What game theory does. "Strategy games": How mathematics helps to understand life. Basic concepts of game theory p. 4

In each situation, we adhere to a certain strategy. This usually happens unconsciously, hence the frequent mistakes. You can avoid them if you learn to guess the actions of another person.

Take dating, for example. We all choose one main strategy: we try to hide the negative traits and show the positive ones.

Until I tell you that every evening I like to lie on the couch with a beer. I'll tell you when she gets to know me better and realizes that otherwise I'm fine.
Pavel, sofa expert

Such a strategy is, rather, not a lie, but a reticence.

Example

Imagine a situation: a man and a woman meet for several months and once. The man has a small apartment, so it is logical that we are talking about moving to the woman's apartment.

I must say that the man works as an economist. He analyzed the situation and realized that it was not profitable to refuse to rent an apartment. Now he pays little money and in the event of a breakdown, he will not find an equally good option. The woman, upon learning of this, immediately abandons the gentleman.

Where did this couple go wrong? The man, having correctly calculated the situation from an economic point of view, did not take into account the psychological factor. The woman perceived the gesture with the apartment as frivolity of intentions. But she did not think that her boyfriend is an economist, therefore, makes decisions primarily from the position of "profitable - unprofitable". Thus, this game was lost by both participants.

What to do

Consider not only your actions, but also the reactions of other people. Ask yourself more often: how can you interpret my action? Advice especially for men: explain your actions and remember that any misunderstanding is a reason for your other half to dream up. Strategic thinking is not only mathematics, but also psychology!

2. Game for 90 points

Riddles, quests, and logic will no longer be a problem after learning game theory. You will learn how to search for all the existing answer options and choose the most suitable one among them.

Example

Two students asked the professor to postpone the exam. They told a heartbreaking story about how they drove to another city for the weekend, but on the way back they had a flat tire. They had to look for help all night, so they did not get enough sleep and do not feel well. (In fact, friends celebrated the end of the session, and this exam was final and not the most difficult.)

The professor agreed. The next day, he seated the students in different classrooms and handed out a sheet of paper containing only two questions. The first was worth only 10 points, and the second - 90 and sounded like this: "Which wheel is flat?"

If we rely on logic, then the answer will be "Right front wheel": it is on the right, closer to the side of the road, most often there is any debris, which is first hit by the front tire. But don't be in a hurry.

In this situation, it is important to give not so much the correct (logical) answer as an answer that will be written on a piece of paper by a friend.

Therefore, it is obvious that both students will speculate on the assumption that the other thinks.

You can argue like this: do the students have something "in common" with one of the wheels? Perhaps a year ago they already had to change a wheel together. Or one tire is smeared with paint and both students know about it. If such a moment is found, it is this option that should be chosen. Even if the other student is not familiar with game theory, he can remember this case and point the right wheel.

What to do

In your reasoning, rely not only on logic, but also on life circumstances. Remember: not everything that is logical for you is also logical for another. Involve friends and family in thinking games more often. This will allow you to understand how people close to you think, and in the future to avoid difficult situations, as in the example above.

3. Play with yourself

Knowledge of strategy games helps you to analyze your own decisions more deeply.

Example

A certain Olga decides whether to try smoking or not.

Game tree

The figure shows the so-called game tree: it is useful to draw it every time you need to make a decision. The branches of this tree are variants of the development of events. The numbers (0, 1 and -1) are the winnings, that is, whether the player will be the winner if he chooses this or that option.

So where to start. First, you need to determine which solution will be the best and the worst. Suppose that Olga's preferred course of action is to try smoking, but not continue to do so. Let us assign a payoff of 1 to this variation (the first digit of the lower left branch). In the worst case, the girl will become addicted to smoking: we assign a payoff of -1 to this option (the first digit of the lower right branch). Thus, the tree branch with the option not to try smoking at all gets 0.

Suppose Olga decided to try smoking. What's next? Will she give up or not? This will already be decided by Future Olga, in the picture she comes into play on the "Try" branch. If she has already formed an addiction, then she will not want to quit smoking, so we set the “Continue” option to win 1 (the second digit of the lower right branch).

What do we get? The current Olga will benefit if she tries to smoke, but does not become addicted. And this, in turn, depends on Future Olga, for whom it is more profitable to smoke (she has been smoking for a long time, which means that she has an addiction, therefore, she will not want to quit). So is it worth the risk? Maybe play a draw: get a win of 0 and not try smoking at all?

What to do

It is possible to calculate a strategy not only in a game with someone, but also in a game with oneself. Try to draw a game tree and you will see if your current decision is winning.

4. Auction game

There are different types of auctions. For example, in the film "Twelve Chairs" there was a so-called English auction. Its scheme is simple: the winner is the one who offers the highest amount for the exposed lot. Usually the minimum step is set for raising the price, otherwise there are no restrictions.

Example

In the episode with the auction from "The Twelve Chairs," Ostap Bender made a strategic mistake. Following the offer of 145 rubles per lot, he immediately raised the price to two hundred.

From the point of view of game theory, Ostap should have raised the rate, but only minimally until there were no competitors left. Thus, he could save money and not get into a mess: Ostap did not have enough 30 rubles to pay the commission.

What to do

There are games like Auction that you only have to play with your head. Decide in advance on your tactics and think about the maximum amount you are willing to give per lot. Give yourself your word not to exceed the limit. This step will help you cope with the excitement if it suddenly overtakes you.

5. Playing in an impersonal market

An impersonal market is banks, insurance companies, contractors, consulates. In general, those participants in the game who do not have names and surnames. They are impersonal, but it is a mistake to believe that the rules of game theory do not apply to them.

Example

Maxim turns to the bank in the hope of getting a loan. His credit history is not perfect: two years ago, he refused to repay another loan for six months. The employee who accepts the documents says that, most likely, Maxim will not receive a loan.

Then Maxim asks for permission to bring the documents. He brings an extract from the hospital confirming that his father was seriously ill during those six months. Maxim writes a statement, where he indicates the reasons for the delay in the payment of the previous loan (the money was needed for the treatment of his father). And after a while he gets a new loan.

What to do

When you do business with impersonal players, always remember that there are personalities behind them. Figure out how to draw your opponents into the game and set your own rules.

Game theory is a new science, but it is already being studied in the best universities in the world. The textbook "Strategic Games" has been published in the publishing house "MYTH". It will come in handy if you want to learn how to analyze your every action, make informed decisions, better understand not only others, but also yourself.

In the 1930s, John and Oscar Morgenstern became the founders of an interesting new branch of mathematics called "game theory". In the 1950s, the young mathematician John Nash became interested in this area. Equilibrium theory became the topic of his dissertation, which he wrote at the age of 21. Thus was born a new one called "Nash Equilibrium", which won the Nobel Prize many years later - in 1994.

The long gap between dissertation writing and general acceptance challenged the mathematician. Genius without recognition resulted in serious mental disorders, but John Nash was able to solve this problem thanks to his excellent logical reason. His theory of "Nash equilibrium" won the Nobel Prize, and his life was adapted in the film "Beautiful mind".

Game theory at a glance

Since Nash's equilibrium theory explains the behavior of people in interaction, it is therefore worth considering the basic concepts of game theory.

Game theory studies the behavior of participants (agents) in terms of interaction with each other like a game, when the outcome depends on the decisions and behavior of several people. The participant makes decisions based on his predictions about the behavior of others, which is called a game strategy.

There is also a dominant strategy in which the participant gets the optimal result for any behavior of the other participants. This is the player's best no-lose strategy.

Prisoner's dilemma and scientific breakthrough

The prisoner's dilemma is a case of play, when the participants are forced to make rational decisions, achieving a common goal in a conflict of alternatives. The question is which of these options he will choose, realizing his personal and general interest, as well as the impossibility of obtaining both. Players seem to be trapped in harsh game conditions, which sometimes makes them think very productively.

This dilemma was explored by an American mathematician. The equilibrium he deduced became revolutionary in its own way. Especially brightly, this new thought influenced the opinion of economists about how market players make their choice, taking into account the interests of others, with close interaction and intersection of interests.

It is best to study game theory with specific examples, since this mathematical discipline itself is not dry-theoretical.

An example of a prisoner's dilemma

For example, two people committed a robbery, fell into the hands of the police and are being interrogated in separate cells. At the same time, the police officers offer each participant favorable conditions under which he will be released if he testifies against his partner. Each of the criminals has the following set of strategies to consider:

Both testify at the same time and receive 2.5 years in prison.
Both are silent at the same time and receive 1 year each, since in this case the evidence base of their guilt will be small.
One gives testimony and gets freedom, while the other is silent and gets 5 years in prison.

Obviously, the outcome of the case depends on the decision of both participants, but they cannot come to an agreement, since they are sitting in different cells. The conflict of their personal interests in the struggle for a common interest is also clearly visible. Each of the prisoners has two options for action and 4 options for outcomes.

Logical inference chain

So, offender A considers the following options:

I am silent and my partner is silent - we both will receive 1 year in prison.
I turn in my partner and he turns me over - we both get 2.5 years in prison.
I am silent, and my partner turns me over - I will get 5 years in prison, and he is free.
I hand over my partner, but he is silent - I get freedom, and he is 5 years in prison.

Here is a matrix of possible solutions and outcomes for clarity.

Table of likely outcomes of the prisoner's dilemma.

The question is, what will each participant choose?

"Silence, you can't speak" or "You can't be silent, you can't speak"

To understand the choice of the participant, you need to go through the chain of his thoughts. Following the reasoning of the offender A: if I remain silent and my partner does not say anything, we will get the minimum term (1 year), but I cannot find out how he will behave. If he testifies against me, then it is also better for me to testify, otherwise I may be imprisoned for 5 years. It is better for me to go to prison for 2.5 years than for 5 years. If he keeps silent, then all the more I need to testify, since this will give me freedom. Participant B argues in the same way.

It is not hard to understand that the dominant strategy for each of the criminals is to testify. The optimal point of this game comes when both criminals testify and receive their "prize" - 2.5 years in prison. Nash game theory calls this equilibrium.

Non-optimal optimal Nash solution

The revolutionary nature of the Nash view is that it is not optimal when considering the individual participant and his personal interest. After all, the best option is to remain silent and be released.

The Nash equilibrium is a point of convergence of interests, where each participant chooses an option that is optimal for him only if other participants choose a certain strategy.

Considering the option when both criminals are silent and receive only 1 year each, we can call it the Pareto-optimal option. However, it is only possible if the criminals could come to an agreement beforehand. But even this would not guarantee this outcome, since the temptation to deviate from the agreement and avoid punishment is great. The lack of complete trust in each other and the danger of getting 5 years makes it necessary to choose the option with recognition. It is simply irrational to speculate that the participants will stick to the silent option, acting in concert. This conclusion can be made if we study the Nash equilibrium. Examples only prove the case.

Selfish or rational

Nash's equilibrium theory yielded startling findings that disproved prior principles. For example, Adam Smith viewed the behavior of each of the participants as completely selfish, which brought the system into balance. This theory was called the "invisible hand of the market."

John Nash saw that if all participants act in pursuit of their own interests, then this will never lead to an optimal group result. Given that rational thinking is inherent in each participant, the choice that the Nash equilibrium strategy offers is more likely.

Purely male experiment

A striking example is the game "blonde paradox", which, although it seems inappropriate, is a vivid illustration of how Nash's theory of games works.

In this game, you need to imagine that a group of free guys came to a bar. Nearby is a group of girls, one of whom is preferable to others, say a blonde. How can guys act to get the best friend for themselves?

So, the guys' reasoning: if everyone starts to get to know a blonde, then most likely nobody will get her, then her friends will not want to meet. Nobody wants to be the second fallback. But if guys choose to avoid the blonde, then the likelihood of each guy finding a good girlfriend among the girls is high.

The Nash equilibrium situation is not optimal for guys, because, pursuing only their own selfish interests, everyone would choose a blonde. It can be seen that the pursuit of only selfish interests will be tantamount to the collapse of group interests. Nash equilibrium will mean that each guy acts in his own interests, which are in contact with the interests of the entire group. This is not an optimal option for everyone personally, but optimal for everyone, based on the overall strategy of success.

Our whole life is a game

Making decisions in real life is very much like a game, when you expect certain rational behavior from other participants. In business, at work, in a team, in a company, and even in relationships with the opposite sex. From big deals to ordinary life situations, everything obeys this or that law.

Of course, the criminals and bar game situations discussed are just great illustrations of the Nash equilibrium. Examples of such dilemmas very often arise in the real market, and this is especially true in cases with two monopolists controlling the market.

Mixed strategies

Often we are involved not in one, but in several games at once. Choosing one of the options for one game, guided by a rational strategy, but you find yourself in another game. After a few rational decisions, you may find that you are not happy with your outcome. What should be done?

Consider two types of strategy:

A pure strategy is the behavior of a participant that comes from thinking about the possible behavior of other participants.
A mixed strategy or random strategy is the alternation of pure strategies in a random way or the selection of a pure strategy with a certain probability. This strategy is also called randomized.

By looking at this behavior, we get a new look at the Nash equilibrium. If earlier it was said that the player chooses a strategy once, then another behavior can be imagined. It can be assumed that the players choose a strategy at random with a certain probability. Games that cannot find Nash equilibria in pure strategies always have them in mixed ones.

The Nash equilibrium in mixed strategies is called mixed equilibrium. It is an equilibrium where each participant chooses the optimal frequency of choosing their strategies, provided that other participants choose their strategies with a given frequency.

Penalties and mixed strategy

An example of a mixed strategy can be found in the game of soccer. The best illustration of mixed strategy is perhaps the penalty shootout. So, we have a goalkeeper who can only jump to one corner, and a player who will take the penalty.

So, if the first time a player chooses a strategy to kick into the left corner, and the goalkeeper also falls into this corner and catches the ball, then how can events unfold the second time? If a player kicks in the opposite corner, this is probably too obvious, but a kick in the same corner is no less obvious. Therefore, both the goalkeeper and the batter have no choice but to rely on a random choice.

So, alternating random choices with a certain clean strategy, the player and the goalkeeper try to get the maximum result.

Game theory is a science that studies the principles of decision making in situations in which several agents interact with each other. Decisions made by one person affect the decisions of others and the outcome of the interaction in general. Interactions of this type are called strategic.

The word "play" should not be misleading. This concept is interpreted in game theory more broadly than in everyday life. The situation of strategic interaction can be described in the form of a model, which is called a game. Thus, in the theory of games, not only playing chess, but also voting in the UN Security Council and bargaining between a seller and a buyer in the market will be considered a game.

Strategic interactions are found in almost every area of our life. An example from economics: several companies competing in the market must look at the actions of competitors when making decisions. If we talk about politics, then the candidates competing in the elections, when announcing their election platform, naturally take into account the positions of other candidates in relation to this issue. And if we study the interaction of people in society, then with the help of game theory you can learn a lot of interesting things about the propensity of people to cooperate.

Social scientists often use game theory as a tool for solving problems of interest to them. Simplifying, game-theoretic modeling can be broken down into two stages.

First, you need to build a formal model based on a real life situation. As a rule, the model needs to reflect three main characteristics of a life situation: who interacts with each other (such agents are called players in game theory), what decisions players can make, and what payments they receive as a result of this interaction. The formal model is called the game.

Once we have built the game, it needs to be solved in some way. At this stage, we completely abstract from reality and study exclusively the formal model. How does the model solution work? We must fix the concept of the behavior of the players in the game, that is, the principles of their decisions. Once we have fixed this concept, we can try with its help to solve the game, that is, to present the outcome that will end the game.

Different game-theoretic concepts can be used to solve different classes of games. One of the most beautiful theoretical results of game theory proves that in a very wide class of models it is guaranteed to find a solution. I mean the result of John Nash in 1950: in any finite game in normal form, you can always find at least one equilibrium in mixed strategies. Chronologically, this was the first universal game-theoretic concept, which allows a guaranteed solution to be found in a very wide class of models.

Unlike representatives of the social sciences, game mathematicians are more interested in the intrinsic properties of games and the concepts of their solution. It is thanks to such theoretical results that we can be sure that by building and solving this or that game-theoretic model, we will eventually get a solution with the necessary properties.

Of course, John Nash is not the sole author of game theory. Game theory as an independent science began to develop a little earlier, at the beginning of the twentieth century. The first attempts to formally define games, player strategies, and game solution concepts go back to the names of Emil Borel and John von Neumann. However, it was Nash who presented the concept of equilibrium, which makes it possible to find a guaranteed solution in finite games. In honor of the author of the theorem on the existence of an equilibrium in mixed strategies in finite games, this equilibrium was called the Nash equilibrium.

The first Nobel Prize awarded in 1994 for results in the field of game theory (to John Nash, Reinhard Zelten and John Harsagni) actually confirmed the status of game theory as an independent scientific direction with its own problems and methods of their solution. Several more Nobel Prizes that followed were awarded both for fundamental game-theoretic results and for applications of game theory to one side or another of our lives. At the world's leading universities in both economics and political science programs, game theory is necessarily included in the standard set of courses. It is often studied by both psychologists and mathematicians.

Today, if you look at the sections of large conferences and articles in leading scientific journals on game theory, the number of works using the apparatus of game theory for solving applied problems is much greater than the number of fundamental game-theoretic results. The current state of the discipline can be described as follows: in game theory, a sufficiently powerful core has been formed, a layer of knowledge that allows researchers from related fields to obtain good and interesting results.

Nevertheless, new and interesting areas of research are always opening up in game theory itself. So, thanks to the development of computing technologies, new game-theoretic concepts have appeared, taking into account the capabilities and limitations of computers. Thanks to them, it became possible to solve new problems. Bowling, Birch, Johansson, and Tammelin's 2015 equilibrium result for a version of poker is a remarkable example of the use of modern theories and technologies.

"Strategy games". This is a textbook on game theory that even people without special education can use to understand mathematical concepts. The Village publishes excerpts from the first chapter, which explores the application of game theory to real life.

What Game Theory Learns

When you use the word "game", you may get the impression that we are talking about a superficial, insignificant subject in a large-scale picture of the world, studying such trivial pursuits as gambling and sports, while there are a lot of more important issues in the world - war, business, education, career and relationships. In fact, the strategy game is not just a game; all of the above questions are examples of games, and game theory helps us understand their essence.

Game theory is the analysis or, if you prefer, the science of such an interactive decision-making process. When you carefully weigh everything before taking anything, that is, you are aware of your goals or preferences, as well as any restrictions or requirements for your actions, and deliberately choose your actions in order to achieve maximum success based on your own criteria, it is considered that you behave rationally. In other words, game theory is the science of rational behavior in interactive situations.

We are not suggesting that game theory will teach you the secrets of the perfect game, or help you never lose. First, your opponent might read the same books; besides, you both cannot win constantly. More importantly, many games contain many complex and subtle nuances, and most real-life situations involve rather peculiar or random factors. Game theory cannot offer an infallible recipe for action; what it does is it provides a set of general principles for analyzing strategic interactions.

We will first offer you a number of simple examples, many of which are borrowed from situations that you have probably encountered in your life. In each example, we point out an important strategic principle.

The Prisoners' Dilemma at the University

You have enrolled in a course that is graded on average. Regardless of how well you succeed in absolute terms, only 40% of students will get A grades and only 40% - B grades. Therefore, you must work hard, not only in absolute terms, but also in terms of how hard you work. schoolmates (in fact, "schoolmates" seems to be a more appropriate expression in this context). All students understand this, so after the first lecture, they gather for an impromptu meeting and agree not to show excessive zeal. After a few weeks, the temptation to gain an advantage over others with a little more effort becomes overwhelming. After all, your fellow students cannot see everything you do and have no real impact on you, and the benefits of increasing your GPA are substantial. As a result, you start to visit the library more often and stay there longer. The problem is, everyone else is doing the same. Therefore, you will receive the same score as if you had adhered to the agreement. The only difference is that you all spent more time studying than you would like.

This is an example of the prisoners' dilemma. In its original version, the two suspects are interrogated separately and each is asked to confess guilt. One of them, say, suspect A, is told the following: “If the other suspect (B) does not confess, then you can make a good deal and mitigate the punishment by admitting your guilt. But if B confesses, then you better do it too, otherwise the court will be especially harsh in relation to you. So you should confess anyway. " Suspect B is convinced using similar arguments. Faced with this choice, A and B confess, although it would be better for both if they were silent, since the police have no hard evidence against them.

Play on the edge with neighbors

Suppose you are sharing an apartment with one or more students and you notice that it is running low on detergent, paper towels, oatmeal, beer, and other essentials. You have an agreement to split the actual costs equally, but shopping takes time. Are you ready to highlight it and go shopping, or will you rely on one of your comrades, leaving yourself more time for study or relaxation? Will you go shopping for soap or will you watch TV so as not to miss the next episode? In many situations like this, the waiting game can go on long enough before the one who actually needs one of these things (usually beer) breaks down and goes to the store. As a result, all this can lead to serious quarrels and even a breakdown in relations between roommates.

This strategic game can be viewed from two points of view. According to the first, each roommate has a simple binary choice - to go shopping or not. Without a doubt, the best option for you is for the neighbor to go to the store and you stay at home, and the worst option is the reverse procedure. If you both shop unbeknownst to each other, say, on your way home from university or work, unnecessary duplication will occur and perhaps even spoilage of some products; If no one makes a purchase, it can cause serious inconvenience, or even a local disaster, if toilet paper runs out at the most inopportune moment.

This situation is analogous to the coward game that American teenagers used to play. Two teenagers were racing towards each other in cars. The one who swerved to the side to avoid a collision was considered the loser (coward), and the one who continued to go straight won. In a second, more interesting and dynamic perspective, the same situation is viewed as a "war of attrition" in which each roommate tries to wait out the others, hoping that someone else will run out of patience sooner. In the meantime, the risk that the apartment will run out of something important, which will lead to serious inconvenience or a major quarrel, increases. Each player allows such an increase up to his point of tolerance; the most unrestrained loses. Everyone is trying to understand how close to the brink of disaster the other participants in the game will allow the situation to develop. Hence the term "edge balancing", which refers to such strategy and play.

Date Screening

When you're going out on a date with someone, you want to present your best side to that person and hide flaws. Of course, you cannot hide them indefinitely, especially if your relationship develops, but you are determined to get better or hope that by then your partner will accept you as you are. You also know that a relationship will be hopeless if you don't make a good first impression - alas, you won't get a second chance.

Of course, you want to know about the person with whom you are on a date, everything (both good and bad). But you also know that if your partner is as good at dating as you are, then he (or she) will also try to show his best side and hide the worst. You will analyze the situation more carefully and try to understand which signs of good qualities are real, and which can be easily imitated in order to make a favorable impression. Even the most unkempt person can show up at an important meeting in neat clothes, but courtesy and good manners, which are manifested in many small details, are difficult to portray all evening if you are not accustomed to them. Flowers are a relatively cheap gift; more expensive gifts may have a certain value, but not in their essence, but as reliable evidence of what this person is willing to sacrifice for you. And the “currency” in which the value of such a gift is calculated can have different meanings depending on the context: a diamond donated by a millionaire may cost less in this case than the time spent by a person to communicate with you or some business performed at your request.

You should be aware that your counterpart will equally carefully analyze the information content of your actions. Therefore, you need to do what will signal your true positive qualities, not those that can be imitated. This is important not only on the first date: disclosing, hiding and collecting information about the deep intentions of another person are relevant throughout the entire period of maintaining a relationship. Here is a story that illustrates this.

In New York, there was a man and a woman who had separate rent-regulated apartments. The couple's relationship reached its climax, and they decided to live together. The woman suggested that the man give up the second apartment, but he, being an economist, explained to her the fundamental principle: it is always better to have more options. Perhaps the chances of them breaking apart are minimal, but given even a small risk, it would be wise to keep a second apartment with a low rent. The woman took this extremely negatively and immediately broke off the relationship with her partner!

Economists, hearing this story, say that it only confirms the principle of the advisability of a wider choice. However, strategic thinking offers a slightly different, more convincing explanation. The woman was not sure of the man's seriousness, and her proposal was a brilliant strategic way to find out the truth. Words are worthless: anyone can say "I love you." If the man backed up his words with deeds and agreed to break the lease, this would be concrete evidence of his love, but his refusal became strong evidence of the opposite, which means that the woman did the right thing by breaking off relations with him.

All of these examples, calculated from your direct experience, belong to a very important class of games in which the main strategic issue is information manipulation. Strategies that allow you to transmit winning information about yourself are called signals; and strategies that encourage people to act so that they reliably disclose personal information, good or bad, are called screening tools. Consequently, the woman's offer to give up one of the apartments was a tool that presented the man with a choice: either to give up the apartment, or to demonstrate the lack of serious intentions.

Cover: Publishing house "MYTH"

A funny example of the application of game theory can be found in the fantasy book by Anthony Pierce "The Brave Golem"

Lots of text

“The point of what I’m going to show you all,” Grundy began, “is to gain the required number of points. The scores can be very different - it all depends on the combination of decisions that are made by the participants in the game. For example, suppose each participant testifies against their playmate. In this case, each participant can be awarded one point!
- One point! - said the Sea Witch, showing an unexpected interest in the game. Obviously, the sorceress wanted to make sure the golem had no chance of the demon Xanth being pleased with it.
- Now let's assume that each of the participants in the game does not testify against his comrade! - continued Grundy. - In this case, each can be awarded three points. I want to especially point out that as long as all participants act in the same way, they will be awarded the same number of points. No one has any advantage over others.
- Three points! Said the second witch.
- But now we have the right to suggest that one of the players began to testify against the second, and the second is still silent! Said Grundy. - In this case, the one who gives these readings gets five points at once, and the one who is silent does not get a single point!
- Aha! - both witches exclaimed in one voice, licking their lips predatory. It was clear that both of them were clearly going to get five points.
- I was losing points all the time! The demon exclaimed. - But you have only outlined the situation so far, and you have not yet presented a way to resolve it! So what is your strategy? No need to waste time!
- Wait, I'll explain everything now! Exclaimed Grundy. - Each of us four - there are two of us golems and two witches - will fight against their opponents. Of course, the witches will try not to yield to anyone in anything ...
- Of course! The two witches exclaimed again in unison. They perfectly understood the golem perfectly!
“And the second golem will follow my tactics,” Grundy continued unperturbed. He looked at his double. - You, of course, know?
- Oh sure! I'm your copy! I perfectly understand everything what you think!
- That is great! In that case, let's make the first move so that the demon can see everything for himself. Each fight will have several rounds so that the whole strategy can manifest itself to the end and give the impression of a coherent system. Perhaps I should start.

- Now each of us must make marks on our sheets of paper! - the golem turned to the witch. - First, draw a smiling face. This will mean that we will not testify against a fellow prisoner. You can also draw a frowning face, which means that we think only of ourselves and give the necessary testimony to our comrade. We both realize that it would be better if no one turned out to be that frowning face, but, on the other hand, a frowning face gains certain advantages over a smiling one! But the bottom line is that each of us doesn't know what the other will choose! We will not know until the playmate reveals his drawing!
- Start you bastard! The witch swore. She, as always, could not do without abusive epithets!
- Ready! - exclaimed Grundy, drawing a large smiling face on his piece of paper so that the witch could not see what he portrayed there. The witch made her move, also depicting a face. Presumably, she certainly portrayed an unkind face!
“Well, now all we have to do is show each other our drawings,” Grandi announced. Looking back, he opened the drawing to the public and showed it in all directions so that everyone could see the drawing. Grunting something displeased, the Sea Witch did the same.
As Grundy had hoped, an angry, displeased face looked from the drawing of the sorceress.
“Now you, dear spectators,” said Grandi solemnly, “you see that the witch chose to testify against me. I'm not going to do that. Thus, the Sea Witch scores five points. And I, accordingly, do not get a single point. And here…
A light noise rolled through the rows of spectators again. Everyone clearly sympathized with the golem and longed for the Sea Witch to fail.
But the game has just begun! If only his strategy was correct ...
- Now we can move on to the second round! - announced Grundy solemnly. - We must repeat the moves again. Everyone draws a face that is closer to him!
And so they did. Grundy now portrayed a gloomy, displeased face.
As soon as the players showed their drawings, the audience saw that both of them were now portrayed with angry faces.
- Two points each! Said Grundy.
- Seven two in my favor! - shouted the witch happily. - You can't get out of here, you bastard!
- Let's start again! Exclaimed Grundy. They made another drawing and showed them to the public. The same angry faces again.
- Each of us repeated the previous move, behaved selfishly, and therefore, it seems to me, it is better not to award points to anyone! - said the golem.
- But I still lead the game! - said the witch, happily rubbing her hands.
- Okay, don't make a noise! Said Grundy. - The game is not over. Let's see what happens! So, dear audience, we are starting the fourth round!
The players made the drawings again, showing the public what they depicted on their sheets. Both sheets again showed the viewers the same evil faces.
- Eight - three! Shouted the witch, bursting into evil laughter. “You dug your own grave with your foolish strategy, golem!
- Round five! Shouted Grundy. The same thing was repeated as in the previous rounds - again angry faces, only the score changed - it became nine or four in favor of the sorceress.
- Now the last, sixth round! - announced Grundy. His preliminary calculations showed that this particular round should become fateful. Now the theory had to be confirmed or refuted by practice.
Several quick and nervous movements of the pencil on the paper - and both drawings appeared before the eyes of the public. Again, two faces, now even with bared teeth!
- Ten - five in my favor! My game! I won! - the Sea Witch cackled.

“You really won,” Grundy agreed grimly. The audience was eerily silent.
The demon moved his lips to say something.

“But our competition is not over yet! - shouted loudly Grundy. - It was only the first part of the game.
- Give you forever! The demon Xant growled in displeasure.
- It's right! Said Grundy calmly. - But one round does not solve anything, only methodicalness indicates the best result.
Now the golem approached the other witch.
- I would like to play this tour with another opponent! He announced. - Each of us will depict faces, as it was the previous time, then will demonstrate the drawn to the public!
And so they did. The result was the same as the last time - Grundy painted a smiling face, and the witch - so generally a skull. She immediately gained an advantage of as much as five points, leaving Grundy behind.
The remaining five rounds ended with the results that could be expected. Again the score was ten or five in favor of the Sea Witch.
- Golem, I really like your strategy! - the witch laughed.
- So, you watched two rounds of the game, dear viewers! Exclaimed Grundy. - Thus, I scored ten points, and my rivals - twenty!
The audience, which was also counting the points, nodded their heads mournfully. Their count was the same as that of the golem. Only a cloud named Frakto seemed quite pleased, although, of course, it did not sympathize with the witch either.
But Rapunzelia smiled approvingly at the golem - she continued to believe in him. She may have remained the only one who believed him now. Grundy hoped that he would justify this boundless trust.
Now Grundy approached his third rival - his double. He was supposed to be his last opponent. Quickly scratching their pencils on the paper, the golems showed them to the public. Everyone saw two laughing faces.
- Note, dear viewers, each of us chose to be a kind cellmate! Exclaimed Grundy. - And therefore, none of us got the necessary advantage over the opponent in this game. Thus, we both get three points and proceed to the next round!
The second round has begun. The result was the same as the previous time. Then the remaining rounds. And in each round, both opponents scored three points again! It was just incredible, but the audience was ready to confirm everything that was happening.

Finally, this tour came to an end, and Grundy, quickly running his pencil over the paper, began to calculate the result. Finally he announced solemnly:
- Eighteen to eighteen! In total, I scored twenty-eight points and my opponents scored thirty-eight!
“So you lost,” announced the Sea Witch happily. - Thus, one of us will become the winner!
- Perhaps! Said Grundy calmly. Now there was another important point. If everything goes as it was intended ...
- We need to see it through to the end! - exclaimed the second golem. “I also have to fight two Sea Witches, too! The game is not over yet!
- Yes, of course, come on! Said Grundy. - But only be guided by the strategy!
- Oh sure! - assured his double.
This golem approached one of the witches and the tour began. It ended with the same result with which Grundy himself came out of a similar round - the score was ten or five in favor of the sorceress. The witch was downright beaming with inexpressible joy, and the audience sullenly fell silent. The Demon Xanth looked somewhat tired, which was not a good omen.
Now it was time for the final round - one witch had to fight against the second. Each had twenty points in her assets, which she could get by fighting with golems.
- And now, if you will allow me to score at least a few extra points ... - conspiratorially whispered the Sea Witch to her double.
Grundy tried to keep calm, at least outwardly, although a hurricane of conflicting feelings raged in his soul. His luck now depended on how correctly he predicted the possible behavior of both witches - after all, their character was, in essence, the same!
Now was perhaps the most critical moment. But if he was wrong!
- Why on earth should I give in to you! Croaked the second witch to the first. - I myself want to score more points and get out of here!
“Well, if you’re behaving so impudently,” the applicant yelled, “then I’ll trim you so that you’ll no longer look like me!”
The witches, giving each other hating glances, drew their drawings and showed them to the public. Of course, nothing else, except for two skulls, simply could not be there! Each of them scored one point.
The witches, cursing each other, proceeded to the second round. The result is again the same - again two roughly drawn skulls. Thus, the witches gained one more point. The audience diligently recorded everything.
This continued in the future. When the tour ended, the weary witches discovered that they each had scored six points. Draw again!
- Now let's calculate the results and compare everything! Said Grandi triumphantly. “Each of the witches scored twenty-six points, and the golems scored twenty-eight points. So what do we have? And we have the result that golems have more points!
A sigh of surprise rolled through the spectators. Excited spectators began to write columns of numbers on their sheets of paper, checking the correctness of the count. Many during this time simply did not count the number of points scored, believing that they already knew the result of the game. Both witches began to growl with indignation, it is not clear who exactly was blaming for what happened. The demon Xant's eyes lit up again with alert fire. His trust was justified!
“I ask you, dear audience, to pay attention to the fact,” Grandi raised his hand, demanding that the audience calm down, “that none of the golems won a single round. But the final victory will still be for one of us, of the golems. The results will be more telling if the competition continues further! I want to say, my dear viewers, that in an eternal duel, my strategy will invariably turn out to be a winning one!
The demon Xanth listened with interest to what Grundy was saying. Finally, emitting a puff of steam, he opened his mouth.
- What exactly is your strategy?
- I call her "Be Hard But Honest"! - explained Grundy. - I start the game honestly, but then I start to lose, because I come across very specific partners. Therefore, in the first round, when it turns out that the Sea Witch begins to testify against me, I automatically remain the loser in the second round - and so it continues until the end. The result may be different if the witch changes her tactics of playing the game. But since she could not even think of such a thing, we continued to play according to the previous pattern. When I started playing with my doppelganger, he treated me well, and I treated him well in the next round of the game. Therefore, our game also went differently and somewhat monotonously, since we did not want to change tactics ...
- But you haven't won a single round! The demon objected in surprise.
- Yes, and these witches have not lost a single round! - confirmed Grundy. - But the victory does not automatically go to the one for whom the tours are left. The victory goes to the one who scored the most points, which is a completely different matter! I managed to score more points when we played with my doppelganger than when I played with witches. Their selfish attitude brought them a momentary victory, but in the longer term it turned out that it was because of this that they both lost the game entirely. This often happens!