally informed pessimists; also the first agent, in the opinion of the third, considers all three to be equally informed pessimists.

We have: D x D, / 2>

These conditions can be written as the following identities, which hold for any a e I(we use the corresponding definitions and statements 2.2.1, 2.2.2 and 2.2.5):

Similar relations hold for equilibrium actions x p.

The left-hand sides of these identities show that any structure 1 P for |oj > 3 is identical to some structure /„ |m| 1, A, /3, /sz /13, /sv /132? hn,/sv-

Thus, the basis is formed by the following pairwise different structures: (/b />, /3, /зз /в, /lb) - The complexity of this awareness structure is equal to six, and the depth is equal to three. The graph of the corresponding reflexive game is shown in Fig. ten.

Rice. ten.

To find the informational equilibrium, it is necessary to solve the following system of equations (see expression (2.3.1)):

Thus, the actions of real agents in a situation of informational equilibrium will be as follows: x, = x 3 =17/35, x 2 * = 12/35.

Having completed the description of the graph of a reflexive game, we continue to study the properties of informational equilibrium.

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Russian Academy Sciences Institute of Management Problems. V.A. Trapeznikova D.A. NOVIKOV, A.G. CHKHARTISHVILI REFLECTIVE GAMES SINTEG Moscow - 2003 UDC 519 BBC 22.18 N 73 Novikov D.A., Chkhartishvili A.G. Reflexive H 73 games. M.: SINTEG, 2003. - 149 p. ISBN 5-89638-63-1 The monograph is devoted to the discussion of modern approaches to the mathematical modeling of reflection. The authors introduce a new class of game-theoretic models - reflexive games that describe the interaction of subjects (agents) that make decisions based on a hierarchy of ideas about essential parameters, ideas about representations, etc. An analysis of the behavior of phantom agents that exist in the representations of other real or phantom agents and the properties of an information structure that reflects the mutual awareness of real and phantom agents allows us to propose an information equilibrium as a solution to a reflexive game, which is a generalization of a number of well-known equilibrium concepts in non-cooperative games. Reflective games make it possible: - to model the behavior of reflective subjects; - to study the dependence of the payoffs of agents on the ranks of their reflection; - set and solve problems of reflexive control; - uniformly describe many phenomena associated with reflection: hidden control, information control through the media, reflection in psychology, works of art and others. The book is addressed to specialists in the field of mathematical modeling and management of socio-economic systems, as well as university students and graduate students. Reviewers: Doctor of Technical Sciences, prof. V.N. Burkov, Doctor of Technical Sciences, prof. A.V. Shchepkin UDC 519 BBC 22.18 N 73 ISBN 5-89638-63-1 Ó D.A. Chkhartishvili, 2 2003 CONTENTS INTRODUCTION .................................................. ................................................. .......... 4 CHAPTER 1. Information in decision-making .................................. ........... 21 1.1. Individual Decision Making: A Model of Rational Behavior.................................................................. ................................................. ............................... 21 1.2. Interactive decision-making: games and equilibria .............................. 24 1.3. General Approaches to Describing Awareness.................................................. 31 CHAPTER 2. Strategic Reflection....... ................................................. 34 2.1. Strategic reflection in two-person games .............................................. 34 2.2. Reflection in bimatrix games .............................................................. ........... 41 2.3. Limitation of the rank of reflection .............................................................. .............. 57 CHAPTER 3. Informational reflection .............................. ...................... 60 3.1. Informational reflection in two-person games .............................................. 60 3.2. Information structure of the game .............................................................. .............. 64 3.3. Information balance .............................................................. ................... 71 3.4. Graph of a reflexive game ............................................................... ........................... 76 3.5. Regular awareness structures.............................................................. 82 3.6. The rank of reflection and informational equilibrium .............................................. 91 3.7. Reflective control .................................................................. ....................... 102 CHAPTER 4. Applied models of reflexive games .................................. 102 ............. 106 4.1. Hidden control .................................................................. .................................. 106 4.2. Mass media and information management .............................................................. ...... 117 4.3. Reflection in psychology .............................................................. ........................... 121 4.3.1. Psychology of chess creativity.............. .................................. 121 4.3.2. Transactional analysis .............................................................. .................. 124 4.3.3. Johari window .................................................. .................................. 126 4.3.4. Ethical Choice Model .................................................................. .............. 128 4.4. Reflection in works of art............................................... 129 CONCLUSION..... ................................................. ...................................... 137 LITERATURE .......... ................................................. ................................................... 142 3 - Minnows frolic freely, this is their joy! - You're not a fish, how do you know what her joy is? “You’re not me, how do you know what I know and what I don’t know?” From a Taoist parable - The point, of course, venerable archbishop, is that you believe in what you believe in because you were brought up that way. - May be so. But the fact remains that you, too, believe that I believe what I believe, because I was brought up that way, for the reason that you were brought up that way. From the book "Social Psychology" by D. Myers essential parameters, representations about representations, etc. Reflection. One of the fundamental properties of human existence is that, along with the natural (“objective”) reality, there is its reflection in consciousness. At the same time, between the natural reality and its image in the mind (we will consider this image as a part of a special - reflective reality) there is an inevitable gap, a mismatch. A purposeful study of this phenomenon is traditionally associated with the term “reflection”, which is defined in the Philosophical Dictionary as follows: “REFLEXION (lat. reflexio – turning back). A term meaning reflection, as well as the study of the cognitive act. The term "reflection" was introduced by J. Locke; in various philosophical systems (J. Locke, G. Leibniz, D. Hume, G. Hegel, etc.) it had a different content. A systematic description of reflection from the point of view of psychology began in the 60s of the XX century (school 4 of V.A. Lefebvre). In addition, it should be noted that there is an understanding of reflection in a different meaning related to the reflex - "the reaction of the body to the excitation of receptors". This paper uses the first (philosophical) definition of reflection. To clarify the understanding of the essence of reflection, let us first consider the situation with one subject. He has ideas about natural reality, but he can also be aware (reflect, reflect) these ideas, as well as be aware of the awareness of these ideas, etc. This is how reflective reality is formed. Reflection of the subject regarding his own ideas about reality, the principles of his activity, etc. is called auto-reflection or reflection of the first kind. It should be noted that in most humanitarian studies we are talking, first of all, about autoreflection, which in philosophy is understood as the process of an individual thinking about what is happening in his mind. Reflection of the second kind takes place regarding ideas about reality, the principles of decision-making, auto-reflection, etc. other entities. Let us give examples of reflection of the second kind, illustrating that in many cases correct one's own conclusions can be made only if one takes the position of other subjects and analyzes their possible reasoning. The first example is the classic Dirty Face Game, sometimes referred to as the wise men and hats problem or the husbands and unfaithful wives problem. We describe it as follows. “Let's imagine that Bob and his niece Alice are in the compartment of a Victorian carriage. Everyone's face is messed up. However, no one blushes with shame, although any Victorian passenger would blush knowing that the other person sees him dirty. From this we conclude that none of the passengers knows that his face is dirty, although everyone sees the dirty face of his companion. At this time, the Conductor looks into the compartment and announces that there is a man with a dirty face in the compartment. After that, Alice blushed. She realized that her face was dirty. But why did she understand this? Didn't the Guide tell her what she already knew? 5 Let's follow the chain of Alice's reasoning. Alice: Suppose my face is clean. Then Bob, knowing that one of us is dirty, should conclude that he is dirty and blush. If he does not blush, then my premise about my clean face is false, my face is dirty and I should blush. The conductor added to the information known to Alice information about Bob's knowledge. Until then, she hadn't known that Bob knew that one of them was dirty. In short, the conductor's message turned the knowledge that there was a man with a dirty face in the compartment into general knowledge ". The second textbook example is the Coordinated Attack Problem; there are problems close to it about the optimal information exchange protocol - Electronic Mail Game, etc. (see reviews in ). The situation is as follows. Two divisions are located on the tops of two hills, and the enemy is located in the valley. You can win only if both divisions attack the enemy at the same time. The general - the commander of the first division - sends the general - the commander of the second division - a messenger with the message: "We attack at dawn." Since the messenger can be intercepted by the enemy, the first general must wait for a message from the second general that the first message has been received. But since the second message can also be intercepted by the enemy, the second general needs to receive confirmation from the first general that he received confirmation. And so on ad infinitum. The task is to determine after what number of messages (confirmations) it makes sense for the generals to attack the enemy. The conclusion is that under the conditions described, a coordinated attack is impossible, and the way out is to use probabilistic models. The third classical problem is the "two broker problem" (see also speculation models in ). Suppose that two stock brokers have their own expert systems that are used to support decision making. It happens that the network administrator illegally copies both expert systems and sells his opponent's expert system to each broker. After that, the administrator tries to sell each of them the following information - "Your opponent has your expert system." Then the administrator tries 6 to sell information - "Your opponent knows that you have his expert system", and so on. The question is how brokers should use the information they receive from the administrator, and what information is relevant at which iteration? Having completed the consideration of examples of reflection of the second kind, we will discuss in what situations reflection is essential. If the only reflective subject is an economic agent that seeks to maximize its target function by choosing one of the ethically acceptable actions, then the natural reality enters the target function as a parameter, and the results of reflection (representations about representations, etc.) are not arguments of the target function. Then we can say that autoreflection is "not needed", since it does not change the action chosen by the agent. Note that the dependence of the subject's actions on reflection can take place in a situation where actions are ethically unequal, that is, along with the utilitarian aspect, there is a deontological (ethical) one - see. However, economic decisions are usually ethically neutral, so let's consider the interaction of several actors. If there are several subjects (the decision-making situation is interactive), then the target function of each subject includes the actions of other subjects, that is, these actions are part of natural reality (although they themselves, of course, are due to reflexive reality). At the same time, reflection (and, consequently, the study of reflective reality) becomes necessary. Let us consider the main approaches to mathematical modeling of reflection effects. Game theory. Formal (mathematical) models of human behavior have been created and studied for more than a century and a half (see review in ) and are increasingly being used both in management theory, economics, psychology, sociology, etc., and in solving specific applied problems. The most intensive development has been observed since the 40s of the XX century - the moment the theory of games appeared, which is usually dated to 1944 (the release of the first edition of the book by John von Neumann and Oscar Morgenstern "Game Theory and Economic Behavior"). 7 Under the game in this work we will understand the interaction of the parties whose interests do not coincide (note that another understanding of the game is possible - as "a type of unproductive activity, the motive of which lies not in its results, but in the process itself" - see also , where the concept of the game is interpreted much more broadly). Game theory is a branch of applied mathematics that explores decision-making models in conditions of conflicting interests of the parties (players), when each party seeks to influence the development of the situation in its own interests. Further, the term "agent" is used to refer to the decision-maker (player). In this paper, we consider non-cooperative static games in normal form, that is, games in which agents choose their actions once, simultaneously and independently. Thus, the main task of game theory is to describe the interaction of several agents whose interests do not coincide, and the results of activity (winning, utility, etc.) of each depend in the general case on the actions of all. The result of such a description is a forecast of a reasonable outcome of the game - the so-called solution of the game (equilibrium). Description of the game consists in setting the following parameters: - set of agents; - preferences of agents (dependencies of payoffs on actions): it is assumed (and this reflects the purposefulness of behavior) that each agent is interested in maximizing his payoff; - sets of admissible actions of agents; - awareness of agents (the information that they have at the time of making decisions about the chosen actions); - the order of functioning (the order of moves - the sequence of choice of actions). Relatively speaking, the set of agents determines who participates in the game. Preferences reflect what agents want, sets of allowed actions what they can do, awareness reflects what they know, and order of operation when they choose actions. 8 The listed parameters define the game, but they are not sufficient to predict its outcome - the solution of the game (or the equilibrium of the game), that is, the set of actions that are rational and stable from one point of view or another. To date, there is no universal concept of equilibrium in game theory - taking certain assumptions about the principles of decision-making by agents, you can get different solutions. Therefore, the main task of any game-theoretic research (including the present work) is the construction of an equilibrium. Since reflexive games are defined as such an interactive interaction of agents in which they make decisions based on the hierarchy of their representations, the awareness of agents is essential. Therefore, we dwell on its qualitative discussion in more detail. The role of awareness. General knowledge. In game theory, philosophy, psychology, distributed systems, and other fields of science (see the review in ), not only the beliefs (beliefs) of agents about essential parameters are essential, but also their ideas about the representations of other agents, etc. The set of these representations is called the hierarchy of beliefs and is modeled in this paper by the information structure tree of a reflexive game (see Section 3.2). In other words, in situations of interactive decision-making (modeled in game theory), each agent must predict the behavior of opponents before choosing his action. To do this, he must have certain ideas about the vision of the game by opponents. But the opponents must do the same, so the uncertainty about which game will be played creates an endless hierarchy of representations of the participants in the game. Let's give an example of a view hierarchy. Suppose that there are two agents - A and B. Each of them can have their own non-reflexive ideas about the indefinite parameter q, which we will further call the state of nature (state of nature, state of the world). We denote these representations by qA and qB, respectively. But each of the agents, within the framework of the process of reflection of the first rank, can think about the ideas of the opponent. These representations (representations of the second order) will be denoted by qAB and qBA, where qAB are agent A's representations of agent B's representations, 9 qBA are agent B's representations of agent A's representations. But the matter is not limited to this - each of the agents, as part of the process of further reflection (reflection of the second rank), can think about what the opponent's ideas about his ideas are. This is how representations of the third order, qABA and qBAB, are generated. The process of generating representations of higher orders can continue indefinitely (there are no logical restrictions on increasing the rank of reflection). The totality of all representations - qA, qB, qAB, qBA, qABA, qBAB, etc. - forms a hierarchy of views. A special case of awareness is when all representations, representations about representations, etc. coincide to infinity - is common knowledge. More correctly, the term "common knowledge" is introduced in to denote a fact that satisfies the following requirements: 1) it is known to all agents; 2) all agents know 1; 3) all agents know 2, and so on. ad infinitum The formal model of general knowledge was proposed in and developed in many works - see . Models of agent awareness - hierarchy of representations and general knowledge - in game theory are devoted, in fact, entirely, this work, so we will give examples illustrating the role of general knowledge in other areas of science - philosophy, psychology, etc. (see also review). From a philosophical point of view, common knowledge was analyzed in the study of conventions. Consider the following example. It is written in the Rules of the Road that each road user must comply with these rules, and also has the right to expect that other road users observe them. But other road users also need to be sure that others follow the rules, and so on. to infinity. Therefore, the agreement to "observe traffic rules" should be common knowledge. In psychology, there is the concept of discourse - “(from Latin discursus - reasoning, argument) - verbal thinking of a person mediated by past experience; acts as a process of connected logical 10 reasoning, in which each subsequent thought is conditioned by the previous one. The role of general knowledge in understanding discourse is illustrated in the following example. Two people leave the cinema. One asks the other: "How do you like the movie?". In order for the second person to understand the question, he must understand that he is being asked about the movie they just watched together. In addition, he must understand that the first one understands this. The questioner, in turn, must be sure that the second one will understand that the question is about the movie they watched, and so on. That is, for adequate interaction (communication) "movie" must be common knowledge (people must reach an agreement on the use of language). Mutual awareness of agents is also essential in distributed computing systems, in artificial intelligence and other areas. In game theory, as a rule, it is assumed that all1 parameters of the game are common knowledge, that is, each agent knows all the parameters of the game, as well as that it is known to all agents, etc. to infinity. Such an assumption corresponds to an objective description of the game and makes it possible to use the concept of Nash equilibrium2 as a predictable outcome of a non-cooperative game (that is, a game in which negotiations between agents are impossible in order to create coalitions, exchange information, joint actions, redistribute payoffs, etc.). Thus, the common knowledge assumption suggests that all agents know what game they are playing, and their ideas about the game are the same. Instead of an agent's action, we can consider something more complex - his strategy, that is, the mapping of the information available to the agent into the set of his allowed actions. Examples are: strategies in a multi-stage game, mixed strategies, strategies in Howard's metagames (see also information) none of them benefits from a one-sided (that is, under the condition that the other agents choose the appropriate equilibrium components) deviation from equilibrium - see the correct definition below. However, even in these cases the rules of the game are common knowledge. Finally, we can consider that the game is chosen randomly according to some distribution, which is common knowledge - the so-called Bayesian games. In the general case, each of the agents can have their own ideas about the parameters of the game, each of which corresponds to some subjective description of the game. In this case, it turns out that the agents participate in the game, but objectively do not know which one, or they represent the game being played in different ways - its rules, goals, roles and awareness of opponents, etc. There are no universal approaches to the construction of equilibria with insufficient general knowledge in game theory today. On the other hand, within the framework of the “reflexive tradition” of the humanities, for each agent the world around him contains (includes) other agents, and ideas about other agents are reflected in the process of reflection (differences in ideas can be due, in particular, to unequal awareness). However, no constructive formal results in this area have been obtained so far. Therefore, there is a need to develop and study mathematical models of games in which the awareness of agents is not common knowledge and agents make decisions based on the hierarchy of their representations. We call this class of games reflexive games (a formal definition is given in Section 3.2 of this paper). It should be recognized that the term "reflexive games" was introduced by V.A. Lefebvre in 1965 in . However, in this work, as well as in the works of the same author, there is mainly a qualitative discussion of the effects of reflection in the interaction of subjects, and no general concept of solution for this class of games has been proposed. The same remark is also true for , in which a number of particular cases of awareness of the participants in the game were considered. Thus, the study of reflexive games and the construction of a unified concept of equilibrium for them is relevant, which motivates the present study. 12 Before proceeding to the presentation of the main content of the work, we will discuss at a qualitative level the main approaches used below. Basic approaches and structure of work. In the first chapter "Information in Decision Making", which is mainly of an overview and introductory nature, models of individual and interactive decision making are presented, an analysis of the awareness necessary for the implementation of certain well-known concepts of equilibrium is carried out, and the famous models common knowledge and hierarchy of representations. As defined above, a reflexive game is one in which agents' awareness is not shared knowledge3 and agents make decisions based on a hierarchy of their representations. From the point of view of game theory and reflexive decision-making models, it is advisable to separate strategic and informational reflection. Informational reflection is the process and result of the agent's thinking about what the values ​​of uncertain parameters are, what his opponents (other agents) know and think about these values. At the same time, the "game" component itself is absent, since the agent does not make any decisions. Strategic reflection is the process and result of the agent's thinking about what decision-making principles his opponents (other agents) use within the framework of the awareness that he ascribes to them as a result of information reflection. Thus, informational reflection is usually associated with insufficient mutual awareness, and its result is used in decision-making (including strategic reflection). Strategic reflection takes place even in the case of complete awareness, anticipating the agent's decision on the chosen action. In other words, informational and strategic reflections can be studied independently, but in conditions of incomplete and insufficient awareness, both of them take place. 3 If in the model under consideration, awareness is common knowledge, then all the results of the study of reflexive games are transferred to the corresponding classical results of game theory - see below. 13 Strategic reflection is discussed in the second chapter of this paper. It turns out that if we assume that the agent, modeling the behavior of his opponents, ascribes to them and himself certain reflexion ranks, then the original game turns into a new game in which the agent's strategy is to choose the reflexion rank. If we consider the process of reflection in a new game, we get a new game, and so on. At the same time, even if in the original game the set of possible actions was finite, then in the new game the set of possible actions - the number of different ranks of reflection - is infinite. Consequently, the main task to be solved in the study of strategic reflection is to determine the maximum expedient rank of reflection. The answer to this question was obtained in the second chapter for bimatrix games (Section 2.2) and models that take into account the limitations of a person's ability to process information (Section 2.3). Let's give an example of strategic reflection - "Penalty" (see. (See also the examples "Hide and seek" and "Demolition on a minuscule" in section 2.2). The agents are the kicker and the goalkeeper. Let's assume for simplicity that the player has two actions - "to hit the left corner of the goal" and "to hit the right corner of the goal". The goalkeeper also has two actions - "catch the ball in the left corner" and "catch the ball in the right corner". If the goalkeeper guesses which corner the player is hitting, then he catches the ball. Let's model the agents' reasoning. Let the goalkeeper know that this player usually shoots in the right corner. Therefore, he needs to catch the ball in the right corner. But, if the goalkeeper knows that the player knows that the goalkeeper knows how the player usually behaves, then the goalkeeper should model the player's reasoning. He may think like this: “The player knows that I know his usual tactics. So he expects me to catch the ball in the right corner and can hit the left corner. In this case, I need to catch the ball in the left corner. If a player has sufficient depth of reflection, then he can guess the goalkeeper's reasoning and try to outwit him by hitting the right corner. The same chain of reasoning can be carried out by the goalkeeper and, on this basis, catch the ball in the right corner. Both the player and the goalkeeper can increase the depth of reflection to infinity by reasoning for each other, and neither of them has rational grounds to stop at some final step. Therefore, within the framework of modeling mutual reasoning, it is impossible to a priori determine the outcome of the game under consideration. The game itself, in which each of the agents has two possible actions, can be replaced by another game in which the agents choose the ranks of reflection assigned to the opponent. But even in this game there is no reasonable solution, since each agent can model the opponent's behavior by considering a "doubly reflexive" game, and so on. to infinity. The only thing that can help agents in the situation under consideration is to limit the depth of their reflection, noting that starting from the second rank of reflection (due to the finiteness of the initial set of possible actions), the situation begins to repeat - being both at zero and at the second (and, in general, at any even) level of reflection, the player will hit in the right corner. Therefore, it remains for the goalkeeper to guess the parity of the player's reflection level. The maximum reflexion rank that an agent should have in order to cover the entire variety of game outcomes (losing sight of some of the opponent's strategies, the agent risks reducing his payoff), we will call the maximum expedient reflexion rank. It turns out that in many cases this rank is finite - the corresponding formal results are given in Sections 2.2 and 3.6). In the "Penalty" example, the maximum expedient rank of agents' reflection is two. If the goalkeeper does not have information about where the striker usually hits, the actions of the latter are symmetrical (the left and right corners are “equivalent”). However, there remain opportunities to artificially introduce asymmetry in order to try to use it for your own purposes. For example, the goalkeeper can move towards one of the corners, as if inviting the attacker to hit the other (and rushes to that “far” corner). A more complex strategy is as follows. A player of the goalkeeper's team comes up to him and shows where the attacker is going to hit, and he does it in such a way that the attacker sees it (after which, at the moment of impact, the goalkeeper catches the ball not in the corner that his teammate defiantly showed him, but in the opposite) . Note that both described techniques are taken "from life" and turned out to be successful. The first took place in the international match of the USSR national team, the second - in the final of the USSR Cup in football in the penalty shootout. 15 The third chapter is devoted to the study of formal models of information reflection. Since the key factor in reflexive games is the awareness of agents - the hierarchy of representations, then for its formal description, the concept of an information structure is introduced - a tree (in the general case, infinite), the vertices of which correspond to information (representations) of agents about essential parameters, representations of other agents, etc. .d. (see the view hierarchy example above). The concept of the structure of awareness (information structure) allows us to give a formal definition of some intuitively clear concepts, such as: adequate awareness of one agent about another, mutual awareness, equal awareness, etc. One of the key concepts used in this work to analyze reflexive games is the concept phantom agent. Let us discuss it at a qualitative level (postponing the rigorous mathematical definition until Section 3.2). Let two agents, A and B, interact in a certain situation. It is quite natural that in the minds of each of them there is a certain image of the other: A has an image of B (let's call it AB), and B has an image of A (let's call it BA). These images may coincide with reality, and may differ from it. In other words, an agent, for example, A, may or may not have an adequate idea of ​​B (this fact can be written as the identity AB = B). Here the question immediately arises - can the identity AB = B be fulfilled in principle, because B is a real agent, and AB is only his image? Without going into a discussion of this essentially philosophical question, we note the following two circumstances. Firstly, we are not talking about a complete understanding of the personality in its entirety, but about its modeling in this particular situation. At the ordinary, everyday level of human communication, we are constantly faced with situations of both adequate and inadequate perception of one person by another. Secondly, in the framework of formal (game-theoretic) modeling of human behavior, an agent - a participant in a situation - is described by a relatively small set of characteristics. And these characteristics can be fully known to another agent to the same extent that they are known to the researcher. 16 Let us consider in more detail the case when there is a difference between B and AB (this difference can stem, speaking formally, from the incompleteness of A's information about B, or from trust in false information). Then A, when deciding on any of his actions, has in mind not B, but the image of him that he has, that is, AB. We can say that subjectively A interacts with AB. Therefore, AB can be called a phantom agent. It does not exist in reality, but it is present in the mind of the real agent A and, accordingly, affects his actions, that is, reality. Let's bring the simplest example . Let A believe that he and B are friends, and B, knowing this, is the enemy of A (this situation can be described by the word "betrayal"). Then, obviously, there is a phantom agent AB in the situation, which can be described as follows: “B, who is a friend of A”; in reality, there is no such entity. Note that in this case, B is adequately informed about A, that is, BA = A. Thus, in addition to real agents actually participating in the game, it is proposed to consider phantom agents, that is, agents that exist in the minds of real and other phantom agents. Real and phantom agents, within the framework of their reflection, endow phantom agents with a certain awareness, which is reflected in the information structure. There can be an infinite number of real and phantom agents participating in the game, which means the potential infinity of the implementation of acts of reflexive reflection (the infinite depth of the awareness structure tree). Indeed, even in the simplest situation, an endless expansion of reasoning of the form “I know ...”, “I know that you know ...”, “I know that you know that I know ...”, “I know that you know that I know that you know…”, etc. However, in practice such a “bad infinity” does not take place, since, starting from a certain moment, representations “stabilize”, and an increase in the rank of reflection does not give anything new. Thus, in real situations, the structure of awareness has a finite complexity: the corresponding tree has a finite number of pairwise distinct subtrees - 17 ev. In other words, the game involves a finite number of real and phantom agents4. The introduction of the concept of phantom agents makes it possible to define a reflexive game as a game of real and phantom agents, as well as to define an informational equilibrium as a generalization of the Nash equilibrium for the case of a reflexive game, in which it is assumed that each agent (real and phantom) when calculating its subjective equilibrium (equilibrium in the game he is playing from his point of view) uses his existing hierarchy of ideas about objective and reflexive reality. A convenient tool for studying information equilibrium is a reflexive game graph in which the vertices correspond to real and phantom agents, and each agent vertex includes arcs (their number is one less than the number of real agents) coming from agent vertices, on whose actions the payoff depends in the subjective equilibrium. this agent. The graph of a reflexive game can be constructed without concretizing the target functions of agents. At the same time, it reflects, if not the quantitative ratio of interests, then the qualitative ratio of awareness of reflective agents, and is a convenient and expressive means of describing the effects of reflection (see Section 3.4). For the example of two agents described above, the reflexive game graph has the form: B ¬ A « AB - the real agent B (the traitor) is adequately informed about the agent A, which interacts with the phantom agent AB (B, who is A's friend). Let us give one more example of a graph that reflects a reflexive interaction (although it is not formally a graph of a reflexive game in the sense of the definition introduced above). On the cover of this book is a painting by E. Burne-Jones "Deadly Head", written in 1886-1887. based on the myth of Perseus and Andromeda. Three real agents are involved in the situation: Perseus (let us denote him by the letter P), Andromeda (A) and the gorgon Medusa (M). In addition, 4 In the limiting case - when there is common knowledge - the phantom agent of the first level coincides with its real prototype and the tree has unit depth (more precisely, all other subtrees repeat trees of a higher level). 18, there are the following "phantom" agents: the reflection of Perseus (OP), the reflection of Andromeda (OA) and the reflection of Medusa (OM). The graph is shown in Figure 1. M P A OP OA OM Fig. 1. Graph of the painting by E. Burne-Jones "Deadly Head" (see cover) 19 The awareness of real agents in the example under consideration is as follows: Perseus sees Andromeda; Andromeda does not see Perseus, but sees his reflection, her own reflection and the reflection of the Gorgon Medusa; the reflection of Perseus sees the reflection of Andromeda; Andromeda's reflection sees all real agents. Fortunately, none of the real agents see the gorgon Medusa herself. The introduction of an information structure, an information equilibrium and a graph of a reflexive game, firstly, makes it possible to describe and analyze various situations of collective decision-making by agents with different awareness from a unified methodological position and with the help of a single mathematical apparatus, to study the influence of reflexion ranks on the payoffs of agents, to study the conditions the existence and feasibility of information equilibria, etc. Numerous examples of applied models are given below. Secondly, the proposed model of the reflexive game makes it possible to study the influence of reflexion ranks (the depth of the information structure) on the payoffs of agents. The results obtained in Sections 2.2, 3.5, and 3.6 of this paper show that, under minimal assumptions, it is possible to show that the maximum expedient reflection rank is limited. In other words, in many cases an unlimited increase in the reflexion rank is inexpedient from the point of view of agents' payoffs. Thirdly, the presence of a model of a reflexive game allows us to determine the conditions for the existence and properties of information equilibrium, as well as constructively and correctly formulate the problem of reflexive control, which consists in searching for such an information structure by the governing body that the information equilibrium realized in it is most beneficial from its point of view. The problem of reflexive control is posed and solved for a number of cases in Section 3.7. The theoretical results of its solution are used in a number of applied models presented in the fourth chapter - hidden control, information control through the media, etc. And, finally, fourthly, the language of reflexive games (information structures, graphs of a reflexive game, etc.) is convenient for describing the effects of reflection both in psychology (which is illustrated by the example of a chess game, transactional analysis, 20 models of ethical choice, etc.), and in works of art - see the fourth chapter of this work. Having completed quality review content of the work, we note that we can offer several approaches to familiarization with the material of this book. The first is linear, consisting in the sequential reading of all four chapters. The second is intended for the reader who is more interested in formal models and consists in reading the second and third chapters and a cursory acquaintance with the examples in the fourth chapter. The third is aimed at a reader who does not want to delve into mathematical subtleties, and consists in reading the introduction, the fourth chapter and the conclusion. CHAPTER 1. INFORMATION IN DECISION-MAKING In the first chapter of this paper, we present a model of individual decision-making (Section 1.1), review the main concepts for solving non-cooperative games, discuss the assumptions used in these concepts about the awareness and mutual awareness of agents (Section 1.2), and analyze known models awareness and general knowledge (section 1.3). 1.1. INDIVIDUAL DECISION-MAKING: A MODEL OF RATIONAL BEHAVIOR Let us describe, following , the model of decision-making by a single agent. Let the agent be able to choose some action x from the set X of allowed actions. As a result of choosing the action x н X, the agent receives payoff f(x), where f: X ® В1 is a real-valued objective function reflecting the agent's preferences. Let us accept the hypothesis of rational behavior, which is that the agent, taking into account all the information available to him, chooses actions that are most preferable in terms of the values ​​of his objective function (this hypothesis is not the only possible one - see, for example, the concept of bounded rationality). In accordance with the hypothesis of rational behavior, the agent chooses an alternative from the set of "best" alternatives. In the case under consideration, this set is the set of alternatives on which the maximum of the objective function is achieved. Therefore, the choice of an action by an agent is determined by the rule of individual rational choice P(f, X) н X, which singles out the set of actions that are most preferable from the point of view of the agent5: P(f, X) = Arg max f(x). xн X Let's complicate the model, namely, we assume that the agent's payoff is determined not only by his own actions, but also by the value of the indefinite parameter q н W – the state of nature. That is, as a result of choosing an action x н X and realizing a state of nature q н W, the agent receives a payoff f(q, x), where f: W ´ X ® Â1. If the agent's gain depends, in addition to his actions, on an indefinite parameter - the state of nature, then in the general case there is no uniquely "best" action - when deciding on the chosen action, the agent must "predict" the state of nature. Therefore, we introduce the hypothesis of determinism, which consists in the fact that the agent seeks to eliminate, taking into account all the information available to him, the existing uncertainty and make decisions in conditions of complete information (in other words, the final criterion that guides the agent making decisions should not contain uncertain parameters). That is, the agent must, in accordance with the hypothesis of determinism, eliminate the uncertainty about parameters independent of him (perhaps by introducing certain assumptions about their values). Depending on the information I that the agent has about uncertain parameters, there are: - interval uncertainty (when only the set W of possible values ​​of uncertain parameters is known); 5 When highs and lows are used, it is assumed that they are reached. 22 - probabilistic uncertainty (when, in addition to the set W of possible values ​​of uncertain parameters, their probability distribution p(q) is known); - fuzzy uncertainty (when, in addition to the set W of possible values ​​of uncertain parameters, the membership function of their values ​​is known). In this paper, we consider the simplest - "point" - case, when agents have ideas about a specific meaning of the state of nature. The possibility of generalizing the obtained results to the case of interval or probabilistic uncertainty is discussed in the conclusion. We introduce the following assumption regarding the uncertainty elimination procedures used by the agent: the interval uncertainty is eliminated by calculating the maximum guaranteed result (MGR), the probabilistic one is the expected value of the objective function, the fuzzy one is the set of maximally non-dominated alternatives. from the objective function f(q, x) to the objective function f(x), which does not depend on the uncertain parameters. In accordance with the introduced assumption, in the case of interval) uncertainty f (x) = min f(q, x), in the case of probabilistic uncertainty f (x) = q нW ò f (x,q) p(q)dq and etc. . W Eliminating the uncertainty, we obtain a deterministic model, that is, the rule of individual rational choice has the form:) P(f, X, I) = Arg max f (x), xн X 6 The assumptions introduced are not the only possible ones. The use of other assumptions (for example, the hypothesis about the use of MHR can be replaced by the hypothesis of optimism, or the “weighted optimism-pessimism” hypothesis, etc. ) will lead to other solution concepts, but the process of obtaining them will follow the general scheme implemented below. 23 where I is the information used by the agent when eliminating uncertainty f Þ f . I So far we have been looking at individual decision making. Let us now consider the game uncertainty, within the framework of which the agent's assumptions about the set of possible values ​​of the game environment (the actions of other agents chosen by them within the framework of certain inaccurate principles of behavior known to the agent under consideration) are essential. 1.2. INTERACTIVE DECISION-MAKING: GAMES AND EQUILIBRIUMS Game model. To describe the collective behavior of agents, it is not enough to determine their preferences and the rules of individual rational choice separately. As noted above, in the case when the system has a single agent, the hypothesis of its rational (individual) behavior assumes that the agent behaves in such a way as to maximize the value of its objective function by choosing an action. In the case when there are several agents, it is necessary to take into account their mutual influence: in this case, a game arises - an interaction in which the payoff of each agent depends both on his own action and on the actions of other agents. If, by virtue of the hypothesis of rational behavior, each of the agents seeks to maximize its objective function by choosing an action, then it is clear that in the case of several agents, the individually rational action of each of them depends on the actions of other agents7. Consider a game-theoretic model of interaction between n agents. Each agent selects an action xi belonging to the admissible set Xi, i н N = (1, 2, …, n) – the set of agents. The choice of actions by agents is carried out once, simultaneously and independently. 7 In game-theoretic models, it is assumed that the rationality of the players, that is, following their hypothesis of rational behavior, is a common knowledge. This assumption is also accepted in the present work. 24 The payoff of the i-th agent depends on his own action xi н Xi, on the vector actions x-i= (x1, x2, …, xi-1, xi+1, …, xn) н Xi= Õ X j opponents N\(i) and from the state of nature8 q н W, and jнN \ (i ) is described by a real-valued payoff function fi = fi(q, x), where x = (xi, x-i) = (x1, x2, …, xn) Î X" = Õ X j is the vector of actions of all jÎN agents. For a fixed value of the state of nature, the set Г = ( N, (Xi)i н N, (fi(×))i н N) sets of agents, sets of their admissible actions and objective functions is called a game in normal form. agents By virtue of the hypothesis of rational behavior, each agent will strive to choose the best actions for him (in terms of the value of his objective function) under a given situation. x-i games н X-i and states of nature q н W. Therefore, the principle of making a decision about the chosen action can be written as follows (BR denotes the best response - best response): (1) BRi(q, x-i) = Arg max fi(q, xi, x-i), i н N. xi н X i In parallel, we will discuss the awareness that is necessary for the implementation of equilibrium. Equilibrium in dominant strategies. If for some agent the set (1) does not depend on the situation, then it constitutes the set of its dominant strategies (the set of dominant strategies of agents is called the equilibrium in dominant strategies - RDS) . If each of the agents has a dominant strategy, then they can make decisions independently, that is, they can choose actions without having any information and without doing anything. 25 assumptions about the situation. Unfortunately, RDS does not exist in all games. For agents to realize equilibrium in dominant strategies, if the latter exists, it is sufficient for each of them to know only its objective function and admissible sets X" and W. all games: (2) xiг н Arg max min min fi(q, xi, x-i), i н N. xi н X i x -i н X -i q нW If for at least one of the agents the set (1) depends on the situation (that is, there is no RDS), then the situation is more complicated. -2), …, BRn(q, x-n)). The Nash equilibrium in the state of nature q (more precisely, the parametric Nash equilibrium) is the point x*(q) н X", satisfying the following condition: (4) x*(q) О BR(q, x*(q)). Embedding (4) can also be written as: " i н N, " yi н Xi fi(q, x*(q)) ³ fi(q, yi, x-* i (q)). The set EN(q) of all points of the form (4) can be described as follows: (5) EN(q) = (x О X’ | xi О BRi(q, x-i), i О N). For the case of two agents, an alternative equivalent way to define the set EN(q) is to specify it as a set of pairs of points (x1* (q), x2* (q)), simultaneously satisfying the following conditional relations: (6) x1* (q) н BR1(q, BR2(q, BR1(q, . ..BR2(q, x2* (q))...))), (7) x2* (q) н BR2(q, BR1(q, BR2(q, ...BR1(q, x1* ( q))...))). Let us consider what information agents must have in order to implement the Nash equilibrium by simultaneously and independently choosing their actions. By definition, the Nash equilibrium is the point from which a one-sided deviation is unfavorable for any of the agents (provided that the other agents choose the corresponding 26 components of the Nash equilibrium action vector). If agents repeatedly choose actions, then the Nash point is in a certain sense (see details in ) stable and can be considered realizable within the framework of knowledge, as in the case of RDS, by each agent only of its own objective function and admissible sets X" and W ( at the same time, however, it is necessary to introduce additional assumptions about the principles for making decisions by agents on the choice of actions depending on the history of the game ). and W to realize the Nash equilibrium is no longer enough. Therefore, we introduce the following assumption, which we will consider fulfilled in the course of the entire subsequent presentation: information about the game Г, the set W, and the rationality of agents is common knowledge. The substantively introduced assumption means that each of the agents is rational, knows the set of game participants, objective functions and admissible sets of all agents, and also knows the set of possible values ​​of the states of nature. In addition, he knows that the other agents know it, and that they know that he knows it, and so on. to infinity (see above). Such awareness can, in particular, be achieved by public (that is, simultaneously to all agents gathered together) communication of relevant information, which ensures possible achievement by all agents of the infinite rank of information reflection. Note that the assumption introduced does not say anything about the awareness of agents regarding the specific value of the state of nature. If the meaning of the state of nature is common knowledge, then this is sufficient to implement the Nash equilibrium. To substantiate this assertion, let us model, using the example of a two-player game, the course of reasoning of the first agent (the second agent argues in a completely similar way, and his reasoning will be considered separately only if they differ from the reasoning of the first agent). He argues as follows (see expression (6)): "My action, by virtue of (1), should be the best response to the action of the second agent in a given state of nature. Therefore, I need to model his behavior. About him (by virtue of assumptions that objective functions and admissible sets are common knowledge), I know that he will act within the framework of (1), that is, he will look for the best answer to my actions for a given state of nature (see (7)). he needs to model my actions, and he will (again, by virtue of the introduced assumptions that objective functions and admissible sets are common knowledge) reason in the same way as I do, and so on ad infinitum (see ( 6))." In game theory, for such reasoning, a successful physical analogy of reflection in mirrors is used - see, for example,. Thus, to implement the Nash equilibrium, it is sufficient that all game parameters, as well as the value of the state of nature, be common knowledge (a weakening of this assumption is considered in ). The reflexive games considered in this paper are characterized by the fact that the value of the state of nature is not common knowledge, and each agent generally has its own ideas about this value, the ideas of other agents, etc. subjective balance. The considered types of equilibrium are special cases of subjective equilibrium, which is defined as a vector of actions of agents, each component of which is the best response of the corresponding agent to the game environment that can be realized from his subjective point of view. Let's consider possible cases. Suppose that the i-th agent counts on the realization of the situation) of the game x-Bi ("B" stands for beliefs; sometimes the terms "assumption", "guess" are used - conjecture) and states) of nature q i , then he will choose)) (8 ) xiB н BRi(q i , x-Bi), i н N. The vector xB is a point subjective equilibrium. Note that this definition of "equilibrium" does not require the validity of the agents' assumptions about the actions of opponents, that is, it may turn out that $ i н N: x-Bi ¹ x-Bi . A justified subjective equilibrium, that is, such that x-Bi = x-Bi , i Î N, is a Nash equilibrium (for this, in particular, it is sufficient that all game parameters are common knowledge, and that each agent at 28 ) constructing x-Bi modeled the rational behavior of opponents). In the special case, if the best response of each agent does not depend on assumptions about the situation, then the subjective equilibrium is an equilibrium in dominant strategies. In a more general case, the i-th agent can count on the choice of actions by opponents from the set X -Bi н X-i and the realization of the state of nature from the set Wi н Wi н N. Then the best answer will be guaranteeing subjective equilibrium:) (9) xi (X -Bi , Wi) О Arg max minB min) fi(q, xi, x-i), i О N. xi О X i B -i x ОX q ОW i -i -i) = X-i, Wi = W, i н N, then xi(X -Bi) = xiг, i н N, that is, the guaranteeing subjective equilibrium is a “classical” guaranteeing equilibrium. A variation of the guaranteeing subjective balance is the P-balance, described in detail in. In an even more general case, as the best response of the i-th agent, we can consider the probability distribution pi(xi), where pi(×) н D(Xi) is the set of all possible distributions on Xi that maximizes the expected payoff of the agent, taking into account his ideas about the distribution probabilities mi(x-i) н D(X-i) of actions chosen by other agents, and the probability distribution qi(q) н D(W) of the state of nature (we get the Bayesian decision principle): (10) pi(mi(×), qi( ×), ×) = = arg max ò fi (q , xi , x-i) pi (xi) qi (q) mi (x-i) dq dx , i Î N. p i ÎD (X i) X ", W Thus, to implement subjective equilibrium, agents must be minimally informed – each of them must know their objective function fi(×) and the admissible sets W and X'. , that is, in order for the assumptions to be justified, additional valid assumptions about the mutual awareness of agents. The strongest is the assumption of common knowledge, which turns the subjective point equilibrium into a Nash equilibrium, and the set of Bayesian decision principles into a Bayes–Nash equilibrium. Bayes–Nash equilibrium. If the game has incomplete information (see ), then the Bayesian game is described by the following set: - a set of N agents; - a set of K possible types of agents, where the type of the i-th agent is ki н Ki, i н N, the vector of types k = (k1, k2, …, kn) н K’ = Õ K i ; - the set X' = Õ Xi iОN of admissible vectors of actions of iОN agents; - a set of utility functions ui: K' ´ X' ® Â1; - representations mi(×|ki) н D(K-i), i н N, agents. The Bayes-Nash equilibrium in a game with incomplete information is defined as a set of strategies of agents of the form si: Ki ® Xi, i н N that maximize the corresponding expected utilities (11) Ui(ki, si(×), s-i(×)) = ò ui (k, si(ki), s-i(k-i)) mi(k-i| ki) dk-i, i Î N. k -i ÎÕ K j j ¹i In Bayesian games, it is usually assumed that representations (mi(×| ×))i н N are common knowledge. For this, in particular, it is sufficient that they are consistent, that is, they are derived by each of the agents according to the Bayes formula from the distribution m(k) Î D(K’), which is common knowledge. For Bayesian games in which (mi(×|×))iн N is common knowledge, the notion of rationalizable strategies Di Н D(Xi), i О N such that Di Н BRi(D-i), i О N. In two-person games, the set of rationalizable strategies coincides with the set of strategies obtained as a result of iterative elimination of strongly dominated strategies9. Generalization of Rationalizable Strategies to the Case of Maximin 9 Recall that an agent's strategy is called strongly dominated such that there is another strategy of the agent that in any situation provides this agent with a strictly greater payoff. Iterative elimination of strictly dominated strategies consists in their sequential (generally infinite) exclusion from the set of considered strategies of agents, which leads to finding the “weakest” solution of the game – the set of non-dominated strategies. 30 (guaranteeing) equilibrium is realized in . It is possible to complicate the constructions of subjective equilibrium by introducing prohibitions on certain combinations of agents' actions, etc. Thus, the implementation of the RDS, guaranteeing and subjective equilibrium (if they exist) requires that each agent has at least information about its objective function and all admissible sets, and the implementation of the Nash equilibrium, if it exists, additionally requires that the values ​​of all essential parameters were common knowledge. Once again, we note that the realizability of the Nash equilibrium implies the ability of agents (and the control body - the center, or the operations researcher, if they have the appropriate information) to a priori and independently calculate the Nash equilibrium and immediately choose Nash equilibrium actions in a one-step game (in this case, a separate issue is in which of the equilibria the agents and the center choose if there are several Nash equilibria). Qualitatively, common knowledge is necessary so that each of the agents (and the center) can model the decision-making principles of other agents, including those taking into account his own decision-making principles, etc. Therefore, we can conclude that the concept of solving the game is closely related to the awareness of agents. Solution concepts such as RDS and Nash equilibrium are, in a sense, limiting cases - the first requires minimal awareness, the second requires the infinity of the rank of information reflection of all agents. Therefore, below we will describe other ("intermediate") cases of agents' awareness - representation hierarchies - and construct the game solutions corresponding to them. Before implementing this program, let's review known shared knowledge models and representation hierarchy. 1.3. GENERAL APPROACHES TO THE DESCRIPTION OF INFORMATION In the concepts of equilibrium considered in the previous section (with the possible exception of Nash and Bayes-Nash equilibria, in which the presence of common knowledge is assumed), there is no reflection, since each agent does not try to take the position of opponents. Reflection takes place when an agent has and uses a hierarchy of representations when making decisions - its own representations of the representations of other agents, their representations of its representations and each other's representations, etc. Analysis of ideas about uncertain factors corresponds to information reflection, and ideas about the principles of decision-making - to strategic reflection. In terms of subjective equilibrium, strategic reflection corresponds to the agent's assumptions that the opponent will calculate one or another specific, for example, subjective guaranteeing equilibrium, and informational reflection - what specific assumptions about the situation the opponent will use. Let us consider the currently known10 approaches to describing the hierarchy of representations and general knowledge. As noted in, there are two approaches to describing awareness - syntactic and semantic (recall that "syntactics is the syntax of sign systems, that is, the structure of the combination of signs and the rules for their formation and transformation, regardless of their meanings and functions of sign systems", "semantics - studies sign systems as a means of expressing meaning, its main subject is the interpretation of signs and symbol combinations "). The foundations of these approaches were laid in mathematical logic. With the syntactic approach, the hierarchy of representations is described explicitly. If representations are given by a probability distribution, then the hierarchies of representations at a certain level of the hierarchy correspond to the distributions on the product of the set of states of nature and distributions that reflect the representations of previous levels. An alternative is to use "formulas" (in the logical sense), that is, the rules for transforming elements of the original set based on the use of logical 10 It should be noted that representation hierarchies and general knowledge have become the subject of research in game theory quite recently - the above-mentioned book by D. Lewis is pioneering. (1969) and an article by R. Aumann (1976). An analysis of the chronology of publications (see bibliography) indicates a growing interest in this problem area. 32 operations and operators of the form "player i believes that the probability of the event ... is not less than a" . At the same time, knowledge is modeled by sentences (formulas) constructed in accordance with certain syntactic rules. Within the framework of the semantic approach, representations of agents are given by probability distributions on the set of states of nature. The hierarchy of representations is then generated based only on these distributions. In the simplest deterministic case, knowledge is represented by a set W of possible values ​​of an uncertain parameter and partitions (Ri)i О N of this set. The partition element Ri, which includes q н W, represents the knowledge of the ith agent, i.e., the set of values ​​of an indefinite parameter that are indistinguishable from his point of view given the known fact q . Correspondence (relatively speaking, "equivalence") between syntactic and semantic approaches is established in . Of particular note are the experimental studies of representation hierarchies in - see the review in . Held short review shows that there are two extremes. The first “extreme” is general knowledge (J. Harshany’s merit is that he reduced all information about the agent that affects his behavior to his only characteristic – type – and built an equilibrium (Bayes-Nash) under the hypothesis that the probability distribution of types is common knowledge). The second "extreme" is an endless hierarchy of consistent or inconsistent views. An example of the latter is the construction given in , which, on the one hand, describes all possible Bayesian games and all possible hierarchies of representations, and, on the other hand, (due to its generality) is so cumbersome that it does not allow constructively setting and solving specific problems. Most studies of awareness are devoted to answering the question: in what cases does the hierarchy of agents' representations describe general knowledge and/or adequately reflect the awareness of agents. The dependence of the game solution on the final hierarchy of consistent or inconsistent representations of agents (that is, the entire range between the two "extremes" noted above) has practically not been studied. Exceptions are, firstly, the work , in which the Bayes–Nash equilibria for three-level hierarchies of inconsistent probability representations of two agents were built on the assumption that the representations at the lower level of the hierarchy coincide with the representations of the previous level – see also assumptions of the Pm type and the corresponding equilibria in . Secondly, the third chapter of this work, which describes arbitrary (finite or infinite, consistent or inconsistent) hierarchies of "point" representations, for which an information equilibrium is constructed and studied - the equilibrium of a reflexive game (the possibility and expediency of generalizing the results obtained to the case of interval or probabilistic representations of agents is discussed in the conclusion). Thus, both the study of strategic reflection (Chapter 2 of this work) and the construction of a solution to a reflexive game, as well as the study of the dependence of this equilibrium on the hierarchy of representations of agents (Chapter 3 of this work), are relevant. CHAPTER 2. STRATEGIC REFLECTION This chapter explores game-theoretic models of strategic reflection. In Section 2.1, we study the model of strategic reflection in a two-person game, which in Section 2.2 allows us to solve the problem of the maximum expedient rank of strategic reflection in bimatrix games. Section 2.3 is devoted to a discussion of the finiteness of the rank of reflection, generated by the limited abilities of a person to process information. 2.1. STRATEGIC REFLEXION IN TWO-PERSON GAMES Let us consider sequentially, in order of increasing awareness, reflexive decision-making models in two-person games. Zero rank of reflection. Let us consider the problem of making a decision by an agent in the case of a complete absence of information about the state of nature (recall that the assumption that the target functions and admissible sets are common knowledge is considered to be satisfied). On the one hand, it seems reasonable to use the decision-making principle based on the maximum guaranteed result, according to which the i-th agent will choose a guaranteeing (according to the state of nature and the action of the opponent) strategy (12) 1 xiг = arg max min min fi(q, xi , x-i). xi н X i q нW x -i н X -i can compute the opponent's guaranteeing strategy). Then the best answer is (13) 2 xiг = arg max min fi(q, xi, 1 x-г i). xi н X i q нW But the opponent of the agent under consideration can argue in a similar way. If the agent under consideration allows such a possibility, then its guaranteeing strategy will be (14) 13) by replacing the index "i" with "i" and vice versa. The chain of increasing the “reflection rank” (the agent’s assumptions about the opponent’s reflexion rank) can be continued further (see analogies in the dynamic models considered in ) by recursively determining (15) x-i i), k = 2, 3, ..., xi н X i q нW g 1 i where x , i = 1, 2, are determined by (12). The set of actions of type (15) will be called the set of reflexive guaranteeing strategies. Let's consider an illustrative example. Example 1. Let the objective functions of agents have the form: f1(x1, x2) = x1 – x12 /2x2, f2(x1, x2) = x2 – x22 /2(x1 + d), where d > 0. As for admissible sets, suppose , that X1 = X2 = , 0< e < 1. Будем считать, что каждая из констант e и 35 d много меньше единицы. Гарантирующие стратегии агентов приведены в таблице 1. Табл. 1. Гарантирующие стратегии агентов в примере 1 k г k x1 1 e 2 e+d 3 e+d 4 e + 2d 5 e + 2d 6 e + 3d 7 e + 3d ... ... x2г e+d e+d e + 2d e + 2d e + 3d e + 3d e + 4d ... k Видно, что, во-первых, значения гарантирующих действий увеличиваются с ростом «ранга рефлексии». Во-вторых, различным «рангам рефлексии» агентов соответствуют в общем случае различные гарантирующие действия (отметим, что равновесием11 Нэша в данном примере является вектор (1; 1)) ·12. Вопрос о том, какое действие следует выбирать агенту, остается открытым. Единственно, можно констатировать, что, обладая информацией только о множестве возможных значений состояния природы, i-ый агент может выбирать одно из действий k xiг, i = 1, 2; k = 1, 2, ..., определяемых выражениями (12) и (15). Доопределить рациональный выбор агента в рассматриваемой модели можно следующим образом. Если агенту неизвестна целевая функция оппонента (что исключено в рамках предположения о том, что целевые функции и допустимые множества являются общим знанием), то единственным его рациональным действием является выбор (12), то есть классический МГР. В рамках введенных предположений агенту известна целевая функция оппонента, а также известно, что оппоненту известен этот факт и т.д. Поэтому с точки зрения агента нерационально использование классического МГР, и ему следует рассчитывать, как минимум, что оппонент будет ис11 В качестве отступления заметим, что, если в рассматриваемом примере целевая 2 функция второго агента имеет вид f2(x1, x2) = x2 + x2 /2x1, то у него существует доминантная стратегия (равная единице), и последовательность гарантирующих стратегий первого агента стабилизируется уже на втором члене: 2 г i x x 2 xiг. Символ «·» здесь и далее обозначает окончание примера или доказательства. 36 = e, = 1/2. Если первый агент может вычислить доминантную стратегию своего оппонента, то представляется рациональным выбор им действия 12 г 1 i пользовать МГР, что приведет к выбору 2 xiг. Но, опять же, в силу того, что целевые функции являются общим знанием, агент может предположить, что такой ход его рассуждений может быть восстановлен оппонентом, что сделает целесообразным выбор 3 xiг и т.д. до бесконечности. Следовательно, с точки зрения агента остается неопределенность относительно «ранга рефлексии» оппонента13. Относительно этого параметра он не имеет никакой информации (если у агента имеются некоторые убеждения по этому поводу, то может реализоваться соответствующее субъективное равновесие), что делает rational use guaranteed result according to the “reflection rank” of the opponent: (16) x'i = arg max min min fi(q, xi, j x-i i). xi н X i j =1, 2,... q нW Note that, firstly, x’i may differ from the classical guaranteeing strategy 1 xiг defined by expression (12). Second, when using strategy (16), the fact that the opponent has a dominant strategy will be taken into account by the agent (see the footnote in example 1). Table 2 shows the values ​​of the objective function of the first agent in example 1, depending on the "rank of reflection" of the opponent and the corresponding actions of the opponent. It can be seen that when using strategy (16), the payoff of the ith agent is equal to e + d, which exceeds the payoff e obtained when using the classical MHR. Tab. 2. The payoffs of the first agent in the example + 4d e+d e+d e + 2d e + 2d e + 3d e + 3d e + 4d 13 In other words, the original game can be replaced by a game in which agents choose their reflection ranks. For new game reflexive analogues can also be constructed, and so on. to infinity (see examples: "Penalty" - in the introduction, "Hide and seek" and "Demolition on a miser" - in section 2.2). One of the possible ways to deal with such "infinity" is to use a guaranteed result according to the opponent's reflection rank. Another possible way, efficient for finite games, is to determine the maximum expedient reflexion rank of agents – see Section 2.2. 37 Thus, in the model under consideration, it can be considered rational to use strategies (15) or (16) by the agent. The first rank of reflection. Suppose now that the agent has certain information about the state of nature, which he considers true, and nothing more is known to him with certainty. Within the framework of the existing uncertainty, due to the principle of determinism, an agent performing strategic reflection has two alternatives - either to assume that his opponent does not have any information, or to assume that the latter has the same information as himself14. If the agent does not introduce any assumptions about the awareness and principles of the opponent’s behavior, then he is forced to apply the principle of maximum guaranteed result (MGR) - no additional (compared to the model of the zero rank of reflection considered above) information about the opponent has been added to the agent15 - that is, to rely on the worst choice for him of the second agent from the set of strategies of type (16). The guarantee strategy will be: (17) xi (qi) = arg max min fi(qi, xi, j x-i i). xi н X i j =1, 2,... Note that, being in the information situation corresponding to the model under consideration, calculating (17), the agent considers the opponent as being in the information situation corresponding to the previous model. This general principle - having some information, the agent can consider the opponent as having either the same or one lower reflexion rank - will be used in a number of other reflexive decision-making models. If the first agent believes that his opponent has the same information as himself (the second agent can reason similarly - see assumption P1 in ), then he calculates the subjective 14 This principle (and its generalizations) will be widely used below in determining finite information structures - indeed, having information Ii, the i-th agent can, in case of uncertainty, attribute to other agents only the awareness that is consistent with Ii. 15 Of course, the agent may assume that the opponent has some information, but since this information does not appear in the model, we will not consider such assumptions. 38 equilibrium (that is, the “Nash equilibrium” for the corresponding subjective* * description of the game) EN(q1) = ((x11 (q1), x12 (q1))) of the following form: * * * (18) " x1 н X1 f1 (q1, x11 (q1), x12 (q1)) ³ f1(q1, x1, x12 (q1)), * * * " x2 Î X2 f2(q1, x11 (q1), x12 (q1)) ³ f1( q1, x11 (q1), x2). Substantially, the above systems of inequalities reflect the calculation by the first agent of “his own” Nash equilibrium and the choice of the corresponding coordinate of this equilibrium. In the general case, the agent and his opponent will calculate different equilibria - a coincidence is possible if the awareness is such that xij* (qi) = x*jj (qj), i, j = 1, 2. Thus, rational in the model of the first rank of reflection can be consider the agent's choice of either a reflexive guaranteeing strategy (17) or a subjective equilibrium (18). Subjective equilibrium (18) determined by the first agent can be conventionally depicted as a graph with two vertices x12 x1 us x1 and x12, corresponding to the first agent and his ideas about the second agent16 (see Fig. 1. Subjective nok 1). The incoming arrows at equilibrium in the model of the first reflect that information, of a strategic rank, which is used by each of the agents' reflections about the opponent. The second rank of reflection. In the model of the second rank of reflection, the ith agent has information about the opponent's ideas qij about the state of nature and about his own ideas qii about the state of nature (we will assume that qi = qii - see the autoinformation axiom below). The agent can expect that his opponent will choose a guaranteeing (within the knowledge of qij) strategy. Then the best answer is 16 Such agents that exist in the representations of other agents are called phantom agents. 39 (19) 2 xiг = arg max fi(qi, xi, x-г i (qij)), xi н X i г -i where x (qi,-i) is determined by (17). In addition to the guaranteeing strategy (19), the first agent can calculate the subjective equilibrium * * EN(q1, q12) = ((x11 (q1, q12), x12 (q1, q12))) of the following form: * * * (q1,q12) , x12 (q1,q12)) ³ f1(q1, x1, x12 (q1,q12)), (20) " x1 н X1 f1(q1, x11 * * * " x2 н X2 f2(q12, x121 (q1, q12), x12 (q1,q12)) ³ f2(q12, x121 (q1,q12), x2), * * * " x1 н X1 f1(q12, x121 (q1,q12), x12 (q1,q12)) ³ f2(q12, x1, x12 (q1,q12))). (19) or subjective equilibrium (20).Note that the first two systems of inequalities in (20) reflect the Nash equilibrium from the point of view x12 x1 of the first agent, and the second and third systems of inequalities reflect the Nash equilibrium, which the second agent must determine from the point of view vision of the first agent - see graph in figure 3, in which the dotted line is circled Fig. 3. The subjective "model" of the second agent, which the first agent uses at equilibrium in the RDM2 decision-making model. The analysis of the simplest models of strategic reflection of the first few ranks shows that in the case of several agents and their insufficient awareness, one can consider the processes of their decision-making independently - each of them models the behavior of their opponents, that is, it seeks to build its own closed model of the game (see the discussion of differences in subjective and an objective description of the game in ). In the case of general knowledge, subjective models coincide. 40 Above we considered the reflection of the zero, first and second ranks. Increasing the ranks of reflection can be done further by analogy. Essential in all models are the agent's assumptions about what reflexion rank his opponent has, that is, in fact, the agent's reflexion rank is determined by what reflexion rank he ascribes to his opponent. No reasonable recommendations limiting the growth of the rank of one's own reflection can be offered a priori to the agent. From this point of view, it can be stated that there is no universal concept of equilibrium for games with strategic reflection. In this case, the only way out is to use either the MHR according to the opponent's reflexion ranks, or subjective equilibrium, in which each agent introduces certain assumptions about the opponent's reflexion rank and chooses his action that is optimal within these assumptions. Therefore, we will concentrate our main attention on the study of cases where the reflexion rank does not grow unboundedly. There are two reasons why the rank of reflection may be finite. First, it is inexpedient to increase the reflexion rank above a certain one from the point of view of the agent's payoff (when a further increase in the reflexion rank certainly does not lead to an increase in the payoff). Secondly, a person's ability to process information is limited, and the infinite rank of reflection is nothing more than a mathematical abstraction. Therefore, in the subsequent sections of this chapter, models are presented that take into account both of these reasons - in Section 2.2, using the example of bimatrix games, the maximum expedient rank of strategic reflection is determined, and in Section 2.3, the role of information constraints is investigated. 2.2. REFLECTION IN BIMATRIX GAMES The main idea developed in this section is that in bimatrix games17 in which there is no Nash equilibrium, or in which, given the existing Nash equilibrium, agents choose subjective guaranteeing strategies (see 17 Recall that two-person finite games are called bimatrix games. 41 of the previous section of this paper), the payoff of each of the agents depends both on his reflexion rank and on the opponent's reflexion rank. In addition, it is shown that an unlimited increase in the rank of strategic reflection does not lead to an increase in payoff. Let's move on to a formal description. Consider a bimatrix game18 in which the payoffs of the first and second agents are given by the matrices A = ||aij|| and B = ||bij|| dimensions n ´ m, respectively. Denote19 I = (1, 2, …, n) – the set of actions of the first agent (selecting a row), J = (1, 2, …, m) – the set of actions of the second agent (selecting a column). In the game under consideration, the agents' guaranteeing strategies are as follows: i0 Î Arg max min aij, j0 Î Arg max min bij. iОI jОJ jОJ iОI Let us introduce the following assumptions. Let the payoff matrices be such that each action of each agent is the best response to some action of the opponent, and let, in addition, the best response to each action of the opponent is unique (if there are several best responses, then we can introduce a rule that further determines the choice of the agent).20 Therefore, when determining the best answers, instead of the expressions “i… О Arg max …” and iОI “j… О Arg max …”, you can use, respectively, the expressions jОJ “i… = arg max …” and “j… = arg max …”. iОI jОJ Let a0 = max min aij, b0 = max min bij be the maximum iОI jОJ jОJ iОI 18 Since matrix games (antagonistic finite games of two persons) are a special case of bimatrix games, all the results presented in this section are also valid for matrix games. 19 Let's hope that the use of the same (historically established) designation for the information structure and the set of actions of the first agent will not lead to confusion. 20 If these assumptions are abandoned, then all the results obtained in this section will remain valid, since the assumptions introduced make it possible to obtain an upper estimate for the maximum expedient rank of strategic reflection. 42 Let us define a reflexive bimatrix game MGkl (matrix game) as a bimatrix game with matrices A and B, in which the first and second agents have reflexion ranks equal to k and l, respectively, k, l О А, where А is the set of natural numbers. Let us clarify what will be understood by the rank of reflection (more precisely, by the rank of strategic reflection) in bimatrix games. In bimatrix (and not only bimatrix - see. ) in games, the choice of actions by agents can be carried out on the basis of knowledge of the opponent's reflexion ranks. Reflection ranks are defined as follows. “An agent has a zero reflection rank if he knows only the payment matrix. An agent has the first reflexion rank if he believes that his opponents have a zero reflexion rank, that is, they know only the payoff matrix. In general, an agent with the kth reflexion rank assumes that his opponents have the k-1st reflexion rank. He does the necessary reasoning for them on the choice of strategy and chooses his strategy based on knowledge of the payoff matrix and extrapolation of the actions of his opponents. Let's take an illustrative example. Example 2 (Hide and Seek) . The first agent hides in one of several rooms of different lighting, and the other agent must choose the room where he will look for him. The degrees of illumination are known to both agents. The agent strategies are as follows. The seeker, ceteris paribus, prefers to look where it is lighter (it is easier to find there). Hiding it is clear that in a darker room, the chances of finding him are less than in a lighted one. An increase in the rank of reflection means that it becomes clear to the agent that this is clear to his opponent, and so on. Let's present the reflexion ranks of agents and the corresponding actions for choosing rooms in the form of Table 3. Tab. 3. Rank of reflexion of agents and corresponding actions for choosing rooms Rank of reflexion of an agent except for the lightest Any, except for the darkest Lightest It can be seen that after the second rank of reflection, the entire set of allowed actions is exhausted, and after the third rank of reflection, the room selection strategies begin to repeat. This fact was an illustration of the fact that in the game of two persons, an increase in the ranks of reflection above a certain objective does not give anything new, although the subjective increase in complexity can continue. The discrepancy between the ranks of the reflection of the success of the activity is as follows. Let the hider have rank 0 (hides in the darkest room). If at the same time the seeker has rank 1, then he always wins (searches in the darkest room). But if the seeker has the 3rd rank (searches in any room except the darkest), then he always loses to the one hiding with the 0th rank, because, as we remember, he hides in this very dark room, where the seeker, after a series of reflective reasoning, will never look. Thus, it is impossible to state unequivocally that a higher reflection rank is better than a lower one. The preference of one or another rank is determined by its interaction with the opponent's reflex rank. · Since in bimatrix games it is assumed that each agent has some belief about the opponent's reflexion rank , this allows using the notion of a subjective guaranteeing strategy. Let us define subjective guaranteeing strategies in the bimatrix game MGkl: (21) ik = arg max aijk -1 , jl = arg max bil -1 j , k, l Î À. iнI jнj Thus, the game MG00 coincides with the original game, and the "equilibrium" in the game MGkl is (aik jl ; bik jl), k, l н └. We note two interesting facts. First, the payoff of any agent in the game MGkl for k ³ 1, l ³ 1 may turn out to be less than the maximum guaranteed payoff (see the “Drift on a miser” example below). Second, assigning44 to an opponent a reflexion rank one less than its own is contradictory, since in the game MGkl for k ³ 1, l ³ 1 this means that l = k – 1 and k = l – 1 must be satisfied simultaneously, which is obviously impossible. Therefore, the equilibrium in a reflexive game is essentially subjective, and a priori the agents do not know what game they are playing (the reflexion ranks of both agents cannot be common knowledge, since this would contradict the very definition of the reflexion rank). Therefore, a promising direction for future research is the study of information reflection on the ranks of reflection of agents in bimatrix games. The internal inconsistency of strategic reflection in bimatrix games can be illustrated by the following scheme: Figure 4a shows a subjective description of the game MGkl in terms of the reflexive game graph from the point of view of the first agent, Figure 4b shows a subjective description of the same game from the point of view of the second agent. i0 j0 i0 j0 i1 j1 i1 j1 … … ik-2 jk-2 il-2 jl-2 ik-1 jk-1 il-1 jl-1 ik ? Rice. 4a. A subjective description of the game MGkl from the point of view of the first agent? jl Fig. 4b. A subjective description of the game MGkl from the point of view of the second agent 45 Looking ahead a bit (see Section 3.4), we note that the graph of a reflexive game has the property that the number of arcs entering each of its vertices must be one less than the number of agents ( that is, in bimatrix games it is equal to one). Subjective equilibrium actions are shown in bold and lead to "equilibrium" (ik, jl). The actions ik-1 for the first agent and jl-1 for the second are not used in the corresponding subjective descriptions of the game (see below). question marks in Figure 4), that is, each of them turns out to be internally open. Having completed a brief discussion of the internal inconsistency in determining the rank of strategic reflection in bimatrix games, let us return to the study of the dependence of the subjective equilibrium and payoffs of agents on their ranks of reflection. Denote IK = ik , JL = jl , K = 0, 1, 2, …, U U k =0 ,1,...,K l =0 ,1,...,L L = 0, 1, 2, … . By I¥ and J¥ we mean the corresponding unions over all reflexion ranks from zero to infinity. If one agent (or both agents) does not know the opponent's reflexion rank, then it is reasonable to consider the game MG¥¥, in which each agent calculates a guaranteed result from the opponent's reflexion rank. We introduce guaranteeing strategies corresponding to complete uncertainty about the opponent's reflexion rank: (22) i¥ = arg max min aij, j¥ = arg max min bij. iОI jОJ ¥ jОJ iОI ¥ Similarly, one can define guaranteeing strategies within the framework of information that the opponent’s reflexion rank does not exceed a known value (i.e., the first agent believes that the reflexion rank of the second one is not higher than L, and the second agent believes that the reflexion rank of the first one is not higher than K ): (23) iL = arg max min aijl , jK = arg max min bik j . iОI lОJ L jОJ kОI K Note that in (23), in contrast to (21), the strategy of each of the agents does not depend on its own reflexion rank, but is determined by information about the opponent's reflexion rank. Expressions (21)-(23) do not exhaust the whole variety of possible situations, since, for example, the first agent may assume that the second one will choose j¥, and then his best answer will be arg max aij¥ , and so on. In addition, although only "strong" agents are capable of increasing the rank of reflection iÎI, it is intuitively clear that with the growth of this rank, that is, with the lengthening of the chain of reasoning "I think that he thinks that I think ..." there is a danger of "to be too smart ". A strong agent with a high reflection rank overestimates the opponent, assuming that he also has a high reflection rank. But, if the opponent's rank is actually low, this leads to a loss to a weaker opponent - see the examples "Hide and Seek" and "Demolition on a miser". Therefore, a systematic study of the ratio of agents' payoffs depending on the type of game being played is necessary. We present the results of this study. Essential for our consideration is the presence or absence of a Nash equilibrium, as well as the choice by agents (and the use in constructing subjective equilibria) of guaranteeing strategies or actions that are Nash equilibrium. Thus, the following four situations are possible. Option 1 (Nash equilibrium in pure strategies exists, and agents are guided by Nash equilibrium actions). Let (i*; j*) be the numbers of Nash equilibrium pure strategies. Then, if by analogy with (21) we assume that in a reflexive game each agent chooses his best response to the opponent's choice of the corresponding equilibrium component, then we get that (24) ik = arg max aij* , jl = arg max bi* j , k, l О А. iнI jнJ From (24), by virtue of the definition of the Nash equilibrium, it follows that ik = i*, jl = j*, k, l н └, i.e., within the framework of variant 1, strategic reflection is meaningless21 (with the possible exception of the case when the best responses are defined in such a way that agents choose components of different Nash equilibria in the case when there are several of them). Option 2 (Nash equilibrium exists in pure strategies, but agents choose guaranteeing strategies (21)). 21 By the senselessness of strategic reflection in bimatrix games, we mean the case when the equilibrium in a reflexive game with any combination of non-zero reflexion ranks of agents coincides with the equilibrium in the original game. 47 If the guaranteeing strategies form a Nash equilibrium (as is the case in antagonistic games with a saddle point), then we fall into the conditions of option 1. Therefore, strategic reflection makes sense only if, in the framework of option 2, the Nash equilibrium does not coincide with the equilibrium in guaranteeing strategies ( i0, j0). Option 3 (there is no Nash equilibrium in pure strategies, and agents are guided by Nash equilibrium mixed strategies22). If agents, when determining their best responses by analogy with (24), rely on the opponent choosing Nash equilibrium mixed strategies, then it is easy to show that the maximum expected payoff of each agent will be achieved when he also chooses the corresponding Nash equilibrium mixed strategy. Therefore, in the framework of option 3, any equilibrium coincides with the Nash equilibrium in mixed strategies, that is, strategic reflection in this case is meaningless. Option 4 (there is no Nash equilibrium in pure strategies, and agents are guided by guaranteeing strategies (21)). In the fourth variant, the analysis of reflection obviously makes sense. Thus, having considered all four possible variants of agents' behavior, we obtain that the validity of the following statement is substantiated. Statement 1. Strategic reflection in bimatrix games makes sense if agents use subjective guaranteeing strategies (21) that are not Nash equilibrium. Denote (25) Kmin = min (K н └ | IK = I¥), (26) Lmin = min (L н └ | JL = J¥). In essence, Kmin and Lmin are the minimum reflexion ranks of the first and second agents, under which their sets of subjective equilibrium actions coincide with the maximum possible sets of subjective guaranteeing strategies in the game under consideration. 22 Recall that in bimatrix games the Nash equilibrium in mixed strategies always exists. 48 By definition " K, L О └ IK Н IK+1, JL Н JL+1. Hence " K ³ Kmin IK = I¥, " L ³ Lmin JL = J¥. If the reflexion rank of the first and second agents does not exceed K and L, respectively, then the sets of subjective guaranteeing strategies of the first and second agents from the point of view of the opponent are equal to IL-1 and JK-1, respectively.So, an increase in the reflexion ranks can lead to an expansion of the set of subjective guaranteeing strategies if (27) L – 1< Kmin, (28) K – 1 < Lmin. Отметим, что с рассматриваемой точки зрения максимальный целесообразный ранг рефлексии23 первого агента зависит от свойств субъективных гарантирующих стратегий второго агента (см. (28)), и наоборот. С другой стороны, агенту не имеет смысла увеличивать ранг своей рефлексии, если он уже «исчерпал» собственное множество возможных субъективных равновесных действий. С этой точки зрения увеличение рангов рефлексии может приводить к расширению множества субъективных гарантирующих стратегий, если (29) K < Kmin, (30) L < Lmin. Объединяя (28) и (29), а также (27) и (30), получаем, что первому агенту не имеет смысла увеличивать свой ранг рефлексии выше (31) Kmax = min {Kmin, Lmin + 1}, а второму агенту не имеет смысла увеличивать свой ранг рефлексии выше (32) Lmax = min {Lmin, Kmin + 1}. Обозначим (33) Rmax = max {Kmax, Lmax}. Таким образом, доказана справедливость следующего утверждения. 23 Под максимальным целесообразным рангом рефлексии агента будем понимать такое его значение, что увеличение ранга рефлексии выше данного не приводит к появлению новых субъективных (с точки зрения данного агента) равновесий. 49 Утверждение 2. Использование агентами в биматричной игре рангов стратегической рефлексии выше, чем (31) и (32), не имеет смысла24. Утверждение 2 дает возможность в каждом конкретном случае (для конкретной разыгрываемой игры) каждому агенту (и исследователю операций) вычислить максимальные целесообразные ранги стратегической рефлексии обоих агентов. Так как величины (31)-(33) зависят от игры (матриц выигрышей), то получим оценки зависимости этих величин от размерности матриц выигрышей (очевидно, что |I¥| £ |I| = n, |J¥| £ |J| = m, а для игр размерности два справедлива более точная оценка – см. утверждение 3). Для этого введем в рассмотрение граф наилучших ответов. Графом наилучших ответов G = (V, E) назовем конечный двудольный ориентированный граф, в котором множество вершин V = I È J, а дуги проведены от каждой вершины (соответствующей действию одного из агентов) к наилучшему на нее ответу оппонента. Опишем свойства введенного графа: 1. Из каждой вершины множества I выходит дуга в вершину множества J (у второго агента есть наилучший ответ на любое действие первого агента), из каждой вершины множества J выходит дуга в вершину множества I (у первого агента есть наилучший ответ на любое действие второго агента). 2. В каждую вершину множества V входит ровно одна дуга (так как каждое действие каждого агента является наилучшим ответом на какое-либо действие оппонента). 3. Если любой путь дважды прошел через одну и ту же вершину, то по определению наилучших ответов его часть является контуром, и в дальнейшем новых вершин в этом пути не появится. 4. Максимальное число попарно various activities of the first agent contained in the path starting at the vertex i0 is equal to min (n; m + 1). 5. The maximum number of pairwise distinct actions of the second agent contained in the path starting at the vertex i0 is equal to min (n; m). 24 That is, for any rank of reflection that exceeds the indicated estimates, there is a rank of reflection that satisfies the indicated estimates and leads to the same subjective balance. 50 6. The maximum number of pairwise distinct actions of the first agent contained in the path starting at the vertex j0 is equal to min (n; m). 7. The maximum number of pairwise distinct actions of the second agent contained in the path starting at the vertex j0 is min (n + 1; m). The revealed properties of the graph of the best answers make it possible to obtain upper bounds for the expedient ranks of strategic reflection in bimatrix games. Statement 3. In 2 ´ 2 bimatrix games where there is no Nash equilibrium, I¥ = I, J¥ = J. Proof. Consider an arbitrary 2 ´ 2 bimatrix game in which there is no Nash equilibrium. Let X1 = (x1, x2), X2 = (y1, y2). Let us calculate the guaranteeing strategies i0 and j0. For definiteness, we set x1 = i0, y1 = j0. There are two mutually exclusive options: j1 = y1 and j1 = y2. If j1 = y1, then i1= i2 = x2 (otherwise (x1, y1) is a Nash equilibrium). Then j2 = j3 = y2 (otherwise (x2, y1) is a Nash equilibrium). Therefore, i3 = i4 = x1 (otherwise (x2, y2) is a Nash equilibrium). That is, in the first case I¥ = I, J¥ = J. If j1 = y2, then i2 = x2 (otherwise (x1, y2) is the Nash equilibrium). Then j3 = y1 (otherwise (x2, y2) is a Nash equilibrium). Therefore, i4 = x1 (otherwise (x2, y1) is a Nash equilibrium). That is, in the second case also I¥ = I, J¥ = J. · Qualitatively, Statement 3 means that in a 2 ´ 2 bimatrix game in which there is no Nash equilibrium, any outcome can be realized as a subjective equilibrium. A promising direction for further applied research can be considered the analysis of subjective equilibria in basic ordinary games of two persons 2 ´ 2 (recall that there are 78 structurally different ordinary games, that is, games in which both agents, each of which has two admissible actions, can strictly order own gains from best to worst). Assertion 3 suggests that, perhaps, in all bimatrix games in which there is no Nash equilibrium, I¥ = I, J¥ = J. where vertices i0 and j0 are shaded. I¥ I J¥ J 5. An example of the graph of the best answers in a 4 ´ 4 bimatrix game in which I¥ Ì I, J¥ Ì J J¥, we investigate how quickly (at what minimum ranks of strategic reflection) these sets are “covered” by the corresponding subjective equilibria. The third property of the graph of the best answers means that in a bimatrix game, an expedient increase in the rank of strategic reflection, starting from the second step, necessarily changes the set of strategies that must be subjective guaranteeing at reflexion ranks less than or equal to this one. Since in bimatrix games the sets of admissible actions are finite, then the sets I¥ and J¥ are finite, therefore, due to properties 4-7 of the graph of the best answers, the values ​​Lmin and Kmin are also finite, that is, in bimatrix games an unlimited increase in the reflexion rank is obviously inexpedient. Again, due to the finiteness of admissible sets, quantities (31) and (32), which determine the maximum expedient reflexion ranks, can be easily calculated for any particular bimatrix game. But the properties of the graph of the best answers make it possible to obtain specific estimates from above of the maximum expedient reflexion ranks. 52 In a bimatrix game n ´ m, the guaranteed estimates25 for the values ​​(31)-(33) will obviously depend on the dimension of the payoff matrices, that is, Kmin = Kmin(n), Lmin = Lmin(m). Therefore, (34) Kmax(n, m) = min (Kmin(n), Lmin(m) + 1), (35) Lmax(n, m) = min (Lmin(m), Kmin(n) + 1 ). Expression (33) will then take the form: (36) Rmax(n, m) = max (Kmax(n, m), Lmax(n, m)). From properties 4-7 of the graph of the best answers and expressions (34)-(36) the following assertion follows. Statement 4. In bimatrix games n ´ m, the maximum expedient ranks of strategic reflection of the first and second agents satisfy the following inequalities (37) Kmax(n, m) £ min (n, m + 1), (38) Lmax(n, m) £ min (m, n + 1), (39) Rmax(n, m) £ max (min (n, m + 1), min (m, n + 1)). Corollary 1. In a bimatrix game n ´ n, n ³ 2, the maximum expedient rank of strategic reflection of any agent26 is Rmax(n, n) £ n. For the case of two admissible actions (due to its prevalence in applied models), we formulate a separate corollary. Corollary 2. In a 2 ´ 2 bimatrix game, the maximum expedient reflexion rank does not exceed two. Once again, we note that estimates (37)-(39) are upper estimates - the existence of several best responses to the same action, the presence of a Nash equilibrium in the original game or dominated strategies can lead to

Russian Academy of Sciences V.A. Trapeznikova D.A. NOVIKOV, A.G. CHKHARTISHVILI REFLECTIVE GAMES SINTEG Moscow - 2003 UDC 519 BBC 22.18 N 73 Novikov D.A., Chkhartishvili A.G. Reflexive H 73 games. M.: SINTEG, 2003. - 149 p. ISBN 5-89638-63-1 The monograph is devoted to the discussion of modern approaches to the mathematical modeling of reflection. The authors introduce a new class of game-theoretic models – reflexive games that describe the interaction of subjects (agents) that make decisions based on a hierarchy of ideas about essential parameters, ideas about representations, etc. An analysis of the behavior of phantom agents that exist in the representations of other real or phantom agents and the properties of an information structure that reflects the mutual awareness of real and phantom agents allows us to propose an information equilibrium as a solution to a reflexive game, which is a generalization of a number of well-known concepts of equilibrium in non-cooperative games. Reflective games make it possible: - to model the behavior of reflective subjects; - to study the dependence of the payoffs of agents on the ranks of their reflection; - set and solve problems of reflexive control; - uniformly describe many phenomena related to reflection: hidden control, information control through the media, reflection in psychology, works of art, etc. The book is addressed to specialists in the field of mathematical modeling and management of socio-economic systems, as well as university students and graduate students. Reviewers: Doctor of Technical Sciences, prof. V.N. Burkov, Doctor of Technical Sciences, prof. A.V. Shchepkin UDC 519 BBK 22.18 N 73 ISBN 5-89638-63-1 Chkhartishvili, 2003 2 CONTENTS INTRODUCTION .................................................. ................................................. .......... 4 CHAPTER 1. Information in decision-making .................................. ........... 21 1.1. Individual Decision Making: A Model of Rational Behavior.................................................................. ................................................. ............................... 21 1.2. Interactive decision-making: games and equilibria .............................. 24 1.3. General Approaches to Describing Awareness.................................................. 31 CHAPTER 2. Strategic Reflection....... ................................................. 34 2.1. Strategic reflection in two-person games .............................. .................. 34 2.2. Reflection in bimatrix games .............................................................. ........... 41 2.3. Limitation of the rank of reflection .............................................................. .............. 57 CHAPTER 3. Informational reflection .............................. ...................... 60 3.1. Informational reflection in two-person games .............................................. 60 3.2. Information structure of the game .............................................................. .............. 64 3.3. Information balance .............................................................. ................... 71 3.4. Graph of a reflexive game ............................................................... ........................... 76 3.5. Regular awareness structures.............................................................. 82 3.6. The rank of reflection and informational equilibrium .............................................. 91 3.7. Reflective control .................................................................. ....................... 102 CHAPTER 4. Applied models of reflexive games .................................. 102 ............. 106 4.1. Hidden control .................................................................. .................................. 106 4.2. Mass media and information management .............................................................. ...... 117 4.3. Reflection in psychology .............................................................. ........................... 121 4.3.1. Psychology of chess creativity............................................... 121 4.3 .2. Transactional analysis .............................................................. .................. 124 4.3.3. Johari window .................................................. .................................. 126 4.3.4. Ethical Choice Model .................................................................. .............. 128 4.4. Reflection in works of art............................................... 129 CONCLUSION..... ................................................. ...................................... 137 LITERATURE .......... ................................................. ................................................... 142 3 - Minnows frolic freely, this is their joy! – You are not a fish, how do you know what its joy is? “You’re not me, how do you know what I know and what I don’t know?” From a Taoist parable - The point, of course, venerable archbishop, is that you believe in what you believe in because you were brought up that way. - May be so. But the fact remains that you, too, believe that I believe what I believe, because I was brought up that way, for the reason that you were brought up that way. From the book “Social Psychology” by D. Myers on the basis of a hierarchy of ideas about essential parameters, ideas about views, etc. Reflection. One of the fundamental properties of human existence is that, along with the natural ("objective") reality, there is its reflection in consciousness. At the same time, between the natural reality and its image in the mind (we will consider this image as a part of a special - reflective reality) there is an inevitable gap, a mismatch. Purposeful study of this phenomenon is traditionally associated with the term “reflection”, which is defined in the “Philosophical Dictionary” as follows: “REFLEXION (lat. reflexio – reversal). A term meaning reflection, as well as the study of a cognitive act. The term "reflection" was introduced by J. Locke; in various philosophical systems (J. Locke, G. Leibniz, D. Hume, G. Hegel, etc.) it had different content. A systematic description of reflection from the point of view of psychology began in the 60s of the XX century (school 4 of V.A. Lefebvre). In addition, it should be noted that there is an understanding of reflection in a different meaning, related to the reflex - “the reaction of the body to the excitation of receptors”. In this paper, we use the first (philosophical) definition of reflection. To clarify the understanding of the essence of reflection, let us first consider the situation with one subject. He has ideas about the natural reality, but he can also be aware (reflect, reflect) these ideas, as well as be aware of the awareness of these ideas, etc. This is how reflective reality is formed. Reflection of the subject regarding his own ideas about reality, the principles of his activity, etc. is called auto-reflection or reflection of the first kind. It should be noted that in the majority of humanitarian studies, we are talking, first of all, about autoreflection, which in philosophy is understood as the process of thinking of an individual about what is happening in his mind. Reflection of the second kind takes place regarding ideas about reality, decision-making principles, self-reflection, etc. other entities. Let us give examples of reflection of the second kind, illustrating that in many cases the correct own conclusions can be made only if we take the position of other subjects and analyze their possible reasoning. The first example is the classic Dirty Face Game, sometimes referred to as the wise men and hats problem or the husbands and unfaithful wives problem. Let us describe it following . “Let's imagine that Bob and his niece Alice are in the compartment of a Victorian carriage. Everyone's face is messed up. However, no one blushes with shame, although any Victorian passenger would blush knowing that the other person sees him dirty. From this we conclude that none of the passengers knows that his face is dirty, although everyone sees the dirty face of his companion. At this time, the Conductor looks into the compartment and announces that there is a man with a dirty face in the compartment. After that, Alice blushed. She realized that her face was dirty. But why did she understand this? Didn't the Guide tell her what she already knew? 5 Let's follow the chain of Alice's reasoning. Alice: Suppose my face is clean. Then Bob, knowing that one of us is dirty, should conclude that he is dirty and blush. If he does not blush, then my premise about my clean face is false, my face is dirty and I should blush. The conductor added information about Bob's knowledge to the information known to Alice. Until then, she hadn't known that Bob knew that one of them was dirty. In short, the conductor's message turned the knowledge that there was a man with a dirty face in the compartment into general knowledge. The second textbook example is the Coordinated Attack Problem; there are problems close to it about the optimal information exchange protocol - Electronic Mail Game, etc. (see reviews in ). The situation is as follows. Two divisions are located on the tops of two hills, and the enemy is located in the valley. You can win only if both divisions attack the enemy at the same time. The general - the commander of the first division - sends the general - the commander of the second division - a messenger with the message: "We attack at dawn." Since the messenger can be intercepted by the enemy, the first general must wait for a message from the second general that the first message has been received. But since the second message can also be intercepted by the enemy, the second general needs to get confirmation from the first general that he received confirmation. And so on ad infinitum. The task is to determine after what number of messages (confirmations) it makes sense for the generals to attack the enemy. The conclusion is as follows: under the described conditions, a coordinated attack is impossible, and the way out is to use probabilistic models. The third classical problem is the "two broker problem" (see also speculation models in ). Suppose that two brokers playing on the stock exchange have their own expert systems that are used to support decision making. It happens that the network administrator illegally copies both expert systems and sells his opponent's expert system to each broker. After that, the administrator tries to sell each of them the following information - "Your opponent has your expert system." Then the administrator tries 6 to sell information - "Your opponent knows that you have his expert system", and so on. The question is, how should brokers use the information they get from the administrator, and what information is relevant at which iteration? Having completed the consideration of examples of reflection of the second kind, let us discuss the situations in which reflection is essential. If the only reflexive subject is an economic agent that seeks to maximize its objective function by choosing one of the ethically acceptable actions, then the natural reality enters the objective function as a parameter, and the results of reflection (representations about representations, etc.) are not elements of the objective function. Then we can say that autoreflection is “not needed”, since it does not change the action chosen by the agent. Note that the dependence of the subject's actions on reflection can take place in a situation where actions are ethically unequal, that is, along with the utilitarian aspect, there is a deontological (ethical) one - see . However, economic decisions are, as a rule, ethically neutral, so let's consider the interaction of several subjects. If there are several subjects (the decision-making situation is interactive), then the target function of each subject includes the actions of other subjects, that is, these actions are part of natural reality (although they themselves, of course, are due to reflexive reality). At the same time, reflection (and, consequently, the study of reflective reality) becomes necessary. Let us consider the main approaches to mathematical modeling of reflection effects. Game theory. Formal (mathematical) models of human behavior have been created and studied for more than a century and a half (see review in ) and are increasingly being used both in control theory, economics, psychology, sociology, etc., and in solving specific applied problems. . The most intensive development has been observed since the 40s of the XX century - the moment of the emergence of game theory, which is usually dated to 1944 (the first edition of the book by John von Neumann and Oskar Morgenstern "Game Theory and Economic Behavior"). 7 Under the game in this work we will understand the interaction of the parties whose interests do not coincide (note that another understanding of the game is possible - as "a type of unproductive activity, the motive of which lies not in its results, but in the process itself" - see also , where the concept of the game is interpreted much more broadly). Game theory is a branch of applied mathematics that studies decision-making models in the conditions of a mismatch of interests of the parties (players), when each party seeks to influence the development of the situation in its own interests. Further, the term "agent" is used to refer to the decision-maker (player). In this paper, we consider non-cooperative static games in normal form, that is, games in which agents choose their actions once, simultaneously and independently. Thus, the main task of game theory is to describe the interaction of several agents whose interests do not coincide, and the results of activity (winning, utility, etc.) of each depend in the general case on the actions of all . The result of such a description is a forecast of a reasonable outcome of the game - the so-called solution of the game (equilibrium). Description of the game consists in setting the following parameters: - set of agents; - preferences of agents (dependencies of payoffs on actions): it is assumed (and this reflects the purposefulness of behavior) that each agent is interested in maximizing his payoff; - sets of admissible actions of agents; - awareness of agents (the information that they have at the time of making decisions about the chosen actions); - the order of functioning (the order of moves - the sequence of choice of actions). Relatively speaking, the set of agents determines who participates in the game. Preferences reflect what agents want, sets of allowed actions what they can do, awareness reflects what they know, and order of operation reflects when they choose actions. 8 The listed parameters define the game, but they are not sufficient to predict its outcome - the solution of the game (or the equilibrium of the game), that is, the set of actions of agents that are rational and stable from one point of view or another. To date, there is no universal concept of equilibrium in game theory – taking certain assumptions about the principles of decision-making by agents, one can obtain various solutions. Therefore, the main task of any game-theoretic research (including the present work) is the construction of an equilibrium. Since reflexive games are defined as such an interactive interaction of agents in which they make decisions based on the hierarchy of their representations, the awareness of agents is essential. Therefore, let us dwell on its qualitative discussion in more detail. The role of awareness. General knowledge. In game theory, philosophy, psychology, distributed systems, and other fields of science (see review in ), not only agents' beliefs about essential parameters are important, but also their beliefs about other agents' beliefs, and so on. The set of these representations is called a hierarchy of beliefs and is modeled in this paper by the information structure tree of a reflexive game (see Section 3.2). In other words, in situations of interactive decision-making (modeled in game theory), each agent must predict the behavior of opponents before choosing his action. To do this, he must have certain ideas about the vision of the game by opponents. But the opponents must do the same, so the uncertainty about which game will be played creates an endless hierarchy of representations of the participants in the game. Let's give an example of a view hierarchy. Suppose that there are two agents, A and B. Each of them can have their own non-reflexive ideas about the indefinite parameter q, which we will call the state of nature (state of the world). We denote these representations by qA and qB, respectively. But each of the agents within the framework of the process of reflection of the first rank can think about the ideas of the opponent. These representations (representations of the second order) are denoted by qAB and qBA, where qAB are agent A's representations of agent B's representations, 9 qBA are agent B's representations of agent A's representations. second rank) can think about what the opponent's ideas about his ideas are. This is how representations of the third order, qABA and qBAB, are generated. The process of generating representations of higher orders can continue indefinitely (there are no logical restrictions on increasing the reflexion rank). The totality of all representations - qA, qB, qAB, qBA, qABA, qBAB, etc. - forms a hierarchy of views. A special case of awareness is when all representations, representations about representations, etc. coincide to infinity – is common knowledge. More correctly, the term "common knowledge" is introduced in to denote a fact that satisfies the following requirements: 1) it is known to all agents; 2) all agents know 1; 3) all agents know 2, and so on. ad infinitum The formal model of general knowledge was proposed in and developed in many works - see . Models of agents’ awareness – the hierarchy of representations and general knowledge – in game theory are, in fact, entirely devoted to this work, so we will give examples illustrating the role of general knowledge in other areas of science – philosophy, psychology, etc. (see also review ). From a philosophical point of view, common knowledge was analyzed in the study of conventions. Consider the following example. It is written in the Rules of the Road that each road user must comply with these rules, and also has the right to expect that other road users observe them. But other road users also need to be sure that others follow the rules, and so on. to infinity. Therefore, the agreement to "observe traffic rules" should be common knowledge. In psychology, there is the concept of discourse - “(from Latin discursus - reasoning, argument) - verbal thinking of a person mediated by past experience; acts as a process of associated logical 10