Basic concepts of game theory and game models. Game theory in economics and other areas of human activity
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Siberian State University of Telecommunications and Informatics
Interregional Center for Retraining Specialists
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Discipline: Institutional Economics
Completed by: Lapina E.N.
Group: EBT-52
Option: 4
Novosibirsk, 2016
INTRODUCTION
Any person around the world performs some kind of action every day, 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 on the basis of their own or group interests, depending on who the decision belongs to (an individual or a group, an organization as a whole).
Institutions are created by humans to maintain order and reduce the uncertainty of exchange. They provide predictability in human behavior. Institutions allow us to save our thinking 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.-С.18.
Institutions are the “rules of the game” in society, or, more formally, human-created constraints that govern relationships between people. Labsker L.G., Yashchenko N.A .: Game theory in economics. Workshop with problem solving. Tutorial. Publisher: Knorus, 2014.-P.21. Institutions emerge to solve problems that arise from repeated interactions between people. At the same time, they must not only solve the problem, but also minimize the resources spent on solving it.
Game theory is a mathematical method for studying optimal strategies in games. A game is understood as a process in which two or more parties are involved in the struggle for the realization of their interests. Each of the parties 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 several factors:
1. considerations about other participants;
2. resources of the participants;
3. the expected 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 generally known, i.e. each player knows his payoff function and the set of strategies at his disposal, 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 economics - at such moments when the theoretical foundations of the theory of choice in classical economic theory do not work, for example, in the fact that the consumer makes his choice rationally, he is fully aware of the situation in this market and about a specific given product.
CHAPTER 1. THEORETICAL FOUNDATIONS OF GAME THEORY
1.1 CONCEPT OF GAME THEORY
As mentioned above, game theory is a branch of mathematics that studies formal models for making optimal decisions in a conflict. In this case, a conflict is understood as a phenomenon in which various parties participate, endowed with different interests and opportunities to choose actions available to them in accordance with these interests. Each side has its own goal and uses a certain strategy that 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 the ideas of other participants, their resources and their possible actions
Game theory has its roots in neoclassical economics. For the first time, the mathematical aspects and applications of the theory were presented in the classic 1944 book by John von Neumann and Oskar Morgenstern "Game theory and economic behavior."
The game is a simplified formalized model of a real conflict situation. Mathematically formalization means that certain rules have been developed for the actions of the parties during the game: options for the actions of the parties; the outcome of the game for this option; 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 the possible situations of the game. Dominance in game theory is a situation in which one of the strategies of some player gives a greater payoff than the other, for any actions of his opponents. Protasov I.D. Game theory and operations research: textbook. allowance. - M .: Helios ARV, 2013.-P.121.
The focal point is the balance in the coordination game, chosen by all participants in the interaction on the basis of common knowledge that helps them coordinate their choice. The concept of a focal point was introduced by the 2005 Nobel laureate economist Thomas Schelling in a 1957 article that became the third chapter of his famous book, Strategy of Conflict (1960).
If a strictly dominant strategy exists 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 single Nash equilibrium. However, this equilibrium will not necessarily be Pareto efficient, i.e. non-equilibrium outcomes can provide greater gains to all players. Prisoner's Dilemma is a classic example of this situation. 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 effective if none of the players can improve their position without making the other player worse off.
It is also worth mentioning the Stackelberg equilibrium. Stackelberg equilibrium is a situation when 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 equilibrium of dominant strategies and the Nash equilibrium, this type of equilibrium always exists.
Game theory can be interpreted in two ways: matrix and graphical. The matrix method will be depicted below, where situations that lead to the emergence of institutions will be considered.
For a graphical example, consider the following situation where there is one pasture for grazing cows. Now let's ask the question: for how many cows, n, would the use of a given pasture be optimal? In accordance with the marginal optimization principle, which assumes the equation of marginal costs and marginal income, the answer should be that the optimal number of cows will be that at which the value of the marginal product from grazing the last cow, VМР, will be equal to the cost of one cow, p. In the conditions of private ownership of this pasture, this principle would be observed, since an individual owner would compare the benefits and costs associated with each additional cow, and would settle on the number of them, Ер, at which the possibility of obtaining a positive rent from grazing cows on the pasture , Rp, would have been exhausted, and, accordingly, the maximum of this rent would have been reached (Fig. 1). This is summarized in the equation below, which maximizes the difference between the value of the total product, VTP, and the total cost, i.e. the value of the cow multiplied by the number of cows, when the margin principle is observed.
VMP (n *) \u003d c maxn VTP (n) - cn (1)
Figure 1. - Graph of the value of the marginal and average grazing of cows
However, in conditions of free access to pasture, i.e., the absence of exclusive rights to it, the marginal principle of optimization will not be observed and the number of cows on the pasture will exceed the optimal value, Ер, and reach the point of equality of the value of the average product from grazing a cow, VAP, and the cost of a cow. ... As a result, there will be a new equilibrium number of cows in conditions of free access, Ес. In this case, the positive rent, Rp, created by grazing cows until their optimal number, Ep, is spent on additional cows, and upon reaching the point Ес, it 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" \u003d c? VTP (n ") - cn" \u003d 0;
1.2 VARIETY OF SITUATIONS AND SPHERES OF HUMAN LIFE IN WHICH GAME THEORY IS APPLICABLE
In life, there are many examples of a collision of opposite sides, taking the form of a conflict with two acting sides pursuing opposite interests.
Such situations arise, for example, when it comes to trust. The compliance of the counterparty's actions with expectations becomes especially important in those situations when the risk of the decisions made by the individual is determined by the actions of the counterparty. Game theory models are the best illustration of this: a player's choice of one strategy or another depends on the actions of another player. Trust consists in "anticipating 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 between transactions on the market and 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 trust exists in a depersonalized form for the implementation of the simplest market transaction using prepayment (Fig. 2).
Picture 2
Suppose that the buyer is facing many sellers and he knows from his previous business experience the probability of fraud (1 - p). Let us calculate such a value of p so that the transaction will take place, that is, “making an advance payment” is an evolutionarily stable strategy.
EU (make a prepayment) \u003d 10p - 5 (1 - p) \u003d 15p - 5,
EU (do not make an advance payment) \u003d 0.15p - -5\u003e 0, p\u003e 1/3.
In other words, if the level of buyer's confidence in sellers is less than 33.3%, prepaid transactions become impossible under the given conditions. In other words, p \u003d 1/3 is the critical, minimum required level of trust.
To generalize the results, we replace the specific values \u200b\u200bof the buyer's gain (10) and loss (--5) with the symbols G and L. Then, with the same structure of the game, the deal will take place at
the higher the value of the loss relative to the gain, the higher the level of trust between the parties to the transaction should be. James Coleman portrayed the dependence of the need for trust on the terms of the transaction as follows (Fig. 3).
Figure 3
The calculated minimum level of confidence is empirically confirmed. 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 the north of Italy and 65% - in the south. The low level of trust in southern Italy, where the mafia is traditionally strong, is indicative. It is no coincidence that one of the researchers of the mafia, D. Gambetta, explains its emergence by the critically low level of trust in the southern regions of Italy and, therefore, by the need for a substitute for trust, which takes the form of “third party” intervention, which both parties to the transaction trust.
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 purchase and sale of chairs, given the fact that the presence of wired treasures in them is in question. We will depict an example taking into account the fact that, in 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 a 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 operator finds its solution in the presence of any non-zero level of Ostap's trust, then the problem of the dependence of Ostap’s payoff on the presence of treasures in the chairs remains insoluble, which, incidentally, confirms the ending of the novel.
1.3 POSSIBLE STRATEGIES IN REPEATING GAMES
1. Mixed strategies. When players find themselves in a certain situation of choice more than once, their interaction becomes much more complicated. They can afford to combine strategies to maximize the overall payoff. Let us show this using a model describing 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, striving to maintain inflation at a fixed level (p0), or on emission and, therefore, an increase in inflation (p1). In turn, the economic agent acts on the basis of its inflationary 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\u003e pe, the Central Bank receives profits from seigniorage and inflation tax. If pe \u003d p1, then both the Central Bank will lose because of the reduction in revenues from seigniorage, and economic agents, which continue to bear the burden of the inflation tax. If pe \u003d p0, then the status quo is preserved and no one loses. Finally, if p\u003e p0, then only economic agents lose: producers - because of the loss of demand for unreasonably expensive products, consumers - because of the creation of unjustified stocks.
In the proposed model, with a single interaction, agents have no dominant strategies, and there is no Nash equilibrium either. With repetitive interactions many times, and it is this interaction that is characteristic of real situations, both participants can use both the one and the other strategy at their disposal. Does the alternation of strategies in a certain sequence allow players to maximize their utility, i.e., to achieve Nash equilibrium in mixed strategies: an outcome in which no participant can increase his payoff by unilaterally changing his strategy? Let us assume that the Central Bank pursues a tight monetary policy with probability P1 (in P1% of cases), and with probability (1 - P1) - inflationary policy. Then, when the economic agent chooses non-inflationary expectations (pe \u003d p0), the Central Bank can expect to receive a gain equal to
theory game strategy
EU (CB) \u003d Р1 0+,
1 (1 - P1) \u003d 1- -P1
In the event of inflationary expectations, the economic agent will gain by the Central Bank
EU (CB) \u003d P10 + (1 - P1) (- 2) \u003d 2P1 - 2.
Now let's assume that the 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) \u003d P2 (1 - P1) + (1 - P2) (2P1-2) \u003d \u003d ZR2-ZR1 P2 + 2P1 - 2 (Fig. 5).
Figure 5
Similar calculations for an economic agent will give
EU (e.a.) \u003d P1 (P2- 1) + (1 - P1) (- P2-2) \u003d 2P1P2 + P1- P2-2.
If we rewrite these expressions in the following form
EU (CB) \u003d Pl (2-3P2) + ZR2-2
EU (e.a.) \u003d \u003d P2 (2P1-1) + P1-2,
then it is easy to see that for
the Central Bank's gain does not depend on its own policy, and if
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 the economic agent in 2/3 of cases of non-inflationary expectations and the conduct of the Central Bank in half of the cases of 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 Central Bank's policy affect the behavior of economic agents only to the extent that they are unexpected and unpredictable. The Central Bank's strategy in 50% of cases to pursue a tough monetary policy, and in 50% - a soft one is the best way to create an atmosphere of unpredictability.
2. An evolutionary stable strategy. An evolutionarily stable strategy is a strategy such that if it is used by the majority of individuals, then no alternative strategy can supplant it through the mechanism of 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 counterpart is not constant, and in each period the individual interacts with a new counterpart. Therefore, the likelihood of a counterparty choosing a particular strategy will depend not so much on the configuration of the mixed strategy as on the preferences of each of the counterparties. In particular, it is assumed that out of the total number N potential counterparties n (n / N%) always choose strategy A, and m (m / N%) - strategy B. Thus, the prerequisites are created for achieving a new type of equilibrium, evolutionarily stable strategies. Evolutionary Stable Strategy (ESS) is a strategy in which if all members of a certain population use it, then no alternative strategy can supplant it through the mechanism of natural selection. Let us consider as an example the simplest variant of the coordination problem: passing two cars on a narrow road. It is assumed that in a given area the left- 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 part. If both cars take to the left, entering the left shoulder in the direction of travel, then they leave without problems. The same happens if both cars take to the right. When one car takes to the right, and the second to the left and vice versa, then 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 - R). The condition for the "take to the right" strategy to become evolutionarily stable for motorist A is formulated as follows: EU (right)\u003e EU (left), or
0P + 1 (1 - P)\u003e 1P + 0 (1 - P),
whence P< 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 terms, 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)\u003e EU (J, p),
which is identical
pU (I, I) + (l -p) U (I, J)\u003e pU (J, I) + (1 - p) U (J, J) (3)
From which it follows:
U (I, I)\u003e U (J, I)
U (I, I) \u003d U (J, I)
U (I, J)\u003e U (J, J),
where - U (I, I) the player's payoff when choosing strategy I, if the counterparty chooses strategy I; U (J, I) - the player's payoff when choosing strategy J, if the counterparty chooses strategy I, and so on.
Figure 7
You can present these conditions in graphical form. Let us postpone the expected utility of choosing one strategy or another along the vertical axis, and the proportion of individuals in the total population of players choosing both strategies along the horizontal axis. Then we get the following graph (the values \u200b\u200bare taken from the model of the crossing of two cars), shown in Fig. 7.
It follows from the figure that both “take to the left” and “take to the right” have equal chances of becoming an evolutionarily stable strategy as long as neither of them covers more than half of the “population” of drivers. If the strategy oversteps this line, then it will gradually but inevitably replace the other strategy and cover the entire population of drivers. The fact is that if a 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 a strict form, this statement will look like this
dp / dt \u003d G, G "\u003e 0 (4)
The main result of the analysis of repetitive games is to increase the number of equilibrium points and, on this basis, to solve the problems of coordination, cooperation, compatibility, and fairness. Even in the prisoners' dilemma, the transition to repetitive interaction allows you to achieve the optimal Pareto result ("deny guilt"), without going beyond the norm of rationality and the prohibition on the exchange of information between players. This is precisely the meaning of the "general theorem": any outcome that suits the individual individually can become equilibrium during the transition to the structure of the repeated game. In a situation of prisoners' dilemma, an equilibrium outcome under certain conditions can be both a simple “do not recognize” strategy, and many 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 in two previous periods in a row the counterparty admitted guilt; DOWING is a strategy based on the assumption that the counterparty is equally likely to use “deny guilt” 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 choosing a strategy to "admit guilt" in the next period; TESTER - start with an admission of guilt, and if the counterparty also admits guilt, then in the next period deny guilt.
CONCLUSION
In conclusion, the essay can be concluded about the need to use game theory in modern economic conditions.
In the context of an alternative (choice), it is very often difficult to make a decision and choose one or another strategy. Operations research allows using appropriate mathematical methods to make an informed decision about the appropriateness of a particular strategy. Game theory, which has in stock an arsenal of methods for solving matrix games, makes it possible to effectively solve these problems by several methods and select the most effective from their set, as well as to simplify the original matrices of games.
The essay illustrated the practical application of the main strategies of game theory and made relevant conclusions, studied the most used and frequently used strategies and basic concepts.
LIST OF USED LITERATURE
1. Petrosyan L.A., Zenkevich N.A., Shevkoplyas E.V .: Game theory: textbook. Publisher: BHV, 2012.-212s.
2. Labsker LG, Yashchenko NA: Theory of games in economics. Workshop with problem solving. Tutorial. Publisher: Knorus, 2014.-125s.
3. Neilbuff, Dixit: Game Theory. The art of strategic thinking in business and life. Publisher: Mann, Ivanov and Ferber, 2015 .- 99s.
4. Oleinik A. N. Institutional economics. Study guide, 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. Maths. Study guide for the section "Elements of game theory", LLC "Resolventa", 2011.-211s.
7. Shikin E.V. Mathematical methods and models in management: textbook. manual for students exercise. 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 are trying to bypass each other in the struggle for primacy. A branch of economic analysis called "game theory" deals with such situations.
"Game theory studies the way in which two or more players choose individual actions or entire strategies. The name of this theory sets in a somewhat abstract way, since it is associated with playing chess and bridge or waging wars. In fact, the conclusions of this discipline are very deep Game theory was developed by a native of Hungary, the brilliant mathematician John von Neumann (1903-1957) This theory is a relatively young mathematical discipline.
Later, 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 if their opponents do not change their strategies. Each player's strategy is the best answer to his opponent's strategy. Sometimes the Nash equilibrium is also called 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), except for their own benefit.
The equilibrium of a perfectly competitive market is also a Nash equilibrium, or noncooperative equilibrium, in which every firm and every consumer makes decisions based on already existing prices as independent of its will. We already know that when every firm seeks to maximize profits and every consumer is utility, equilibrium arises when prices are equal to marginal costs, and profits are zero. "Mamaeva LN Institutional Economics: A Course of Lectures - M .: Publishing and Trade Corporation" Dashkov and K ", 2012. - 200 p.
Let us recall the concept of the "invisible hand" of Adam Smith: "Pursuing his own interests, he (the individual) often contributes to the prosperity of society to a greater extent than if he consciously aspired to it" Smith A. Research on the nature and causes of the wealth of nations // Anthology of economic classics ... - M .: Ekonov-Klyuch, 19931. The paradox of the "invisible hand" is that, although each one acts as an independent force, in the end, society remains a winner. At the same time, 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 completely competitive economy, non-cooperative behavior is cost-effective from the point of view of the interests of society.
On the contrary, when members of a group decide to cooperate and jointly arrive at a monopoly price, this behavior will damage economic efficiency. The state is forced to create antimonopoly legislation and thereby reason with those who are trying to inflate prices and divide the market. However, disunity in behavior is not always cost effective. Competition between firms leads to low prices and competitive production volumes. The "invisible hand" has an almost magical effect on perfectly competitive markets: efficient allocation of resources occurs as a result of the actions of individuals seeking to maximize profits.
However, in many cases, non-cooperative behavior leads to economic inefficiency or even poses a threat to society (for example, an arms race). Non-cooperative behavior on the part of both the US and the USSR forced both sides to invest huge amounts of money in the military field and led to the creation of an arsenal of almost 100,000 nuclear warheads. There is also concern that America's excessive availability of weapons could trigger a kind of internal arms race. Some people arm themselves against others - and this "race race" can continue indefinitely. Here a completely "visible hand" comes into play, directing this destructive competition and having nothing to do with the "invisible hand" of Adam Smith. Another important economic example is the "pollution game" (of the environment). Here the object of our attention will be such kind of side effects as pollution. If firms never asked anyone what to do, they would rather create pollution than install expensive cleaners. If any firm, out of noble motives, decided to reduce harmful emissions, then costs, and consequently, prices for its products, would increase, and demand would fall. It is quite possible that this firm would have simply gone bankrupt. Living in a brutal world of natural selection, firms are more likely to choose to remain in the Nash equilibrium. No firm can increase profits by reducing pollution.
By entering a deadly economic game, every profit-maximizing and uncontrolled steel firm will produce water and air pollution. If a firm tries to clean up its emissions, it will be forced to raise prices and suffer losses. Non-cooperative behavior will establish a Nash equilibrium in a high-emission environment. The government can take steps to shift the balance. In this position, the pollution will be negligible, 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.
The pollution games are one of the cases when the mechanism of action of the "invisible hand" does not work. This is a situation where the Nash equilibrium is ineffective. 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 induce firms to choose a low-pollution outcome. Firms earn exactly the same as before, with large emissions, but the world is becoming somewhat cleaner.
Game theory also applies to macroeconomic policy. Economists and politicians in the United States often scold current monetary and fiscal policies: the federal deficit is too large and reduces national savings, while monetary policy generates interest rates that constrain investment. Moreover, this "fiscal and monetary syndrome" has been a feature of the macroeconomic landscape for over a decade. Why is America so stubbornly pursuing both types of policies, although neither of them is desirable?
One can try to explain this syndrome in terms of game theory. It has become customary in modern economics to separate these kinds of politics. The central bank of America - the Federal Reserve System - determines monetary policy independently of the government by setting interest rates. Fiscal policy, taxes and expenditures are in charge of the legislative and executive branches. However, each of these policies has different goals. The central bank seeks to limit the growth of money supply and keep inflation low.
Arthur Berne, an expert on economic cycles and a former head of the Federal Reserve, wrote: “Central bank officials tend to keep prices in check, by tradition, perhaps by personal makeup. Their hatred of inflation is even more heated by talking to like-minded private financial circles ". The authorities in charge of fiscal policy are more concerned with issues such as full employment, their own popularity, keeping taxes low and upcoming elections.
Fiscal policy makers prefer the lowest possible unemployment, higher government spending coupled with tax cuts, and don't care about inflation and private investment.
In the money-budget game, the cooperative strategy leads to moderate inflation and unemployment, coupled with a large volume of investment that stimulates economic growth. However, the desire to reduce unemployment and implement social programs prompts the country's leadership to resort to increasing the budget deficit, while aversion to inflation forces the central bank to raise interest rates. Non-cooperative equilibrium means the smallest possible investment.
They choose the "big budget deficit". On the other hand, the central bank tries to reduce inflation, is unaffected 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 "fiscal 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 player strategies are considered and payments are determined that correspond to any possible combination of player strategies. The game described in this way is called playing in normal form.
Normal form of two-player playconsists of two payment 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 amount of the first player's gain, and the second - the amount of the second player's payoff. The first player (state) chooses one of m strategies, with each strategy corresponding to a row of the matrix I (i \u003d 1,…, m). The second player (business) chooses one of n strategies, with each strategy corresponding to a column of the matrix j (j \u003d 1,…, n). A pair of numbers at the intersection of a row and a column, which correspond to the strategies chosen by the players, shows the amount of winnings for each of them. In general, if player I chooses the strategy iand player II is strategy j, then the payoffs of the first and second players are, respectively, and (i \u003d 1,…, m; j \u003d 1,…, n), where m, n is the number of final strategies of players I and II, respectively. It is assumed that each of the players knows all the 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 variants of the opponent's strategies (matrix elements), then the game is called indefinite and can have an infinite number of variants (strategies).
There are other classes of games where players win and lose at the same time.
Antagonistic games of two personsassociated with 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 opposite to each other.
As an example, consider a game in which two players participate, each of them has two strategies. The winnings 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 loss of business); if both players choose different strategies (player I - і 1 player II - j 2 then the first loses, and the second wins (the state raises taxes on business - business evades them; the state's loss is a business gain).
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 the participants in the game 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 the results, it is necessary to consider certain classes of games distinguished by some restrictive assumptions. Such restrictions can be imposed in several ways.
Can be distinguished several ways (ways) of imposing restrictions.
1. Limitations of the possibilities of player relationships with each other. The simplest case is when the players are completely disconnected and cannot knowingly help or interfere with each other by action or inaction, information or misinformation. Such a state of affairs inevitably occurs when only two players (the state and business) participate in the game, having diametrically opposed interests: an increase in the gain of one of them means a decrease in the gain of the other, and, moreover, by the same amount, provided that the gains of both players are expressed in the same units of measurement. Without breaking the generality, we can take the total payoff of both players to be zero and treat the payoff 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 no relationships between players, no compromises, exchanges of information and other resources can exist by the very nature of things, in the essence of the game, since every message received by a player about the intentions of another can only increase the payoff of the first player and thereby increase losing to his opponent.
Thus, we conclude that in antagonistic games, players need not have direct relationships and at the same time be in a state of play (opposition) in relation to each other.
2. Restrictions 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 sets of strategies, eliminates the need to introduce any technology on the sets.
Games in which the sets of strategies of each of the players are finite are called end games.
3. Suggestions about the internal structure of each strategy, ie. about its content. So, for example, as strategies can be considered functions of time (continuous or discrete), the values \u200b\u200bof which are the actions of the player at the appropriate moment. These and similar games are usually called dynamic (positional) games.
The constraints of the players' strategies 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 restrictions on the strategy are also related to the ways of achieving these goals in certain time intervals, for example, the desire of a business to achieve a reduction in the size of mandatory sales of foreign exchange earnings within the next three months (or one year). If no assumptions are made about the nature of strategies, then they are considered some abstract set. Games of this kind in the simplest formulation of the question are called games in normal form.
Final 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 not the rows or columns of the matrix themselves, but their numbers. Then the situations of the game are the cells of this matrix standing at the intersections of each row with each of the columns. Having filled these situation cells with numbers describing the payoffs of player I in these situations, we complete the task of the game. The resulting matrix is \u200b\u200bcalled the payoff matrix of the game,or the game matrix.Due to 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 it only in sign. Therefore, no further guidance on the payoff function of player II in the matrix game is required.
A matrix with m rows and n columns is called an (m * n) - matrix, and a game with this matrix is \u200b\u200ban (m * n) - game.
The process of (m * n) - games with a matrix can be represented as follows:
Player I fixes the number of row i, and player II fixes 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 (state) choose some of its strategies i. Then, in the worst case, he will receive a payoff min. In game theory, players are assumed to be cautious, counting on the least favorable turn of events for themselves.
Such a state of affairs that is least favorable for player I may occur, for example, in the case when strategy i becomes known to player II (business). Anticipating such a possibility, player I must choose his strategy so as to maximize this minimum gain:
min \u003d max min (I)
The value on the right-hand side of the equality is the guaranteed payoff of player I. Player II (business) must choose a strategy such that
max \u003d min max (II)
The value on the right side of the equality is the payoff of player I, more than which he cannot receive with the correct actions of the opponent.
The actual payoff of player I must, with reasonable actions of partners, be in the interval between the payoff values \u200b\u200bin the first and second cases. If these values \u200b\u200bare 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 (strategies correspond to the rows and columns of the matrix). A player's random choice of his strategies is called mixed countrythis player's tags. In (m * n) - the game, the mixed strategies of player I are determined by the sets of probabilities: X \u003d (, ...), with which this player chooses his initial, pure strategies.
The theory of matrix games is based on the Neumann theorem for active strategies: “If one of the players adheres to his optimal strategy, then the payoff remains unchanged and equal to the game price regardless of what the other player does, if he does not go beyond his active strategies (i.e. That is, he uses any of them in its pure form or mixes them in any proportions "Neumann J. Contributions to the theory of games. 1995 .. - 155 p.). Note that activeis the pure strategy of a player included in his optimal mixed strategy with nonzero probability.
The main goal of the game isfinding 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 gives 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 the player only needs to know one of them for a successful game. Therefore, in relation to matrix games, the question of finding at least one optimal strategy for each of the players is relevant.
The main theorem on matrix games states the existence of a game value and optimal mixed strategies for both players. The optimal strategy does not have to be one-off. This is a very important conclusion drawn from game theory.
The subject playing the matrix game is characterized by followingquality:
matrix elements interpretedas cash payments and, accordingly, their winnings and losingevaluated in monetaryform;
each of playersapplies the function to these elements usefulness;
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 "with their own bell towers ".
These assumptionslead to zero-sum games in which relations of cooperation, bargaining, and other types of interactions arise between playersas before games,and in its 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 to include othersanalysis capabilities, leads tointeresting, but rather difficult tasks. In the development of game theory, it is necessary to apply the utility function not only to monetary outcomes, but also to the amounts with the expected futureoutcomes. These the assumptions are controversial, but they exist.In this case, we proceed from the fact that this assumption about a similar operationit has likenesswith behavior players incertain decision-making situations and admits the possibility that the method playing the gamethis player depends on the state of his capital during conducting themgames.
Consider this in the following example. Let bethe first player at the start of the game G has capital x dollars.Then his capital at the end games willis equal to + x, where is the actual winnings he gets from the game. The usefulness he attributes to such the outcome,is equal to f (+ x), where f is the utility function.
These few examples illustrate only a fraction of the vast variety of results that can be obtained using game theory. This section of economic theory is an extremely useful (for economists and other social scientists) tool for analyzing 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 emerged in the forties of the XX century is most often applied in economics. But how can we model the behavior of people in society using the concept of games? Why do economists study in which corner football players are more likely to shoot penalties, and how to win in Rock, Scissors, Paper, 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 standpoint of an economist and game theory, it is not a game. It implies a mandatory conflict of interest.
A Beautiful Mind, about John Nash, a Nobel laureate in economics, has a scene with a blonde in a bar. It shows the idea for which the scientist received the prize - this is the idea of \u200b\u200bNash equilibrium, which he himself called 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, the players in this situation are only men, 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 cannot see where the others went, and then walk yourself). If a girl rejects a man, the game ends: it is impossible to return to her or choose another.
What is the likely ending of this game situation? That is, what is its stable configuration, from which everyone will understand that they have made the best choice? First, as Nash correctly observes, if everyone goes to the blonde, it won't end well. Therefore, the scientist further suggests that everyone needs to go to 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 the real balance - the outcome, in which one goes to the blonde, and the rest to the brunettes. It might seem like it's unfair. But in a situation of equilibrium, 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 realizes that even knowing how others are like, he would have done the same. In another way, you can call it an outcome, where each participant responds in an optimal way to the actions of the others.
"Rock Paper Scissors"
Consider other balance games. For example, in "Rock, Scissors, Paper" 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 the World Championship and the World Rock Paper Scissors Society collecting game statistics. Obviously, you can improve your chances of winning by knowing something about the normal behavior of people in this game.
A 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 played move (37.8%). Paper is favored by 32.6%, scissors by 29.6%. Now you know to choose paper. However, if you are playing with someone who also knows this, you no longer have 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 were deciding 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, Scissors, Paper, and the representatives of the houses sent him their options by e-mail. Sotheby's, as they later said, without much hesitation, 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, so most people choose it. But if we are not playing with a completely stupid beginner, he will not throw a stone, he will expect us to do it, and he will throw out the paper himself. But we will think ahead and throw away the scissors. "
Thus, you can think ahead, but this does 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 at random. So, in "Rock, Scissors, Paper" the balance, which we have not found before, is precisely in mixed strategies: to choose each of the three options for a move with a probability of one third. If you choose a stone more often, your opponent will adjust his choice. Knowing this, you will correct yours, and balance will not come out. But none of you will begin to change behavior if everyone simply chooses rock, scissors, or paper with equal probability. This is because in mixed strategies, it is impossible to predict your next move based on previous actions.
Mixed strategies and sports
There are a lot of more serious examples of mixed strategies. For example, where to serve in tennis or to beat / receive a penalty in football. If you don't know anything about your opponent or you are just constantly playing against different opponents, the best strategy is to act more or less randomly. Professor of the London School of Economics Ignacio Palacios-Huerta published a paper in the American Economic Review in 2003, the essence of which was to find the Nash equilibrium in mixed strategies. Palacios Huerta chose football as the subject of his research and, as a result, watched over 1400 penalty kicks. Of course, in sports everything is more cunning than in "Rock, Scissors, Paper": it takes into account the athlete's strong leg, hitting different angles when hitting with full force, and the like. Nash equilibrium here consists in calculating options, that is, for example, determining the angles of the goal, which must be kicked in order to win with a greater probability, knowing your strengths and weaknesses. The statistics for each footballer and the equilibrium found in it in mixed strategies showed that footballers do something like economists predict. It hardly needs to be argued that the people who take penalties read game theory textbooks and did some pretty tricky math. Most likely, there are different ways to learn how to behave optimally: you can be a brilliant football player 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 coach who was then playing in the Champions League final in Moscow. The scientist wrote a note to the coach with penalty shootout recommendations regarding the behavior of the opponent's goalkeeper, Edwin van der Sar of Manchester United. For example, according to statistics, he almost always hit the middle level and more often rushed in the natural direction for a penalty kick. As we defined above, it is more correct to randomize your behavior taking into account knowledge about your opponent. When the penalty was already 6-5, Chelsea striker Nicolas Anelka should have scored. Pointing to the right corner before hitting, van der Sar seemed to ask Anelk if he was going to hit there.
The bottom line is that all of Chelsea's previous shots were to the right of the breaker. We do not know exactly why, perhaps because of the consultation of an economist, to beat in an unnatural direction for them, because according to statistics, van der Sar is less ready for this. Most of Chelsea's players were right-handed: hitting an unnatural right-hand corner, all but Terry scored. Apparently, the strategy was for Anelka to hit the same spot. But van der Sar seems to have figured it out. He acted brilliantly: he pointed to the left corner saying, "Are you going to hit there?", From which Anelka was probably horrified, because he was solved. At the last moment, he decided to act differently, hitting his natural side, which was what van der Sar needed, who took this blow and ensured Manchester's victory. This situation teaches random choice, because otherwise your decision can be calculated and you will lose.
The Prisoner's Dilemma
Perhaps the most famous game that kicks off college game theory courses is Prisoner's Dilemma. According to legend, two suspects in 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 a short time. 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 prison for three years. If one confesses, and the accomplice is silent, the confessed one will leave immediately, and the other will be imprisoned for five years. If, on the contrary, the first does not confess, and the second surrenders him, the first will sit for five years, and the second will leave immediately. If no one confesses, both will go to prison for a year for keeping weapons.
The Nash equilibrium here lies in the first combination, when both suspects are not silent and both are imprisoned 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 other is silent, I would also better say: not sitting down is better than sitting down for a year. " This is the dominant strategy: speaking is beneficial no matter what the other is doing. However, there is a problem in it - there is a better option, because sitting for three years is worse than sitting 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 be 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 make money for himself, it will be better for everyone: he will make tasty meat, which the baker will buy with money from the sale of buns, which, in turn, he 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 situations where everyone acts for himself, but everyone is bad.
Therefore, sometimes economists and game theorists do not think about the optimal behavior of each player, that is, not about the Nash equilibrium, but about the outcome in which the whole society will be better (in "Dilemma" the society consists of two criminals). From this point of view, the outcome is effective when there is no Pareto improvement in it, that is, it is impossible to make someone better without making others worse. If people just exchange goods and services, this is a Pareto improvement: they do it voluntarily, and it is unlikely that anybody is bad for it. But sometimes, if you just let people interact and not even interfere, then what they come to will not be Pareto optimal. This is what happens in The Prisoner's Dilemma. In it, if we allow everyone to act as it suits them, it turns out that everyone is bad from this. It would be better for everyone if everyone did not act optimally for themselves, that is, they were silent.
Community tragedy
The Prisoner's Dilemma is a stylized toy story. You might not expect to find yourself in a similar situation, but similar effects are everywhere around us. Consider the “Dilemma” with a large number of players, sometimes referred to as a community tragedy. For example, there are traffic jams on the roads, and I decide how to go to work: by car or by bus. Others 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 particularly fast, so this outcome is even worse. If, on average, everyone travels by bus, then I, having done the same, will get there pretty quickly without a traffic jam. But if under such conditions go by car, I will also get there quickly, but also with comfort. So, the presence of a plug does not depend on my actions. The Nash equilibrium is here - in a situation where everyone chooses to go by car. Whatever the others do, I'd better choose a car, because there will be a traffic jam or not, it is unknown, but in any case I will get there with comfort. It's the dominant strategy, so everyone ends up driving and we have what we have. The task of the state is to make a bus ride 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 rational ignorance of the voter. Imagine that you do not know the outcome of an election in advance. You can study the program of all candidates, listen to the debates and then vote for the best. The second strategy is to come to the polling station and vote at random or for someone who is more often shown on TV. What is the best behavior if my vote never determines who will win (and in a 140 million country, 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 study the candidates' programs carefully. Therefore, not wasting time on this is the dominant strategy of behavior.
When you are invited to come to the Saturday clean-up, it will not depend on anyone separately whether the yard becomes clean or not: if I go out alone, I cannot remove everything, or, if everyone comes out, I will not go out, because everything is without me removed. Another example is shipping in China, which I learned about in Stephen Landsburg's excellent book The Economist on the Couch. 100-150 years ago, a method of transporting goods was widespread 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 make an effort and pull with all your might, and if everyone does that, the load will arrive on time. If someone 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 cannot change anything." As a result, over the time of delivery, everything was very bad, and the movers themselves found a way out: they began to hire a seventh and pay him money so that he would whip lazy people. The very presence of such a person forced everyone to work with all their might, because otherwise everyone would fall into a bad balance, from which no one individually could profitably escape.
The same example can be observed in nature. A tree growing in a garden differs from that 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 trees agreed and grew the same way, they would equally distribute the number of photons, and everyone would be better off. But it is unprofitable for anyone to do so. Therefore, each tree wants to grow slightly taller than those around it.
Сommitment device
In many situations, one of the players in the game may need a tool that will convince others that they are not bluffing. It's called a commitment device. For example, the law of some countries prohibits paying ransom 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 opportunity to save him by circumventing the law, you will. Imagine a situation that the law can be circumvented, but the relatives turned out to be poor and they have nothing to pay the ransom. The perpetrator has two options in this situation: 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 was punished, or remain silent. The best outcome for the offender is to release the victim who will not surrender him. 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 an option 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 be silent. Somewhere I read an option when she can ask a terrorist to arrange an erotic photo session. If the criminal is arrested, his accomplices will post photos on the Internet. Now, if the kidnapper remains free, that's bad, but open-access photos are even worse, so a balance is obtained. It's a way for the victim to stay alive.
Other examples of games:
Bertrand model
While we're on the subject of economics, let's look at 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 profit is approximately the same, because then buyers choose a store by chance. The only Nash equilibrium here is to sell the product at cost. But shops want to make money. Therefore, if one puts 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.
Exit on a narrow road
Let's 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 to the side. If they decide to turn left or right from themselves, 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 move out? To help find balance in such games, there are, for example, traffic rules. In Russia, everyone needs to turn right.
In the fun of Chiken, when two people are driving at high speed towards each other, there are also two balances. If both swerve to the side of the road, a situation arises called Chiken out, if both do not swerve, they die in a terrible accident. If I know that my opponent is going straight, it is beneficial for me to move out in order to survive. If I know that my opponent will move out, then it is beneficial for me to go straight in order to get $ 100 afterwards. It is difficult to predict what will actually happen, however, each of the players has their own method of winning. Imagine that I secured the steering wheel so that it cannot be turned, and showed this 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 benefits for everyone. The QWERTY layout was created to slow down your typing speed. Because if everyone typed too fast, the typewriter heads that hit the paper would cling to each other. Therefore, Christopher Scholes placed the letters often standing next to each other as far away as possible. If you go to 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 right now. Dvorak expected the world to move 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 work hard and retrain, and the result would be a balance in which everyone types quickly. Now we are also in balance - in bad balance. But it is not beneficial for anyone to be the only one who will retrain, because it will be inconvenient to work at any computer, except for a personal one.
In recent years, the importance of game theory has increased significantly in many areas of the economic and social sciences. In economics, it is applicable not only for solving general economic problems, but also for analyzing the strategic problems of enterprises, developing organizational structures and incentive systems.
Already at the moment 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 the economic sciences through the use of a new approach. These forecasts could not be considered too bold, since from the very beginning this theory claimed to describe rational behavior in decision-making in interrelated situations, which is typical for most pressing problems in the economic and social sciences. Subject areas such as strategic behavior, competition, cooperation, risk and uncertainty are key in game theory and are directly related to management tasks.
The first works on game theory were distinguished by simplified assumptions and a high degree of formal abstraction, which made them of little use 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 transaction cost and patron-agent theories, will be perceived as the most economically justified element of organization theory. It should be noted that already in the 80s M. Porter introduced into use some key concepts of the theory, in particular, such as “strategic move” and “player”. True, there was no explicit analysis associated with the concept of equilibrium in this case.
Fundamentals of game theory
To describe a game, you must first identify its participants. This condition is easily met when it comes to ordinary games such as chess, canasta, etc. The situation is different with “market games”. It is not always easy to recognize all players here, i.e. current or potential competitors. Practice shows that it is not necessary to identify all the players, it is necessary to find the most important ones.
Games usually cover several periods during which players take consecutive or simultaneous actions. These actions are referred to as “move”. Actions can be related to prices, sales volumes, research and development costs, etc. The periods during which players make their moves are called game stages. The moves chosen at each stage ultimately determine the “payments” (gain or loss) of each player, which can be expressed in material values \u200b\u200bor money (mostly discounted profit).
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, which seems to him to be the “best response” 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 arise in the course of this game.
The form of the game is also important. Usually, a normal, or matrix, shape and an expanded, 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 to the realm of control, the game can be described as follows. Two factories producing homogeneous products face 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 competitive struggle, both receive profit P W. If one of the competitors sets a high price, and the other sets a low price, then the latter realizes the monopoly profit P M, while the other incurs losses P G. A similar situation may, for example, arise when both firms must announce their prices, which cannot be subsequently revised.
In the absence of strict conditions, it is beneficial for both companies to set a low price. The “low price” strategy is dominant for any firm: no matter what price the competing firm chooses, it is always preferable to set a low price. But in this case, the firms face a dilemma, since the profit P K (which is higher for both players than the profit P W) is not achieved.
The strategic combination of “low prices / low prices” with the corresponding payments is a Nash equilibrium, in which it is not profitable for any of the players to separately deviate from the chosen strategy. This concept of equilibrium is fundamental in resolving strategic situations, but under certain circumstances it still needs improvement.
As for the above dilemma, its solution depends, in particular, on the originality of the players' moves. If an enterprise has the ability to revise its strategic variables (in this case, price), then a cooperative solution to the problem can be found even without a rigid agreement between the players. Intuition dictates that with repeated contacts of players, there are opportunities to achieve acceptable "compensation". So, under certain circumstances, it is inappropriate to strive for short-term high profits through price dumping if a “price war” may arise in the future.
As noted, both figures characterize the same game. The rendering of the game in normal form normally reflects “synchronicity”. However, this does not mean “simultaneity” of events, but indicates that the choice of a strategy by the player is carried out in conditions of ignorance about the choice of strategy by the opponent. In the expanded form, this situation is expressed through the oval space (information field). In the absence of this space, the game situation takes on a different character: first, one player should make a decision, and the other could do it after him.
Application of game theory to strategic management decisions
As examples, we can name decisions about conducting a principled pricing policy, entering new markets, cooperation and joint ventures, identifying leaders and performers in the field of innovation, vertical integration, etc. The provisions of this theory, in principle, can be used for all types of decisions, if other actors influence their adoption. These individuals, or players, do not have to be market competitors; they can be subsuppliers, lead customers, organizational staff, and work colleagues.
Quadrants 1 and 2 characterize a situation where the reaction of competitors does not significantly affect the payments of the firm. This happens when the competitor has no motivation (field 1 ) or opportunity (field 2 ) to strike back. Therefore, there is no need for a detailed analysis of the strategy of competitors' motivated actions.
A similar conclusion follows, albeit for a different reason, and for the situation reflected by the quadrant 3 ... This is where the competitor's reaction could have a significant impact on the firm, but since its own actions cannot greatly affect the competitor's payments, its reaction should not be feared. An example is decisions to enter a market niche: under certain circumstances, large competitors have no reason to react to such a decision of a small company.
Only the situation shown in the quadrant 4 (the possibility of reciprocal steps of market partners), requires the use of the provisions of game theory. However, it reflects only the necessary but insufficient conditions to justify the application of the base of game theory to fight competitors. There are situations when one strategy will definitely dominate all others, regardless of what actions the competitor takes. If we take, for example, the drug market, then it is often important for a company to be the first to announce a new product on the market: the profit of the “pioneer” turns out to be so significant that all other “players” only need to step up their innovation faster.
The same game situation can be presented in normal form (Fig. 4). There are two states designated here - "entry / friendly reaction" and "non-entry / aggressive reaction". Obviously, the second equilibrium is untenable. It follows from the expanded form that it is impractical for a company already established in the market to react aggressively to the emergence of a new competitor: in case of aggressive behavior, the current monopolist receives 1 (payment), and in case of a friendly one - 3. The outsider company also knows that it is not rational for a monopolist initiate actions to displace it, and therefore it decides to enter the market. The outsider company will not suffer threatening losses in the amount of (-1).
This kind of rational equilibrium is characteristic of a "partially improved" game that deliberately excludes absurd moves. In practice, such equilibrium states are in principle quite easy to find. 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 choice of the “best” move at the last stage of the game is made, then the “best” move at the previous stage is chosen, taking into account the choice at the last stage, and so on, until the starting node of the tree is reached games.
How can companies benefit from game theory analysis? For example, there is a known conflict of interests between IBM and Telex. In connection with the announcement of the latter's preparatory plans for entering the market, a “crisis” meeting of the IBM management was held, at which measures were analyzed to force the new competitor to abandon the intention to enter the new market.
Telex apparently became aware of these events. Analysis based on game theory has shown that the high costs of IBM's threats are unfounded.
This shows that it is useful for companies to explicitly think about the possible reactions of their partners in the game. Isolated business calculations, even based on decision-making theory, are often limited, as in the situation described. Thus, an outsider company could have chosen the “non-entry” course if preliminary analysis had convinced it that market penetration would cause an aggressive reaction from the monopolist. In this case, in accordance with the criterion of the expected value, it is reasonable to choose the “non-entry” move with the probability of an aggressive response of 0.5.
For the enterprise 1 it would be best if the enterprise 2 abandoned competition. His benefit would then be 3 (payments). Most likely the enterprise 2 the rivalry would win when the enterprise 1 would accept a curtailed investment program, and the enterprise 2 - wider. This position is reflected in the upper right quadrant of the matrix.
Analysis of the situation shows that equilibrium occurs at high costs of 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 an enterprise 1 a reduced budget is preferable if the company 2 refuses to participate in the rivalry; 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 edge can resort to game-theory analysis of the situation to ultimately achieve its optimal outcome. With the help of a certain signal, it should show that it is ready to carry out large expenditures on research and development. If such a signal is not received, then for the enterprise 2 it is clear that the enterprise 1 chooses the low cost option.
The reliability of the signal must be evidenced by the commitment of the enterprise. In this case, it may be an enterprise decision 1 on the purchase of new laboratories or the recruitment of additional research personnel.
From a game theory perspective, such commitments are tantamount to changing the course of the game: a situation of simultaneous decision making is replaced by a situation of successive moves. Company 1 firmly demonstrates the intention to go to large costs, the enterprise 2 registers this step and he no longer has a reason to participate in the rivalry. The new equilibrium arises from the “non-participation of the enterprise 2 "And" high costs of 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 laboratory conditions, and the results obtained serve as an impetus for practitioners. Theoretically, it was found out under what conditions it is advisable for two selfish partners to cooperate and achieve the best results for themselves.
This knowledge can be used in business practice to help two firms achieve a win / win situation. Today, game-trained consultants quickly and unambiguously identify opportunities that businesses can take advantage of to conclude stable and long-term contracts with customers, sub-suppliers, development partners, and the like.
Practical problems
in management
However, it should be pointed out that there are certain limits to the application of the analytical tools of game theory. In the following cases, it can be used only on condition of additional information.
First, this is the case when businesses have different ideas about the game in which they participate, 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 by comparing such cases, taking into account certain differences.
Second, game theory is difficult to apply to many equilibrium situations. This problem can arise even in the course of simple games with a simultaneous choice of strategic decisions.
Thirdly, if the situation for making strategic decisions is very difficult, then players often cannot choose the best options for themselves. It is easy to imagine a more complex market penetration situation 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 difficult than being 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.
The fundamental assumption underlying game theory about the so-called "common knowledge" is by no means indisputable. 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 rest of the game partners know. And this situation remains until the end of the game.
But in order for an enterprise in a particular case to make a preferred decision for itself, this condition is not always required. For this, less rigid prerequisites are often sufficient, such as “mutual knowledge” or “rationalized strategies”.
In conclusion, it should be emphasized that game theory is a very complex area of \u200b\u200bknowledge. When referring to it, one must observe a certain caution and clearly know the limits of application. Too simple interpretations, adopted by the firm on its own or with the help of consultants, are fraught with hidden danger. Because of its complexity, game theory analysis and advice is 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 planning strategic decisions, including when preparing large cooperation agreements.
Game theory - a mathematical method for studying optimal strategies in games. A game is understood as a process in which two or more parties are involved in the struggle for the realization of their interests. Each side has its own goal and uses a certain strategy that can lead to a win or a loss, depending on the behavior of other players. Game theory helps you choose the best strategies, taking into account the perceptions of other participants, their resources and their possible actions.
Game theory is a branch of applied mathematics, more precisely, operations research. Most often, game theory methods 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 very important for artificial intelligence and cybernetics, especially with the manifestation of interest in intelligent agents.
History.
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. Cermelo, E. Borel put forward the idea of \u200b\u200bthe mathematical theory of conflict of interest.
Mathematical game theory originates from neoclassical economics... For the first time, the mathematical aspects and applications of the theory were presented in the classic 1944 book by John von Neumann and Oskar Morgenstern "Game Theory and Economic Behavior" (eng. Theory of Games and Economic Behavior ).
This area of \u200b\u200bmathematics has found some reflection in social culture. In 1998, American writer and journalist Sylvia Nazar published a book about the fate of John Nash, Nobel laureate in economics and scientist in the field of game theory; and in 2001, based on the book, the film "A Beautiful Mind" was shot. Some American television shows, such as Friend or Foe, Alias, or NUMB3RS, occasionally reference the theory in their episodes.
J. Nash in 1949 wrote a dissertation on game theory, 45 years later he received the Nobel Prize in economics. J. Nash, after graduating from Carnegie Polytechnic Institute with two diplomas - bachelor's and master's - entered Princeton University, where he attended lectures by John von Neumann. In his writings, J. Nash developed the principles of "management dynamics". The first concepts of game theory analyzed antagonistic games where there are losers and winners at their expense. Nash develops methods of analysis in which all participants either win or fail. 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 players to maintain this balance, since any change will worsen their situation. These works by J. Nash made a significant 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 everyone is for himself, is not optimal. More optimal strategies are when everyone tries to do better for himself, doing better for others.
Although game theory originally looked at economic models, it remained a formal theory within mathematics until the 1950s. But since the 1950s. Attempts began to apply the methods of game theory not only in economics, but in biology, cybernetics, technology, and anthropology. During the Second World War and immediately after it, the military became seriously interested in game theory, who saw it as a powerful apparatus for researching strategic decisions.
In 1960-1970. interest in game theory is waning, despite the significant mathematical results obtained by that time. Since the mid-1980s. begins active practical use of game theory, 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 expounded without the application of game theory.
A major contribution to the application of game theory was the work of Thomas Schelling, 2005 Nobel laureate in economics, "The Strategy of Conflict." T. Schelling examines various "strategies" of behavior of the parties to the conflict. These strategies coincide with the tactics of conflict management and the principles of conflict analysis in conflict management (this is a psychological discipline) and in managing conflicts 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 meanings that have developed earlier. The cultural concept of play was given in the work of Johan Huizing "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 man himself, since animals also play. The concept of a game is found in Eric Byrne's concept "Games that people play, people who play games." These are purely psychological games based on transactional analysis. J. Hösing's concept of play differs from the interpretation of play in conflict theory and mathematical game theory. Games are also used for training in business cases, seminars by G.P.Schedrovitsky, the founder of the organizational-activity approach. During Perestroika in the USSR, G.P.Schedrovitsky played many games with Soviet managers. In terms of psychological intensity, ODIs (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 movement of ODI. Critics point to the artificial uniqueness of ODI. The Moscow Methodological Circle (MMK) became the basis of the ODI.
The mathematical theory of games is now developing rapidly, dynamic games are being considered. However, the mathematical apparatus of game theory is expensive. It is used for justified tasks: politics, economics of monopolies and the distribution of market power, etc. A number of well-known scientists became 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 the Cold War, which confirms the scale of the problems involved in game theory.
Nobel laureates in economics for achievements in the field of game theory and economic theory were: Robert Aumann, Reinhard Zelten, John Nash, John Harsagni, 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 field of application 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 theory analysis was first used to describe animal behavior by Ronald Fischer in the 1930s (although even Charles Darwin used the ideas of game theory without formal justification). The term "game theory" does not appear in Ronald Fischer's work. Nevertheless, the work is essentially done in the mainstream of game-theoretic analysis. The developments in economics were applied by John Maynard Smith in his 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 applied game theory to better understand good (decent) behavior. Generally speaking, the first game-theoretic arguments explaining correct behavior were expressed by Plato.
Description and modeling.
Originally, game theory was used to describe and model the behavior of human populations. Some researchers believe that by determining equilibrium in appropriate 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 modeling are often violated in real life. Researchers may assume that players choose behaviors that maximize their net benefit (the economic man model), but in practice human behavior often fails to meet this premise. There are many explanations for this phenomenon - irrationality, modeling discussion, and even different motives of the players (including altruism). Game-theoretic authors object to this, 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 on game theory when experiments revealed that people did not follow equilibrium strategies in practice. For example, in the games "Centipede", "Dictator", the participants often do not use the strategy profile that makes up the Nash equilibrium. There is an ongoing debate about the significance of such experiments. According to another 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 come to the Nash equilibrium remains open. Some researchers in search of an answer to this question have switched to the study of evolutionary game theory. Evolutionary game theory models imply limited rationality or irrationality of the players. Despite the name, evolutionary game theory deals with more than just natural selection of species. This branch of game theory examines models of biological and cultural evolution as well as models of the learning process.
Normative analysis (identifying the best behavior).
On the other hand, many researchers view 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 are the best response to the behavior of another player, the use of the Nash equilibrium concept to select behavior seems quite reasonable. However, this use of game-theoretic models has been criticized. First, in some cases, it is beneficial for a player to choose a strategy that is not in equilibrium if he expects that other players will not follow equilibrium strategies either. Secondly, the famous game " The Prisoner's Dilemma”Allows one more counterexample. IN " The Prisoner's Dilemma»Pursuing personal interests puts both players in a worse situation in comparison with the one in which they would sacrifice their personal interests.
Game types
Cooperative and non-cooperative.
The game is called cooperative, or coalition, if players can join in groups, taking on some obligations to other players and coordinating their actions. This is different from non-cooperative games in which everyone is obliged to play for themselves. Recreational 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 the players pursue personal goals, and vice versa.
Of the two types of games, non-cooperative games describe situations in great detail and produce more accurate results. Cooperatives look at the whole process of the game. 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 of noncooperative games.
Hybrid games include elements of co-op and non-co-op 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 payments. In other words, if the players can exchange places and their winnings for the same moves will not change. Many of the two-player games under study are symmetrical. In particular, these are: "Prisoner's Dilemma", "Deer Hunt", "Hawks and Pigeons". As asymmetrical games, you 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 second player's payoff 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 - a 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 winnings is equal to the sum of all losses on any move. Look to the right - the numbers represent payments to the players - and their total in each cell is zero. Examples of such games are poker, where one wins all the bets of others; reverse, where the opponent's pieces are captured; or banal theft.
Many games studied by mathematicians, including the already mentioned "Prisoner's Dilemma", are of a different kind: non-zero-sum games the gain of one player does not necessarily mean the loss of 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 playerwho "appropriates" the surplus or makes up for the lack of funds.
Another game with a nonzero sum is tradewhere every member benefits. A well-known example where it decreases is war.
Parallel and sequential.
In parallel games, players move at the same time, or at least they are not aware of the choice of others until everything will not make their move. In sequential, or dynamicIn games, participants can make moves in a predetermined or random order, but at the same time they receive some information about the previous actions of others. This information may even be not quite complete, for example, a player can find out that his opponent from his ten strategies definitely didn't choose fifth, knowing nothing about others.
The differences in the presentation of parallel and sequential games were discussed above. The former are usually presented in normal form, and the latter in extensive.
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. Complete information is not available in parallel games, since they do not know the current moves of the opponents. Most of the games studied in mathematics are with incomplete information. For example, all the "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 something similar - perfect information ... For the latter, only knowledge of all strategies available to opponents is sufficient, knowledge of all their moves is not necessary.
Games with an infinite number of steps.
Real-world games or games studied in economics tend to last the final number of moves. Mathematics is not so limited, and in particular, set theory deals with games that can go on 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" - no player has such a strategy. The existence of winning strategies for some specially designed games has an important role in descriptive set theory.
Discrete and continuous games.
Most games studied 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 these elements are often called differential games. They are associated with some kind of material scale (usually a time scale), although the events occurring in them may be discrete in nature. Differential games are also considered in optimization theory and 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 purpose of metagames is to increase the usefulness of the set of rules produced. Metagame theory is related to theory of optimal mechanisms .
based on materials from wikipedia.org
