Games with non-opposing interests. Nash equilibrium

February 10th, 2015

Let's split the $100 quickly. You and I decide how many of the hundred we demand and at the same time announce the amounts. If our total is less than a hundred, everyone gets what they wanted. If the total is over one hundred, the one who asked for the least amount gets the desired amount, while the greedier person gets what's left. If we ask for the same amount, each gets $50. How much will you ask? How will you split the money?

There is only one winning move.

To start scientifically:

Nash equilibrium(English) Our equilibrium) is named after John Forbes Nash - this is the name in game theory of a type of solution for a game of two or more players, in which no participant can increase the payoff by changing his decision unilaterally, when other participants do not change their decision. Such a set of strategies chosen by the participants and their payoffs is called the Nash equilibrium.

The concept of Nash equilibrium (NE) was not first used by Nash; Antoine Auguste Cournot showed how to find what we call the Nash equilibrium in the Cournot game. Accordingly, some authors call it Nash-Cournot equilibrium. However, Nash was the first to show in his dissertation on non-cooperative games in 1950 that such equilibria must exist for all finite games with any number of players. Prior to Nash, this was only proven for 2-player zero-sum games by John von Neumann and Oskar Morgenstern (1947).

And now the solution to the problem that was presented at the beginning of the post:

The $51 requirement will give you the maximum amount no matter what your opponent chooses. If he asks for more, you will receive $51. If he asks for $50 or $51, you will get $50. And if he asks for less than $50, you will get $51. In any case, there is no other option that will bring you more money than this one. The Nash equilibrium is a situation in which we both choose $51.

And now a little about this man:

John Nash was born June 13, 1928 in Bluefield, Virginia to a strict Protestant family. My father worked as an engineer at Appalachian Electric Power, my mother managed to work for 10 years as a school teacher before marriage. I studied average at school, but I didn’t like mathematics at all - it was taught boringly at school. When Nash was 14, Eric T. Bell's The Great Mathematicians fell into his hands. “After reading this book, I managed myself, without outside help, to prove Fermat's little theorem,” Nash writes in his autobiography. So his mathematical genius declared itself.

Studies

This was followed by studies at the Carnegie Polytechnic Institute (now private Carnegie Mellon University), where Nash tried to study chemistry, took a course in international economics, and then finally established himself in the decision to take up mathematics. In 1948, after graduating from the institute with two diplomas - a bachelor's and a master's degree - he entered Princeton University. Nash Institute professor Richard Duffin provided him with one of the most concise letters of recommendation. It contained a single line: "This man is a genius!"

Works

At Princeton, John Nash heard about game theory, then only introduced by John von Neumann and Oscar Morgenstein. Game theory struck his imagination, so much so that at the age of 20, John Nash managed to create the foundations scientific method which played a huge role in the development of the world economy. In 1949, the 21-year-old scientist wrote his dissertation on game theory. Forty-five years later, he received the Nobel Prize in Economics for this work. Nash's contribution has been described as a fundamental analysis of equilibrium in the theory of non-cooperative games.

Neumann and Morgenstein were engaged in so-called zero-sum games, in which the victory of one side inevitably means the defeat of the other. In 1950 - 1953 Nash published four, without exaggeration, revolutionary papers in which he provided an in-depth analysis of "non-zero-sum games" - a special class of games in which all participants either win or lose. An example of such a game would be negotiations on wage increases between the trade union and the management of the company. This situation can end either in a long strike in which both sides suffer, or in reaching a mutually beneficial agreement. Nash was able to see the new face of competition by simulating what came to be known as the "Nash equilibrium" or "non-cooperative equilibrium" in which both sides use ideal strategy, which leads to the creation of a stable equilibrium. It is beneficial for the players to maintain this balance, since any change will only worsen their situation.

In 1951, John Nash joined the Massachusetts Institute of Technology (MIT) in Cambridge. Colleagues did not particularly like him, because he was very selfish, but they treated him patiently, because his mathematical abilities were brilliant. There, John developed a close relationship with Eleanor Stier, who was soon expecting a child from him. So Nash became a father, but he refused to give his name to the child to be recorded on the birth certificate, and also refused to provide any financial support. In the 1950s Nash was famous. He collaborated with the RAND Corporation, an analytical and strategic research firm that employed leading American scientists. There, again through his research in game theory, Nash became one of the leading experts in the field of cold war". In addition, while working at MIT, Nash wrote a number of papers on real algebraic geometry and the theory of Riemannian manifolds, highly appreciated by his contemporaries.

Disease

Soon John Nash met Alicia Lard and in 1957 they got married. In July 1958, Fortune magazine named Nash America's rising star in "new mathematics." Soon Nash's wife became pregnant, but this coincided with Nash's illness - he fell ill with schizophrenia. At this time, John was 30 years old, and Alicia was only 26. At the beginning, Alicia tried to hide everything that was happening from friends and colleagues, wanting to save Nash's career. However, after several months of insane behavior, Alicia forcibly placed her husband in a private psychiatric clinic in the suburbs of Boston, McLean Hospital, where he was diagnosed with paranoid schizophrenia. After being discharged, he suddenly decided to leave for Europe. Alicia left her mother's newborn son and followed her husband. She brought her husband back to America. Upon their return, they settled in Princeton, where Alicia found work. But Nash's illness progressed: he was constantly afraid of something, spoke of himself in the third person, wrote meaningless postcards, called former colleagues. They patiently listened to his endless discussions about numerology and the state of political affairs in the world.

The deterioration of her husband's condition depressed Alicia more and more. In 1959 he lost his job. In January 1961, a completely depressed Alicia, John's mother and his sister Martha took difficult decision: to put John in Trenton State Hospital in New Jersey, where John underwent insulin therapy - harsh and risky treatment, 5 days a week for a month and a half. After his release, Nash's colleagues from Princeton decided to help him by offering him a job as a researcher, but John again went to Europe, but this time alone. He sent only cryptic letters home. In 1962, after 3 years of confusion, Alicia divorced John. With the help of her mother, she raised her son by herself. Later it turned out that he also had schizophrenia.

Despite the divorce from Alicia, fellow mathematicians continued to help Nash - they gave him a job at the University and arranged a meeting with a psychiatrist, whom he prescribed anti-psychotic drugs. Nash's condition improved and he began to spend time with Eleanor and his first son, John David. “It was a very encouraging time,” recalls John's sister Martha. - It was quite a long period. But then everything started to change.” John stopped taking his medications, fearing that they might have a depressing effect on mental activity, and the symptoms of schizophrenia reappeared.

In 1970, Alicia Nash, convinced that she had made a mistake by betraying her husband, accepted him again, and now as a boarder, this may have saved him from a state of homelessness. In later years, Nash continued to go to Princeton, writing strange formulas on blackboards. Princeton students nicknamed him "The Phantom". Then in 1980. Nash became noticeably better - the symptoms receded and he became more involved in the life around him. The disease, to the surprise of the doctors, began to recede. More precisely, Nash began to learn to ignore her and again took up mathematics. “Now I think quite sanely, like any scientist,” Nash writes in his autobiography. “I won’t say that it gives me the joy that anyone who recovers from a physical illness experiences. Sound thinking limits man's ideas about his connection with the cosmos.

Confession

In 1994, at the age of 66, John Nash received the Nobel Prize for his work on game theory. However, he was deprived of the opportunity to give the traditional Nobel lecture at Stockholm University, as the organizers feared for his condition. Instead, a seminar was organized (with his participation) to discuss his contributions to game theory. After that, Nash was invited to give a lecture at the University of Uppsala, since he did not have such an opportunity in Stockholm. According to Krister Kiselman, professor at the Mathematical Institute of the University of Uppsala, who invited him, the lecture was devoted to cosmology.

In 2001, 38 years after their divorce, John and Alicia remarried. Nash has returned to his office at Princeton, where he continues to explore mathematics and explore this world - the world in which he was so successful in the beginning; the world that forced him to go through a very difficult disease; and yet this world accepted him again.

"Mind games"

In 1998, American journalist (and Columbia University economics professor Sylvia Nazar) wrote a biography of Nash called A Beautiful Mind: The Life of Mathematical Genius and Nobel Laureate John Nash. The book became an instant bestseller.

In 2001, under the direction of Ron Howard, based on the book, the film "A Beautiful Mind" was filmed, in the Russian box office "A Beautiful Mind". The film won four Oscars (for Best Adapted Screenplay, Best Director, Best Supporting Actress and, finally, Best Picture), a Golden Globe Award, and won several Bafta Awards (British Film Achievement Award).

As you can see, the film is almost true. Of course, with some "literary" distortions.

• Robert Redford was offered to direct the film, but he was not satisfied with the filming schedule.
• Tom Cruise auditioned for the role of John Nash and Salma Hayek for the role of Alicia. It is curious that she was born in the same town of El Salvador as her failed heroine.
• When Nash first sees Parker, he refers to him as "big brother" (an allusion to Orwell's 1984). Another reference to Orwell comes later, when we see the number on the door of Nash's office - 101.
• The manuscript that young John Nash shows to his curator, Professor Helinger, is a genuine copy of an article published in the journal Econometrica under the heading "The Dealing Problem."
• The screenwriter of the film, Akiva Goldsman, had considerable experience in dealing with mentally ill people: in his time as a doctor, he personally developed methods for restoring the mental health of children and adults.
• The curator of the film on the mathematical part was the professor at Barnard College, Dave Byer - it was with his hand that Russell Crowe "displays" tricky formulas on the board.
• "Wise formulas" upon closer examination are just a meaningless set of Greek letters, arrows and mathematical signs.
• Unlike his on-screen counterpart, who was distinguished by rare devotion to his "half", the real John Nash was married several times in his life, and at the age of twenty he adopted an illegitimate child.
• In the 1994 Nobel Prize section of the film, Nash talks about taking a new type of antipsychotic, but in reality, John Nash stopped taking them back in 1970, and his remission was not related to taking antipsychotics. .

Where are Nash's discoveries applied today?

Having experienced a boom in the seventies and eighties, game theory has taken a strong position in some branches of social knowledge. Experiments in which the Nash team at one time recorded the behavior of the players in the early fifties were regarded as a failure. Today they formed the basis of "experimental economics". "Nash equilibrium" is actively used in the analysis of oligopolies: the behavior of a small number of competitors in a particular market sector.

In addition, in the West, game theory is actively used when issuing licenses for broadcasting or communications: the issuing authority mathematically calculates the most optimal variant of frequency distribution.

In the same way, a successful auctioneer determines what information about the lots can be provided to specific buyers in order to obtain optimal income. With the theory of games successfully work in jurisprudence, social psychology, sports and politics. For the latter, a characteristic example of the existence of a "Nash equilibrium" is the institutionalization of the concept of "opposition".

However, game theory has found its application not only in the social sciences. Modern evolutionary theory would not be possible without the concept of the "Nash equilibrium", which mathematically explains why wolves never eat all hares (because otherwise they will starve to death in a generation) and why animals with defects contribute to the gene pool of their species (because in such case, the species may acquire new useful characteristics).

Now Nash is not expected to make grandiose discoveries. It doesn't seem to matter anymore, because he managed to do two of the most important things in his life: he became a recognized genius in his youth and defeated an incurable disease in his old age.

And a few more scientific theories: here's an example, and here. Let's also remember about , and . And yet there is also The original article is on the website InfoGlaz.rf Link to the article from which this copy is made -

What should the agents participating in the game do? How can they determine which strategy is better than others?

Let's start with a more modest goal: to determine which strategies will definitely not work.

Definition 1.2. An agent's strategy is said to be dominated if there exists a strategy such that

In this case, they say that it dominates over.

In other words, a strategy is dominated if there exists another strategy that is no worse at every point, for any possible combination of other agents' strategies. Hence, there is no reason to prefer at all, and it can simply be discarded in the analysis.

Example 1.4. Recall example 1.2, in which Colonel Blotto was about to deploy troops on the field. If we analyze the matrix from Example 1.2, it becomes obvious that the strategies , and are dominated by others: for example, the strategy will be better than any of them. Of course, the same is true for Blotto's adversary. Thus, the matrix will be significantly reduced.

End of Example 1.4.

Example 1.5. In Example 1.3, where we discussed Cournot competition, there were a lot of dominated strategies. All strategies were like that: they were guaranteed to bring non-positive profit, while the zero strategy (produce nothing) guarantees zero profit. Therefore, it was immediately possible to confine ourselves to the analysis of the square as a set of strategies.

End of Example 1.5.

However, it is worth noting that it is easy to construct an example in which any strategy is dominated. This will mean that some strategies are equivalent, that is, they dominate each other. In such cases, at least one of them should be left, otherwise there will be nothing to choose from at all.

We continue the conversation. After dominated strategies, it would be logical to introduce dominant strategies.

Definition 1.3. The agent's strategy is called dominant, if every other strategy is dominated by it, that is,

Dominant strategy for an agent - a real happiness. He doesn’t need to think at all: it’s enough to choose a dominant strategy, anyway, no other will give anything better under any outcome.

Moreover, if all agents have dominant strategies, then the analysis of such a game ends before it starts. It is safe to say that all agents will choose their dominant strategies.

Definition 1.4. Equilibrium in dominant strategies for a strategic game, this is a strategy profile such that for any agent the strategy is dominant.

This balance is the most stable of all. In the next lecture, we will give an example from the theory of economic mechanisms in which such an equilibrium arises - the so-called Vickrey auction (see Theorem 2.1.

But, unfortunately, happiness is not always achievable. Neither in Example 1.1, nor in Example 1.2, nor in Example 1.3 is there any equilibrium in dominant strategies did not work. For each player's strategy, there was a profile of strategies of other players, in which it would be beneficial for the player to change to one or another.

Nash equilibrium

In the previous paragraph, we discussed that if an agent has dominant strategy, then he has nothing to think about and worry about: he can simply choose this strategy. But what should the agents participating in the game do when such strategies do not exist and are not expected?

Then one has to take into account not only one's own strategies, but also the strategies of other agents. This account will lead to the concept of equilibrium, formulated in 1950 by John Nash.

Definition 1.5. Nash equilibrium in pure strategies for a strategic game, this is a profile of strategies such that the following condition is satisfied for any agent:

In other words, as before, it is unprofitable for the agent to deviate from the chosen strategy. But now it is unprofitable for him to do this not in the abstract, with any choice of strategies for other agents, but only in a specific profile of strategies.

Example 1.6. We continue to consider poor Blotto. The colonel's game matrix without dominated strategies was given in Example 1.4. It is easy to see from the matrix that if one player chooses strategy , then nothing depends on the choice of the other, that is, we can say that there is no reason for the other to deviate from strategy . All this means that for this game the strategy profile is in Nash equilibrium.

End of Example 1.6.

Let's give a continuous example - believe me, we are still waiting for such reasoning, and it's time to get used to a slightly more serious analysis.

Example 1.7. Let us return to the Cournot analysis of competition from Example 1.3. This time we will not simplify anything: let the price be given by an unknown function , and the cost of production for each firm is an unknown function of . To find the Nash equilibrium, we find the best answer function. The company's profit is defined as

To determine the maximum of a function for fixed , you just need to find the derivative

and equate it to zero. Accordingly, the Nash equilibrium is reached where both firms give the optimal response to the opponent's strategy, that is, on the decisions of the following system differential equations:

We leave it to the reader to verify that in the particular case considered in Example 1.3, the Nash equilibrium will indeed be the point of intersection of the lines in Fig. 1.1.

End of Example 1.7.

Definition 1.5 mentioned the strange term " pure strategies": what else are they? It turns out that strategies are not only pure, but also mixed. Mixed strategies are a logical extension of the concept of strategy: let's allow the player not only to choose one of them, but also to make a more or less random choice from them.

Definition 1.6. Mixed Strategy for a player in a strategy game is probability distribution, where is the set of all probability distributions above .

A mixed strategy can also be thought of as setting the weights for each strategy so that the sum (in the continuous case, the integral) of all weights is equal to 1.

There are games where there are no Nash equilibria for pure strategies. But it is always (ultimately) in mixed strategies.

Example 1.8. Recall the game "rock-paper-scissors", the matrix of which we have already written out in example 1.1.

Obviously, no Nash equilibria in pure strategies not here: for any strategy there is someone to refute it. But the Nash equilibrium in mixed strategies available here. Suppose that the second player chooses rock, scissors or paper with probability , and the first player chooses them with probabilities , and . Then the first player wins with probability

and also loses and draws with the same probability. In other words, if the opponent chooses a strategy with equal probability, all strategies are equivalent for the player. Since the game is symmetric, it turns out that the mixed strategy profile

is in balance.

End of example 1.8.

Proof that the equilibrium in mixed strategies always exists, follows from the Kakutani fixed point theorem [ , ].

Theorem 1.1(Kakutani) Let be a non-empty convex compact subset euclidean space, a - multivalued function on with a closed graph such that the set is non-empty, closed, and convex for all . Then u have

In games with non-zero sum all participants in the game can win or lose. Bimatrix game is a finite game of two players with a non-zero sum. In this case, for each game situation A i B j, each player has his own payoff a ij for the first player and b ij for the second player. For example, the behavior of producers in markets of imperfect competition is reduced to a bimatrix game. Use the online calculator to find the solution bimatrix game, as well as situations Pareto optimal and Nash stable situations.

Consider a conflict situation in which each of the two participants has the following options for choosing their own line of behavior:

• player A can choose any of the strategies А 1 ,…,А m ,
• player В – any of the strategies В 1 ,…,В n .

At the same time, their joint choice is evaluated quite definitely: if player A chooses i-th strategy A i , and player B is the k -th strategy B k , then as a result the payoff of player A will be equal to some number a ik , and the payoff of player B to some, generally speaking, another number b ik .
Sequentially going through all the strategies of player A and all the strategies of player B, we can fill two tables with their payoffs.

The first of the tables describes the payoff of player A, and the second - the payoff of player B. Usually these tables are written in the form of a matrix.
Here A is the payoff matrix of player A, B is the payoff matrix of player B.

Thus, in the case when the interests of the players are different (but not necessarily opposite), two payoff matrices are obtained: one is the payoff matrix for player A, the other is the payoff matrix for player B. Therefore, the name that is usually assigned to such a game sounds quite natural - bimatrix.

Nash equilibrium- equilibrium, when each participant in the game chooses a strategy that is optimal for him, provided that the other participants in the game adhere to a certain strategy.
The Nash equilibrium is not always the most optimal for the participants. In this case, we say that the equilibrium is not Pareto optimal.
Pure Strategy- a certain reaction of the player to the possible behavior of other players.
Mixed Strategy- probabilistic (not exactly defined) reaction of the player to the behavior of other players.

Example #1. Fight for markets.
Firm a intends to sell a consignment of goods in one of the two markets controlled by the larger firm b. To this end, she preparatory work associated with certain costs. If firm b guesses in which of the markets firm a will sell its product, it will take countermeasures and prevent the “capture” of the market (this option means the defeat of firm a); if not, firm a wins. Suppose that for firm a, penetration into the first market is more profitable than penetration into the second, but the struggle in the first market also requires large funds from it. For example, the victory of firm a in the first market brings her twice as much profit as a victory in the second, but a defeat in the first market completely ruins her.
Let's compose mathematical model of this conflict, considering firm a as player 1 and firm b as player 2. Player 1's strategies are: A 1 - market penetration 1, A 2 – market penetration 2; player 2 strategies: V 1 - countermeasures in market 1, V 2 - countermeasures in the market 2. Let for the company and its victory in the 1st market is estimated at 2 units, and the victory in the 2nd market - at 1 unit; the defeat of firm a in the 1st market is estimated at -10, and in the 2nd - at -1. For firm b, its victory is 5 and 1, respectively, and its loss is -2 and -1. As a result, we get a bimatrix game Г with payoff matrices
.
By the theorem, this game can have either pure or completely mixed equilibria. There are no equilibrium situations in pure strategies here. Let us now verify that this game has a completely mixed equilibrium situation. We find , .
Thus, the game under consideration has a unique equilibrium situation , where , . It can be implemented by repeating the game many times (that is, by repeatedly reproducing the described situation) as follows: firm a should use pure strategies 1 and 2 with frequencies 2/9 and 7/9, and firm b should use pure strategies 1 and 2 with frequencies 3/14 and 11/14. Any of the firms, deviating from the specified mixed strategy, reduces its expected payoff.

Example #2. Find Pareto optimal situations and Nash stable situations for a bimatrix game.

Example #3. There are 2 firms: the first one can produce one of the two products A 1 and A 2 , the second can produce one of the two products B 1 , B 2 . If the first firm produces products A i (i = 1, 2), and the second - B j (j = 1, 2), then the profit of these firms (depending on whether these products are complementary or competitive) is determined by table No. 1 :

	IN 1	IN 2
A 1	(5, 6)	(3, 2)
A 2	(2, 1)	(5, 3)

Assuming that the firms enter into an agreement among themselves, determine the fair distribution of profits using the Nash arbitrage solution.

Scholars have been using game theory for nearly sixty years to extend the analysis of the strategic decisions that firms make, in particular to answer the question: why do firms tend to collude in some markets while competing aggressively in others; using firms to keep potential competitors from invading; how price decisions should be made, when demand or cost conditions change, or when new competitors enter the market, etc.

The first to conduct research in the field of game theory were J.-F. Neumann and O. Morgenstern and described the results in the book "Game Theory and Economic Behavior" (1944). They extended the mathematical categories of this theory to the economic life of society, introducing the concept of optimal strategies, maximizing expected utility, dominance in the game (on riyku), coalition agreements, and the like.

Scientists sought to formulate the fundamental criteria for the rational behavior of a participant in the market in order to achieve favorable results. They distinguished two main categories of games. The first is a "zero-sum game" where the payoff consists solely of losing other players. In this regard, the benefit of some must necessarily be formed at the expense of the losses of other players, so that the total sum of benefits and losses is always equal to zero. The second category is "winning games" where individual players compete for a win made up of their own stakes. Sometimes it is formed due to the presence of an "output" (a term from card game in bridge, which means one of the players who, when making a bet, does not participate in the game), is completely passive and often serves as an object of exploitation. In both cases, the game is inevitably fraught with risk, since each of its participants, as the researchers believed, "seeks to maximize the function, the variables of which are not controlled by him." If all players are skillful, chance is the deciding factor. But this rarely happens. Cunning is almost always an important part of the game, with the help of which attempts are made to reveal the intentions of opponents and veil their intentions, and then to take advantageous positions that would force these opponents to act to their own detriment. Much depends on the "counter-cunning".

important during the game rational behavior player, i.e. thoughtful selection and implementation of the optimal strategy. An important contribution to the development of a formalized (in the form of models) description conflict situations, especially in the definition of the "equilibrium formula", i.e. the stability of the decisions of opponents in the game, was introduced by the American scientist J.-F. Nash.

Nash John Forbes was born in 1928 (G. Vluefild, USA). He studied at Carnegie Mellon University with a degree in chemical engineering, mastered the course "international economics". He received a bachelor's degree and at the same time a master's degree in mathematics.

In 1950, at the University of Iriaston, he defended his doctoral dissertation on "non-cooperative games". Since 1951 And for nearly eight years, Nash was a professor at the Massachusetts Institute of Technology while simultaneously conducting active research activities.

Since the spring of 1959, the scientist fell ill and lost his ability to work. In the 70s he was able to return to his mathematical hobbies, but it was difficult for him to produce scientific results. The Nobel Committee in 1994 actually awarded a work written in 1949

Member of the US National Academy of Sciences, the Bkonometric Society, and the American Academy of Arts and Sciences.

Having thoroughly studied various games, creating a series of new mathematical games and observing the actions of participants in various game situations, Nash tried to gain a deeper understanding of how the market works, how companies make risk-related decisions, why buyers act in certain ways. In economics, as in a game, company managers must take into account not only the latest, but also the previous steps of competitors, as well as the situation on the entire economic (game, for example, chess) field, and many other important factors.

Subjects of economic life- its active participants who take risks in the market in a competitive environment, and it must be justified. Therefore, each of them, as a player, must have their own strategy. This is what Nash had in mind when he developed the method that was later named after him (Nash equilibrium).

His understanding of strategy as the basic concept of game theory J.-F. Nash explains in terms of a "zero-sum game" (he calls it a "symmetric game") where each player has a certain number of strategies. The payoff of each player depends on what strategies he and his opponent have chosen. Based on this, a matrix is ​​constructed to find the optimal strategy, which, after multiple repetitions of the game, provides this player with the maximum possible average gain (or the maximum possible average loss). Since the player does not know what strategy the opponent will choose, it is better for him (rationally) to choose a strategy that is designed for the worst behavior of the opponent for him (the principle of the so-called "guaranteed result"). Acting carefully and considering the opponent as a strong competitor, our player will choose the minimum possible payoff for each of his strategies. Then, from all the minimum winning strategies, he will choose the one that will provide the maximum of all the minimum payoffs - the maximin.

But the enemy will probably think the same. He will find for himself the greatest losses in all strategies of the player, and then from these maximum losses he will choose the minimum - minimax. If maximin is equal to minimax, the decisions of the players will be stable, and the game will have an equilibrium. The stability (balance) of decisions (strategies) is that it will be unprofitable for both participants in the game to deviate from the chosen strategies. In the case when the maximin is not equal to the minimax, the decisions (strategies) of both players, if they somehow guessed the choice of the opponent's strategy, turn out to be unstable, neuro-important.

General short definition Nash equilibrium is the result in which the strategy of each of the players is the best among the others adopted by the rest of the participants in the strategy game. This definition is based on the fact that none of the players, by changing their own role, can achieve the greatest benefit (maximization of the utility function), if the other participants firmly adhere to their line of conduct.

J.-F. Nash has repeatedly strengthened, including in it, as an indispensable factor for developing strategies, the indicator of the optimal amount of information. He derived this indicator of optimality from the analysis of situations (1) with the player fully informed about his opponents and (2) with incomplete information about them. Having translated this postulate from mathematical language into the language of economics, Nash introduced the uncontrolled variables of market relations as an important information element of knowledge of environmental conditions. After that, the Nash equilibrium became a method used in almost all branches of economic science to better understand complex relationships, noted in October 1994 during the announcement of new Nobel Prize winners in economics A. Lindbeck, a member of the Royal Swedish Academy and chairman of the Nobel Committee in Economics.

The application of the Nash equilibrium was an important step in microeconomics. its use contributed to an in-depth understanding of the development and functioning of markets, the rationale for strategic decisions made by managers of various firms. The Nash equilibrium can be used to study the process of political negotiations and economic behavior, including in oligopolistic markets.

For pioneering equilibrium analysis in non-cooperative games, the 1994 Nobel Prize in Economics was awarded to J.-F. Nash in, R. Selten and J. Harshani. Starting with the classic work by J. Neumann and O. Morgenstern "Game Theory and Economic Behavior", an integral part of economic analysis has become the study of the strategy of interaction between economic entities in conditions when, in order to develop one's own line of behavior, it is necessary to take into account the actions of another sub "object (as this occurs, in particular, in chess, preference and other games).These three Nobel laureates made a great contribution to the offshoot of game theory - the theory of non-cooperative games (that is, games when an agreement is reached between the participants).The fundamental point of this theory is the concept of equilibrium, is used to predict the results of the interaction.

The Nash equilibrium has become a fundamental concept of game theory.

Discrete Choice Analysis

By the last quarter of the twentieth century. the opinion that the main role in the behavior of consumers is played by common sense and calculation dominated. It is with the common sense of consumers in mind that liberal economic theories are formulated. Economists of this scientific direction believe that the market as a system of relations between economic entities is able to self-regulate and set fair prices for goods and services based on common sense.

Although the liberal economic school has given the world more scientific achievements than the competitive conservative one, its theories have limited application, which its supporters admit. For example, monetarists (they are also liberals) have not yet been able to reasonably explain the behavior of investors in international financial markets and the huge fluctuations in prices for world raw materials.

The liberal market approach turned out to be too simplistic for reliable forecasting consumer demand for services and goods in an environment where consumers have a huge selection of such goods and at the same time are not limited in the volume of purchases, since now in developed countries consumer credit is extremely common. Moreover, liberal theory cannot explain, for example, an American family (or an English family) buying an American (or English) car, while a Korean one costs less. That is, this theory does not take into account national and other characteristics of consumer behavior, which are difficult to explain from the point of view of common sense.

Therefore, in recent years, eco-Jarmist scientists have been talking more and more often about the emergence of a new economic theory that has developed directly on the basis of data on consumer behavior, which must be studied using statistical methods. This theory offers a description of how utility is measured. Despite the fact that such assessments are subjective, it is subjectivity that determines their value for the implementation of economic policy. Many economists even predict that it is the theory of consumer behavior (well-known author - D. - L. McFedden) that will be in the XXI century. basis for determining the economic and political strategy of developed countries.

McFedden Daniel Little was born in 1937. (Raleigh, Carolina State, USA). Studied and worked at the University of Minnesota. In 1962 he defended his doctoral dissertation, worked as an assistant professor of economics at the University of Pittsburgh, then a professor of economics at the University of California, where since 1991 he has been in charge of the econometric laboratory.

Published in co-authorship such works: "Essays on economic behavior in conditions of instability" (1974), "Demand for urban movement: a behavioral analysis" (1976), "Economics of production: a dual approach to theory and practice" (1978), "Structural analysis of discrete data with econometric applications" (1981), " Microeconomic modeling and numerical analysis: A study of demand in public utilities "(1984)," Handbook of Econometrics "(Vol. 4, 1994), and many scientific articles.

During 1983-1984. He was Vice-President, and in 1985 - President of the Econometric Society. In 1994, he was elected Vice President of the American Economic Association. Member of the US National Academy of Sciences, the American Econometric Society and the Academies of Arts and Sciences, the American Economic Association awarded him the J.-B. Clark, Econometric Society - R. Frisch medal.

It is known that quite often microdata reflect discrete choices - choices among a finite set of alternative solutions. In economic theory, traditional demand analysis assumed that individual choice should be represented by a continuous variable, but this treatment is not consistent with studying the behavior of discrete choice. By the previous achievements of many scientists empirical researches of such elections were not substantiated in the economic theory.

Methodology for Discrete Choice Analysis D.-l. McFadden is rooted in microeconomic theory, according to which each individual chooses a certain alternative that maximizes his utility. Utility functions are ways of describing consumer choice: if a set of services X is chosen, while a set of services B is available, then X must have a greater utility than B. By studying the choice made by consumers, one can derive an estimated utility function that would adequately describe their behavior . Obviously, it is impossible to investigate the whole complex of facts influencing the choice of an individual, but an analysis of the dynamics of changes among individuals with approximately the same characteristics allows us to draw fairly objective conclusions.

D.-l. McFedden, in collaboration with T. Domenick, studied consumer behavior in relation to regular transport trips1. In most major cities, commuters have a choice between using public transport or driving to work. Each of these alternatives can be viewed as a set various characteristics A: travel time, waiting time, available costs, comfort, convenience and the like. Thus, one can denote the length of travel time for each type of trip as x (, the length of waiting time for each type of trip as x 2, etc.

If (xx, x2, xx) represents the value of n different characteristics of car trips, and (y1, y2 ... .. y n) - the values ​​of the characteristics of bus trips, then we can consider a model in which the consumer decides whether to go him by car or bus, based on the preference for one set of specified characteristics over another. More specifically, it can be assumed that the benefits of the average consumer in relation to these characteristics can be represented by a utility function of the form:

where the coefficients b and, b 2 i etc. D - unknown parameters. Any monotonic transformation of this utility function can describe consumer choice, but from a statistical point of view, it is much easier to work with a linear function.

Let's assume that there is a group of similar consumers who choose whether to travel by car or bus, based on specific data on the length of travel time, expenses and other characteristics of travel that they encounter. In statistics, there are techniques that can be used to find the values ​​of the coefficients D, when and - 1, n, the most suitable for the research structure of the choice made by a given plurality of consumers. These statistical techniques allow us to derive an estimated utility function for various ways transport movement.

McFadden and Domenick proposed a utility function of the form:

where TW is the total walking time to or from the bus or car; TT - total travel time in minutes; C is the total cost of the trip in dollars.

Using the estimated utility function, it was possible to correctly describe the choice between automobile and bus transport for 93% of households in the sample taken by the authors. The coefficients for the variables in the above equation show the marginal utility of each such characteristic. The ratio of one coefficient to another shows the marginal rate of substitution of one characteristic for another. For example, the ratio of the marginal utility of walking time to the marginal utility of total trip duration does not indicate that the average consumer considers walking time to be about 3 times slower than travel time. That is, the consumer would be willing to spend 3 additional minutes on the trip to save 1 minute of walking. Similarly, the ratio of trip cost to total trip duration indicates the choice of the average consumer with respect to these two variables. In the study, an average passenger estimated a minute of travel time by transport at 0.0411 x x 2.24 = $0.0183 per minute, which is $1.10 per hour. (For comparison, the average passenger's hourly wage in 1967 was $2.85 an hour in sen.)

Such estimating utility functions can be valuable in determining whether any changes should be made to the public transport system. For example, in the above utility function, one of important factors that explain what consumers are guided by in their choice is the duration of the trip. The city's transport authority could, at little cost, increase the number of buses to reduce this total journey time, but the additional number of passengers needs to be figured out to justify the increase in costs.

Using the utility function and the sample of consumers, it is possible to predict which consumers will want to travel by car and which will prefer the bus. This will give some idea of ​​whether the revenue will be sufficient to cover the additional costs. In addition, the marginal rate of substitution can be used to provide insight into each consumer's estimate of the reduction in travel time. According to McFadden and Domenick's research, the average passenger in 1967 estimated travel time at a rate of $1.10 per hour, he was willing to pay 37 cents to reduce travel time by 20 minutes. This number represents the dollar value of more timely bus service. The presence of a quantitative measure of gain certainly contributes to the adoption rational decisions in the field of transport policy.

Another significant contribution of McFedden is the development in 1974 of the so-called conditional logit analysis. The model assumes that each person faces a number of alternatives in life. Let us denote as X the characteristics associated with each alternative, and as 2 the characteristics of individuals that the researcher can observe using the available data. For example, for a travel choice study where the alternative could be car, bus, or subway, X could include information on time and expenses, while X could include data on age, income, and education. But the differences between individuals and alternatives to the folder, as between X \%, although they are invisible to the researcher, but they determine the individual's most useful choice. Such characteristics are represented by random error vectors. McFadden suggested that these random errors have a certain statistical distribution (distribution) among the population, calling it an extreme value distribution. Under these conditions (plus some technical predictions), he demonstrated that the probability that a person chooses the alternative / can be written as polynomials of the logit model:

where e is the base of the natural logarithm; b and b are parameters (vectors). In his database, the researcher can observe variables X and Z in fact, as the individual chooses an alternative. As a result, the scientist is able to estimate the parameters p and<5, использовав известные статистические методы. Мак-Федденивське дифференцировки логит-модели осталось новацией и признается фундаментальным достижением.

Models are commonly used in urban transportation demand studies. They can also be used in transport when it is planned to study the effectiveness of political measures, as well as social or environmental changes. For example * these models can explain how changes in the price of goods improve their availability, they affect the demographic situation, the volume of travel using alternative modes of transportation. The models are also applicable to many other areas, in particular in studies of the choice of housing, place of residence or education. McFadden used the developed methods to analyze many social problems, such as the demand for household energy, telephone services and housing for the elderly, and the like.

As a result of his research, the scientist came to the conclusion that conditional logit models have a certain feature regarding the probability of choosing between two alternatives, for example, travel by bus or train, independent of the cost of other travel options. This feature, called the independence of unrelated alternatives (NNA), is unrealistic for statistical consumption. D.-l. McFadden not only invented statistical tests for matching HNA, but also proposed general models, called by the prisoner logit models, which assume that the choices of individuals can be made in a certain sequence. For example, when examining decisions regarding place of residence and type of housing, it is assumed that a citizen first chooses a microdistrict, and then the type of housing.

Even with these generalizations, the models are quite sensitive to certain predictions about the distribution of unobservable characteristics across the population. During the last decade D.-l. McFadden developed simulation models (simulated moment methods) for the statistical evaluation of a discrete selection of models that make much more basic assumptions. Powerful computers have extended the practical applicability of these numerical methods. As a result, individuals' discrete choices can now be described more realistically and their decisions more accurately envisaged. Based on his new theory, McFadden developed microeconometric models that can be used, for example, to predict the intentions of the part of the population that will choose different alternatives. For the development of methods for the formal processing of individual statistical and economic data, McFedden was awarded the Nobel Prize.

D.-l. McFedden in the 1960s also invented econometric methods for evaluating production technology and explored factors that indirectly affect a firm's need for capital and labor. During the 90s, a talented scientist scientifically developed the economics of environmental management, enriched the methodological literature on estimating the value of natural resources, in particular, he studied the loss of social wealth due to the environmental damage caused in 1989 by an oil slick moving from the tanker "Exxon Valdez" damaged in the accident * along the coast of Alaska.

The leitmotif of the research of Professor D.-l. McFedden's attempts to combine economic theory, statistical and empirical methods to solve social problems with their help. His scientific developments also help sociologists and politicians evaluate the choice of voters, based on the money in their income, etc.

McFadden was the first to propose a methodology for discrete choice analysis, according to which each individual chooses a certain alternative that maximizes his utility. Utility functions are ways of describing consumer choice. By studying the choices made by consumers, it is possible to derive an estimated utility function that would adequately describe their behavior.

It manifests itself in reality in order to show that this concept is not just an abstract term, but a generalization of a real-life pattern. However, despite the clarity of the example, on the basis of only one it may seem that we have stumbled upon some kind of degenerate case. Therefore, it makes sense to consider a more general description of this rule.

Many readers may be familiar with the Nash equilibrium from one very common special case of it - the so-called "prisoner's dilemma". Its essence is about the following.

There are two prisoners in the prison, who were caught red-handed separately, but are still suspected of more serious crimes. If participation is proven, then the term of prisoners will increase to ten years. Now they sit for a year each. The investigation invites each of them to make a deal and testify against the second. In this case, the first term will be knocked off to six months, and the second will sit down for ten. However, the prisoners understand that if they slander each other, they are unlikely to be spared both of them - rather, they will add another five years to each.

The layout can be displayed using the following table.

It is easy to see that the "green" options (1, 2) and (2, 1) are symmetrical, while in the other two the position of the prisoners will be identical. Therefore, it is possible to consider the logic of the situation from the point of view of only one of the prisoners - for the second it will be the same.

The prisoner, of course, wants the shortest possible sentence for himself. But if he remains silent, then perhaps his colleague will testify against him, which will increase his sentence to ten years. If it were not for the promised reduction in the term, then one could console oneself with the thought “why should I?”, but the temptation to reduce the term is too great. In addition, the second prisoner, as the first understands, will suspect him, the first, of testifying against the second and thereby increasing his sentence.

“It will be a shame to be extreme and thunder for ten years,” the first one thinks. But “the second one probably thinks the same way, and also suspects me,” he understands, “and therefore there are very few chances that a colleague will not lay me down. It turns out that it is necessary to testify: if the second one by some miracle remains silent, then there will be six months, if he speaks out - five. Well, at least not ten, which I will inevitably get because of my accomplice who opened up with the investigation!

The "orange" option (1, 1) is digestible for both and in some sense it is the optimum in this situation. However, everyone has an even better option - the corresponding "green" (1, 2) or (2, 1). As a result, the “red” version (2, 2) will be implemented in practice.

We can say that for each of the prisoners it is not so bad: only five years against ten in the "green" version in favor of an accomplice. However, let's imagine that in the "red" version, both will be given ten. The logic in this case will change a little: “if I hand him over, then at least there is a chance to get out of ten years, and if I keep silent, there is no chance, he will probably lay me for the same reasons.” However, here the system pushes the prisoners to choose the worst option possible. Acting, which is typical, strictly for its own benefit.

Let's now consider another situation. There are two firms - A and B. Each of them can use the strategy - X or Y. However, the results are influenced not only by the strategy chosen by the firm itself, but also by the strategy of the second firm. We will present the gain or loss of each of the firms in the form of the following table.

To heighten the intensity of passions, I chose the numbers so that the unprofitable state for both firms would only slightly differ from the “neighboring” ones: it is all the more surprising that it will be realized. Firms, acting strictly in their own interests, will most likely want to get a thousand rubles instead of a hundred, and thus will not receive anything, but, on the contrary, will even lose. The transition of one of the firms to strategy X will worsen its position even more - the other firm will get richer, and the second one will lose even more, although slightly more.

Let's write the above matrices in a more general form, abstracting from "firms", "prisoners", "terms" and "rubles". Let's assume that we just have two players A and B playing some game where on each move one of two moves can be made - X or Y. The winnings are simply some "points", the largest number of which each player seeks to score.

	A makes a move X	And Ygrek makes a move
B makes X's move	A: a 0 B: b 0	A: a 1 > a 0 B: b 1< b 3
B makes Y's move	A: a 2< a 3 B: b 2 > b 0	A: b 3 B: a 3

The rules of the game, represented by this matrix, will "push" the players to implement the "red" option (2, 2), even if the players' payoffs in this case are significantly less than in all other options. True, depending on the ratio of wins (which can be negative as well - that is, losses), denoted by the letters "a" and "b" with indices, the frequency of implementation of each of the options will be different.

In particular, the choice can be influenced by the arithmetic mean of the payoffs when choosing each of the strategies, as well as the estimated probability with which the player will make one or another move (which, by the way, can be approximated by the frequency of moves made in previous rounds). So, in the simplest case, player A adds a 0 and a 2 to evaluate the move X and divides the result by two, assuming the choice of move by B is equally probable. He does the same for move Y - adds a 1 to a 3 , then divides the result by two - and compares the results. In a more complex case, the player calculates the sum a 0 *p x + a 2 *p y , where p x and p y are the probabilities of moves X and Y made by player B. The result is compared with a 1 *p x + a 3 *p y .

One could, of course, again divide the result by two, but since division by two takes place for both variants of the move, this operation is not necessary to compare the values, as, however, in the case of “equiprobable moves”.

Also, the player can focus on the values ​​themselves. For example, if one of the moves means a probable loss - especially a big one that the player cannot afford - the player will probably choose another move, even if the expected payoff on the other move is on average lower, but in both cases positive .

Finally, we must remember that people often, let's say, "remember the other player." If the second player is a competitor or even an enemy, then there may be a tendency to choose a move that will hurt the other player, even if the first player gains little, and perhaps even loses because of this. If the second player is a friend, then more often a move will be chosen that allows him to win a little too - in the event that the “game” is not a pre-declared competition, but some kind of process from real life. The possibilities of revenge and indulgence, of course, depend on the ratios in the matrix - with some of them, they will rather forget that the opponent is your friend than begin to play along with him a little.

In other words, the principle we are considering reflects precisely that trend, and not determinism. The stronger the ratios of wins and losses are similar to those featured in the "prisoner's dilemma", the more often and faster the system will lead players to the "worst" option, and the "worst" this option will be.

There is, as it were, an “invisible hand of the market”, which, as it were, invisibly pushes the players ... well, you know. More precisely, no, maybe you don't know. In the classic version, the “hand of the market” seems to push where everyone needs it, but here it pushes in the wrong direction at all. Not for the common good, but in a permanent crisis, which in other scenarios could have been avoided, which is illustrated by the “prisoner’s dilemma”, and a hypothetical example of competition between firms, and a real example with the inevitable overestimation of software development time, which was discussed in previous article.

The market pushes players towards a Nash equilibrium, which can be arbitrarily far from their common and personal good.

In this case, we considered only two players and a game with two moves, but a wider generalization is possible, which is precisely the formulation of the Nash equilibrium:

If in some game with an arbitrary number of players and a payoff matrix there is such a state that if any of the players individually chooses a move that does not correspond to it, his personal payoff will decrease, then this state will turn out to be “equilibrium” for this game.

In addition, in some cases, the players' moves will tend to move towards this state, even if there are other states in this game in which the payoff of the players as a whole and/or individually is higher.

It is noticeably more difficult to give examples of this general case in a manner similar to the one previously used, since adding each player will add another dimension to the payoff matrix. However, more on that later.