Reflexive games for children. Reflexive games and Bayesian games. Development of group rules of conduct

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Final report on the work done on the implementation of the "Reflexive Circle" plan in the framework of socialization

Reflection is a person's thinking aimed at analyzing himself (introspection) - his own states, his actions and past events.(PHOTO 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 to make assumptions, to draw the simplest conclusions.

In daily reflective circles in preschool groups, the educator asks questions that the children actively answer.

(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.

(PHOTO)

The children learned to use the rules depicted in the pictograms and are in each group at the level of the children's eyes.

(PHOTOS of pictograms)

Starting with the younger group, a “reflective circle” is held every day before breakfast with all the children in the group. The purpose of this circle is to discuss plans for the day or any problems of the group. If circumstances require it, for example, an 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 children are 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 children pass to each other while answering questions, which helps children to concentrate on listening to answers and not interrupt each other.

Reflexive circles are also held after club hours. In these circles you can find out and understand what the children liked and what they didn’t like.during the club hour.

(PHOTOS FROM SPACE AND PHOTOS 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 an 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. Formed skills to listen to each other. Most children want to express their feelings and experiences.

September

Situation of the month "My kindergarten"

p / p	Participants	date holding
		4.09.2017	Who do we call friends? What friend are you dreaming of?
		18.09.2017	What color is friendship?
	Medium groups	11.09.2017	Who would I like to be friends with in a group? How do we share toys?
		25.09.2017	What 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? My parents' favorite place in Moscow. And why?
	Medium groups	11.10.2017	And 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 resident of the Globe"

p / p	Participants	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 a foreigner?
	Medium 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 what can you decorate a Christmas tree in 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?
	Younger and middle groups	6.12.2017	How and what can you decorate a Christmas tree in 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	Participants	date holding
	Senior and preparatory groups	24.01.2018	Who is this girl? Who is a boy? Difference of signs.
		7.02.2018	What influences our mood?
	Medium groups	31.01.2018	Why do we eat?
		14.01.2018	What kind things can you do towards boys? What kind things can you do to 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	What do parents do?
	Medium groups	28.02.2018	What does a friendly family mean?
		14.03.2018	Who lives at home with you?

2018 academic year of the year)

Situation of the month "Spring is red"

p / p	Participants	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?
	Medium groups	Senior and preparatory groups	10.04.2018	What do we know about space?
			18.04.2018	What do we know about planet Earth?
	Medium groups	11.04.2018	Who is the first astronaut?
		25.04.2018	The planet we live on.8.05.2018	Great holiday "Victory Day". What is our Motherland - Russia?
23.05.2018	What is our Motherland - Russia?
	Medium groups	2.05.2018	What do you know about the Great Victory holiday?
		16.05.2018	Who are we residents of the country of Russia?

The result of "Reflexive circles" for the year:

Children know how to communicate politely with each other and with the surrounding adults. They know how to conduct a dialogue, while using various means of expression. Children listen to each other carefully and understand.

Along with reflexive games, a possible method of game-theoretic modeling under conditions of incomplete information is bayesian games, proposed in the late 60s of the XX century. J. Harshani. In Bayesian games, all private (i.e., not shared knowledge) information available to an agent at the time of his choice of 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 ’\u003d J - [ X x admissible vectors of actions of the agent

• -set of target functions /: R'x X’-\u003e 9? 1 (the objective function of the agent generally depends on the types and actions of all agents);
• - representations F, (- | r,) e D (/? _,), / "e N, agents (here by /? _, denotes the set of all possible sets of types of all agents, except for the / th, R.j \u003d P R t, and D (/? _,) denotes the set

in all possible probability distributions on /? _,). The Bayesian game solution is bayes-Nash equilibrium,defined as a set of strategies of agents of the form x*: R, -\u003e 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 the j-th. We emphasize that in a 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. Harshani's model can be interpreted in different ways (see). According to one interpretation, all agents know the prior distribution of types F (r) f D (R ') and, having recognized their own type, they calculate the conditional distribution from it using the Bayes formula Fj (r.i| d,). In this case, the representations of agents (F, (- | -)), sW are called agreed (and, in particular, are common knowledge - each agent can compute them, knows what others can do, etc.).

Another interpretation is as follows. Let there be a certain set of potential participants in the game of all possible types. Each such "potential" agent chooses his strategy depending on his type, after which he is randomly selected p "Actual" participants in the game. In this case, the representations of the agents, generally speaking, are not necessarily consistent (although they are common knowledge). Note that this interpretation is called selten's game (R. Selgen - 1994 Nobel Prize Laureate in Economics, together with J. Nash and J. Harshany).

Now consider a situation where conditional allocations are not necessarily common knowledge. It is convenient to describe it as follows. Let the payoffs of agents depend on their actions and on some parameter in e 0 ("state of nature", which can also be interpreted as a set of types of agents), the meaning of which is not common knowledge, that is, the objective function of the ith agent has the form f i (0, x x, ..., x n): 0 x X ' - "" L 1, / "e N. As noted in the second chapter of this work, the agent's choice of his strategy is logically preceded by information 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 structure of the agent's awareness, reflecting his awareness of an unknown parameter, of the views of other agents, etc.

Within the framework of probabilistic awareness (representations of agents include the following components: probabilistic distribution on the set of states of nature; probabilistic distribution on the set of states of nature and distributions on the set of states of nature that characterize the representations of other agents, etc.), a universal space of possible mutual representations (universal beliefs space). In this case, the game is formally reduced to a kind of "universal" Bayesian game, in which the type of agent is his entire structure of information. However, the proposed construction is so cumbersome that it seems impossible to find a solution to the “universal” Bayesian game in the general case.

In this section, we restrict ourselves to considering games of two persons, while the representations of the agents are specified by the point structure of awareness (agents have quite definite ideas about the value of an indefinite parameter; about what are the representations (also quite definite) of the opponent, etc.) From these simplifications, finding the Bayes-Nash equilibrium is reduced to solving a system of two relations defining 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 is common knowledge. The first agent has the following representations: undefined parameter is 0 e 0; the second agent considers the undefined parameter to be at 2 e 0; the second agent thinks that the first agent considers the undefined parameter to be at 2 e 0, etc. Thus, the point structure of the 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's now look at reflexive play (2) - (3) from the Bayesian point of view. The type of the agent in this case is its awareness structure /, / \u003d 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 will be in this case the distributions F, (- | -) 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, 012u 0N2,) and the probability O for the rest of the types. Accordingly, if the type of the second agent ^ 2 \u003d (02\u003e $ 2b Fig *)\u003e then the distribution F 2 (- | / 2) assigns probability 1 to the type of opponent 1 \u003d (in 2, 0 212, 02: 2i) and probability 0 for the rest of the types.

To simplify the notation, we will use the following notation:

We also introduce the notation

In these designations point Bayes-Nash equilibrium (1) is written as a pair of functions ((pi-), i // (-)) satisfying the conditions

Note that within the framework of the point structure of awareness, the 1st agent is sure that the value of the undefined 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 awareness structures can be of finite or infinite depth. Let us show that the application of the Bayes-Nash equilibrium concept to agents with an infinite depth of information structure gives a paradoxical result - for them any admissible action is equilibrium.

Let us define the concept of finiteness of the depth of the structure of information in relation to the case of a game with two participants, when the structure of information of each of them is an infinite sequence of elements from 0.

Given a sequence T \u003d (t j) " =[ elements of 0 and a non-negative integer to. Sequence (о к (Т) \u003d (t t ) / \u003d i + 1

will call to-end sequences T.

We will say that the sequence T It has endless depth, if for any p there will be k\u003e n such that the sequence co k (T) does not match (meaning a normal element-wise match) with any of the set sequences a\u003e u (T) \u003d T, (0 (T), ..., (o n (T). Otherwise the sequence Tit has final depth.

In other words, a sequence of finite depth has a finite number of pairwise different endings, while a sequence of infinite depth has infinitely many of them. For example, the sequence (1, 2, 3, 4, 5, ...) has infinite depth, and the sequence (1, 2, 3, 2, 3, 2, 3, ...) is finite.

Consider game (2), in which the objective functions f, f 2 and many X, X 2, 0 have the following properties:

(5) for any Л "| e X, x 2 e X 2, in e 0 sets

Conditions (5) mean that for any in e © and any action Xi е 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; similarly, any action

X 2 G X 2.

It turns out that under conditions (5) in game (2) anythe action of an agent with an information structure of infinite depth is equilibrium (ie, is a component of some equilibrium (4)). The ego is true for both agents; to be definite, we formulate and prove the assertion for the first.

Statement 2.10.1. Suppose that in the game (2), in which conditions (5) are satisfied, there exists at least one point Bayes-Nash equilibrium (4). Then for any structure of information of infinite depth 1 and any % e X there is an equilibrium (*, * ()\u003e x * (-)), in which x * (/,) \u003d x-

The idea of \u200b\u200bthe proof is to constructively construct the corresponding equilibrium. We fix an arbitrary equilibrium (1. By virtue of conditions (4), the value of the function ф () took on the structure 1 value x-

We precede the proof of Statement 2.10.1 by four lemmas, for the formulation of which we introduce the notation: if p \u003d (p, ..., /\u003e „) is finite, and T \u003d (/.) ", - an infinite sequence of elements

from 0, then pT \u003d 0, h, ...)

Lemma 2.10.1. If the sequence T has infinite depth, go for any finite sequence r and any tosequence pso k (t) also has infinite depth.

Evidence. Insofar as T has infinite depth, it has an infinite set of pairwise different endings. When moving from T to s k (t) their number decreases by no more than to, still remaining infinite. When moving from co k (T) to ry k (T) the number of pairwise distinct endings obviously does not decrease.

Lemma 2.10.2. Let the sequence T representable as T \u003d pprwhere r - some non-empty finite sequence. Then T has finite depth.

Evidence. Let be r has the form p \u003d (p, Then the elements of the sequence T are related by the relations t i + nk \u003d t, for all integers /\u003e 1 and to\u003e 0. Take an arbitrary y-ending, y\u003e p. Number jcan be uniquely represented in the form j \u003d i + n k, where / e (1, ..., "), A"\u003e 0. It is easy to show that a\u003e (T) \u003d (o, (T) for any whole m \u003e 0 is executed \u003d t i + „k + m \u003d

Given the arbitrariness j we have shown that the sequence T no more p pairwise different endings, i.e., its depth is finite.

Lemma 2.10.3. Let for the sequence T the identity holds T \u003d p T, Where r - some non-empty finite sequence. Then T has finite depth.

Evidence. Let be p \u003d (/? b ...,R"). We have:

T \u003d p T \u003d pp T \u003d ppr T \u003d pppp T \u003d .... Thus, for any whole to\u003e 0 fragment (/ „* +, ..., /„ * + „) matches (p b therefore

T representable as T \u003d ppr ... and, according to Lemma 2.10.2, has finite depth.

Lemma 2.10.4. Let for the sequence T the identity holds p T \u003d q T, Where r and q - some non-identical nonempty finite sequences. Then T has finite depth.

Evidence. Let be r \u003d (/ ;,. and q \u003d (q b ..., q k). If a n \u003d k,th, obviously, the identity pT \u003d q T cannot be executed. Therefore, consider the case pfk. Let for definiteness n\u003e k. Then p \u003d (q u ..., q k, p k + , ...,R"), and from the condition pT \u003d q Tfollows that d T \u003d T,where d \u003d (j) k + 1, ..., p n). Applying Lemma 2.10.3, we obtain that the depth of the sequence T is finite.

Proof of statement 2.YUL. Let there be an arbitrary information structure of the first agent of infinite depth - for consistency with Lemmas 2.10L-2L0.4, we will denote it not by /, but Т \u003d (t, t 2, ... By the condition of the statement, there is at least one pair of functions! // ()) satisfying relations (4); fix any of these pairs. We put the value of the function f () on the sequence T equal

X ". F (T) \u003d x (hereinafter, for "newly defined" functions, we will use the notation f () and f()) Substituting T as a function argument f () in relation (4), we obtain that the value φ (T) \u003d x is related (due to (4)) with the values \u200b\u200bof the function f () on the sequence (0 (T), and also on all such sequences 7 ",

FOR WHICH CO (T ') \u003d T.

Let us choose the values \u200b\u200bof the function f () on these sequences so that conditions (4) are satisfied:

where t e Q; it follows from (5) that the ego can be done. If the set BR "(t, x) or BR 2 (t, x) contains more than one element, take any of them.

p (* 3, / 4, ...) € BR 2 "(t 2, and, substituting (t, t 2, t 2, ...), choose

Continuing to substitute the already obtained values \u200b\u200binto relations (4), one can sequentially determine the values \u200b\u200bof 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). In what follows, we will assume that in (6) for t\u003e 1 running Ф t m., - then the representation in the form (6) is

unambiguous.

The algorithm for determining the values \u200b\u200bof functions on sequences of the form (6) consists of two stages. At the first stage, we put φ (T) \u003d x and determine the values \u200b\u200bof the corresponding functions on the sequences ω, n (Γ) \u003d ( t „„ t m + 1, ...), m \u003e 1 (i.e., for k \u003d 0), alternately applying the mappings DD, 1 and 5 / ?, 1.

At the second stage, to determine the values \u200b\u200bof the corresponding functions on sequences (6) at to\u003e 1 we proceed from the value determined at the first stage in the sequence (t „„ t „, +1, ...), applying alternately the mappings BR and BR 2.

According to Lemma 1, all sequences of the form (6) have infinite depth. According to Lemma 4, they are all pairwise different (if any two sequences of the form (6) coincided, the ego would contradict the infinity of depth). Therefore, determining the values \u200b\u200bof the functions f () and f (), we do not run the risk of assigning different values \u200b\u200bto the function to the same argument.

Thus, we have determined the values \u200b\u200bof the functions f () and f () on sequences of the form (6) in such a way that these functions still satisfy conditions (4) (that is, they are a Bayes-Nash point equilibrium) and, at the same time, f (T) \u003d%. Statement 2. K). 1 is proven.

So, the concept of Bayes-Nash point equilibrium was introduced above. It has been proved that under additional conditions (5), any admissible action of an agent with an infinite depth information structure is in equilibrium. (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 in terms of information equilibrium and in terms of the Bayes-Nash equilibrium.

More generally, it can be noted that the proven statement is an argument (and not the only one, see, for example, Sections 2.6 and 3.2) in favor of the inevitable limited rank of information reflection of decision-making subjects.

SUMMARIZING THE CASE (REFLEXION)

"MOOD TREE"

A tree is drawn on a sheet of Whatman paper - each branch is a separate day. In the evening, a child on today's twig can draw a leaf of one of three colors. green means that the child's mood is excellent, yellow means good, red means so-so. By the end of the shift, you will have a complete picture of how it went for your children.

"THE STAR FALLED FROM THE SKY"

Children are told that when the stars are falling, one can make a wish and many people, seeing a shooting star, make the most cherished wish and it will certainly come true. The guys write on their star (cut out of cardboard) what they expect from this shift. The counselor collects all the stars and hangs them on the wall. At the end of the shift they are removed, the wishes are read and together they discuss what has come true and what has not

"SUITCASE ON THE ROAD"

You can put anything in a magic suitcase and it will remain unchanged. Everyone chooses three things that he would like to take away from the class: a good mood, a friend, a chair on which he sits.

"FINISH THE OFFER"

Today is:

My mood today:

I would like tomorrow:

Based on the statements of the children, the counselor sums up the day.

Let's say goodbye to the Adaman Islands in the Pacific Ocean.

Place your right palm under the palm of your right neighbor; and the left - on the palm of the neighbor on the left. And with the kindest and brightest wishes, with the most positive energy, we blow on the palm of the neighbor on the right.

“Envelope of revelations "

The counselor prepares an envelope in advance with a lot of questions. It is desirable that the questions are of a moral and ethical nature, such as:

what do you value most in people?

what is your biggest goal in life?

what traits of a person's character are especially unpleasant for you?

which of the famous heroes of the past (film, book) would you like to be like and why? etc.

"They meet by their clothes ..."

This stage of work arouses interest and mild excitement among the group members. In the process of searching for their "portrait" they have to read not one, but several sheets of paper, "argue" with one of the applicants for the same characteristic, and be able to defend their right to it. During the discussion, the moderator suggests answering several questions:

Are you satisfied with what is written on the piece of paper you received?

What caused the surprise, did the "discoveries" occur?

What generated the most interest during the work?

What difficulties did you experience while doing the exercise?

Horoscope

2-3 day shift.

Children are divided into groups - zodiac signs. Before the statement - a brief description of the sign. An unusual characteristic of children, a form of memorizing a personality through the allocation of unusual qualities. Can be compared by seasons of the year, eye color, etc.

Unit level is low.

Situational

(Electric chair)

One participant has his back to the audience, everyone writes notes with a brief description of this person, which are then read out by the presenter (correcting the text if it is incorrect in relation to the person).

It makes it possible to assess the behavior of a child in the detachment without ambition, offense, insults to his personal dignity.

A candle is a day gone by.

The guys sit in a circle and, passing the candle to each other, tell in turn how the day went for them and their assessment.

The candle is desire.

Passing the candle around, you can add your wishes for tomorrow to the usual assessment of the day. You can start with the words: "I would like tomorrow ..."

Cobweb.

All sit in a circle. The counselor takes a ball of thread, winds the thread on his finger and passes this ball to any child from the circle ("I would like to pass this ball to Katya, because ..."). Further, the second participant winds his thread on his finger, and passes the ball to the next, explaining his choice. And so on, until everyone is connected by one thread. You can see the connections that have arisen between the guys in the unit. Everyone can cut a thread wound around a finger as a keepsake.

"Five minutes of revelation"

The entire squadron throughout the day puts notes in a box in the form of a mailbox with questions that they would like to ask their counselors at the final "light". Counselors answer these questions on the "spark".

"Notes"

Envelopes and small notes are made. The number of envelopes is equal to the number of children in the squad. On the notes, each child writes wishes, words of gratitude, impressions of each person from the squad. These scraps are enclosed in envelopes that are creatively designed and signed. These envelopes are then creatively presented to their owners. But only the next day is it allowed to open the envelopes and read the notes.

"Post-it therapy"

Purpose. Create an environment for participants in which they can receive positive feedback from each other.

Materials Required: A large number of Post-it * s - a small colored paper with a sticky layer.

Process. Agree with the guys that they will distribute 10 Post-it * s a day to their friends. The message on the note is a personal confession, admiration for a friend's work, his active participation in the life of a class, squad, group, diligence, etc.

• I missed you when you were sick;
• Congratulations on a goal scored this weekend;
• I love your sense of humor;
• You worked very hard. I just want you to know that I noticed this.

These Post-it * s can be attached to a notebook, desk, computer, textbook, etc. Such notes are different from notes in a notebook and can be collected.

Methodology for carrying out "ARI"

"ORZ" is a frank conversation of experts, which allows children to get answers to the most intimate questions, to express their opinion in a free form. A group of children of at least 10 people is participating. All sit in a circle. It starts with the "Acquaintance" exercise, the next exercise is "Impression in a circle", where the guys share their impressions of the past week. And then the guys are invited to write their own question on the cards (you can not sign), which worries him, and hand over to the presenter. Candles are lit and the questions are laid out in a circle with the blank side of the card facing up.

The moderator chooses the first question and read it out loud. Those interested can answer it and even discuss it. The facilitator should monitor the situation, give everyone an opportunity to speak and move on to the next question. The conversation ends when all questions have been asked and answers have been received. The last questions are asked by the presenter “Did you like this method of communication or not?”, “Are you satisfied with the answers?”, “What are your wishes for the future?”.

"Color painting" (modifications of M. Luscher's test)

Purpose: identifying mood and reasons that influenced the mood through color.

Training

Prepare two posters on Whatman sheets:

a) Color - mood:

• red - enthusiastic
• orange - joyful
• yellow - calm,
• green - balanced,
• blue - sad
• purple - alarming
• black - despondency.

b) Reasons:

1. Very personal.
2. Collective evening.
3. Creative workshop;
4. Relationship with adults.
5. Successes.
6. Failures.
7. There was no freedom of action.
8. Simply tired.
9. Dissatisfaction with yourself.
10. I'm interested here.
11. I learned something new, learned something.
12. I found many friends.
13. They were poorly fed.
14. What else?

(reasons correspond to the topic of the case)

Prepare cards of different colors for writing the reasons for the mood.

Methodology

After the case, evening, camp shift, it is proposed to choose a card of the color that matches the mood and write only the numbers of the reasons.

• The predominant color of the cards determines the overall mood.
• The numbers on the cards explain the reasons for this mood.
• Provides an opportunity to reveal individuality in the mood of the participants.

"Hungry or full."

Several sessions before the end of the group's work, it makes sense to ask the participants to think about how satisfied they are with what they received in the group or thanks to it. As a group leader, you will receive specific feedback from each participant and will be able to take into account the wishes of those who are still unsatisfied with the work in the remaining time. This procedure will visually show the participants how successfully the group has worked so far. The exercise can be done in any group.

Instruction: I would like to suggest an exercise in which you can understand how happy you are with what you have received in the group so far and with what you have achieved here. I want those of you who now feel that he is already "full" to stand near the door, and those who still feel "hungry" gather at the window. Decide for yourself what suits you best, and stand in accordance with this in this or that part of the room. In this case, please, and talk until you determine your place ...

Now I would like everyone to briefly tell us what influenced his decision. At the same time, you can communicate how you feel about the decision of other group members. I suggest starting to speak out "well fed" ...

Record for yourself, if possible, any important needs expressed by the “hungry”, and finally discuss with the group what you can do together to satisfy their “hunger”.

Issues for discussion:

• Who greatly surprised me with their choice?
• How satisfied am I with the results of the group's work?
• How much time and energy would I like to devote to "hungry"?
• What else could we really do?
• What am I feeling now?

In another version of the exercise, the group members line up: the participant who feels the most "hungry" becomes at the beginning of the row, and at the end of it is the one who feels the most "satiated".

"List of events".

At the end of any team program, it is very important that the group determines what activities it intends to undertake after the work is completed.

Instructions: Announce the topic of the lesson: "What do we as a team want to do after the workshop? What do we want to do differently?"

Participants break down into fours and make a list of specific events, for example, monthly discuss the state of affairs with all employees, etc. Give them 30 minutes to do this. During the general discussion, a list of events with four subheadings is compiled and written on a large sheet of Whatman paper: "What?", "Who?", "Starting with ...", "Until ..."

Each of the proposed activities is discussed and recorded only if everyone agrees with it. After the workshop, the list of events is copied and handed over to all team members.

"I like you"

Objectives: This is a great exercise to help develop good relationships between children. Some children can easily express their emotions, for others it is a problem. In this game, all participants get a real opportunity to develop this important skill. The "web" is a great metaphor for the interconnectedness of all the children of the squad. Material: Ball of colored wool.

Instructions: Please, all sit in one common circle. I would like to invite you to take part in one very interesting game. All of us together will make one big flower web that connects us to each other. In addition, each of us can express our kind thoughts and feelings that he has for his comrades. Now I will show you how this game should proceed.

Wrap the free end of the woolen thread a couple of times around your palm and roll the ball towards one of the children. Try to choose a member who is not the most popular in the group.

You see what I have done now. I have chosen a student who should be next on the web. After we have passed the tangle to someone, we say to that student a phrase starting with the same words; "Kolya (Masha, Petya)! I like you because ..." For example, I say: "Kolya! I like you, because today, before the start of the lesson, you politely opened the door to the class for me." After listening to the words addressed to him, Kolya wraps the thread around his palm so that the "web" is more or less stretched. After that, Kolya must think and decide who to transfer the ball to. When the next student has the ball, Kolya turns to him with a phrase that begins with the same words as mine. For example: "Yana, I like you, because yesterday you helped me solve a difficult problem in mathematics." At the same time, you can talk about what this person made you happy, what you like about him, for which you would like to thank him. And so our whole game continues further and further ... Try to remember well what they will tell you when they will pass the ball. Make sure that all children get a ball during the game. Explain to the children that we love not only our own; closest friends, but also every member in the group. After all, everyone has something that is worthy of respect and love. It is very important to constantly repeat and emphasize these thoughts in a modern society filled with competition for a place in the sun. Not a single family, not a single collective can be complete and effective as long as there are "scapegoats" and "outsiders" in them. If some children have difficulty in pronouncing the initial phrase "I like you because ...", then let them replace it with the words "I liked how you ...".

Gradually, the "web" will grow and fill. The child who received the ball last begins to wind it in the opposite direction. At the same time, each child smears his part of the thread on the ball and pronounces the words he said and the name of the one who said, gave the ball back to him.

Exercise analysis:

• Is it easy for you to say nice things to other children?
• Who has said something nice to you before this game?
• Is our squad friendly enough?
• Why is every child worthy of love?
• Did anything surprise you about this game?

"Heart of the group"

Objectives: This game, due to the symbology used in it, is great for Valentine's Day or the 8th of March. But, of course, you can spend it on other days. On a large cardboard heart, something positive is written about each participant, so that everyone feels respected and worthy of love and understanding.

Material: Paper, pencils, felt-tip pens and a large heart cut out of red cardboard.

Instruction: Did you know that our group has its own heart? I want you to do something nice for each other now. Write your name on a piece of paper and fold it so that each of you can then draw lots with someone else's name. Be sure to check if you pulled out your own name, in which case you can change the piece of paper.

And now I will tell you what you are going to do now. I brought with me a large heart, which will become the heart of our class. Come up with a friendly and pleasant phrase to address the person whose name you drew by lot. Take a piece of paper and write down what you come up with on it. Maybe you will write: “I like the fact that Petya is so funny”, “Yulia always has very interesting thoughts”, “Zhenya is always ready to help.” If you like your phrase, take a felt-tip pen and write it down on red heart of the class.

Place the red heart on the table so that the children can approach it from all sides. After you have analyzed the exercise, the heart of the class can be a great decoration for a room.

Exercise analysis:

• Did you like this game?
• What do you like about atom heart?
• Was it easy to say something nice about another child?
• Do you like what is written on your heart about you?

"What I almost forgot ..."

Instruction: Before we all part, I would like to give you the opportunity to speak out what you did not have time to say, or discuss during the group work. Close your eyes for a minute and sit comfortably ...

Imagine that you are returning home and on the way you remember the group ... The faces of the participants and the situations they experienced rush through your head, and suddenly you realize that for some reason you did not do or did not express something ... You regret about this ... What is left unspoken or undone? (1 minute.)

Now open your eyes ... Now you have the opportunity to express what you did not have time before.

"Letter to the Host".

This exercise will always be useful to you as a facilitator. It will provide an opportunity to get important information about the group, maintain personal relationships with members even in large groups and show each member of the group that his opinion is interesting and important to you. In this case, it is necessary that after completing this exercise, the group receives feedback from you. Directly or indirectly comment on the messages received in the next session and make it clear to the participants that you are taking them into account. It makes sense to use this exercise only when you really want it.

Instruction: I want to ask you to write me a short letter. I often think about our group and about each of you. It may turn out that in these reflections I am somewhat not quite right. Help me to correct my idea of \u200b\u200bthe group. Write about what you think I should know….

"My life".

Everyone sits around a chair with a burning candle on it. The light is out.

Any of you can go out now and take this candle. You sit down and bring it up to your eye level. You will only look at the candle, not at those who will ask you questions. And you will be asked questions - about you. Anyone can sit on this chair - everyone who undertakes to answer any question frankly. If you are afraid of any questions and are not sure that you will answer any sincerely, you do not need to go out and sit in this chair.

It is very important for the teacher to create a thoughtful, somewhat mysterious atmosphere where empty entertainment is inappropriate.

It is worth prompting the group that the questions should not be random, but a little confessional. For example, "What time do you get up in the morning?" or "What's your favorite color?" - empty questions, they say little about the person and they can be asked under other circumstances. It is more interesting and difficult when questions such as are asked: When was the last time you cried with happiness? If you had a magic wand, what would you like to change in the outside world or in yourself? Mind? Soul? Body? What are you mad about? Who do you like here? What hopes and dreams do you have? Under what circumstances cannot you be happy? As a rule, everyone's confession should end with the question: “What do we not know about you? What is the secret of your life? "

Light "Tell me about me"

This is a fairly well-known form and always new, unpredictable content. It is good to carry out such a light during the main period of the camp change, since by this time the guys and the counselors got to know each other well enough.

To carry out the light, you can use a closed darkened room in the center of which one or several candles are burning, and the romantic atmosphere of a dying day in nature around a small fire can create comfort and coziness.

Instruction. Probably no one knows you better than yourself. And yet, each has features and abilities that you will not see in the mirror, but they are easily revealed to the people around us. Let's try to look like in a mirror into the eyes of our comrades. Each of you can choose 3 people from those sitting here and ask them one single question - the request: "Tell me about me ?!" You can't refuse a request, but you shouldn't be cunning either: be sincere and still try to take care of each other.

Spark "Bouquet of flowers"

To conduct the light, the teacher should make colorfully decorated paper flowers or petals in advance, in which questions are written about the most memorable things and events. For example, "What is your brightest childhood impression?", "Remember the most touching moment of your life?", "What book are you ready to reread endlessly?" etc. etc. This type of questions was not chosen by chance, as it allows to reveal the level of value orientations and life incentives of the participants.

In order to create the right atmosphere, the presenter tells a parable about that. What if not very attentive and mature people gather around the candle fire, then the candle quickly melts as if crying from misunderstanding and resentment. And if everything is in order and kind and attentive people have gathered around her, then the candle burns brightly and slowly, trying to warm everyone. After this legend, everyone can choose their own flower and answer the question stored in it.

To find a solution to a bimatrix game, a game model can be used in the form of the so-called reflective play , i.e. a game in which the player models the behavior of an opponent.

Consider a reflexive game using the example of the above game "competing firms" (see Table 4.3) under the assumption that the player ANDsimulates the behavior (choice) of the player IN. Corresponding game matrix G(24) is presented in table. 4.5.

Table 4.22

Bj A i				IN 4 –

Player IN(unlike Table 4.3), two more "supposed" strategies have been added - IN 3 + - answer with the same number strategy that the player chose AND, and IN 4 – - respond with the opposite strategy.

It has been proven that in a reflexive game, the winner is the player whose reflection rank is one higher than that of the opponent. If the rank of reflection differs by more than one, then the outcome of the game is not clear.

1. Practical example

Let there be a firm consisting of two departments - production (P), whose task is to produce some product, and transport (T), which must deliver the produced product to the consumer. It is known that the income of department P from the production of products in the volume of one machine is amonetary units, the costs of department T for sending one car with cargo to the consumer is equal to cmonetary units, and the cost of storing non-exported products in the volume of one machine is bmonetary units and are divided equally between departments P and T.

Let it also be known that in the period of interest (for example, for a working day), department P can produce products in the amount of 5 or 10 cars, and department T for its transportation allocate a small convoy (4 cars), a large convoy (7 cars), two small convoys (8 vehicles) or one large and one small convoys (11 vehicles).

The model of the described situation can be a bimatrix game presented in Table. 4.6.

Table 4.23

	T 1 (4)	T 2 (7)	T 3 (8)	T 4 (11)
P 1 (5 cars)	(4a - b / 2; – 4c - b / 2)	(– 5a; – 7c)	(5a; – 8c)	(5a; – 11c)
P 2 (10 cars)	(4a -3b; – 4c -3b)	(7a -1,5b; – 7c -1,5b)	(8a - b; – 8c - b)	(10a; – 11c)

It is necessary to give recommendations to the head of department P about the most profitable volume of production for him (i.e., about choosing a strategy P 1 or P 2), given that department P is interested in maximizing its income, and department T is interested in minimizing its costs.

To obtain numerical results, we take a= 10,b= 6,c\u003d 2. Then tab. 4.6 will take the form of Table 4.7.

Table 4.24

	T 1 (4)	T 2 (7)	T 3 (8)	T 4 (11)
P 1 (5 cars)
P 2 (10 cars)

Let us first use the maximin method, which orientates the head of department P to the most careful behavior. In this case, the optimal strategy is P 1, which guarantees department P income of 37 monetary units (see the last column of Table 4.7). Taking into account the interests of department T (as can be seen from Table 4.7, the minimum costs for T will be when choosing strategy T 1), it is this income that will be received by department P.

Note, however, that the choice of strategy P 1 is hardly the best for department P. So, if he chooses strategy P 2 and informs the head of department T about his choice, then, guided by the interests of his department, he will have to choose strategies T 3 or T 4, which guarantees the income of department P in 74 or 100 currency units. Moreover, it is possible to "stimulate" department T to choose strategy T 4 by sharing with it in this case a part of the income, for example, 10 currency units (in this case, the income of department P will be 90 currency units, and the costs of department T - only 12 units). It is in this way that the head of the department P.

Let's change a little the initial situation by increasing the cost of storage of not exported products: a= 10,b= 10,c\u003d 2. We get the corresponding table table. 4.8.

Table 4.25

	T 1 (4)	T 2 (7)	T 3 (8)	T 4 (11)
P 1 (5 cars)
P 2 (10 cars)

Although in this case the minimum possible income for department P when choosing strategy P 1 is 3.5 times greater than when choosing strategy P 2 (35 and 10, respectively), however, in this case, it is better to choose strategy P 2, informing about your decision the head of department T. Toth, guided by the interests of his department, will have to choose the strategy T 4 (corresponding to the minimum costs of department T), which guarantees the income of department P in 100 monetary units. Note that in this situation there is no need to "stimulate" the T department.