Let us study the mechanism for establishing market equilibrium, when, under the influence of changes in supply or demand factors, the market leaves the ϶ᴛᴏth state. There are two main variants of the disproportion between supply and demand: excess and shortage of goods.
Excess(surplus) of goods - ϶ᴛᴏ such a situation in the market when the supply of goods at a given price exceeds the demand for it. In this case, competition arises between producers, a struggle for buyers. The winner is the one who offers more favorable conditions for the sale of goods. Thus, the market tends to return to a state of equilibrium.
deficit goods - in this case, the demand for goods at a given price exceeds the quantity of goods offered. In this situation, competition already arises between buyers for the opportunity to purchase a scarce product. The one who offers the highest price for this product wins. The increased price attracts the attention of producers to it, which begin to expand production, thereby increasing the supply of goods. As a result, the system returns to a state of equilibrium.
Based on all of the above, we come to the conclusion that the price implements a balancing function, stimulating the expansion of production and supply of goods with a shortage and restraining supply, ridding the market of surpluses.
The balancing role of price will be through both demand and supply.
We will proceed from the assumption that the equilibrium established in our market was disturbed - under the influence of any factors (for example, income growth) there was an increase in demand, as a result, its curve shifted from D1 V D2(Fig. 4.3 a), and the proposal remained unchanged.
If the price of a given product did not change immediately after the shift in the demand curve, then following the growth in demand, a situation will arise when, at the previous price, P1 the quantity of goods that each of the buyers can now purchase (QD) exceeds the volume that can be offered at a given price by the producers of a given Goods (QS). The amount of demand will now exceed the amount of supply of this product, which means that shortage of goods at the rate of Df = QD – Qs in this market.
The shortage of goods, as we already know, leads to competition between buyers for the opportunity to purchase this product, which leads to an increase in market prices. In ϲᴏᴏᴛʙᴇᴛϲᴛʙii with the law of supply, the reaction of sellers to an increase in price will be an increase in the volume of the goods offered. On the chart, ϶ᴛᴏ will be expressed by moving the market equilibrium point E1 along the supply curve until it intersects with the new demand curve D2 where the new equilibrium of the given market will be reached E2 s equilibrium quantity of goods Q2 and equilibrium price P2.
Rice. 4.3. Equilibrium price point shift.
Let us study the situation when the equilibrium state is disturbed by the supply side.
We will proceed from the assumption that under the influence of some factors there was an increase in supply, as a result, its curve shifted to the right from the position S1 V S2 and demand remained unchanged (Fig. 4.3 b).
As long as the market price remains the same (R1) an increase in supply will lead to excess goods in size Sp = Qs–QD. As a result, there is vendor competition, leading to a decrease in the market price (with P1 before P2) and an increase in the volume of goods sold. On the chart, ϶ᴛᴏ will be reflected by moving the market equilibrium point E1 along the demand curve until it intersects with the new supply curve, resulting in a new equilibrium E2 with parameters Q2 And P2.
Similarly, it is possible to identify the effect on the equilibrium price and the equilibrium quantity of goods of a decrease in demand and a decrease in supply.
In the educational literature, four rules for the interaction of supply and demand are formulated.
An increase in demand causes an increase in the equilibrium price and the equilibrium quantity of goods.
A decrease in demand causes a fall in both the equilibrium price and the equilibrium quantity of goods.
An increase in supply entails a decrease in the equilibrium price and an increase in the equilibrium quantity of goods.
A decrease in supply entails an increase in the equilibrium price and a decrease in the equilibrium quantity of goods.
It is worth saying - using these rules, you can find an equilibrium point for any changes in supply and demand.
The following circumstances can mainly prevent the price from returning to the market equilibrium level:
administrative regulation of prices;
monopolism producer or consumer, allowing to keep the monopoly price, which can be both artificially high and low.
In an antagonistic game, it is natural to consider the optimal outcome to be one in which it is unprofitable for any of the players to deviate from it. Such an outcome (x*,y*) is called an equilibrium situation, and the principle of optimality based on finding an equilibrium situation is called the equilibrium principle.
Definition. In a matrix game with a matrix of dimensions, the outcome is equilibrium situation or a saddle point if
At a saddle point, a matrix element is both the minimum in its row and the maximum in its column. In the game from the example, element 2 a 33 is a saddle point. Optimal in this game are the third strategies for both players. If the first player deviates from the third strategy, then he starts to win less than a 33. If the second player deviates from the third strategy, then he begins to lose more than a 33. Thus, for both players, there is nothing better than consistently sticking to the third strategy.
The principle of optimal behavior: if there is a saddle point in a matrix game, then the optimal strategy is the choice corresponding to the saddle point. What happens if there is more than one saddle point in the game?
Theorem. Let 
two arbitrary saddle points in a matrix game. Then:
Proof. From the definition of the equilibrium situation, we have:
Let us substitute into the left side of inequality (2.8) , and into the right - , into the left side of inequality (2.9) - , into the right - . Then we get:
Where does the equality come from:
It follows from the theorem that the payoff function takes the same value in all equilibrium situations. That is why the number is called at the cost of the game. And the strategies corresponding to any of the saddle points are called optimal strategies players 1 and 2, respectively. By virtue of (2.7), all the player's optimal strategies are interchangeable.
The optimality of the players' behavior will not change if the set of strategies in the game remains the same, and the payoff function is multiplied by a positive constant (or a constant number is added to it).
Theorem. For a saddle point (i*,j*) to exist in the matrix game, it is necessary and sufficient that the maximin be equal to the minimax:
(2.10) 
Proof. Necessity. If (i*,j*) is a saddle point, then, according to (2.6) :
(2.11) 
However, we have:
(2.12) 
From (2.11) and (2.12) we get:
(2.13) 
Arguing similarly, we arrive at the equalities:
Thus,
On the other hand, the reverse inequality (2.5) is always satisfied, so (2.10) is true.
Adequacy. Let (2.10) be true. Let us prove the existence of a saddle point. We have:
According to equality (2.10), inequalities (2.15) and (2.16) turn into equalities. After which we have:
The theorem has been proven. Along the way, it is proved that the total value of maximin and minimax is equal to the price of the game .
Mixed Game Expansion
Consider a matrix game G. If there is an equilibrium situation in it, then the minimax is equal to the maximin. Moreover, each of the players can tell the other player information about his optimal strategy. His opponent will not be able to derive any additional benefit from this information. Now suppose that there is no equilibrium situation in the game G. Then:
In this case, the minimax and maximin strategies are not stable. Players may have incentives to deviate from their prudent strategies related to the possibility of getting more payoff, but also the risk of losing, i.e., getting a payoff less than using a prudent strategy. When using risky strategies, the transfer of information about them to the opponent has detrimental consequences: the player automatically receives a smaller payoff than when using a cautious strategy.
Example 3. Let the game matrix look like:
For such a matrix , i.e. equilibrium does not exist. The cautious strategies of the players are i*=1, j*=2. Let player 2 follow strategy j*=2, and player 1 choose strategy i=2. then the latter will receive a payoff of 3, which is two units more than the maximin. If, however, player 2 guesses about the plans of player 1, he will change his strategy to j=1, and then the first one will receive a payoff of 0, that is, less than his maximin. Similar reasoning can be carried out for the second player. In general, we can conclude that the use of an adventurous strategy in a separate game of the game can bring a result greater than guaranteed, but its use is associated with risk. The question arises, is it possible to combine a reliable cautious strategy with an adventurous one in such a way as to increase your average payoff? Essentially, the question is how to divide the payoff (2.17) between the players?
It turns out that a reasonable solution is to use a mixed strategy, that is, a random choice of pure strategies. Recall that player 1's strategy is called mixed, if the choice of the i-th row is made by him with some probability p i . Such a strategy can be identified with the probability distribution 
on multiple lines. Suppose the first player has m pure strategies and the second player has n pure strategies. Then their mixed strategies are probability vectors:
(2.18) 
Consider two possible mixed strategies for the first player in Example 3: 
. These strategies differ in the probability distributions between pure strategies. If in the first case the rows of the matrix are chosen by the player with equal probabilities, then in the second case - with different ones. When we talk about a mixed strategy, we mean by random choice not a “random” choice, but a choice based on the operation of a random mechanism that provides the probability distribution we need. So for the implementation of the first of the mixed strategies, a coin toss is well suited. The player chooses the first line or the second, depending on how the coin falls out. On average, the player will choose both the first row and the second row equally often, but the choice at a particular iteration of the game is not subject to any fixed rule and has the maximum degree of secrecy: before the implementation of the random mechanism, it is unknown even to the very first player. To implement the second mixed strategy, the draw mechanism is well suited. The player takes seven identical pieces of paper, marking three of them with a cross, and throws them into the hat. Then, at random, he extracts one of them. According to classical probability theory, he will pull out a piece of paper with a cross with a probability of 3/7, and a clean piece of paper with a probability of 4/7. Such a draw mechanism is capable of realizing any rational probabilities.
Let the players adhere to mixed strategies (2.18). Then the payoff of the first player at a single iteration of the game is a random variable: v(X,Y). Since players choose strategies independently of each other, then, according to the probability multiplication theorem, the probability of choosing an outcome (i, j) with a win is equal to the product of probabilities . Then the distribution law of the random variable v(X,Y) given by the following table
Now let the game play out indefinitely. Then the average payoff in such a game is equal to the mathematical expectation of the value v(X,Y).
(2.19) 
For a finite but sufficiently large number of iterations of the game, the average payoff will differ slightly from the value (2.19).
Example 4. Calculate the average payoff (2.19) for the game from example 3 when the players use the following strategies: 
. The payoff matrix and the probability matrix are as follows:
Let's find the average:
Thus, the average payoff (2.20) is intermediate between the maximin and minimax.
Since for any pair of mixed strategies X and Y it is possible to calculate the average value of the game, then the problem of finding the optimal strategy arises. It is natural to start by exploring cautious strategies. The cautious strategy of the first player provides him with a maximin. The cautious strategy of the second player does not allow the first to win more than the minimax. The most significant result in the theory of games with opposite interests can be considered the following:
Theorem. Every matrix game has an equilibrium situation in mixed strategies. The proof of this theorem is not easy. It is omitted in this course.
Consequences: The existence of an equilibrium situation means that the maximin is equal to the minimax, and therefore any matrix game has a price. The optimal strategy for the first player is the maximin strategy. The optimal strategy of the second is minimax. Since the problem of finding optimal strategies has been solved, we say that any matrix game solvable on a set of mixed strategies.
Solution of the game 2x2
Example 5. Solve the game. It is not difficult to verify that there is no saddle point. Denote the optimal strategy of the first player (x, 1-x) is a column vector, but for convenience we write it as a string. Denote the optimal strategy of the second player (y,1-y).
The payoff of the first player is a random variable with the following distribution:
| v(x,y) | 2 | -1 | -4 | 7 |
| p | xy | x(1-y) | (1x)y | (1-x)(1-y) |
We find the average payoff for the iteration of the first player - the mathematical expectation of a random variable v(x,y):
Let's transform this expression:
This mathematical expectation consists of a constant (5/7) and a variable part: 14(x-11/14)(y-8/14). If the value y different from 8/14, then the first player can always choose X in such a way as to make the variable part positive, increasing your winnings. If the value X different from 11/14, then the second player can always choose y so as to make the variable part negative, reducing the payoff of the first player. Thus, the saddle point is defined by the equalities: x*=11/14, y*=8/14.
2.5 Game solving
An example will show how to solve such games.
Example 6. Solve the game 
. We make sure that there is no saddle point. Denote the mixed strategy of the first player X=(x, 1-x) is a column vector, but for convenience we write it as a string.
Let the first player apply the strategy X, and the second - his j-th pure strategy. Let us denote the average payoff of the first player in this situation as . We have:
Let us draw the graphs of functions (2.21) on the segment .
The ordinate of a point located on any of the line segments corresponds to the payoff of the first player in a situation where he uses a mixed strategy (x, (1-x)), and the second player the corresponding pure strategy. The guaranteed result of the first player is the lower envelope of the family of lines (broken ABC). The highest point of this broken line (point B) is the maximum guaranteed result of player 1. The abscissa of point B corresponds to the optimal strategy of the first player.
Since the desired point B is the intersection of the lines and, then its abscissa can be found as a solution to the equation:
Thus, the optimal mixed strategy of the first player is (5/9, 4/9). The ordinate of point B is the price of the game. It is equal to:
(2.22) 
Note that the line corresponding to the second strategy of the second player passes above point B. This means that if the first player applies his optimal strategy, and player 2 uses the second one, then the loss of the second player increases compared to applying strategies 1 or 3. Thus, the second the strategy must not participate in the optimal strategy of the second player. The optimal strategy for player 2 should be: 
. Pure strategies 1 and 3 of the second player that have nonzero components in the optimal strategy are usually called significant. Strategy 2 is called insignificant. From the figure above, as well as from equality (2.22), it can be seen that when the first player applies his optimal strategy, the payoff of the second player does not depend on which of his essential strategies he uses. He can also apply any mixed strategy consisting of essential (in particular, optimal), the payoff will not change in this case either. A completely analogous statement is also true for the opposite case. If the second player uses his optimal strategy, then the payoff of the first player does not depend on which of his essential strategies he uses and is equal to the cost of the game. Using this statement, we find the optimal strategy for the second player.
Optimal strategies in the theory of conflicts are those strategies that lead the players to stable equilibria, i.e. some situations that satisfy all players.
The optimality of a solution in game theory is based on the concept equilibrium situation:
1) it is not profitable for any of the players to deviate from the equilibrium situation if all the others remain in it,
2) the meaning of equilibrium - with repeated repetition of the game, the players will reach a situation of equilibrium, starting the game in any strategic situation.
In each interaction, the following types of equilibria can exist:
1. equilibrium in cautious strategies . Determined by strategies that provide players with a guaranteed result;
2. equilibrium in dominant strategies .
Dominant strategy is such a plan of action that provides the participant with the maximum gain, regardless of the actions of the other participant. Therefore, the equilibrium of the dominant strategies will be the intersection of the dominant strategies of both participants in the game.
If the players' optimal strategies dominate all their other strategies, then the game has an equilibrium in the dominant strategies. In the prisoner's dilemma game, the Nash equilibrium set of strategies will be ("admit - admit"). Moreover, it is important to note that for both player A and player B "recognize" is the dominant strategy, while "not recognize" is dominated;
3. equilibrium Nash . Nash equilibrium is a type of decision of a game of two or more players, in which no participant can increase the payoff by changing his decision unilaterally, when other participants do not change their decision.
Let's say the game n faces in normal form, where is the set of pure strategies and is the set of payoffs.
When each player selects a strategy in the strategies profile, the player receives a payoff. Moreover, the payoff depends on the entire profile of strategies: not only on the strategy chosen by the player himself, but also on other people's strategies. The strategy profile is a Nash equilibrium if a change in its strategy is not beneficial to any player, that is, to any
A game can have a Nash equilibrium in both pure and mixed strategies.
Nash proved that if allowed mixed strategies, then in each game n players will have at least one Nash equilibrium.
In a Nash equilibrium situation, the strategy of each player provides him with the best response to the strategies of other players;
4. Balance Stackelberg. Stackelberg model– game-theoretic model of an oligopolistic market in the presence of information asymmetry. In this model, the behavior of firms is described by a dynamic game with complete perfect information, in which the behavior of firms is modeled using static games with complete information. The main feature of the game is the presence of a leading firm, which first sets the volume of output of goods, and the rest of the firms are guided in their calculations by it. Basic prerequisites of the game:
The industry produces a homogeneous product: the differences in the products of different firms are negligible, which means that the buyer, when choosing which firm to buy from, focuses only on the price;
The industry has a small number of firms.
firms set the quantity of products produced, and the price for it is determined based on demand;
There is a so-called leader firm, on the volume of production of which other firms are guided.
Thus, the Stackelberg model is used to find the optimal solution in dynamic games and corresponds to the maximum payoff of the players, based on the conditions that have developed after the choice already made by one or more players. Stackelberg equilibrium.- a situation where none of the players can increase their winnings unilaterally, and decisions are made first by one player and become known to the second player. In the prisoner's dilemma game, the Stackelberg equilibrium will be reached in the square (1; 1) - "admit guilt" by both criminals;
5. Pareto optimality- such a state of the system, in which the value of each particular criterion describing the state of the system cannot be improved without worsening the position of other players.
The Pareto Principle states: “Any change that does not cause loss, but that benefits some people (in their own estimation), is an improvement.” Thus, the right to all changes that do not bring additional harm to anyone is recognized.
The set of system states that are Pareto optimal is called the "Pareto set", "the set of alternatives that are Pareto optimal", or the "set of optimal alternatives".
A situation where Pareto efficiency has been achieved is a situation where all the benefits from the exchange have been exhausted.
Pareto efficiency is one of the central concepts for modern economics. Based on this concept, the first and second fundamental welfare theorems are constructed.
One of the applications of Pareto optimality is the Pareto distribution of resources (labor and capital) in international economic integration, i.e. economic union of two or more states. Interestingly, the Pareto distribution before and after international economic integration was adequately described mathematically (Dalimov R.T., 2008). The analysis showed that the added value of the sectors and the income of labor resources move in opposite directions in accordance with the well-known heat conduction equation, similar to a gas or liquid in space, which makes it possible to apply the analysis technique used in physics in relation to economic problems of migration of economic parameters.
Pareto optimum states that the welfare of society reaches its maximum, and the distribution of resources becomes optimal if any change in this distribution worsens the welfare of at least one subject of the economic system.
Pareto-optimal state of the market- a situation where it is impossible to improve the position of any participant in the economic process without simultaneously reducing the well-being of at least one of the others.
According to the Pareto criterion (criterion for the growth of social welfare), movement towards the optimum is possible only with such a distribution of resources that increases the welfare of at least one person without harming anyone else.
Situation S* is said to be Pareto dominant situation S if:
for any player his payoff in S<=S*
· there is at least one player for whom his payoff in the situation S*>S
In the "prisoners' dilemma" problem, Pareto equilibrium, when it is impossible to improve the position of any of the players without worsening the position of the other, corresponds to the situation of the square (2; 2).
Consider example 1.
Combining the supply and demand lines in a single graph, we obtain a graphical representation of the equilibrium in the coordinates P, Q(Fig. 2.6). The point of intersection of the lines has coordinates (P * , Q*), Where R* - equilibrium price, Q*- equilibrium volume of production and consumption.
Market equilibrium- this is a state of the market in which, for a given price level, the quantity demanded is equal to the quantity supplied.
Only at the point of balance E the market is balanced, none of the market agents have incentives to change the situation. This means that the market equilibrium has the property sustainability - in the event of a non-equilibrium state, market agents are motivated to return the market to equilibrium. To prove stability, the logic of L. Walras or A. Marshall is usually used.
According to L. Walras, at too high prices, there is an excess supply - overproduction (segment A-B in fig. 2.6i), such a market is called buyer's market since the buyer has the opportunity to demand a price reduction when concluding transactions. In such a situation, first of all, the seller is not interested, who is forced to reduce prices and reduce production volumes. As prices fall, the quantity demanded increases A-B shrinks until it becomes an equilibrium point E.
At low prices, there is an excess of demand - a shortage (segment CFna Fig. 2.6a), develops seller's market. The buyer is forced
If a consumer cuts consumption and overpays for a scarce good, as the price rises, the quantity supplied rises, and the scarcity shrinks until the market balances.
According to A. Marshall (Fig. 2.66), for small volumes of production, the demand price exceeds the seller's price, for large volumes - vice versa. In any case, the imbalance situation stimulates a shift in price or volume of supply and demand towards equilibrium. Equilibrium (A) according to Walras - the price regulates the imbalance of supply and demand, (b) according to Marshall - the prices of the buyer and the seller are balanced by a change in volumes.
Rice. 2.6. Establishment of market equilibrium: c) according to L. Walras; b) according to A. Marshall
A change in market demand or supply leads to a change in equilibrium (Fig. 2.7). If, for example, market demand increases, then the demand line shifts to the right, then the equilibrium price and volume increase. If market supply decreases, the supply line shifts to the left, resulting in an increase in price and a decrease in volume.
This market model is static, since time does not appear in it.
"Spider" model
As an example of a dynamic model of market equilibrium, we present the simplest "cobweb" model. Suppose the quantity demanded depends on the price level of the current period t, and the supply volume - from the prices of the previous period t-1:
Q d i = Q d i (P t) , Q s i = Q s i (P t -1) ,
where t = 0.1….T is the discrete value of the time period.
Rice. 2.7. Change in market equilibrium:
a) due to an increase in demand; b) due to a decrease
offers
Market price P t may not match the equilibrium price R*, and there are three possible dynamics P t(Fig. 2.8).
The variant of the development trajectory in this model depends on the ratio of the slopes of the supply and demand lines.
Rice. 2.8. "Spider" model of market equilibrium:
a) the deviation from equilibrium decreases; 5) deviation
increases from equilibrium (the “catastrophe” model); c) the market
oscillates cyclically around the equilibrium point, but the equilibrium
Topic 4. Game theory and interaction modeling.
1. Basic concepts of game theory.
2. Types of equilibrium: Nash equilibrium, Stekelberg, Pareto-optimal equilibrium, equilibrium of dominant strategies.
3. Basic models of game theory.
Basic concepts of game theory.
The use of mathematical methods, which include game theory, in the analysis of economic processes makes it possible to identify such trends, relationships that remain hidden when using other methods, and even obtain very unexpected results.
Note that game theory is one of the youngest mathematical disciplines. Its emergence as an independent branch of mathematics is attributed to the mid-1950s, when the well-known monograph by F. Neumann and O. Morgenstern "The Theory of Games and Economic Behavior" was published. The origins of game theory associated with the work of E. Porel (1921)."
To date, game theory has turned into a whole mathematical direction, rich in interesting results and having a large number of practical recommendations and applications.
Let us consider the main assumptions and concepts of the game model of interpersonal interactions.
1. The number of interacting individuals is two. Individuals are called players. The concept of a player allows one to model the social roles of an individual: seller, buyer, husband, wife, etc. A game is a simplified representation of the interactions of two individuals with different or similar social roles, for example, buyer - seller, seller - seller, etc.
2. Each individual has a fixed set of behaviors, or alternatives. The number of behavior options for different players may not be the same.
3. Interpersonal interaction is considered to be realized if both players simultaneously choose options for their behavior and act in accordance with them. A single act of interpersonal interaction is called the course of the game. The duration of the act of interaction is assumed to be zero.
4. The course of the game is given by two integers - the chosen number of the behavior option (move) of the first player and the chosen number of the behavior option (move) of the second player. The maximum possible number of different moves in the game is equal to the product of the total number of moves of the first player and the total number of moves of the second player.
5. Each interaction of individuals, or the course of the game, receives its serial number: 1, 2, 3, etc. The concept of "game move" (a pair of numbers) and "game move number" (single number) should not be confused. It is assumed that interactions occur regularly at regular intervals, so the number of the game turn indicates the length of the period of time during which these individuals interact with each other.
6. Each player strives to achieve the maximum value of some target indicator, which is called utility, or payoff. Thus, the player has the features of an "economic man". The player's payoff can be either positive or negative. A negative win is also called a loss.
7. Each move of the game (a pair of alternatives chosen by the players) corresponds to a unique pair of payoffs of the players. The dependence of the payoffs of the players on the moves they have chosen is described by the game matrix, or the payoff matrix. The rows of this matrix correspond to the alternatives (moves) of the first player, and the columns correspond to the alternatives (moves) of the second player. The elements of the game matrix are pairs of payoffs corresponding to the corresponding row and column (players' moves). The payoff of the first player (the first number in the cell of the game matrix) depends not only on his move (row number), but also on the move of the second player (column number). Therefore, before the implementation of the interaction, the individual does not know the exact amount of his gain. In other words, the player's choice of behavior is carried out under conditions of uncertainty, i.e., the player has the features of an "institutional person".
8. A player's strategy is a habitual stereotype of behavior that a player follows when choosing an alternative behavior for a certain period of time. The player's strategy is given by the probabilities (or frequencies) of choosing all possible behaviors. In other words, the player's strategy is a vector whose number of coordinates is equal to the total number of possible alternatives, and the i-th coordinate is equal to the probability (frequency) of choosing the i-th alternative. It is clear that the sum of the values of all coordinates of a given vector is equal to one.
If the player during the considered period of time chooses only one variant of behavior, then the player's strategy is called clean.
All coordinates of the corresponding pure strategy vector are equal to zero, except for one, which is equal to one.
A strategy that is not pure is called mixed.
In this case, the player's strategy vector has at least two non-zero coordinates. They respond to active behaviors. A player following a mixed strategy alternates active behaviors in accordance with given probabilities (frequencies) of choice. In what follows, for simplicity of presentation of the material, we will assume that the player always follows some pure strategy, i.e., in the considered period of time, he invariably chooses the only variant of behavior from a given set of alternatives.
An institutional person is characterized by the variability of his behavior, which depends on his internal state, life experience, external social environment, etc. In the framework of a game approach to the study of institutions, this property of an institutional person is expressed in the possibility of a player changing his strategy. If among the strategies of the player there always existed an objectively best one, then he would invariably follow it, and changing the strategy would be meaningless. But in real life, a person usually considers several strategies of behavior. It is impossible to single out the best among them objectively. The game model of interpersonal interactions allows us to explore this feature of institutional behavior, since it covers a number of behavioral strategies that do not exclude each other and reflect various aspects of the behavior of an institutional person. Let's take a look at these behaviors.
game matrix
| First Player | Second player | ||
| 6; 15 | 2; 13 | 3; 11 | |
| 1; 10 | 5; 14 | 4; 12 | |
| 4; 12 | 4; 13 | 3; 13 |
Distinguish solidary And non-solidary behavior strategies. The former are most typical for the "institutional man", and the latter - for the "economic man".
non-solidary behavioral strategies are characterized by the fact that an individual chooses a variant of his behavior independently, while he either does not take into account the behavior of another individual at all, or, on the basis of existing experience, suggests a possible variant of his behavior.
The main types of non-solidarity behavior include the following: irrational, careful, optimizing, deviant And innovative.
1) Irrational behavior. Denote the two strategies of the first player as A and B, respectively. Strategy A is called dominant in relation to strategy B if, for any move of the second player, the payoff of the first player, corresponding to strategy A, is greater than his payoff, corresponding to strategy B. Thus, strategy B is objectively worse in relation to strategy A.
If strategy A can always be freely chosen by the player, then strategy B should never be chosen at all. If, nevertheless, strategy B is chosen by the first player, then his behavior in this case is called irrational. To identify the irrational behavior of a player, it is enough to analyze the matrix of his payoffs: the payoff matrix of another player is not used in this case.
Note that the term "irrational behavior" is borrowed from neoclassical theory. It only means that the choice of this strategy is obviously not the best in a situation where both players are in an antagonistic confrontation, which is typical for an "economic man". But for an “institutional person” who enters into interpersonal interactions with other people, irrational behavior is not only possible, but may turn out to be the most reasonable behavior option. An example of this is the Prisoner's Dilemma game.
2) Cautious Behavior. The "institutional man", unlike the "economic man", is not absolutely rational, i.e. he does not always choose the best behavior that maximizes the gain. The limited rationality of the "institutional person" is expressed in his inability to choose the best option of behavior due to the large number of alternatives, the complex algorithm for determining the optimal alternative, the limited time for making a decision, etc. At the same time, the notion of bounded rationality suggests that, given all the complexities of choice, a person is able to choose a reasonably good alternative.
In the game approach to the study of institutions, the limited rationality of the individual is illustrated by the cautious behavior of the player.
Precautionary strategy- this is a player's strategy that guarantees him a certain amount of payoff, regardless of the choice (move) of the other player. The cautious strategy is also called maximin because it is calculated by finding the maximum value from several minimum values.
The cautious strategy of the first player is defined as follows. In each row of the matrix of his payoffs, the minimum element is found, and then the maximum, or maximin of the first player, is selected from such minimum elements. The line of the game matrix, on which the maximin of the first player is located, corresponds to his cautious strategy. The cautious strategy of the second player is obtained similarly. In each column of the matrix of its payoffs, the minimum element is found, and then the maximum element is determined from such minimum elements. The column of the game matrix, in which the maximin of the second player is located, corresponds to his cautious strategy. Each player can have several cautious strategies, but they all share the same value maximin (maximum minimum strategy), or a guaranteed win. Cautious strategies exist in any matrix game. To identify a player's cautious strategy, it is sufficient to analyze his payoff matrix, while the payoff matrix of another player is not used. This feature is common to irrational and cautious behavior.
3) Optimizing Behavior. In economic practice, situations often arise when economic agents (for example, a seller and a regular buyer) in the course of long-term interaction with each other find behavior strategies that suit both parties, and therefore are used by “players” for a long period of time. In the game approach to the study of institutions, the described situation is modeled using the concept of equilibrium strategies. A pair of such strategies is characterized by the following property: if the first player deviates from his equilibrium strategy (chooses some other one), and the second player continues to follow his equilibrium strategy, then the first player suffers damage in the form of a decrease in the payoff. A cell of the game matrix located at the intersection of a row and a column corresponding to a pair of equilibrium strategies is called an equilibrium point. The game matrix may have several equilibrium points, or it may not have them at all.
The behavior of a player following an equilibrium strategy is called optimizing ( minimax behavior or minimum-maximum strategy).
It is different from maximizing behavior. First, the equilibrium payoff of the player is not the maximum of all possible payoffs. It corresponds not to the global maximum, but to the local optimum. Thus, the global maximum of a function given on a numerical interval exceeds each of its local maxima. Secondly, following the equilibrium strategy by one player entails the achievement of a local maximum by him only if the equilibrium strategy is maintained by the other player. If the second player deviates from the equilibrium strategy, then further use of the equilibrium strategy by the first player will not give him a maximizing effect.
Equilibrium strategies are determined according to the following rule: a cell of the game matrix is considered to be equilibrium if the payoff of the first player corresponding to it is the maximum in the column, and the payoff of the second player corresponding to it is the maximum in the row. Thus, in the algorithm for finding equilibrium strategies, the payoff matrices of both players are used, and not one of them, as in cases of irrational and cautious behavior.
4) Deviant behavior. The institutionalization of an equilibrium strategy as a basic norm of behavior occurs as a result of a person's generalization of his experience of interpersonal interactions, including the experience of deviant behavior. A person's awareness of the negative consequences of such behavior, based on the choice of non-equilibrium alternatives, is the decisive argument when choosing an optimizing strategy of behavior. Thus, deviant behavior serves as an integral part of the life experience of the "institutional person", acting as an empirical justification for optimizing behavior. The experience of deviant behavior gives a person confidence that the other participant in the game will invariably adhere to the equilibrium strategy. Thus, such experience serves as proof of the rationality of the behavior of the other player and the predictability of future interactions with him.
5) Innovative behavior. Above, deviant behavior was considered, the main purpose of which is the empirical substantiation and consolidation of the initial equilibrium strategy. However, the goal of deviation from the equilibrium strategy may be fundamentally different. Innovative behavior is a systematic deviation from the usual equilibrium strategy in order to find another equilibrium state that is more beneficial for the innovator player.
Within the framework of the game model of interpersonal interactions, the goal of innovative behavior can be achieved if the game matrix has a different equilibrium point at which the payoff of the innovator player is greater than in the initial equilibrium state. If there is no such point, then innovative behavior is likely to be doomed to failure, and the innovator player will return to the original equilibrium strategy. At the same time, his losses from the innovative experiment will be equal to the total effect of the deviation for the entire period of the experiment.
In real life, interacting individuals often agree to follow certain behavioral strategies in the future. In this case, the behavior of the players is called solidary.
The main reasons for solidarity behavior:
a) the profitability of solidary behavior for both players. Within the framework of the game model of interaction, this situation is illustrated by a game matrix, in one cell of which the payoffs of both players are maximum, but at the same time it is not equilibrium and does not correspond to a pair of cautious strategies of the players. Strategies corresponding to this cell are unlikely to be chosen by players who implement non-solid behavior patterns. But if the players come to an agreement on the choice of appropriate solidarity strategies, then subsequently it will be unprofitable for them to violate the agreement, and it will be carried out automatically;
b) the ethical behavior of solidarity often serves as an "internal" mechanism to ensure compliance with the agreement. The moral cost, in the form of social condemnation, that an individual incurs if he violates an agreement, may be of greater importance to him than the increase in gain achieved by this. The ethical factor plays an important role in the behavior of an "institutional person", but it is not actually taken into account in the game model of interpersonal interactions;
c) coercion to solidarity behavior serves as an "external" mechanism to ensure compliance with the agreement. This factor of institutional behavior is also not adequately reflected in the game model of interactions.
Types of equilibrium: Nash equilibrium, Stekelberg, Pareto-optimal equilibrium, equilibrium of dominant strategies.
In each interaction, there can be different types of equilibria: the dominant strategy equilibrium, the Nash equilibrium, the Stackelberg equilibrium, and the Pareto equilibrium. A dominant strategy is a plan of action that provides the participant with maximum utility, regardless of the actions of the other participant. Accordingly, the equilibrium of the dominant strategies will be the intersection of the dominant strategies of both participants in the game. A Nash equilibrium is a situation in which each player's strategy is the best response to the other player's actions. In other words, this equilibrium provides the player with maximum utility depending on the actions of the other player. Stackelberg equilibrium occurs when there is a time lag in the decision-making of the participants in the game: one of them makes decisions, already knowing how the other acted. Thus, the Stackelberg equilibrium corresponds to the maximum utility of the players under conditions of non-simultaneous decision-making by them. Unlike the dominant strategy equilibrium and the Nash equilibrium, this kind of equilibrium always exists. Finally, a Pareto equilibrium exists under the condition that it is not possible to increase the utility of both players at the same time. Let us consider on one of the examples the technology of searching for equilibria of all four types.
Dominant strategy- such a plan of action that provides the participant with maximum utility, regardless of the actions of the other participant.
Nash equilibrium- a situation in which none of the players can increase their winnings unilaterally by changing their action plan.
Stackelberg equilibrium- a situation where none of the players can increase their winnings unilaterally, and decisions are made first by one player and become known to the second player.
Paretto equilibrium- a situation where it is impossible to improve the position of one of the players without worsening the position of the other and without reducing the total payoff of the players.
Let firm A seek to break firm B's monopoly on the production of a particular product. Firm A decides whether to enter the market, and firm B decides whether to reduce output in the event that A still decides to enter. In the case of unchanged output at firm B, both firms lose, but if firm B decides to reduce output, then it "shares" its profit with A.
Equilibrium of dominant strategies. Firm A compares its payoff under both scenarios (-3 and 0 if B decides to start a price war) and (4 and 0 if B decides to reduce output). She does not have a strategy that ensures the maximum gain regardless of B's actions: 0 > -3 => "do not enter the market" if B leaves output at the same level, 4 > 0 => "enter" if B reduces output (see . solid arrows). Although firm A does not have a dominant strategy, B does. It is interested in reducing output regardless of A's actions (4 > -2, 10 = 10, see dotted arrows). Therefore, there is no equilibrium of dominant strategies.
Nash equilibrium. Firm A's best response to firm B's decision to keep output the same is not to enter, but to a decision to reduce output is to enter. Firm B's best response to firm A's decision to enter the market is to reduce output; if firm B decides not to enter, both strategies are equivalent. Therefore, two Nash equilibria (A, A2) are at points (4, 4) and (0, 10) - A enters and B reduces output, or A does not enter and B does not reduce output. It is quite easy to verify this, since at these points none of the participants is interested in changing their strategy.
Stackelberg equilibrium. Suppose firm A is the first to make a decision. If it chooses to enter the market, then it will eventually end up at point (4, 4): the choice of firm B is unambiguous in this situation, 4 > -2. If it decides to refrain from entering the market, then the result will be two points (0, 10): firm B's preferences allow for both options. Knowing this, firm A maximizes its payoff at points (4, 4) and (0, 10) by comparing 4 and 0. Preferences are single-valued, and the first Stackelberg equilibrium StA will be at point (4, 4). Similarly, the Stackelberg equilibrium StB, when Firm B makes the first decision, will be at (0, 10).
Pareto equilibrium. To determine the Pareto optimum, we must iterate through all four outcomes of the game in sequence, answering the question: “Does the transition to any other outcome of the game provide an increase in utility simultaneously for both participants?” For example, from the outcome (-3, -2) we can go to any other outcome by fulfilling the specified condition. Only from the outcome (4, 4) we cannot move on without reducing the utility of any of the players, this will be the Pareto equilibrium, R.
