Games for the reflective circle card index (preparatory group) on the topic. Reflective games provide an opportunity Reflective game all in a circle
Consider the set N={1, 2, … , n) agents. If there is an indefinite parameter in the situation (we will assume that the set is common knowledge), then awareness structure I i(as a synonym we will use the terms information structure and view hierarchy) i th agent includes the following elements. First, presentation i-th agent about the parameter – denote it . Secondly, representations i-th agent about the representations of other agents about the parameter – let's designate them . Thirdly, representations i th agent about submission j th agent about submission k- agent, we denote them by . And so on.
Thus, the structure of awareness I i i-th agent is given by a set of possible values of the form , where l runs through the set of non-negative integers, , and .
Similarly, the structure of awareness of the I game as a whole - a set of values , where l runs through the set of non-negative integers, , and . We emphasize that the structure of awareness I"inaccessible" to the observation of agents, each of which knows only some of its part (namely - I i).
Thus, the structure of awareness is infinite n- tree (that is, the type of structure is constant and is n-tree), the vertices of which correspond to the specific awareness of real and phantom agents.
Reflexive game G I the game described by the following tuple is called:
where N- many real agents, X i i-th agent, - its objective function, , - the set of possible values of an indefinite parameter, I- awareness structure.
Thus, a reflexive game is a generalization of the notion of a game in normal form given by a tuple , in the case when the awareness of agents is reflected by the hierarchy of their representations (information structure I). Within the framework of the accepted definition, a "classical" game in normal form is a special case of a reflexive game - a game with common knowledge. In the "limiting" case - when the state of nature is common knowledge - the concept of solving a reflexive game (information equilibrium - see below) proposed in this paper goes over to the Nash equilibrium.
The set of connections between the elements of agents' awareness can be represented as a tree (see Fig. 6.2). At the same time, the structure of awareness i-th agent is represented by a subtree emanating from the vertex .
Let's make an important remark: in this lecture we will confine ourselves to consideration of the "point" structure of awareness, the components of which consist only of elements of the set . (A more general case is, for example, interval or probabilistic awareness.)
Strategic and informational reflection. So, a reflexive game is one in which the knowledge of the players is not common knowledge. From the point of view of game theory and reflexive decision-making models, it is advisable to separate strategic and informational reflection.
Information reflection- the process and result of the player's thoughts about what the values of the uncertain parameters are, what his opponents (other players) know and think about these values. At the same time, the “game” component itself is absent, since the player does not make any decisions.
In other words, informational reflection refers to the agent's awareness of natural reality (what the game is like) and reflexive reality (how others see the game). Information reflection logically precedes reflection of a somewhat different kind - strategic reflection.
Strategic reflection- the process and result of the player's thinking about what decision-making principles his opponents (other players) use within the framework of the awareness that he ascribes to them as a result of informational reflection. Thus, information reflection takes place only under conditions of incomplete 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 player's decision to choose an action (strategy). In other words, informational and strategic reflections can be studied independently, but in conditions of incomplete awareness, both of them take place.
is the set of all possible finite sequences of indices from N;
– union with an empty sequence;
– the number of indices in the sequence (for an empty sequence it is taken equal to zero), which was called the length of the index sequence above.
If a - representation i-th agent about an indefinite parameter, and - representations i th agent about its own representation, it is natural to assume that . In other words, i The th agent is correctly informed about his own ideas, and also believes that other agents are, and so on. Formally, this means that axiom of self-information, which we will further assume to be satisfied:
This axiom means, in particular, that knowing for all such that , can be uniquely found for all such that .
Along with awareness structures I i, , structures of awareness can be considered I ij(structure of awareness j-th agent in the view i-th agent), Iijk etc. Identifying the structure of awareness with the agent characterized by it, we can say that, along with n real agents ( i-agents, where ) with awareness structures I i, participate in the game phantom agents(-agents, where , ) with awareness structures . Phantom agents, existing in the minds of real agents, influence their actions, which will be discussed below.
Let us define the fundamental concept for further considerations of the identity of awareness structures.
The structures of awareness are called identical if two conditions are met
1) for any ;
2) the last indices in sequences and coincide.
We will denote the identity of awareness structures as follows: .
The first of the two conditions in the definition of the identity of structures is transparent, while the second requires some explanation. The fact is that further we will discuss the action of the -agent depending on its awareness structure and objective function fi, which is just determined by the last index of the sequence . Therefore, it is convenient to assume that the identity of the awareness structures means, among other things, the identity of the target functions.
Let's call -agent -subjectively adequately informed about representations of the -agent (or, in short, about the -agent), if
We will designate -subjective adequate awareness of -agent about -agent as follows: .
The concept of the identity of awareness structures allows us to determine their important property - complexity. Note that, along with the structure I there is a countable set of structures , among which classes of pairwise non-identical structures can be distinguished using the identity relation. It is natural to count the number of these classes the complexity of the awareness structure.
I It has finite complexity v=v(I), if there exists a finite set of pairwise non-identical structures such that for any structure , there is a structure identical to it from this set. If such a finite set does not exist, we will say that the structure I has infinite complexity: .
An awareness structure of finite complexity will be called ultimate(we note once again that in this case the tree of the awareness structure still remains infinite). Otherwise, the awareness structure will be called endless.
It is clear that the minimum possible complexity of the awareness structure is exactly equal to the number of real agents participating in the game (recall that, by the definition of the identity of awareness structures, they differ in pairs for real agents).
Any set (finite or countable) of pairwise nonidentical structures such that any structure identical to one of them is called basis awareness structures I.
If the awareness structure I has finite complexity, then it is possible to determine the maximum length of the index sequence such that, knowing all structures , one can find all other structures. This length, in a certain sense, characterizes the rank of reflection necessary to describe the structure of awareness.
We will say that the structure of awareness I, , It has final depth, if: . 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 (6.6) (that is, the definition of informational equilibrium), while its solution may not exist.
So the Count G I reflexive game G I(see the definition of a reflexive game above), whose information structure has finite complexity, is defined as follows:
1) graph vertices G I correspond to real and phantom agents participating in the reflexive game, that is, pairwise non-identical structures of awareness;
2) graph arcs G I 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 I represent the representations of the corresponding agent about the state of nature, then the reflexive game G I with a finite awareness structure I can be given as a tuple , where N- many real agents, X i- set of allowed actions i-th agent, - its objective function, , G I is the graph of a reflexive game.
Note that in many cases it is more convenient (and visual) to describe a reflexive game in terms of the graph G I, rather than an information structure tree (see examples of reflexive game graphs below).
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 modern approaches to 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. Information reflection in games of two persons. ............................................... 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 subjects. 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 in a carriage compartment Victorian era are Bob and his niece Alice. 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 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 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
Along with reflexive games possible method game-theoretic modeling in conditions of incomplete awareness are bayes games, proposed in the late 1960s. J. Harshanyi. In Bayesian games, all private (i.e., not general knowledge) information that an agent has at the time he chooses his action is called type agent. Moreover, each agent, knowing its type, also has assumptions about the types of other agents (in the form of a probability distribution). Formally, a Bayesian game is described by the following set:
- - many N agents;
- - sets /?, possible types of agents, where the type of the /th agent
many X' = J-[ X x admissible action vectors of the agent
- -a set of objective functions /: R'x X'-> 9? 1 (the objective function of an agent generally depends on the types and actions of all agents);
- - representations F, (-|r,) e D(/?_,), /" e N, agents (here, /?_ denotes the set of possible sets of types of all agents, except for the /-th, R.j= P R t , and D(/?_,) denotes the set
in all possible probability distributions on /?_,). The solution to the Bayesian game is Bayes-Nash Equilibrium, defined as a set of strategies of agents of the form X*: R, -> X h i e N,
which maximize the mathematical expectations of the corresponding objective functions:
where jc denotes the set of strategies of all agents, except for the j-th one. We emphasize that in the Bayesian game, the agent's strategy is not an action, but a function of the dependence of the agent's action on its type.
J. Harshanyi's model can be interpreted in different ways (see). According to one interpretation, all agents know the a priori distribution of types F(r) e D (R') and, having learned their own type, they calculate the conditional distribution from it using the Bayes formula Fj(r.i| G,). In this case representations of agents (F,(-|-)), sW are called agreed(and, in particular, are common knowledge - each agent can calculate them, knows what the others can do, etc.).
Another interpretation is as follows. Let there be some set of potential participants in the game of various types. Each such “potential” agent chooses his strategy depending on his type, after which he randomly chooses P"actual" participants in the game. In this case, the representations of agents, generally speaking, are not necessarily consistent (although they are common knowledge). Note that this interpretation is called playing Selten(R. Zelgen - Nobel Prize in Economics 1994, together with J. Nash and J. Harshanyi).
Now consider a situation where conditional distributions are not necessarily common knowledge. It is convenient to describe it as follows. Let agents' payoffs depend on their actions and on some parameter in e 0 (“states of nature”, which can also be interpreted as a set of types of agents), the value of which is not common knowledge, i.e., the objective function of the /th agent has the form f i (0,x x ,...,x n): 0 x X'- ""L 1, /" e N. As was noted in the second chapter of this work, the agent's choice of his strategy is logically preceded by informational reflection - the agent's thoughts about what each agent knows (assumes) about the parameter 0, as well as about the assumptions of other agents, etc. Thus, we come to the concept the agent's awareness structure, which reflects his awareness of the unknown parameter, the representations of other agents, etc.
Within the framework of probabilistic awareness (representations of agents include the following components: a probabilistic distribution on a set of states of nature; a probabilistic distribution on a set of states of nature and distributions on a set of states of nature that characterize the representations of other agents, etc.), a universal space of possible mutual representations (universal beliefs space). At the same time, the game is formally reduced to a kind of "universal" Bayesian game, in which the agent's type is his entire structure of awareness. However, the proposed construction is so cumbersome that it is apparently impossible to find a solution to the "universal" Bayesian game in the general case.
In this section, we will confine ourselves to considering two-person games, where agents' representations are given by a point structure of awareness (agents have well-defined ideas about the value of an indefinite parameter; about what the opponent's (also well-defined) representations are, etc.) Taking into account These simplifications, finding the Bayes-Nash equilibrium is reduced to solving a system of two relations that define two functions, each of which depends on a countable number of variables (see below).
So, let two agents with objective functions participate in the game
and the functions f and many X b 0 are common knowledge. The first agent has the following representations: the undefined parameter is equal to 0 e 0; the second agent believes that the undefined parameter is equal to in 2 e 0; the second agent thinks that the first agent thinks that the undefined parameter is in 2 e 0, etc. Thus, the point structure of awareness of the first agent /, is given by an infinite sequence of elements of the set 0; let, similarly, the second agent also has a point structure of awareness 1 2:
Let us now look at the reflexive game (2)-(3) from the "Bayesian" point of view. The agent's type in this case is its awareness structure /, /=1, 2. To find the Bayes-Nash equilibrium, it is necessary to find the equilibrium actions of agents of all possible types, and not just some fixed types (3).
It is easy to see what the distributions F,(-|-) will be in this case from the definition of equilibrium (1). If, for example, the type of the first agent 1={6, 0 !2 , 0w, ...), then the distribution Fi(-|/i) assigns the probability 1 type of opponent / 2 =(0 | 2 , 012b 0W2, ) and the probability 0 for other types. Accordingly, if the type of the second agent ^2 = (02> $2b Fig*)> then the distribution F 2 (-|/ 2) assigns probability 1 to the opponent 1=(in 2 , 0 212 , 02:2i ) and probability 0 for other types.
To simplify the notation, we will use the following notation:
Let us also introduce the notation
In these notations point the Bayes-Nash equilibrium (1) is written as a pair of functions ((pi-), i//(-)) satisfying the conditions
Note that within the point structure of awareness, the 1st agent is sure that the value of the indefinite parameter is 0 (regardless of the opponent's ideas).
Thus, to find equilibrium, it is necessary to solve the system of functional equations (4) to determine the functions (R(-) and!//( ), each of which depends on a countable number of variables.
Possible structures of awareness may have a finite or infinite depth. Let us show that the application of the Bayes-Nash equilibrium concept to agents with an infinite depth awareness structure gives a paradoxical result - any admissible action is equilibrium for them.
Let us define the concept of finiteness of the depth of the awareness structure in relation to the case of a game with two participants, when the awareness structure of each of them is an infinite sequence of elements from 0.
Let the sequence T= (t j) " =[ elements from 0 and a non-negative integer to. Subsequence (o k (T) = (t t) /=i+1
we will call k-ending sequences T.
We will say that the sequence T It has endless depth if for any P there will be k>n such that the sequence with to (T) does not match (meaning the usual element-wise match) with any of the sequences in the set a>u(T)=T, (0 (T),..., (o n (T). Otherwise, the sequence T It has final depth.
In other words, a sequence of finite depth has a finite number of pairwise distinct endings, while a sequence of infinite depth has an infinite number of them. For example, the sequence (1, 2, 3, 4, 5, ...) has infinite depth, while the sequence (1, 2, 3, 2, 3, 2, 3, ...) has finite depth.
Consider the game (2) in which the objective functions f, f2 and many X, X 2, 0 have the following property:
(5) for any A" | e X, x 2 e X 2, in e 0 sets
Conditions (5) mean that for any in e© and any action Xi e X the second agent has at least one best answer and, in turn, the action itself X is the best response to some action of the second agent; likewise, any action
X 2 G X 2 .
It turns out that under conditions (5) in game (2) any the action of an agent with an infinite depth awareness structure is equilibrium (i.e., it is a component of some equilibrium (4)). The ego is true for both agents; for definiteness, we formulate and prove the assertion for the first one.
Statement 2.10.1. Let the game (2) in which conditions (5) be satisfied, have at least one point Bayes-Nash equilibrium (4). Then for any information structure of infinite depth 1 and any % e X there is an equilibrium (*,*( ) > x*(-)), in which x*(/,) =x-
The idea of the proof is to construct the corresponding equilibrium constructively. Let us fix an arbitrary equilibrium (1. By virtue of conditions (4), the value of the function φ ( ) took on the structure 1 meaning X-
We preface the proof of Assertion 2.10.1 with four lemmas, for the formulation of which we introduce the notation: if p=(p,...,/>„) is finite, and T=(/.)", - an infinite sequence of elements
from 0, then pT= 0, h, ...)
Lemma 2.10.1. If the sequence T has infinite depth, but for any finite sequence R and any to subsequence rso k (T) also has infinite depth.
Proof. Because the T has infinite depth, it has an infinite number of pairwise different endings. When moving from T to s k (t) their number is reduced by no more than to, still remaining infinite. When moving from with to (T) to ry to (T) the number of pairwise distinct endings obviously does not decrease.
Lemma 2.10.2. Let the sequence T represent in the form T=rrr where R - some non-empty finite sequence. Then T has finite depth.
Proof. Let R has the form p=(p, Then the elements of the sequence T related by the relations t i+nk = t, for all integers / > 1 and to > 0. Take an arbitrary y-ending, y > P. Number j uniquely representable in the form j = i + p k, where /e(1, ..., "), A" > 0. It is easy to show that a>(T) = (o,(T) for any whole m> 0 running = t i+ „ k+m =
Given the arbitrariness j we have shown that the sequence T no more P pairwise distinct endings, i.e., its depth is finite.
Lemma 2.10.3. Let for the sequence T the identity T = p T, where R is some non-empty finite sequence. Then T has finite depth.
Proof. Let p =(/? b ...,R"). We have:
T=r T=rr T=rrr T=rrrr T=... . Thus, for any integer k> 0 fragment (/„*+, ..., /„*+„) matches (p b That's why
T represent in the form T = prr... and, according to Lemma 2.10.2, has finite depth.
Lemma 2.10.4. Let the sequence T the identity p T = q T, where R and q are some non-identical non-empty finite sequences. Then T has finite depth.
Proof. Let R= (/;, . and q = (qb ..., qk). If a n = k, th, obviously, identity pT=q T cannot be executed. Therefore, consider the case pFc. Let for definiteness n > k. Then p = (q u ..., q k ,p k+ , ...,R"), and from the condition pT=q T follows that d T \u003d T, where d = (j) k+ 1 , ...,p p). Applying Lemma 2.10.3, we get that the depth of the sequence T finite.
Proof of Statement 2.Yu.L. Let there be an arbitrary structure of information awareness of the first agent of infinite depth - for uniformity with Lemmas 2.10-2L0.4, we will denote it not /, but T \u003d (t, t 2,. By the condition of the assertion, there is at least one pair of functions!//( )) satisfying relations (4); fix any of these pairs. We set the value of the function f( ) on the sequence T equal
X". φ(T) = x(hereinafter, for "newly defined" functions we will use the notation f( ) and f( )) Substituting T as function argument f( ) in relations (4), we obtain that the value f(t) = x is related (due to (4)) with the values of the function f( ) on the sequence (0 (T), and also on all such sequences 7”,
FOR WHICH CO(T')= T.
We choose the values of the function f( ) on these sequences in such a way that conditions (4) are satisfied:
where t e Q; from (5) it follows that the ego can be made. If the set BR"(t,x) or BR2(t,x) contains more than one element, take any of them.
p(* 3 ,/ 4 ,...) € BR 2 "(t 2, a, substituting (t, t2, t2,...), choose
Continuing to substitute the already obtained values into relations (4), we can successively determine the values of the function f( ) on all sequences of the form
where (t + k)- odd, and function values f(?) on sequences of the form (6) with even (t + k). Further, we will assume that in (6) at t> 1 in progress Ф t m ., - then the representation in the form (6) is
unambiguous.
The algorithm for determining the value of functions on sequences of the form (6) consists of two stages. At the first stage, we assume f(T)=x and determine the values of the corresponding functions on the sequences w,n(r) = ( t„„ t m+ 1, ...), m> 1 (i.e. at k= 0) by alternately applying the mappings DD, 1 and 5/?, 1 .
At the second stage, to determine the value of the corresponding functions on sequences (6) with to > 1 we proceed from the value determined at the first stage on the sequence (t„„ t„,+ 1, ...), applying alternately the mappings BR and BR2.
According to Lemma 1, all sequences of the form (6) have infinite depth. According to Lemma 4, they are all pairwise distinct (if any two sequences of the form (6) coincided, this would contradict the infinity of depth). Therefore, determining the values of the functions f( ) and f( ), we do not risk assigning different function values to the same argument.
Thus, we have determined the values of the functions f( ) and f( ) on sequences of the form (6) in such a way that these functions still satisfy conditions (4) (i.e., they are a point Bayes-Nash equilibrium) and, moreover, f(T) =%. Assertion 2. K). 1 is proven.
So, the notion of point Bayes-Nash equilibrium was introduced above. It is proved that if additional conditions (5) are met, any admissible action of an agent with an infinite depth awareness structure is an equilibrium one. (All considerations were carried out for a game with two participants, however, it can be hypothesized that the result obtained can be generalized to the case of a game with an arbitrary number of participants.) This circumstance, apparently, indicates the inexpediency of considering structures of infinite depth as in terms of information equilibrium , and in terms of the Bayes-Nash equilibrium.
More generally, it can be noted that the proved statement is an argument (and not the only one, see, for example, sections 2.6 and 3.2) in favor of the inevitable limitation of the rank of information reflection of decision-making subjects.
Polina Astanakulova
Games for children 5–7 years old. Reflective circles "Mystery of my Self"
GAMES FOR CHILDREN 5-7 years old
REFLEXIVE CIRCLES
« THE SECRET OF MY SELF»
"Me and Others".
Target:
1. Develop self-confidence, the ability to express your opinion, the ability to listen carefully to your comrades.
2. Develop imagination.
3. Cultivate a friendly attitude towards each other
Material: A ball of thread, calm music.
Content: Children in circle. In the hands of the teacher is a ball of thread. caregiver: Let's find out what you love the most. Music sounds and the teacher says that I like to walk in the forest. Then he passes the ball to the child and everyone expresses his opinion, then the ball returns to the teacher. It turned out such a cobweb. The web wove us into a single whole. Now we are one with you. It is very thin and can break at any moment. So let's make sure that no one can ever quarrel with each other and break our friendship. Children close their eyes and imagine that they are one (the cobweb is wound into a ball).
"I am through the eyes of others".
Target: To give children an idea of individuality. The uniqueness of each of them, develop self-confidence, form the ability to accept a different point of view.
Material: pebble, rugs.
With words: "I'm giving you a stone because you..."
Outcome: with the help of a pebble, you said a lot of good and good things.
« The secret of my "I"» .
Target: Create a trusting environment in the group that allows children to express their feelings and talk about them, develop empathic communication skills, the ability to accept and listen to another person; develop the ability to understand yourself.
Material: candlestick with candles, matches, mirror, classical music.
The queen took out a magic mirror and ordered to him: “My light is a mirror, tell me, but tell the whole truth. Am I sweeter than everyone in the world, all blush and whiter? The teacher shows the children "magic mirror" and He speaks: I also have a magic mirror with which we can also learn a lot of interesting things about each other and answer question: "Who am I?". Let's look at the flame of a candle. It will help us to remember feelings - successes and failures. Music sounds and the teacher talks about himself, then the children talk. So we talked about our advantages and disadvantages and we can correct them. Let's take better care of each other. The children join hands and blow out the candle.
"Me and my emotions".
Target: Learn children talk about your feelings, develop the ability to identify emotions from schematic images, enrich vocabulary children.
Material: pictogram, mat, music.
Content: Children sit in circles on rugs. In the center of the card with the image of different shades of mood. The teacher offers to take the cards that best suit your mood. After the children take a suitable card for themselves. The teacher makes a conclusion about the mood children - sad, funny, thoughtful. What do you need to improve your mood? Let's laugh and forget about the bad mood.
"Me and Others".
Target: to form a friendly attitude towards each other,
To develop in children the ability to express their attitude towards others, (if necessary critically, but tactfully.)
Material: a ball of thread, calm music.
Content: Children in circle. The teacher has a ball of thread in his hands. caregiver A: You've been friends for many years and you all know each other. You are all different, you know each other's strengths and weaknesses. And what could you wish each other to become better? Music sounds, children say wishes to each other. The teacher says a wish to a child sitting next to him (example: so that he would cry less and play more with children.) Then the adult passes the ball to the child (the child says a wish to the person sitting next to him) etc., then the ball returns to the teacher. Children close their eyes and imagine that they are one.
"World of My Fantasy".
Target: Develop imagination, looseness, communication skills, develop a friendly attitude towards each other.
Material: a high chair for each child, a flower - a seven-flower.
Fly, fly, petal,
Through the west to the east
Through the north, through the south,
Come back by doing a circle,
As soon as you touch the ground
To be in my opinion led!
caregiver: Imagine that there is a magician who will fulfill any wishes. To do this, you need to tear off one petal and make a wish and tell about your dream. “Children take turns tearing off the petals and telling what they would like”.
caregiver: Children, what wish did you like the most?
Everyone had different desires, some about themselves, for others they are connected with friends, with parents. But all your wishes will surely come true.
"How can I change the world for the better?"
Target: Develop at children's imagination, the ability to listen to the opinion of another, to take a different point of view, different from one's own, to form group cohesion.
Material: "Magic" glasses.
Content: children sit in circle. The teacher shows "Magic" glasses: “The one who puts them on will see only the good in other people, even what is not always immediately noticeable. Each of you will try on glasses and examine the others. Children take turns putting on glasses and calling each other's advantages. caregiver: “And now we will put on glasses again and look at the world with different eyes. What would you like to change in the world to make it a better place? (Children answer)
It all helps us to see something good in others.
"What is joy?"
Target: To develop the ability to adequately express one's emotional state, to understand the emotional state of another person.
Material: Photos of joyful faces children, pictogram "joy", sun, red felt-tip pen.
caregiver:
What feeling is depicted on them? (Smile)
What needs to be done for this? (smile)
Say hello to each other. Each child turns to the friend on the right, calls him by name and says that he is glad to see him.
caregiver: Now tell me, what is joy? finish sentence: "I'm glad when...". (Children complete sentences). The teacher writes down wishes on pieces of paper and attaches them to the rays. Everyone has their own joy, but it is transmitted to each other.
Which "I"»
Target: creating a positive emotional mood, forms a group and increases personal self-esteem.
Material: mirror.
What color are the eyes?
What are they (large, small);
What color is the hair?
What are they (long, short, straight, wavy);
What shape is the face (round, oval).
"My name"
Target: the game helps to remember the names of your comrades, calls positive emotions and creates a sense of group unity.
Content: children sit in circle. The host chooses one child, the rest come up with affectionate derivatives on his behalf. Then the child says what name he was most pleased to hear. So they come up with names for each child. Further, the presenter talks about the fact that names grow with children. “When you grow up, your name will also grow and become full, you will be called by name and patronymic. Word "patronymic" came from the word "father", it is given by the name of the father. Children give their first and last name.
"Do as I do"
Target
"Understand me"
Target: development of imagination, expressive movements, group cohesion.
"I'm in the future"
Target: development of group cohesion, imagination.
"We are different"
Target: the game makes you feel your importance, causes positive emotions, increases self-esteem.
Which one of us is the tallest?
Who among us is the lowest?
Which one of us has the darkest (light) hair?
Who has a bow, etc.
The host sums up that we are all different, but all are very good, interesting and most importantly - we are together!
To use the preview, create an account for yourself ( account) Google and sign in: https://accounts.google.com
Preview:
Final report on the work done on the implementation of the "Reflexive circle" plan in the framework of socialization
Reflection is a person's reflection aimed at analyzing himself (self-analysis) - his own states, his actions and past events.(PHOTOGRAPHY FROM SPACE)
"Reflexive circle" is a technology that allows you to develop the speech of preschoolers, the thoughts of children. The circle contributes to the improvement of speech as a means of communication, helps children make assumptions, draw the simplest conclusions.
On daily reflective circles in groups preschool age The teacher asks questions, which the children actively answer.
(A PHOTO)
During the daily reflective circles throughout the year, the children learned to listen carefully to the teacher and their peers, not to interrupt each other.
(A PHOTO)
Children have learned to use the rules that are shown in the pictograms and are in each group at the level of the children's eyes.
(PHOTOS of pictograms)
Beginning with junior group A "reflective circle" is held every day before breakfast with all the children present in the group. The purpose of this circle is to discuss plans for the day or any group problems. If circumstances require it, for example, some event has occurred in the group, then the “reflexive circle” can be carried out again immediately after the incident.
The circle is held in the same place, so that in the future the children will get used to discussing their problems in a circle without the presence of a teacher, in this case the circles were held in a group on the carpet. For effective discussion during the circles, we use a candle, which is placed in the center of the circle, and any object that the children pass to each other during the answers to questions, which helps the children concentrate on listening to the answers and not interrupt each other.
Reflective circles are also held after club hours. On these circles, you can find out and understand what the children liked and what they did not like.during club hours.
(PHOTO FROM SPACE AND PHOTO OF CIRCLES)
In addition to the planned ones, the topics of the "Circles of reflection" were determined by the teacher according to the circumstances, for example, if some event occurred in the group.
As a result, by the end of the school year, many children have mastered the skills of coherent speech, the ability to express their thoughts. The skills to listen to each other have been formed. Most children want to express their feelings and experiences.
September
Situation of the month "My Kindergarten»
p/p | Members | the date holding | |
4.09.2017 | Who do we call friends? What friend do you dream of? |
||
18.09.2017 | What color is friendship? |
||
middle groups | 11.09.2017 | Who would I like to be friends with in a group? How do we share toys? |
|
25.09.2017 | Who is an educator? |
||
October Situation of the month "My Motherland" |
|||
Senior and preparatory groups | 4.10.2017 | How well do I know my city? Why do I love my city? |
|
18.10.2017 | |||
31.10.2017 | Playground in my city. What to do on the weekend? Favorite place in Moscow of my parents. And why? |
||
middle groups | 11.10.2017 | What about in our yard? Playground in my city. |
|
25.10.2017 | Where do I go with my parents? |
||
November
Situation of the month "I am a citizen of the globe"
p/p | Members | the date holding | |
Senior and preparatory groups | 8.11.2017 | What countries do I know? Which country would you like to visit? |
|
22.11.2017 | How to behave when meeting with a foreigner? |
||
middle groups | 15.11.2017 | The country where I live. |
|
29.11.2017 | My favorite songs, games, cartoons. Dreamland. |
2017-18 academic year of the year)
Situation of the month New Year. Magic Gifts»
Senior and preparatory groups | 6.12.2017 | How and with what can you decorate a Christmas tree for the New Year? My New Year's wish. What is a miracle? |
|
20.12.2017 | How should you behave at matinees? How to organize your leisure time? |
||
10.01.2018 | How to help birds in winter? |
||
Junior and middle groups | 6.12.2017 | How and with what can you decorate a Christmas tree for the New Year? My New Year's wish. |
|
20.12.2017 | How should you behave at matinees? |
2018 academic year of the year)
Situation of the month "Boys and Girls"
p/p | Members | the date holding | |
Senior and preparatory groups | 24.01.2018 | Who is this girl? Who is this boy? Distinguishing features. |
|
7.02.2018 | What influences our mood? |
||
middle groups | 31.01.2018 | Why do we eat? |
|
14.01.2018 | What good deeds can be done towards boys? What kind deeds can be done towards girls? |
||
2018 academic year of the year) Situation of the month “My family. My roots" |
|||
Senior and preparatory groups | 21.02.2018 | What is family? |
|
28.02.2018 | Why do I love my family? |
||
7.03.2018 | Who are the parents? |
||
middle groups | 28.02.2018 | What does friendly family mean? |
|
14.03.2018 | Who lives with you at home? |
||
2018 academic year of the year)
Situation of the month "Spring is red"
p/p | Members | the date holding | ||
Senior and preparatory groups | 21.03.2018 | What changes occur in nature in spring? |
||
4.04.2018 | What happens to trees in spring? |
|||
middle groups | Senior and preparatory groups | 10.04.2018 | What do we know about space? |
|
18.04.2018 | What do we know about planet Earth? |
|||
middle groups | 11.04.2018 | Who is the first astronaut? |
||
25.04.2018 | The planet we live on. 8.05.2018 | The great holiday "Victory Day". What is our Motherland - Russia? |
||
23.05.2018 | What is our Motherland - Russia? |
|||
middle groups | 2.05.2018 | What do you know about the Great Victory holiday? |
||
16.05.2018 | Who are we the inhabitants of the country of Russia? |
The result of the "Reflexive Circles" for the year:
Children are able to communicate politely with each other and with surrounding adults. They are able to conduct a dialogue, while using various means of expression. Children listen carefully and understand each other.
