Basic concepts of game theory and game models. Game theory in economics and other areas of human activity
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INTRODUCTION
Any person all over the world daily performs some actions, makes a choice for himself in something. In order to perform any actions, a person needs to think about their consequences, choose the most correct, rational of all possible decisions. The choice must be made based on the interests of one's own or group, depending on who the decision refers to (an individual or a group, the organization as a whole).
Institutions are created by people to maintain order and reduce the uncertainty of exchange. They provide predictability of human behavior. Institutions allow us to save our mental abilities, since having learned the rules, we can adapt to the external environment without trying to comprehend and understand it. Petrosyan L.A., Zenkevich N.A., Shevkoplyas E.V.: Game theory: textbook. Publisher: BHV, 2012.-p.18.
Institutions are the "rules of the game" in society, or, more formally, the man-made limits that organize relationships between people. Labsker L.G., Yashchenko N.A.: Game theory in economics. Practice with problem solving. Tutorial. Publisher: Knorus, 2014.-p.21. Institutions appear to solve problems that arise from the repeated interaction of people. At the same time, they not only have to solve the problem, but also minimize the resources spent on solving it.
Game theory is called mathematical method study of optimal strategies in games. The game is understood as a process in which two or more parties participate, fighting for the implementation of their interests. Each side has its own goal and uses some strategy that can lead to a win or a loss - depending on their behavior and the behavior of other players. Game theory helps to choose the most profitable strategies, taking into account some factors:
1. considerations about other participants;
2. resources of participants;
3. the intended actions of the participants.
In game theory, it is assumed that the payoff functions and the set of strategies available to each of the players are well known, i.e. each player knows his payoff function and the set of strategies available to him, as well as the payoff functions and strategies of all other players, and in accordance with this information forms his behavior.
The relevance of the topic lies in a wide range of applications of game theory in practice (biology, sociology, mathematics, management, etc.). Specifically, in the economy - at such moments when the theoretical foundations of the theory of choice in classical economic theory do not work, which, for example, is that the consumer makes his choice rationally, he is fully aware of the situation in this market and about this particular product.
CHAPTER 1. THEORETICAL FOUNDATIONS OF GAME THEORY
1.1 GAME THEORY CONCEPT
As mentioned above, game theory is a branch of mathematics that studies formal models for making optimal decisions in a conflict. At the same time, conflict is understood as a phenomenon in which various parties participate, endowed with various interests and opportunities to choose actions available to them in accordance with these interests. Each of the parties has its own goal and uses some strategy, which can lead to a win or a loss - depending on the behavior of other players. Game theory helps to choose the best strategies, taking into account ideas about other participants, their resources and their possible actions.
Game theory has its origins in neoclassical economics. The mathematical aspects and applications of the theory were first presented in the classic 1944 book by John von Neumann and Oscar Morgenstern, Game Theory and Economic Behavior.
The game is a simplified formalized model of a real conflict situation. Mathematically, formalization means that certain rules for the actions of the parties in the course of the game have been developed: options for the actions of the parties; the outcome of the game with this variant of action; the amount of information each party has about the behavior of all other parties.
Situations in which the interests of two parties collide and the result of any operation carried out by one of the parties depends on the actions of the other party are called conflict situations.
The player is one of the parties in the game situation. The player's strategy is his rules of action in each of possible situations games. Dominance in game theory is a situation in which one of the strategies of a certain player gives a greater payoff than another, for any actions of his opponents. Protasov I.D. Game theory and operations research: textbook. allowance. - M.: Helios ARV, 2013.-S.121.
The focal point is the equilibrium in the coordination game, chosen by all participants in the interaction on the basis of a common knowledge that helps them coordinate their choice. The focal point concept was introduced by the 2005 Nobel Prize-winning economist Thomas Schelling in a 1957 article that became the third chapter of his famous book The Strategy of Conflict (1960).
If there is a strictly dominant strategy for one of the players, he will use it in any of the Nash equilibria in the game. If all players have strictly dominant strategies, the game has a unique Nash equilibrium. However, this equilibrium will not necessarily be Pareto efficient, i.e. disequilibrium outcomes can provide all players with a greater payoff. A classic example This situation is the Prisoner's Dilemma game. A Nash equilibrium is a set of strategies (one for each player) such that none of the players has an incentive to deviate from their strategy. A situation will be Pareto efficient if neither player can improve their position without making the other player worse off.
We should also mention the Stackelberg equilibrium. Stackelberg equilibrium is a situation where none of the players can increase their payoff unilaterally, and decisions are made first by one player and become known to the second player. Unlike the dominant strategy equilibrium and the Nash equilibrium, this kind of equilibrium always exists.
Interpretation of game theory can be carried out in two ways: matrix and graphic. The matrix method will be depicted below, where situations leading to the emergence of institutions will be considered.
For example graphic image Consider the following situation, where there is one pasture for cows to graze. Now let's ask the question: for what number of cows, n, would the use of this pasture be optimal? In accordance with the marginal principle of optimization, which assumes the equation of marginal cost and marginal income, it should be answered that the optimal number of cows will be the one at which the value of the marginal product from grazing the last cow, VMP, will be equal to the cost of one cow, c. Under the conditions of private ownership of this pasture, this principle would be observed, since the individual farmer would compare the benefits and costs associated with each additional cow, and would stop at the number of them, Ep, at which the possibilities of obtaining a positive rent from grazing cows on the pasture, Rp, would be exhausted, and, accordingly , the maximum of this rent would be reached (Fig. 1). This is summarized in the equation below, which, while respecting the margin principle, maximizes the difference between the value of the total product, VTP, and the total cost, i.e. the cost of a cow times the number of cows.
VMP (n*) = c maxn VTP (n) - cn (1)
Figure 1. Graph of the value of the marginal and average grazing of cows
However, in the conditions of free access to the pasture, i.e., the absence of exclusive rights to it, the marginal optimization principle will not be observed and the number of cows on the pasture will exceed the optimal value, Ep, and reach the point of equality of the value of the average cow grazing product, VAP, and the cost of a cow . As a result, there will be a new equilibrium number of cows in free access conditions, Ec. In this case, the positive rent, Rp, created by grazing cows until reaching their optimal number, Еp, will be spent on additional cows and, when the point Еc is reached, will become equal to zero as a result of the accumulation of negative rent equal to it in absolute value. This is summarized in the equations below:
VTP(n")/n"=c?VTP(n")-cn"=0;
1.2 VARIETY OF SITUATIONS AND AREAS OF HUMAN LIFE IN WHICH GAME THEORY IS APPLICABLE
In life, there are many examples of a clash of opposite sides, taking the form of a conflict with two acting parties pursuing opposing interests.
Such situations arise, for example, when it comes to trust. Compliance of the counterparty's actions with expectations becomes especially important in situations where the risk of decisions made by the individual is determined by the actions of the counterparty. Game theory models are the best illustration of what has been said: the choice of a particular strategy by a player depends on the actions of another player. Trust consists in "the expectation of certain actions of others that influence the choice of the individual, when the individual must begin to act before the actions of others become known." Let us emphasize the connection of transactions in the market with trust in a depersonalized form (trust as a norm governing relations between individuals), since the circle of participants in transactions should not be limited to personally familiar people. The following model helps to make sure that there is a need for trust in a depersonalized form for the implementation of the simplest market transaction using an advance payment (Fig. 2).
Figure 2
Let's assume that the buyer is opposed by many sellers and he knows from his previous business experience the probability of deception (1 - p). Let's calculate the value of p so that the transaction takes place, i.e., "make an advance payment" is an evolutionarily stable strategy.
EU (make an advance payment) = 10p - 5 (1 - p) = 15p - 5,
EU(do not prepay) = 0.15p - -5 > 0, p>1/3.
In other words, if the level of confidence of the buyer in the sellers is less than 33.3%, prepaid transactions become impossible under the given conditions. In other words, p=1/3 is the critical, minimum required level of confidence.
To generalize the results, we replace the specific values of the buyer's gain (10) and loss (--5) with the symbols G and L. Then, with the previous structure of the game, the transaction will take place at
the higher the loss relative to the gain, the higher should be the level of trust between the participants in the transaction. James Coleman depicted the dependence of the need for trust on the terms of the deal being concluded as follows (Fig. 3).
Figure 3
Estimated data on the minimum required level of confidence are confirmed empirically. Thus, the level of depersonalized trust in countries with developed market economies, measured by answering the question: “Based on your personal experience, do you think that people around you can be trusted? ”, was 94% in Denmark 24, 90 in Germany, 88 in Great Britain, 84 in France, 72 in northern Italy and 65% in the south. The low level of trust in the south of Italy, where the mafia is traditionally strong, is indicative. It is no coincidence that one of the mafia researchers, D. Gambetta, explains its emergence by a critically low level of trust in the southern regions of Italy and, consequently, by the need for a substitute for trust, which takes the form of the intervention of a “third party” trusted by both participants in the transaction.
Another striking example of game theory is contracts between an investor and the state for the development of mineral deposits.
To illustrate this example, let's take a contract for the sale of chairs, given that the presence of wired treasures in them is in question. We will depict an example taking into account the fact that, within the framework of game theory, factors external to the intentions of the parties to the contract are taken into account by introducing a third player, “nature”, into the game with two participants (Fig. 4).
Figure 4
As follows from the presentation of the game in expanded form, instead of four outcomes, there are six of them in the game. And if the problem of the dependence of Ostap's payoff on the actions of the stage engineer finds its solution in the presence of any non-zero level of Ostap's confidence, then the problem of the dependence of Ostap's payoff on the presence of treasures in the chairs remains unsolvable, which, however, confirms the ending of the novel.
1.3 POSSIBLE STRATEGIES IN REPEATED GAMES
1. Mixed strategies. When players find themselves in a certain choice situation repeatedly, their interaction becomes significantly more complicated. They can afford to combine strategies to maximize the overall payoff. We will show this with the help of a model that describes the relationship between the Central Bank (CB) and an economic agent in connection with the monetary policy pursued by the Central Bank.
The Central Bank focuses either on a tight monetary policy, seeking to maintain inflation at a fixed level (р0), or on emission and, consequently, an increase in inflation rates (р1). In turn, the economic agent acts on the basis of its inflation expectations re (sets prices for its products, decides on the purchase of goods and services, etc.), which can either be confirmed or not confirmed as a result of the policy pursued by the Central Bank. If p1 > pe, the Central Bank receives profit from seigniorage and inflation tax. If pe = p1, then both the Central Bank losers because of the reduction in seigniorage revenues, and economic agents that continue to bear the burden of the inflation tax. If pe = p0, then the status quo is maintained and no one loses. Finally, if pe > p0, then only economic agents lose: producers - because of the loss of demand for products that have risen unreasonably in price, consumers - because of the creation of unjustified stocks.
In the proposed model, with a single interaction, agents do not have dominant strategies, and there is no Nash equilibrium. With repeated repeated interactions, and it is precisely such interaction that is typical for real situations, both participants can use one or the other strategy at their disposal. Does alternating strategies in a certain sequence allow players to maximize their utility, i.e., achieve Nash equilibrium in mixed strategies: an outcome in which no player can increase his payoff by unilaterally changing his strategy? Assume that the Central Bank pursues a tight monetary policy with probability Р1 (in P1 % of cases), and with probability (1 - Р1) - an inflationary policy. Then, when an economic agent chooses non-inflationary expectations (pe = p0), the Central Bank can expect to receive a gain equal to
theory game strategy
EU(CB) = Р1 0+,
1 (1 - P1) = 1- -P1
In the case of inflationary expectations of an economic agent, the gain of the Central Bank will be
EU(CB) = Р10 + (1 - Р1)(-2) = 2Р1 - 2.
Now let's assume that an economic agent has non-inflationary expectations with probability P2 (in P2% of cases), and inflationary expectations with probability (1 - P2). Hence, the expected utility of the Central Bank will be
EU(CB) = Р2(1 - Р1) + (1 - Р2)(2Р1-2) = =ЗР2-ЗР1 Р2+2Р1 - 2 (Fig. 5).
Figure 5
Similar calculations for an economic agent will give
EU (e.a.) = P1(P2-1) + (1 - P1)(-P2-2) = 2P1P2 + P1-P2-2.
If we rewrite these expressions in the following form
EU(CB) = Pl(2-3P2) + ЗР2-2
EU(e.a.)==P2(2P1-1) + P1-2,
it is easy to see that when
the gain of the Central Bank does not depend on its own policy, and when
the economic agent's payoff does not depend on his expectations.
In other words, the Nash equilibrium in mixed strategies will be the formation by an economic agent of non-inflationary expectations in 2/3 of the cases and the implementation of the Central Bank in half of the cases of a tight monetary policy. The found equilibrium is achievable provided that economic agents form expectations in a rational way, and not on the basis of inflationary expectations in the previous period, adjusted for the forecast error of the previous period8. Consequently, changes in the policy of the Central Bank affect the behavior of economic agents only to the extent that they are unexpected and unpredictable. The strategy of the Central Bank in 50% of cases to pursue a tight monetary policy, and in 50% - soft is the best way to create an atmosphere of unpredictability.
2. Evolutionary-stable strategy. An evolutionarily stable strategy is one such that if the majority of individuals use it, then no alternative strategy can supplant it through natural selection, even if the latter is more Pareto efficient.
A kind of repetitive games are situations when an individual repeatedly finds himself in a certain situation of choice, but his counterparty is not constant, and in each period the individual interacts with a new counterpart. Therefore, the probability of a counterparty choosing one or another strategy will depend not so much on the configuration of the mixed strategy, but on the preferences of each of the counterparties. In particular, it is assumed that out of the total number N of potential counterparties, n (n/N%) always choose strategy A, and m (m/N%) - strategy B. This creates the prerequisites for achieving a new type of equilibrium, evolutionarily stable strategies. An Evolutionary Stable Strategy (ESS) is one in which if all members of a particular population use it, then no alternative strategy can displace it through the mechanism of natural selection. Consider, as an example, the simplest variant of the coordination problem: narrow road two cars. It is assumed that in a given area, left-hand and right-hand traffic standards are equal (or the Rules of the Road are simply not always followed). Car A is moving towards several cars with which it needs to pass. If both cars take to the left, entering the left side of the road in the direction of travel, then they part without problems. The same thing happens if both cars take to the right. When one car takes to the right, and the second - to the left and vice versa, they will not be able to part (Fig. 6).
Figure 6
So, motorist A knows the approximate percentage of motorists B who systematically take to the left (P), and the percentage of motorists B who take to the right (1 - P). The condition for the “take right” strategy to become evolutionarily stable for motorist A is formulated as follows: EU(right) > EU(left), or
0P+ 1(1 - P) > 1P+ 0(1 - P),
whence R< 1/2. Таким образом, при превышении доли автомобилистов во встречном потоке, принимающих вправо, уровня 50% эволюционно-стабильной стратегией становится «принять вправо» -- сворачивать на правую обочину при каждом разъезде.
In general, the requirements for an evolutionarily stable strategy are written as follows. Strategy I, used by counterparties with probability p, is evolutionarily stable for the player if and only if the following conditions are satisfied
EU(I, p) > EU(J, p),
which is identical
pU(I, I) + (l -p)U(I,J)>pU(J,I) + (1 - p)U(J,J) (3)
From what it follows:
U(I, I)> U(J, I)
U(I, I) = U(J, I)
U(I, J) > U(J, J),
where -- U(I, I) is the player's payoff when choosing strategy I, if the counterparty chooses strategy I; U(J, I) is the player's payoff when choosing strategy J, if the counterparty chooses strategy I, etc.
Figure 7
These conditions can also be represented graphically. Let us plot the expected utility of choosing one or another strategy along the vertical axis, and the proportion of individuals in the total population of players choosing both strategies along the horizontal axis. Then we will get the following graph (the values are taken from the model of two cars passing), shown in Fig. 7.
It follows from the figure that both “take left” and “take right” have an equal chance of becoming an evolutionarily stable strategy as long as neither of them covers more than half of the “population” of drivers. If the strategy crosses this threshold, then it will gradually but inevitably crowd out the other strategy and cover the entire population of drivers. The fact is that if the strategy crosses the 50% mark, it becomes profitable for any driver to use it in maneuvers, which, in turn, further increases the attractiveness of this strategy for other drivers. In strict form, this statement will look like this:
dp/dt = G, G">0 (4)
The main result of the analysis of repeated games is an increase in the number of equilibrium points and, on this basis, the solution of the problems of coordination, cooperation, compatibility and justice. Even in the prisoner's dilemma, the transition to repetitive interaction makes it possible to achieve a Pareto optimal result ("deny guilt"), without going beyond the norm of rationality and the prohibition on the exchange of information between players. This is the meaning of the “general theorem”: any outcome that suits an individual individually can become an equilibrium in the transition to the structure of a repeated game. In the situation of the prisoners' dilemma, the equilibrium outcome under certain conditions can be both a simple strategy "not to recognize", and a set of mixed strategies. Among the mixed and evolutionary strategies, we note the following: Tit-For-Two-Tats - start with a denial of guilt and admit guilt only if the counterparty admitted guilt in the previous two periods in a row; DOWING is a strategy based on the assumption that the counterparty is equally likely to use the "deny" and "admit" strategies at the very beginning of the game. Further, each denial of guilt on the part of the counterparty is encouraged, and each confession is punished by the choice of the “admit guilt” strategy in the next period; TESTER - start with an admission of guilt, and if the counterparty also admits guilt, then deny guilt in the next period.
CONCLUSION
At the end of the essay, we can conclude that it is necessary to use game theory in modern economic conditions.
In conditions of alternative (choice) it is very often not easy to make a decision and choose this or that strategy. Operations research allows using appropriate mathematical methods to make an informed decision on the appropriateness of a particular strategy. Game theory, which has an arsenal of methods for solving matrix games, allows you to effectively solve these problems by several methods and choose the most effective from their set, as well as simplify the original game matrices.
In the essay, the practical application of the main strategies of game theory was illustrated and the corresponding conclusions were drawn, the most used and frequently used strategies and basic concepts were studied.
LIST OF USED LITERATURE
1. Petrosyan L.A., Zenkevich N.A., Shevkoplyas E.V.: Game theory: textbook. Publisher: BHV, 2012.-212p.
2. Labsker L.G., Yashchenko N.A.: Game theory in economics. Practice with problem solving. Tutorial. Publisher: Knorus, 2014.-125p.
3. Nailbuff, Dixit: Game Theory. The art of strategic thinking in business and life. Publisher: Mann, Ivanov and Ferber, 2015 .- 99p.
4. Oleinik A.N. Institutional economics. Textbook, Moscow INFRA-M, 2013.-78s.
5. Protasov I.D. Game theory and operations research: textbook. allowance. - M.: Helios ARV, 2013.-100s.
6. Samarov K.L. Mathematics. Teaching aid for the section "Elements of game theory", Resolventa LLC, 2011.-211p.
7. Shikin E.V. Mathematical methods and models in management: textbook. allowance for students ex. specialist. universities. - M.: Delo, 2014.-201s.
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In economic reality, at every step there are situations when individuals, firms or entire countries try to outdo each other in the struggle for primacy. The branch deals with such situations economic analysis called "game theory".
"Game theory is the study of how two or more players choose individual actions or entire strategies. The name of this theory is somewhat abstract, as it is associated with the game of chess and bridge or waging wars. In fact, the implications of this discipline are very deep Game theory was developed by the Hungarian mathematician genius John von Neumann (1903-1957) and is a relatively young mathematical discipline.
In the future, game theory was supplemented by such developments as the Nash equilibrium (named after the mathematician John Nash). A Nash equilibrium occurs when none of the players can improve their position unless their opponents change their strategies. Each player's strategy is the best response to his opponent's strategy. Sometimes the Nash equilibrium is also called a non-cooperative equilibrium, since the participants make their choice without entering into any agreements with each other and without taking into account any other considerations (the interests of society or the interests of other parties) than their own benefit.
The equilibrium of a perfectly competitive market is also a Nash equilibrium, or non-cooperative equilibrium, in which each firm and each consumer makes decisions based on already existing prices as independent of his will. We already know that under conditions where every firm seeks to maximize profits and every consumer maximizes utility, equilibrium occurs when prices equal marginal cost and profits zero. " Mamaeva L.N. Institutional Economics: A Course of Lectures - M .: Publishing and Trade Corporation "Dashkov and K", 2012. - 200 p.
Recall the concept of the "invisible hand" of Adam Smith: "In pursuing his own interests, he (the individual) often contributes to the prosperity of society to a greater extent than if he consciously sought it" Smith A. A study on the nature and causes of the wealth of nations // Anthology of economic classics . - M.: Ekonov-Klyuch, 19931. The paradox of the "invisible hand" lies in the fact that, although everyone acts as an independent force, in the end society remains the winner. At the same time, the competitive equilibrium is also a Nash equilibrium in the sense that no one has a reason to change their strategy if everyone else adheres to theirs. In a perfectly competitive economy, non-cooperative behavior is cost-effective from the point of view of the interests of society.
On the contrary, when the members of a certain group decide to cooperate and jointly arrive at a monopoly price, such behavior will be detrimental to economic efficiency. The state is forced to create antimonopoly legislation and thereby reason with those who are trying to raise prices and divide the market. However, disunity in behavior is not always cost-effective. Rivalry between firms leads to low prices and competitive output. The "invisible hand" has an almost magical effect on perfectly competitive markets: the efficient allocation of resources occurs as a result of the actions of individuals striving to maximize profits.
However, in many cases, non-cooperative behavior leads to economic inefficiency or even poses a threat to society (for example, the arms race). Non-cooperative behavior on the part of both the US and the USSR forced both sides to invest heavily in the military field and led to the creation of an arsenal of almost 100,000 nuclear warheads. There is also the fear that America's excessive availability of weapons could spark a kind of internal arms race. Some people arm themselves against others - and this "running race" can continue indefinitely. Here comes into play a completely "visible hand" that directs this destructive contest and has nothing to do with the "invisible hand" of Adam Smith. Another important economic example is the "games of pollution" (environment). Here the object of our attention will be such a side effect as pollution. If firms never asked anyone what to do, any of them would rather create pollution than install expensive cleaners. If, on the other hand, some firm, out of noble motives, decided to reduce harmful emissions, then the costs, and consequently, the prices of its products, would increase, and demand would fall. It is quite possible that this company would simply go bankrupt. Living in a brutal world of natural selection, firms would rather stay in Nash equilibrium. No firm can increase profits by reducing pollution.
Engaged in a deadly economic game, every uncontrolled, profit-maximizing steel firm will produce water and air pollution. If any firm tries to clean up its emissions, it will thereby be forced to raise prices and incur losses. Non-cooperative behavior will establish a Nash equilibrium under high outlier conditions. The government can take steps to shift the balance. In this position, the pollution will be negligible, but the profits will remain the same. Mamaeva L.N. Institutional Economics: A Course of Lectures - M.: Publishing and Trade Corporation "Dashkov and K", 2012. - 203 p.
Pollution games are one of the cases where the "invisible hand" mechanism does not work. This is a situation where the Nash equilibrium is inefficient. Sometimes these out-of-control games become threatening and the government can intervene. By establishing a system of fines and emission quotas, the government can encourage firms to choose a low pollution outcome. Firms earn exactly the same as before, with large emissions, and the world becomes somewhat cleaner.
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One can try to explain this syndrome in terms of game theory. It has become customary in the modern economy to separate these types of policies. The Central Bank of America - the Federal Reserve System - determines monetary policy independently of the government by setting interest rates. Fiscal policy, taxes and spending are in charge of the legislative and executive branches. However, each of these policies has different objectives. The central bank seeks to limit the growth of the money supply and ensure low inflation.
Arthur Berne, specialist in economic cycles and a former head of the Fed, wrote: "Central bank officials tend, by tradition, and perhaps by virtue of their personal make-up, to keep prices in check. Their hatred of inflation flares up even more after interacting with like-minded people in private financial circles." The authorities in charge of fiscal policy, however, are more concerned with issues such as full employment, their own popularity, keeping taxes low, and upcoming elections.
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They choose "big budget deficit". On the other hand, the central bank is trying to reduce inflation, is not influenced by trade unions and lobbying groups, and chooses "high interest rates." The result is a non-cooperative equilibrium with moderate inflation and unemployment but low investment.
It is possible that it was thanks to the "budget-money game" that President Clinton put forward an economic program to reduce the budget deficit, lower interest rates and expand investment.
There are different ways to describe games. One of them is that all possible strategies of the players are considered and the payoffs corresponding to any possible combination of strategies of the players are determined. A game described in this way is called game in normal form.
The normal form of the game of two participants consists of two payoff matrices showing how much each player will receive for any of the possible pairs of strategies. Usually these matrices are expressed in the form of a single matrix, which is called bimatrix. The elements of the bimatrix are pairs of numbers, the first of which determines the payoff of the first player, and the second determines the payoff of the second. The first player (the state) chooses one of m strategies, and each strategy corresponds to a row of the matrix I (i= 1,…,m). The second player (business) chooses one of n strategies, with each strategy corresponding to a column of matrix j (j= 1,…,n). A pair of numbers at the intersection of a row and a column that correspond to the strategies chosen by the players, shows the amount of payoff for each of them. In general, if player I chooses the strategy i and player II - strategy j, then the payoffs of the first and second players are respectively equal to and (i= 1,…,m; j= 1,…,n), where m,n is the number of finite strategies of players I and II, respectively. It is assumed that each of the players knows all elements of the payoff bimatrix. In this case, their strategy is called definite and has a finite number of options.
If the player does not know any options for the opponent's strategies (matrix elements), then the game is called indeterminate and can have an infinite number of options (strategies).
There are other classes of games where players win and lose at the same time.
Antagonistic games of two persons are related to the fact that one of the players wins exactly as much as the other loses. In such games, the interests of its players are directly opposed to each other.
As an example, consider a game in which two players participate, each of them has two strategies. The payoffs of each of the players are determined by the following rules: if both players choose strategies with the same numbers (player I - , player II -), then the first player wins, and the second loses (the state raises taxes - business pays them, i.e. the state's gain determines business loss) if both players choose different strategies (player I - i 1 player II - j 2 then the first loses and the second wins (the state raises taxes on business - business evades them; state loses - business wins).
Game theory is the theory of mathematical models of such phenomena in which the participants ("players") have different interests and have more or less freely chosen paths (strategies) to achieve their goals. In most works on game theory, it is assumed that the interests of game participants are quantifiable and are real functions of situations, i.e. a set of strategies obtained when each of the players chooses some of his strategies. To obtain results, it is necessary to consider one or another class of games, distinguished by certain restrictive assumptions. Such restrictions can be imposed in several ways.
Can be distinguished several ways (ways) of imposing restrictions.
1. Restrictions on the possibilities of the relationship between players. The simplest case is when the players act completely disconnected and cannot consciously help or hinder each other by action or inaction, information or disinformation. This state of affairs inevitably occurs when only two players (state and business) participate in the game, having diametrically opposed interests: an increase in the payoff of one of them means a decrease in the payoff of the other, and, moreover, by the same amount, provided that the payoffs of both players are expressed in the same units of measurement. Without loss of generality, we can take the total payoff of both players zero and treat the gain of one of them as the loss of the other.
These games are called antagonistic (or zero-sum games, or two-person zero games). They assume that there can be no relationships between players, no compromises, exchanges of information and other resources, by the very nature of things, in the essence of the game, since each message received by a player about the intentions of another can only increase the payoff of the first player and thereby increase losing his opponent.
Thus, we conclude that in antagonistic games, players can not have direct relationships and, at the same time, be in a state of play (confrontation) in relation to each other.
2. Constraints or simplifying assumptions on the set of player strategies. In the simplest case, these sets of strategies are finite, which eliminates situations associated with possible coincidences (convergences) in the sets of strategies and eliminates the need to introduce any technology on the sets.
Games in which the sets of strategies of each player are finite are called end games.
3. Suggestions about the internal structure of each strategy, i.e. about its content. So, for example, as strategies, one can consider functions of time (continuous or discrete), whose values are the actions of the player at the corresponding moment. These and similar games are usually called dynamic (positional).
The limitations of the strategies of the players can also be their objective functions, i.e. determination of the goals to be achieved by this or that strategy. It can be assumed that the limitations on the strategy are also related to the ways to achieve these goals in certain time intervals, for example, the desire of businesses to achieve a reduction in the amount of mandatory sales of foreign exchange earnings over the next three months (or one year). If no assumptions about the nature of the strategies are made, then they are considered to be some abstract set. Such games in the simplest formulation of the question are called games in normal form.
Finite antagonistic games in normal form are called matrix. This name is explained by the possibility of the following interpretation of games of this type. We will understand the strategies of the first player (player I - the state) as rows of some matrix, and the strategies of the second player (player II - business) - as its columns. For brevity, the strategies of the players are called not the rows or columns of the matrix themselves, but their numbers. Then the game situations are the cells of this matrix, which are located at the intersections of each row with each of the columns. Filling in these cells-situations with numbers describing the payoffs of player I in these situations, we complete the task of the game. The resulting matrix is called game payoff matrix, or game matrix. In view of the antagonism of the matrix game, the payoff of player II in each situation is completely determined by the payoff of player I in this situation, differing from him only in sign. Therefore, no additional indications about the payoff function of player II in the matrix game are required.
A matrix with m rows and n columns is called an (m*n) matrix, and a game with this matrix is called an (m*n) game.
Process (m * n) - games with a matrix can be represented as follows:
Player I fixes the number of row i, and player II - the number of column j, after which the first player receives from his opponent the sum
The goal of player I in the matrix game is to get the maximum payoff, the goal of player II is to give player I the minimum payoff.
Let player I (the state) choose some strategy i of his own. Then, in the worst case, he will get the payoff min . In game theory, players are assumed to be cautious, counting on the least favorable turn of events for themselves.
This state of affairs, which is the least favorable for player I, can occur, for example, when strategy i becomes known to player II (business). Anticipating this possibility, player I must choose his strategy in such a way as to maximize this minimum payoff:
min = max min (I)
The value on the right side of the equality is the guaranteed payoff of player I. Player II (business) must choose a strategy such that
max = min max (II)
The value on the right side of the equality is the payoff of player I, more than which he right action can't get an enemy.
The actual payoff of player I should, with reasonable actions of the partners, be in the interval between the payoff values in the first and second cases. If these values are equal, then the payoff of player I is a well-defined number, the games themselves are called quite definite. The payoff of player I is called the value of the game, and it is equal to an element of the matrix.
Players may have additional options - choosing their strategies randomly and independently of each other (the strategies correspond to the rows and columns of the matrix). The random choice of strategies by the player is called mixed country this player's tags. In the (m*n) game, the mixed strategies of player I are determined by the sets of probabilities: X = (,…), with which this player chooses his initial, pure strategies.
The theory of matrix games is based on Neumann's active strategies theorem: "If one of the players adheres to his optimal strategy, then the payoff remains unchanged and equal to the cost of the game, regardless of what the other player does, if he does not go beyond his active strategies (i.e., e. uses any of them in their pure form or mixes them in any proportions "Neumann J. Contributions to the theory of games. 1995 .. - 155 p.). Note that active is the pure strategy of the player that is included in his optimal mixed strategy with non-zero probability.
The main goal of the game is finding the optimal strategy for both players, if not with the maximum gain for one of them, then with the minimum loss for both. The method of finding optimal strategies often yields more than is necessary for practical purposes. In a matrix game, it is not necessary for the player to know all of his optimal structures, since they are all interchangeable and for a successful game, it is enough for the player to know one of them. Therefore, in relation to matrix games, the question of finding at least one optimal strategy for each of the players is topical.
The Fundamental Theorem on Matrix Games establishes the existence of a game value and optimal mixed strategies for both players. The optimal strategy need not be singular. This is a very important conclusion derived from game theory.
The subject playing the matrix game is characterized by the following qualities:
matrix elements interpreted as cash payments and, accordingly, their gain and loss evaluated in monetary form;
each of players applies the function to these elements utility;
in the game, each player acts as if his opponent's utility function had exactly the same effect on the matrix, i.e. everyone looks at the game "from their own bell towers".
These assumptions lead to zero-sum games in which there are relations of cooperation, bargaining and other types of interactions between players like before the start games, as well as in the process. Mamaeva L.N. Institutional Economics: A Course of Lectures - M.: Publishing and Trade Corporation "Dashkov and K", 2012. - 210 - 211s.
A generalization of game theory aimed at including others analysis capabilities, leads to interesting but difficult tasks. When developing game theory, it is necessary to apply the utility function not only to monetary outcomes, but also to amounts with expected future outcomes. These the assumptions are debatable, but they exist. In this case, we proceed from the fact that this assumption about such an operation It has resemblance with behavior players in certain decision situations and allows for the possibility that the way playing the game by this player depends on the state of his capital at the time conducting them games.
Let's look at it next example. Let be the first player at the start of the game G has a capital of x dollars. Then his capital at the end games will equals + x, where is the actual payoff he receives from the game. The utility he ascribes to such exodus, equals f (+ x), where f is the utility function.
These few examples illustrate only part of the vast variety of results that can be obtained using game theory. This branch of economics is an extremely useful tool (for economists and other social scientists) to analyze situations in which a small number of people are well-informed and try to outwit each other in the markets, in politics, or in military operations.
The mathematical theory of games that arose in the forties of the XX century is most often used in economics. But how can we use the concept of games to model the behavior of people in society? Why do economists study what angle football players take more often, and how to win at Rock, Paper, Scissors, Danil Fedorovykh, Senior Lecturer at the HSE Department of Microeconomic Analysis, told in his lecture.
John Nash and the blonde at the bar
A game is any situation in which the agent's profit depends not only on his own actions, but also on the behavior of other participants. If you play solitaire at home, from the point of view of an economist and game theory, this is not a game. It implies that there must be a conflict of interest.
In the movie A Beautiful Mind about John Nash, the Nobel laureate in economics, there is a scene with a blonde woman in a bar. It shows the idea for which the scientist received the award - this is the idea of the Nash equilibrium, which he himself called the control dynamics.
A game- any situation in which the payoffs of agents depend on each other.Strategy - a description of the player's actions in all possible situations.
The outcome is a combination of the chosen strategies.
So, from the point of view of theory, only men are the players in this situation, that is, those who make the decision. Their preferences are simple: a blonde is better than a brunette, and a brunette is better than nothing. You can act in two ways: go to the blonde or to "your" brunette. The game consists of a single move, decisions are made simultaneously (that is, you can’t see where the others went, and then be like yourself). If a girl rejects a man, the game ends: it is impossible to return to her or choose another one.
What is the likely outcome of this game situation? That is, what is its stable configuration, from which everyone will understand that they made the best choice? First, as Nash correctly points out, if everyone goes to the blonde, it will not end well. Therefore, further the scientist suggests that everyone needs to go to the brunettes. But then, if it is known that everyone will go to brunettes, he should go to the blonde, because she is better.
This is where the real balance lies - an outcome in which one goes to the blonde, and the rest to the brunettes. This may seem unfair. But in a situation of balance, no one can regret their choice: those who go to brunettes understand that they would not get anything from a blonde anyway. Thus, the Nash equilibrium is a configuration in which no one individually wants to change the strategy chosen by everyone. That is, reflecting at the end of the game, each participant understands that even knowing how others are like, he would do the same. In another way, you can call it an outcome, where each participant optimally responds to the actions of the others.
"Rock Paper Scissors"
Consider other games for balance. For example, in "Rock, Paper, Scissors" there is no Nash equilibrium: in all its probable outcomes, there is no option in which both participants would be happy with their choice. However, there is a World Championship and a World Rock Paper Scissors Society that collects game statistics. Obviously, you can increase your chances of winning if you know something about the usual behavior of people in this game.
Pure strategy in a game is a strategy in which a person always plays the same way, choosing the same moves.
According to the World RPS Society, stone is the most frequently chosen move (37.8%). Paper put 32.6%, scissors - 29.6%. Now you know that you need to choose paper. However, if you are playing with someone who also knows this, you no longer need to choose paper, because the same is expected of you. There is a famous case: in 2005, two auction houses Sotheby's and Christie's decided who would get a very large lot - a collection of Picasso and Van Gogh with a starting price of $ 20 million. The owner invited them to play Rock, Paper, Scissors, and representatives of the houses sent him their options by email. Sotheby's, as they later said, without much thought, chose paper. Won Christie's. Making a decision, they turned to an expert - the 11-year-old daughter of one of the top managers. She said: “The stone seems to be the strongest, which is why most people choose it. But if we play with a not completely stupid beginner, he will not throw the stone, he will expect us to do it, and he will throw the paper. But we will think ahead and throw away the scissors.”
In this way, you can think ahead, but it will not necessarily lead you to victory, because you may not know about the competence of your opponent. Therefore, sometimes, instead of pure strategies, it is more correct to choose mixed ones, that is, to make decisions randomly. So, in "Rock, Paper, Scissors" the equilibrium, which we have not found before, is precisely in mixed strategies: choose each of the three options with a probability of one-third. If you choose a stone more often, the opponent will adjust his choice. Knowing this, you will correct yours, and the balance will not come out. But none of you will begin to change behavior if everyone simply chooses rock, scissors, or paper with the same probability. This is because in mixed strategies it is impossible to predict your next move based on previous actions.
Mixed strategy and sports
There are many more serious examples of mixed strategies. For example, where to serve in tennis or take / take a penalty in football. If you don't know anything about your opponent, or you're just constantly playing against different opponents, the best strategy is to do it more or less randomly. Professor of the London School of Economics Ignacio Palacios-Huerta in 2003 published a paper in the American Economic Review, the essence of which was to find the Nash equilibrium in mixed strategies. Palacios-Huerta chose football as the subject of his research and, in connection with this, watched more than 1,400 penalty kicks. Of course, in sports, everything is arranged more cunningly than in Rock, Paper, Scissors: it takes into account the strong leg of the athlete, hitting different angles when hit with full force and the like. Nash equilibrium here consists in calculating options, that is, for example, determining the corners of the goal that you need to hit in order to win with a greater probability, knowing your weaknesses and strengths. The statistics for each football player and the equilibrium found in it in mixed strategies showed that football players act approximately as economists predict. It is hardly worth arguing that people who take penalties have read textbooks on game theory and dealt with rather difficult mathematics. Most likely, there are different ways to learn how to behave optimally: you can be a brilliant footballer and feel what to do, or you can be an economist and seek balance in mixed strategies.
In 2008, Professor Ignacio Palacios-Huerta met Abraham Grant, the Chelsea manager who was then playing in the Champions League final in Moscow. The scientist wrote a note to the coach with recommendations for a penalty shootout, which concerned the behavior of the opponent's goalkeeper - Edwin van der Sar from Manchester United. For example, according to statistics, he almost always parried shots at an average level and more often rushed to the natural side for a penalty shooter. As we have defined above, it is still more correct to randomize your behavior taking into account knowledge about the opponent. When the score was already 6-5 on penalties, Nicolas Anelka, the Chelsea striker, had to score. Pointing to the right corner before hitting, van der Sar seemed to ask Anelka if he was going to hit there.
The bottom line is that all of Chelsea's previous shots have been delivered to the right of the puncher. We don't know exactly why, perhaps because of an economist's advice to hit in an unnatural direction for them, because according to statistics, van der Sar is less ready for this. Most of the Chelsea players were right-handed: hitting the unnatural right corner for themselves, all of them, except Terry, scored. Apparently, the strategy was to Anelka struck there too. But van der Sar seems to understand this. He acted brilliantly: he pointed to the left corner, saying, “Is he going to beat him there?”, from which Anelka, probably, was horrified, because he was guessed. At the last moment, he decided to act differently, hit in a natural direction for himself, which was what Van der Sar needed, who took this blow and ensured Manchester victory. This situation teaches random choice, because otherwise your decision can be calculated, and you will lose.
"Prisoner's Dilemma"
Probably the most famous game, with which university courses on game theory begin, is the Prisoner's Dilemma. According to legend, two suspects of a serious crime were caught and locked in different cells. There is evidence that they kept weapons, and this allows them to be imprisoned for some short period. However, there is no evidence that they committed this terrible crime. The investigator tells each individual about the conditions of the game. If both criminals confess, both will go to jail for three years. If one confesses, and the accomplice remains silent, the one who confesses will come out immediately, and the second will be imprisoned for five years. If, on the contrary, the first does not confess, and the second turns him in, the first will sit in jail for five years, and the second will be released immediately. If no one confesses, both will go to jail for a year for possession of weapons.
The Nash equilibrium here is in the first combination, when both suspects are not silent and both sit down for three years. The reasoning of each is as follows: “If I speak, I will sit for three years, if I remain silent, for five years. If the second one is silent, it’s better for me to say too: it’s better not to sit down than to sit down for a year. This is the dominant strategy: it is profitable to speak, regardless of what the other is doing. However, it has a problem - the presence of a better option, because sitting down for three years is worse than sitting down for a year (if we consider the story only from the point of view of the participants and do not take into account moral issues). But it is impossible to sit down for a year, because, as we understood above, it is unprofitable for both criminals to remain silent.
Pareto improvement
There is a famous metaphor about the invisible hand of the market, which belongs to Adam Smith. He said that if the butcher tries to earn money for himself, it will be better for everyone: he will make delicious meat that the baker will buy with money from the sale of rolls, which he, in turn, will also have to make tasty so that they are sold . But it turns out that this invisible hand does not always work, and there are a lot of such situations when everyone acts for himself, and everyone is bad.
Therefore, sometimes economists and game theorists think not about the optimal behavior of each player, that is, not about the Nash equilibrium, but about the outcome that will be better for the whole society (in the "Dilemma" society consists of two criminals). From this point of view, the outcome is effective when there is no Pareto improvement, that is, it is impossible to make someone better without making others worse. If people simply exchange goods and services, this is a Pareto improvement: they do it voluntarily, and it is unlikely that anyone will feel bad about it. But sometimes, if you just let people interact and not even interfere, what they end up with will not be Pareto optimal. This is what happens in the Prisoner's Dilemma. In it, if we allow everyone to act in a way that is beneficial to them, it turns out that everyone is bad for it. It would be better for everyone if everyone acted not optimally for themselves, that is, they kept silent.
Tragedy of the community
Prisoner's Dilemma is a toy stylized story. It is unlikely that you would expect to be in a similar situation, but similar effects are everywhere around us. Consider the "Dilemma" with a large number of players, it is sometimes called the tragedy of the community. For example, there are traffic jams on the roads, and I decide how to go to work: by car or by bus. The rest do the same. If I go by car and everyone decides to do the same, there will be a traffic jam, but we will get there in comfort. If I go by bus, there will still be a traffic jam, but I will be uncomfortable and not very fast, so this outcome is even worse. If, on average, everyone takes the bus, then I, having done the same, will get there pretty quickly without a traffic jam. But if under such conditions I go by car, I will also get there quickly, but also with comfort. So, the presence of a traffic jam does not depend on my actions. The Nash equilibrium here is in a situation where everyone chooses to drive. Whatever the rest do, it’s better for me to choose a car, because it’s not known whether there will be a traffic jam or not, but in any case I will get there in comfort. This is the dominant strategy, so in the end everyone drives a car, and we have what we have. The task of the state is to make traveling by bus the best option for at least some, so there are paid entrances to the center, parking lots and so on.
Another classic story is the voter's rational ignorance. Imagine that you do not know the outcome of the elections in advance. You can study the program of all candidates, listen to the debate and then vote for the best one. The second strategy is to come to the polling station and vote at random or for the one who was shown on TV more often. What behavior is optimal if my vote never determines who wins (and in a country of 140 million people, one vote will never decide anything)? Of course, I want the country to have a good president, but I know that no one else will scrutinize candidate programs carefully. Therefore, do not waste time on this - the dominant strategy of behavior.
When you are called to come to a subbotnik, it will not depend on anyone individually whether the yard will be clean or not: if I go out alone, I will not be able to clean everything, or if everyone comes out, then I will not go out, because everything is without me removed. Another example is shipping in China, which I learned about in Steven Landsburg's excellent book The Couch Economist. 100-150 years ago, a method of transporting goods was common in China: everything was folded into a large body, which was dragged by seven people. Customers paid if the goods were delivered on time. Imagine that you are one of these six. You can push hard and pull as hard as you can, and if everyone does that, the load will arrive on time. If someone alone does not do this, everyone will also arrive on time. Everyone thinks: “If everyone else is pulling properly, why should I do it, and if everyone else is not pulling with all their strength, then I can’t change anything.” As a result, with the delivery time, everything was very bad, and the movers themselves found a way out: they began to hire a seventh and pay him money for whipping lazy people with a whip. The very presence of such a person forced everyone to work hard, because otherwise everyone would fall into a bad balance, from which no one could get out of profitably.
The same example can be observed in nature. A tree growing in a garden differs from one growing in a forest in its crown. In the first case, it surrounds the entire trunk, in the second, it is only at the top. In the forest, this is the Nash equilibrium. If all the trees agreed and grew equally, they would equally distribute the number of photons, and everyone would be better off. But it is unprofitable for anyone in particular to do so. Therefore, each tree wants to grow a little higher than the others.
Commitment device
In many situations, one of the participants in the game may need a tool that will convince the others that he is not bluffing. It's called a commitment device. For example, the law of some countries prohibits the payment of ransoms to kidnappers in order to reduce the motivation of criminals. However, this legislation often does not work. If your relative has been captured and you have the ability to save him by circumventing the law, you will. Imagine a situation where the law can be circumvented, but the relatives turned out to be poor and they have nothing to pay the ransom. The perpetrator in this situation has two options: release or kill the victim. He doesn't like to kill, but he doesn't like prison anymore. The released victim, in turn, can either testify so that the kidnapper is punished, or remain silent. The best outcome for the perpetrator is to let go of the victim who won't turn him in. The victim wants to be released and testify.
The balance here is that the terrorist does not want to be caught, which means that the victim dies. But this is not a Pareto equilibrium, because there is a variant in which everyone is better - the victim at large remains silent. But for this it is necessary to do so that it would be beneficial for her to remain silent. Somewhere I read the option when she can ask the terrorist to arrange an erotic photo shoot. If the criminal is imprisoned, his accomplices will post photos on the Internet. Now, if the kidnapper stays free, that's bad, but the photos in the public domain are even worse, so it's a balance. It's a way for the victim to stay alive.
Other game examples:
Bertrand model
Since we are talking about economics, consider an economic example. In Bertrand's model, two stores sell the same product, buying it from the manufacturer at the same price. If the prices in stores are the same, then their profits are approximately the same, because then buyers choose the store randomly. The only Nash equilibrium here is to sell the product at cost. But stores want to make money. Therefore, if one sets the price of 10 rubles, the second will reduce it by a penny, thereby doubling his revenue, since all buyers will go to him. Therefore, it is beneficial for market participants to reduce prices, thereby distributing profits among themselves.
Passage on a narrow road
Consider examples of choosing between two possible equilibria. Imagine that Petya and Masha are driving towards each other along a narrow road. The road is so narrow that they both need to pull over. If they decide to turn left or right away from them, they will simply disperse. If one turns to the right and the other to the left, or vice versa, an accident will happen. How to choose where to go? To help find balance in such games, there are, for example, rules of the road. In Russia, everyone needs to turn right.
In the Chiken game, when two people drive towards each other at high speed, there are also two equilibria. If both turn to the side of the road, a situation arises called Chiken out, if both do not turn off, then they die in a terrible accident. If I know that my opponent is driving straight ahead, it is beneficial for me to move out in order to survive. If I know that my opponent will move out, then it is profitable for me to go straight in order to receive 100 dollars later. It is difficult to predict what will actually happen, however, each of the players has their own method to win. Imagine that I fixed the steering wheel so that it cannot be turned, and showed it to my opponent. Knowing that I have no choice, the opponent will bounce.
QWERTY effect
Sometimes it can be very difficult to move from one balance to another, even if it means benefiting everyone. The QWERTY layout was created to slow down typing speed. Because if everyone was typing too fast, the typewriter heads that hit the paper would cling to each other. Therefore, Christopher Sholes posted frequently standing side by side letters as far away as possible. If you go into the keyboard settings on your computer, you can select the Dvorak layout there and type much faster, since there is no problem with analog presses now. Dvorak expected the world to switch to his keyboard, but we still live with QWERTY. Of course, if we switched to the Dvorak layout, the future generation would be grateful to us. We would all put in the effort and relearn, and the result would be an equilibrium in which everyone types fast. Now we are also in balance - in a bad one. But it is not beneficial for anyone to be the only one who retrains, because it will be inconvenient to work on any computer other than a personal one.
In recent years, the importance of game theory has increased significantly in many areas of economic and social sciences. In economics, it is applicable not only to solve general business problems, but also to analyze the strategic problems of enterprises, develop organizational structures and incentive systems.
Already at the time of its inception, which is considered the publication in 1944 of the monograph by J. Neumann and O. Morgenstern "Game Theory and Economic Behavior", many predicted a revolution in economic sciences through the use of a new approach. These predictions could not be considered too bold, since from the very beginning this theory claimed to describe rational decision-making behavior in interrelated situations, which is typical for most current problems in economic and social sciences. Thematic areas such as strategic behavior, competition, cooperation, risk and uncertainty are key in game theory and are directly related to managerial tasks.
Early work on game theory was characterized by simplistic assumptions and a high degree of formal abstraction, which made them unsuitable for practical use. Over the past 10-15 years, the situation has changed dramatically. Rapid progress in the industrial economy has shown the fruitfulness of game methods in the applied field.
Recently, these methods have penetrated into management practice. It is likely that game theory, along with the theories of transaction costs and “patron-agent”, will be perceived as the most economically justified element of organization theory. It should be noted that already in the 80s, M. Porter introduced some key concepts of the theory, in particular, such as “strategic move” and “player”. True, an explicit analysis associated with the concept of equilibrium was still absent in this case.
Fundamentals of game theory
To describe a game, you must first identify its participants. This condition is easily fulfilled when it comes to ordinary games such as chess, canasta, etc. The situation is different with “market games”. Here it is not always easy to recognize all the players, i.e. existing or potential competitors. Practice shows that it is not necessary to identify all the players, it is necessary to identify the most important ones.
Games cover, as a rule, several periods during which players take consecutive or simultaneous actions. These actions are denoted by the term "move". Actions can be related to prices, sales volumes, research and development costs, and so on. The periods during which the players make their moves are called game stages. The moves chosen at each stage ultimately determine the “payoff” (win or loss) of each player, which can be expressed in wealth or money (predominantly discounted profits).
Another basic concept of this theory is the player's strategy. It is understood as possible actions that allow the player at each stage of the game to choose from a certain number of alternative options such a move that seems to him to be the “best answer” to the actions of other players. Regarding the concept of strategy, it should be noted that the player determines his actions not only for the stages that a particular game has actually reached, but also for all situations, including those that may not occur in the course of this game.
The form in which the game is presented is also important. Usually, a normal, or matrix, form and an expanded one, given in the form of a tree, are distinguished. These forms for a simple game are shown in Fig. 1a and 1b.
To establish the first connection with the sphere of control, the game can be described as follows. Two enterprises producing homogeneous products are faced with a choice. In one case, they can gain a foothold in the market by setting a high price, which will provide them with an average cartel profit P K . When entering into a tough competition, both make a profit П W . If one of the competitors sets a high price, and the second sets a low price, then the latter realizes monopoly profit P M , while the other incurs losses P G . A similar situation can, for example, arise when both firms have to announce their price, which cannot subsequently be revised.
In the absence of stringent conditions, it is beneficial for both enterprises to charge a low price. The “low price” strategy is dominant for any firm: no matter what price a competing firm chooses, it is always preferable to set a low price itself. But in this case, firms face a dilemma, since profit P K (which for both players is higher than profit P W) is not achieved.
The strategic combination “low prices/low prices” with the corresponding payoffs is a Nash equilibrium, in which it is unprofitable for any of the players to deviate separately from the chosen strategy. Such a concept of equilibrium is fundamental in resolving strategic situations, but under certain circumstances it still needs to be improved.
As for the above dilemma, its resolution depends, in particular, on the originality of the players' moves. If the enterprise has the opportunity to revise its strategic variables (in this case, the price), then a cooperative solution to the problem can be found even without a rigid agreement between the players. Intuition suggests that with repeated contacts of players, there are opportunities to achieve acceptable “compensation”. Thus, under certain circumstances, it is inappropriate to seek short-term high profits through price dumping if a “price war” may arise in the future.
As noted, both figures characterize the same game. Presenting the game in normal form usually reflects “synchronicity”. However, this does not mean “simultaneity” of events, but indicates that the choice of strategy by the player is carried out in the absence of knowledge about the choice of strategy by the opponent. With an expanded form, such a situation is expressed through an oval space (information field). In the absence of this space, the game situation acquires a different character: first, one player should make the decision, and the other could do it after him.
Application of game theory for making strategic management decisions
Examples here are decisions regarding the implementation of a principled pricing policy, entry into new markets, cooperation and the creation of joint ventures, identifying leaders and performers in the field of innovation, vertical integration etc. The provisions of this theory can, in principle, be used for all types of decisions, if their adoption is influenced by others. characters. These persons, or players, need not be market competitors; their role can be sub-suppliers, leading customers, employees of organizations, as well as colleagues at work.
quadrants 1 and 2 characterize a situation where the reaction of competitors does not have a significant impact on the company's payments. This happens when the competitor has no motivation (field 1 ) or opportunities (field 2 ) strike back. Therefore, there is no need for a detailed analysis of the strategy of motivated actions of competitors.
A similar conclusion follows, although for a different reason, for the situation reflected by the quadrant 3 . Here, the reaction of competitors could have a great effect on the firm, but since its own actions cannot greatly affect the payments of a competitor, one should not be afraid of his reaction. Niche entry decisions can be cited as an example: under certain circumstances, large competitors have no reason to react to such a decision of a small firm.
Only the situation shown in the quadrant 4 (the possibility of retaliatory steps of market partners), requires the use of the provisions of game theory. However, only the necessary but not sufficient conditions are reflected here to justify the application of the base of game theory to the fight against competitors. There are times when one strategy unquestionably dominates all others, no matter what the competitor does. If we take the drug market, for example, it is often important for a firm to be the first to introduce a new product to the market: the profit of the “pioneer” turns out to be so significant that all other “players” just have to step up innovation activity faster.
The same game situation can also be represented in normal form (Fig. 4). Two states are designated here – “entry/friendly reaction” and “non-entry/aggressive reaction”. It is obvious that the second equilibrium is untenable. It follows from the detailed form that it is inappropriate for a company already established in the market to react aggressively to the emergence of a new competitor: with aggressive behavior, the current monopolist receives 1 (payment), and with friendly behavior - 3. The outsider company also knows that it is not rational for the monopolist start actions to oust it, and therefore it decides to enter the market. The outsider company will not suffer the threatened losses in the amount of (-1).
Such a rational balance is characteristic of a "partially improved" game, which deliberately excludes absurd moves. Such equilibrium states are, in principle, fairly easy to find in practice. Equilibrium configurations can be identified using a special algorithm from the field of operations research for any finite game. The decision maker proceeds as follows: first, the “best” move in the last stage of the game is chosen, then the “best” move in the previous stage is selected, taking into account the choice in the last stage, and so on, until the initial node of the tree is reached games.
How can companies benefit from game theory-based analysis? There is, for example, a case of a conflict of interests between IBM and Telex. In connection with the announcement of the latter's preparatory plans to enter the market, a "crisis" meeting of the IBM management was held, at which measures were analyzed aimed at forcing the new competitor to abandon its intention to penetrate the new market.
Telex apparently became aware of these events. Game theory based analysis showed that the threats of IBM due to high costs are unfounded.
This shows that it is useful for companies to think explicitly about possible reactions game partners. Isolated economic calculations, even based on the theory of decision-making, are often, as in the situation described, limited. For example, an outsider company might choose the “no-entry” move if preliminary analysis convinced it that market penetration would provoke an aggressive response from the monopolist. In this case, in accordance with the criterion of the expected cost, it is reasonable to choose the “non-entry” move with the probability of an aggressive response being 0.5.
For the enterprise 1 it would be best if the company 2 abandoned competition. His benefit in this case would be 3 (payments). It is highly likely that the company 2 would win the competition when the enterprise 1 would accept a cut investment program, and the enterprise 2 - wider. This position is reflected in the upper right quadrant of the matrix.
An analysis of the situation shows that equilibrium occurs at high costs for research and development of the enterprise 2 and low enterprises 1 . In any other scenario, one of the competitors has a reason to deviate from the strategic combination: for example, for the enterprise 1 a reduced budget is preferable if the business 2 refuse to participate in the competition; at the same time the enterprise 2 It is known that at low costs of a competitor it is profitable for him to invest in R&D.
An enterprise with a technological advantage may resort to situation analysis based on game theory in order to ultimately achieve an optimal result for itself. By means of a certain signal, it must show that it is ready to carry out large expenditures on R&D. If such a signal is not received, then for the enterprise 2 it is clear that the company 1 chooses the low cost option.
The reliability of the signal should be evidenced by the obligations of the enterprise. In this case, it may be the decision of the enterprise 1 about purchasing new laboratories or hiring additional research staff.
From the point of view of game theory, such obligations are tantamount to changing the course of the game: the situation of simultaneous decision-making is replaced by the situation of successive moves. Company 1 firmly demonstrates the intention to make large expenditures, the enterprise 2 registers this step and has no more reason to participate in the rivalry. The new equilibrium follows from the scenario “non-participation of the enterprise 2 ” and “high costs for research and development of the enterprise 1 ”.
An important contribution to the use of game theory is made by experimental work. Many theoretical calculations are worked out in the laboratory, and the results obtained serve as an impulse for practitioners. Theoretically, it was found out under what conditions it is expedient for two selfish partners to cooperate and achieve better results for themselves.
This knowledge can be used in the practice of enterprises to help two firms achieve a win-win situation. Today, gaming-trained consultants quickly and unambiguously identify opportunities that businesses can take advantage of to secure stable and long-term contracts with customers, sub-suppliers, development partners, and more.
Problems of practical application
in management
However, it should also be pointed out that there are certain limits to the application of the analytical tools of game theory. In the following cases, it can only be used if additional information is obtained.
Firstly, this is the case when enterprises have different ideas about the game they are participating in, or when they are not sufficiently informed about each other's capabilities. For example, there may be unclear information about a competitor's payments (cost structure). If not too complex information is characterized by incompleteness, then it is possible to operate with a comparison of similar cases, taking into account certain differences.
Second, game theory is difficult to apply to many equilibria. This problem can arise even during simple games with simultaneous choice of strategic decisions.
Thirdly, if the situation of making strategic decisions is very complex, then players often cannot choose the best options for themselves. It's easy to imagine more difficult situation market penetration than the one discussed above. For example, several enterprises may enter the market at different times, or the reaction of enterprises already operating there may be more complex than aggressive or friendly.
It has been experimentally proven that when the game is expanded to ten or more stages, the players are no longer able to use the appropriate algorithms and continue the game with equilibrium strategies.
Nor is the principle underlying assumption of the so-called “common knowledge” underlying game theory, by any means. It says: the game with all the rules is known to the players and each of them knows that all players are aware of what the other partners in the game know. And this situation remains until the end of the game.
But in order for an enterprise to make a decision that is preferable for itself in a particular case, this condition is not always required. Less rigid assumptions, such as “mutual knowledge” or “rationalizable strategies”, are often sufficient for this.
In conclusion, it should be emphasized that game theory is a very complex field of knowledge. When referring to it, one must observe certain caution and clearly know the limits of application. Too simple interpretations, adopted by the firm itself or with the help of consultants, are fraught with hidden danger. Because of their complexity, game theory-based analysis and consultations are only recommended for critical problem areas. The experience of firms shows that the use of appropriate tools is preferable when making one-time, fundamentally important planned strategic decisions, including when preparing large cooperation agreements.
Game theory- a mathematical method for studying optimal strategies in games. The game is understood as a process in which two or more parties participate, fighting for the realization of their interests. Each side has its own goal and uses some strategy, which can lead to a win or a loss - depending on the behavior of other players. Game theory helps to choose the best strategies, taking into account ideas about other participants, their resources and their possible actions.
Game theory is a branch of applied mathematics, more precisely, operations research. Most often, the methods of game theory are used in economics, a little less often in other social sciences - sociology, political science, psychology, ethics and others. Since the 1970s, it has been adopted by biologists to study animal behavior and the theory of evolution. It is of great importance for artificial intelligence and cybernetics, especially with an interest in intelligent agents.
Story.
Optimal solutions or strategies in mathematical modeling were proposed as early as the 18th century. The problems of production and pricing in an oligopoly, which later became textbook examples of game theory, were considered in the 19th century. A. Cournot and J. Bertrand. At the beginning of the XX century. E. Lasker, E. Zermelo, E. Borel put forward the idea mathematical theory conflict of interest.
Mathematical game theory originates from neoclassical economics. The mathematical aspects and applications of the theory were first presented in the classic 1944 book by John von Neumann and Oscar Morgenstern, Game Theory and Economic Behavior. Theory of Games and Economic Behavior ).
This area of mathematics has found some reflection in public culture. In 1998, the American writer and journalist Sylvia Nazar published a book about the fate of John Nash, a Nobel laureate in economics and a scientist in the field of game theory; and in 2001, based on the book, the film A Beautiful Mind was made. Some American television shows, such as "Friend or Foe", "Alias" or "NUMB3RS", periodically refer to the theory in their episodes.
J. Nash in 1949 writes a dissertation on game theory, after 45 years he receives Nobel Prize on economics. J. Nash, after graduating from the Carnegie Polytechnic Institute with two diplomas - a bachelor's and a master's degree - entered Princeton University, where he attended lectures by John von Neumann. In his writings, J. Nash developed the principles of "managerial dynamics". The first concepts of game theory analyzed antagonistic games, when there are losers and players who won at their expense. Nash develops methods of analysis in which all participants either win or lose. These situations are called "Nash equilibrium", or "non-cooperative equilibrium", in a situation the parties use the optimal strategy, which leads to the creation of a stable equilibrium. It is beneficial for the players to maintain this balance, since any change will worsen their situation. These works by J. Nash made a serious contribution to the development of game theory, the mathematical tools of economic modeling were revised. J. Nash shows that A. Smith's classical approach to competition, when it's every man for himself, is not optimal. More optimal strategies are when everyone tries to do better for themselves while doing better for others.
Although game theory originally dealt with economic models, until the 1950s it remained a formal theory within mathematics. But since the 1950s attempts begin to apply the methods of game theory not only in economics, but in biology, cybernetics, technology, and anthropology. During World War II and immediately after it, the military became seriously interested in game theory, who saw it as a powerful tool for investigating strategic decisions.
In 1960-1970. interest in game theory is fading, despite the significant mathematical results obtained by that time. From the mid 1980s. the active practical use of game theory begins, especially in economics and management. Over the past 20 - 30 years, the importance of game theory and interest has grown significantly, some areas of modern economic theory cannot be described without the use of game theory.
A great contribution to the application of game theory was the work of Thomas Schelling, Nobel laureate in economics in 2005, "Strategy of Conflict". T. Schelling considers various "strategies" of the behavior of the participants in the conflict. These strategies coincide with the tactics of conflict management and the principles of conflict analysis in conflictology (this is a psychological discipline) and in conflict management in an organization (management theory). In psychology and other sciences, the word "game" is used in other senses than in mathematics. Some psychologists and mathematicians are skeptical about the use of this term in other senses that have developed before. The culturological concept of the game was given in the work of Johan Huizinga "Homo Ludens" (articles on the history of culture), the author talks about the use of games in justice, culture, ethics .. says that the game is older than the person himself, since animals also play. The concept of a game is found in the concept of Eric Burne "Games that people play, people who play games." These are purely psychological games based on transactional analysis. The concept of the game by J.Hözing differs from the interpretation of the game in the theory of conflicts and the mathematical theory of games. Games are also used for training in business cases, seminars by G. P. Shchedrovitsky, the founder of the organizational and activity approach. During Perestroika in the USSR, G. P. Shchedrovitsky played many games with Soviet managers. In terms of psychological intensity, ODI (organizational-activity games) were so strong that they served as a powerful catalyst for changes in the USSR. Now in Russia there is a whole ODI movement. Critics note the artificial uniqueness of the ODI. The basis of the ODI was the Moscow Methodological Circle (MMC).
Mathematical game theory is now rapidly developing, dynamic games are being considered. However, the mathematical apparatus of game theory is expensive. It is used for legitimate tasks: politics, the economics of monopolies and the distribution of market power, etc. A number of famous scientists have become Nobel laureates in economics for their contribution to the development of game theory, which describes socio-economic processes. J. Nash, thanks to his research in game theory, has become one of the leading experts in the field of conducting " cold war”, which confirms the magnitude of the tasks that game theory deals with.
Nobel laureates in economics for achievements in the field of game theory and economic theory are: Robert Aumann, Reinhard Selten, John Nash, John Harsanyi, William Vickrey, James Mirrlees, Thomas Schelling, George Akerlof, Michael Spence, Joseph Stiglitz, Leonid Hurwitz, Eric Maskin , Roger Myerson, Lloyd Shapley, Alvin Roth.
Application of game theory.
Game theory, as one of the approaches in applied mathematics, is used to study the behavior of humans and animals in various situations. Initially, game theory began to develop within the framework of economic science, making it possible to understand and explain the behavior of economic agents in various situations. Later, the scope of game theory was extended to other social sciences; Currently, game theory is used to explain human behavior in political science, sociology, and psychology. Game-theoretic analysis was first used to describe the behavior of animals by Ronald Fisher in the 1930s (although even Charles Darwin used the ideas of game theory without formal justification). The term "game theory" does not appear in the work of Ronald Fisher. Nevertheless, the work is essentially carried out in line with the game-theoretic analysis. The developments made in economics were applied by John Maynard Smith in the book Evolution and Game Theory. Game theory is not only used to predict and explain behavior; attempts have been made to use game theory to develop theories of ethical or reference behavior. Economists and philosophers have used game theory to better understand good behavior. Generally speaking, the first game-theoretic arguments explaining correct behavior, were expressed by Plato.
Description and modeling.
Initially, game theory was used to describe and model the behavior of human populations. Some researchers believe that by determining the equilibrium in the corresponding games, they can predict the behavior of human populations in a situation of real confrontation. This approach to game theory has recently been criticized for several reasons. First, the assumptions used in simulations are often violated in real life. Researchers may assume that players choose behaviors that maximize their total benefit (the economic man model), but in practice, human behavior often does not match this premise. There are many explanations for this phenomenon - irrationality, discussion modeling, and even different motives of the players (including altruism). The authors of game-theoretic models object to this by saying that their assumptions are analogous to similar assumptions in physics. Therefore, even if their assumptions are not always fulfilled, game theory can be used as a reasonable ideal model, by analogy with the same models in physics. However, a new wave of criticism fell upon game theory when, as a result of experiments, it was revealed that people do not follow equilibrium strategies in practice. For example, in the Centipede and Dictator games, participants often do not use the strategy profile that constitutes the Nash equilibrium. Debate continues about the significance of such experiments. According to another point of view, Nash equilibrium is not a prediction of expected behavior, but only explains why populations already in Nash equilibrium remain in this state. However, the question of how these populations arrive at the Nash equilibrium remains open. Some researchers in search of an answer to this question switched to the study of evolutionary game theory. Evolutionary game theory models assume bounded rationality or irrationality of players. Despite the name, evolutionary game theory deals not only and not so much with the natural selection of biological species. This branch of game theory studies models of biological and cultural evolution, as well as models of the learning process.
Normative analysis (identification of the best behavior).
On the other hand, many researchers consider game theory not as a tool for predicting behavior, but as a tool for analyzing situations in order to identify the best behavior for a rational player. Since the Nash equilibrium includes strategies that best respond to the behavior of another player, using the concept of Nash equilibrium to select behavior seems to be quite reasonable. However, this use of game-theoretic models has also been criticized. First, in some cases it is advantageous for a player to choose a strategy that is not in the equilibrium if he expects other players to also not follow the equilibrium strategies. Second, the famous game The Prisoner's Dilemma” allows us to give one more counterexample. AT " Prisoner's dilemma» Following self-interest leads to the fact that both players are in a worse situation compared to the one in which they would have sacrificed self-interest.
Game types
cooperative and non-cooperative.
The game is called cooperative, or coalition, if the players can unite in groups, taking on some obligations to other players and coordinating their actions. In this it differs from non-cooperative games in which everyone is obliged to play for themselves. Entertainment games are rarely cooperative, but such mechanisms are not uncommon in everyday life.
It is often assumed that cooperative games differ precisely in the ability of players to communicate with each other. In general, this is not true. There are games where communication is allowed, but players pursue personal goals, and vice versa.
Of the two types of games, non-cooperative ones describe situations in the smallest details and give more accurate results. Cooperatives consider the process of the game as a whole. Attempts to combine the two approaches have yielded considerable results. So-called Nash program has already found solutions to some cooperative games as equilibrium situations for non-cooperative games.
Hybrid games include elements of cooperative and non-cooperative games. For example, players can form groups, but the game will be played in a non-cooperative style. This means that each player will pursue the interests of his group, while at the same time trying to achieve personal gain.
Symmetrical and asymmetrical.
The game will be symmetrical when the corresponding strategies of the players are equal, that is, they have the same payoffs. In other words, if the players can change places and at the same time their payoffs for the same moves will not change. Many of the studied games for two players are symmetrical. In particular, these are: "Prisoner's Dilemma", "Deer Hunt", "Hawks and Doves". As asymmetric games, one can cite "Ultimatum" or "Dictator".
In the example on the right, the game at first glance may seem symmetrical due to similar strategies, but this is not so - after all, the payoff of the second player with the strategy profiles (A, A) and (B, B) will be greater than that of the first.
Zero-sum and non-zero-sum.
Zero sum games- special variety constant sum games, that is, those where players cannot increase or decrease the available resources, or the fund of the game. In this case, the sum of all wins is equal to the sum of all losses in any move. Look to the right - the numbers mean payments to the players - and their sum in each cell is zero. Examples of such games are poker, where one wins all the bets of others; reversi, where enemy chips are captured; or banal theft.
Many games studied by mathematicians, including the Prisoner's Dilemma already mentioned, are of a different kind: in non-zero sum games A win for one player does not necessarily mean a loss for another, and vice versa. The outcome of such a game can be less than or greater than zero. Such games can be converted to zero sum - this is done by introducing fictitious player, which "appropriates" the surplus or makes up for the lack of funds.
Another game with a non-zero sum is trade where each participant benefits. A well-known example where it decreases is war.
Parallel and serial.
In parallel games, the players move at the same time, or at least they are not aware of the choices of the others until all won't make their move. in succession, or dynamic In games, participants can make moves in a predetermined or random order, but in doing so they receive some information about the previous actions of others. This information may even not quite complete, for example, a player can find out that his opponent from ten of his strategies definitely didn't choose fifth, without knowing anything about the others.
Differences in the representation of parallel and sequential games were discussed above. The former are usually presented in normal form, while the latter are in extensive form.
With complete or incomplete information.
An important subset of sequential games are games with complete information. In such a game, the participants know all the moves made up to the current moment, as well as the possible strategies of the opponents, which allows them to predict to some extent the subsequent development of the game. Full information is not available in parallel games, since the current moves of the opponents are not known in them. Most of the games studied in mathematics are with incomplete information. For example, all "salt" Prisoner's dilemmas or Coin comparisons lies in their incompleteness.
At the same time, there are interesting examples of games with complete information: Ultimatum, Centipede. This also includes chess, checkers, go, mancala and others.
Often the concept of complete information is confused with similar - perfect information . For the latter, it is sufficient only to know all the strategies available to opponents; knowledge of all their moves is not necessary.
Games with an infinite number of steps.
games in real world or games studied in economics, as a rule, last final number of moves. Mathematics is not so limited, and in particular, set theory deals with games that can continue indefinitely. Moreover, the winner and his winnings are not determined until the end of all moves.
The problem that is usually posed in this case is not to find the optimal solution, but to find at least winning strategy. Using the axiom of choice, one can prove that sometimes even for games with complete information and two outcomes - "win" or "lose" - none of the players has such a strategy. The existence of winning strategies for some specially designed games plays an important role in descriptive set theory.
Discrete and continuous games.
Most studied games discrete: they have a finite number of players, moves, events, outcomes, etc. However, these components can be extended to a set of real numbers. Games that include such elements are often called differential games. They are associated with some real scale (usually - the time scale), although the events occurring in them may be discrete in nature. Differential games are also considered in optimization theory, they find their application in engineering and technology, physics.
Metagames.
These are games that result in a set of rules for another game (called target or game-object). The goal of the metagames is to increase the utility of the rule set that is given out. Metagame theory is associated with theory of optimal mechanisms .
based on wikipedia.org
