Application of game theory simulation in the management of an agro-industrial enterprise

. In the context of climate change, the agro-industrial complex certainly belongs to the class of complex production and economic systems, and in practice experiments to improve the management of such systems are dangerous because they can lead to unexpected and irreversible changes. According to the proposed scenario, agriculture in Uzbekistan will be extremely dangerous, as agricultural production depends on a number of risks, such as geopolitical situation, water scarcity, soil, weather, climatic conditions, pesticides, seeds, price differences, flow of organisms and diseases. mentions several of these game theory theories. This article discusses one of the decision-making methods in choosing the main type of agricultural product. The main goal is to determine the lowest expected outcome and the highest return in the shortest possible time with minimal investment, and the author used mathematical methods of game theory to analyze decision-making options to achieve the expected goal. especially games with nature, ie. maximum and minimum convolutions, Baes criteria, Vald criteria, Hurwitz criteria, Sevige criteria. The results confirm that, in addition to the effectiveness of the described methodology, the use of a game theory model is effective in developing and selecting the best production decisions in conditions of uncertainty where the production process is highly dependent on random factors. for the decision - making process. In order to increase the efficiency of management decision-making in the agro-industrial complex, the prospects for further development of these approaches are justified.


Introduction
It is obvious that the agricultural sector of the economy of the Republic of Uzbekistan is going through difficult times.Falling oil prices, sanctions, and hence the difficulties with import substitution, adversely affect our country's agriculture and the economy as a whole.Also, one should not forget that at the moment Western countries are wary of the Republic of Uzbekistan: now the former economic partners are not cooperating, the supply of food and raw materials has decreased, and interest in national products has been lost .
Against the background of all of the above, it is obvious that the agro-industrial complex of the Republic of Uzbekistan faces a number of intractable problems [12].The main ones are: -insufficient funding of state programs to support agriculture; -washing out working capital: the money of agricultural enterprises that come from the sale of products produced in the reporting year is not enough for the purchase of expensive equipment, seeds, fuels and lubricants and mineral fertilizers; -shortage of credit resources.More precisely, credit resources are available to farmers, but interest rates are unbearable; -the lack of financial resources will lead to a reduction in the procurement of agricultural machinery.
There is still one positive point for agriculture in the whole prevailing situation.In the segment of the real sector of the economy, agriculture remains one of the most attractive sectors for investment [34; 35].You can postpone the purchase of, say, a new car for a year, but you cannot postpone the purchase of meat, bread and vegetables for an equally long period.Also, according to the forecasts of analysts, one can make an assumption that in the future this industry expects an upswing, and therefore, in order to get big profits in the future, it is necessary to develop or start production of agricultural products now [36; 37].
Game theory is a relatively new area in mathematics, which owes much to John von Neumann (1903Neumann ( -1957) ) and uses matrix methods and probability theory as a foundation.The task of game theory is to find optimal strategies for the behavior of participants in a certain conflict in order to maximize their "wins" or minimize "losses".The participants of the game have to choose from a number of alternative behaviors (pure strategies), each of which leads to certain consequences, more or less preferable for the players.The game model describes in detail the potential results that players can count on, and indicates how the player should act in order to get the best result, taking into account the possible actions of opponents [38].
The history of the development of game theory is such that in the 20th century the economic theories that existed at that time did not fully describe the thoughts and actions of a particular market participant, there was no answer to the question on what basis does one or another market participant act.There was a certain spherical entrepreneur in a vacuum who tried to maximize his own profit, not paying attention to other market participants.But this contradicts the theory of a market economy [3; 4].
Therefore, there was a need to find an actual decision-making model.Researchers have concluded that market actions are similar to players competing with each other.Therefore, the concept of "game" was introduced as the activity of two or more players, the conditions of "winning" and "losing"; within the game, all participants manage some resources and interact among themselves, with the goal of "winning" and making decisions based on behavior other players.It also described a mathematical method for finding the optimal strategy for players.This was published in the book Game Theory and Economic Behavior by Oscar Morgenstern and John von Neumann in 1944 [21].
After 5 years in 1949, John Nash went further in his dissertation and defined situations in which participants not only compete with each other, but cooperate to achieve a common goal (cooperative goals).
Nash introduced the concept of "games with a nonzero sum" -the gain was not a constant (games with a zero sum), but could change from the actions of the players [5].
For its time, it was a real breakthrough in the study of game interaction, which clearly showed the decay of the "classical" concept of competition (when everyone is for himself).For this dissertation, John Nash, on October 11, 1994, at the age of 66, received the Nobel Prize in Economics "For the analysis of equilibrium in the theory of non-cooperative games" [22].
Half a century later, the theory grew and was repeatedly applied in practice.Models of game theory (games) can describe economic, legal, class, military conflicts, human interaction with nature.
The classification of games can be carried out: by the number of players, the number of strategies, the nature of the interaction of players, the nature of the win, the number of moves, the state of information, etc.
In this paper, we consider the question of the production of which products it is better to start the creation and development of small business in the agricultural industry (namely, growing vegetables) and what behavior helps to reduce losses [7; 8].
This business has been popular before.Without losing its relevance today, on the contrary, it fits perfectly into the "basket of consumer interests" due to the environmental friendliness of its final products.The fashion for a healthy lifestyle, proper nutrition and rejection of bad habits favorably wraps the eyes of potential customers in the direction of such a group of products as fresh herbs and vegetables of domestic production [9; 10].

Materials and method
The main goal is to determine the maximum expected return on the least expected product and the highest possible production in the short term at the expense of the minimum financial inflow, based on the Bayes criteria, Wald criteria, Hurvis criteria, Sevig criteria.Mathematical methods of game theory, such as games with nature, maximum and minimum convolutions, were used to achieve the expected goal.
Based on the set goal, the following tasks were identified: -to determine whether the production process is effective in the development and selection of the best production solutions, which are highly dependent on random factors, in the conditions of uncertainty in the use of the game theory model; -Substantiate the prospects for further development of these approaches in order to increase the efficiency of management decision-making in the agro-industrial complex

Results and discussion
Belousova, M., Danilina, O., [20] substantiate the types and varieties of plantations that bring maximum benefit, taking into account the attractiveness of the variety and the impact of weather conditions on management results, using game theory, analysis and comparison methods.a game theoretical model of species and varietal composition of fruit plantations was developed.The results of the presented study suggest that the use of a game theory model to develop and select the best production solutions in the face of uncertainty is effective.
In Hendrarini's work [18], a LEACh-based routing model was proposed in the results of the presented study, which was modified by the distribution of cluster heads to prevent neighboring cluster heads.
In a study by Nurimbetov et al. [16], the choice of strategic alternatives is a choice in determining which horticultural products will be grown: potatoes, cabbage or green onions.However, this work did not reflect the features of the species and varietal composition of fruit plantations (Umarov) [2].
The developed mathematical model of factors leading to product success not only studies specific products and scales, but also uses a number of standardized success factors to create a mathematical model of success multiplication factors influencing the success of different products (Setyaningrum) [23].
Using game theory in his work, the author found that it is better to start with the production of spicy vegetables to create a small business in agriculture, but also simulated a situation of losses and possible behavior of competitors.
In the development of the economy of the Republic of Uzbekistan, the game theory of the agricultural sector has found wide application in the innovative economy.In economics, it has been used not only to solve general economic problems, but also to analyze the strategic problems of enterprises, helping to select the main production area and make optimal management decisions.Using this theory, he determined using mathematical models that the country could maintain food security, which would adequately provide citizens with greenhouse crops, covering the costs of planting certain types of crops.
Today, state bodies of the Republic of Uzbekistan are actively using the game theory toolkit in order to find "dependencies" in the direction of making various kinds of payments in the greenhouse industry [1].The figure shows all kinds of situations with industry competitors.The first and second quadrants define a situation in which the reaction of competitive agro-industrial companies cannot have a significant impact on the payments of a stand-alone firm (Fig .1) This happens when competitors have no motivation (1st quadrant), or the ability (2nd quadrant) to strike back.Thus, the importance of the implementation of a detailed analysis of the strategy of motivated actions of competitors is lacking.The investigated conclusion is also formed relative to the 3rd quadrant.In this case, the reaction of competitors can pretty much affect the company, but its forces cannot significantly affect the payments of the competing company, the reaction should not be feared.Only the situation that is reflected in the 4th quadrant can require the application of game theory.But here only the most necessary, insufficient conditions are reflected that justify the use of game theory tools for making decisions regarding the situation of struggle with competitors.Situations often arise in which one state strategy dominates others, regardless of what kind of actions the competitor will take.The greenhouse economy is reflected in one of the important sectors of the economy of the Republic of Uzbekistan.The state of this industry is influenced by many factors, it is important to take them into account when manufacturing products.

Source: [Compiled by the authors]
Consider the following alternatives, the production of which is suitable for the initial stage of development of small business [11; 13].
The first alternative is the production of tubers, which include potatoes, sweet potatoes, Jerusalem artichoke, etc. Potatoes are the most common vegetable crop, occupying one of the first places in the diet.It is rightly called the second bread.
The second alternative is the production of tomato vegetables (tomatoes, eggplant, pepper).These plants are thermophilic and require good watering.There are two ways to grow this product: on the open ground and in greenhouses.Outdoor production is a seasonal process and quickly pays for itself, but there is a risk that it will not be possible to conclude a contract for the supply of goods and it will disappear, that is, there is a risk of loss instead of profit.Growing tomatoes in greenhouses is more profitable, but this requires a large startup capital, which, of course, is a big minus.
And the third alternative is spiced vegetables (dill, parsley, green onions, celery, etc.).These plants are not particularly fastidious to the environment, easily transported and do not require large initial capital to start production.Also their distinctive features: environmental friendliness, quick payback, lack of seasonality and accessibility for the consumer.
To determine the choice of an alternative, we turn to mathematics, namely, to game theory.Economic and mathematical methods and, in particular, game theory are widely used in economics in market research.Previously, the authors managed to apply economic and mathematical methods in the problem of modeling and forecasting demand for cars in Uzbekistan [14; 15].In the present work, based on the theory of games, optimal strategies are found in the problem of agricultural production.
We introduce a number of concepts.
A game is a mathematical model of a conflict situation.Players in a game are parties to a conflict.Winning is the outcome of the conflict.Usually it is quantified, for example, 1; -5; 0. Player strategies -a set of rules that determine the player's choice for a personal move, depending on the situation.
The solution of the game is the choice by each player of a strategy that satisfies the optimality condition, that is, one player must get the maximum gain, while the other adheres to his strategy.At the same time, the other player should have minimal loss if the first adheres to his strategy.Such strategies are called optimal.
Solve the game -this means finding the price of the game and its optimal solution.Building a model and examples of its implementation.We return to our problem and introduce the notation.We are player A, we have three strategies: A1 -choice of alternative number one, A2 -two, A3 -three.Since the cultivation of vegetables will take place on the open ground, one of the influencing factors will be the state of nature.We introduce the second player B, nature.His strategies: B1 -dry summers, B2 -rainy summers, B3 -summers with variable precipitation.Thus, we are one of the parties, an agricultural enterprise that is interested in earning the greatest income, and on the other hand, nature, which can harm us to the maximum extent (weather conditions depend on it).As a gain for player A, we take the profit from the implementation of one or another option (million UZS (sum of Uzbekistan)) and we assume that profit calculations depend on the state of nature.We write them in the form of a matrix, where the rows are the strategies of player A, the columns are the strategies of player B: Find the price of the game.Select the minimum matrix value in each row and choose the maximum one from them.This will be the minimum price of the game (this process is reflected in formula ( 1)) [24], it is denoted as where, α = 4 is the guaranteed win of player A. We also find the maximum price of the game β.To do this, find the maximum value in each column and select the minimum value from them This process is determined by formula (2) [25].
(2) where β = min (4, 9, 8) = 4 where, β = 4 is the guaranteed win for player B. If α = β, as in our case, then their value is the net price of the game.The strategies corresponding to it are called optimal, and their combination is called the optimal solution to the game.
Thus, we can conclude that the optimal solution for player A is to choose a third strategy, namely, the production of spicy vegetables.
But, as you know, nature acts by chance, and there is no certainty that this or that strategy will be chosen.To confirm or refute our choice, we calculate optimality criteria that take into account that the choice of player B (nature) is probabilistic.
To determine the criteria for optimality, we need additional data, namely the probability of the occurrence of a particular alternative and a risk matrix.
U Using the expert assessment method, we determined the probability of the onset of player B's (nature) strategies (given in formula 3) [26]:  (3) In this case, the risk value is determined by formula (4).[27]: where, thus, we obtain a risk matrix: Now we turn directly to the calculation of optimality criteria.

Bayes criterion
We write all the necessary data in the form of a table (Table 1).

Source: [Compiled by the authors]
Here ai is the average income of player A, (1) is calculated according to formula [28]. .
According to the Bayesian criterion, it is necessary to choose the maximum value of the average payoff.In our case, the A3 strategy will be optimal, since with it the average gain reaches (determined by formula (1)) [29]

Vald criterion
According to Wald's criterion, the optimal strategy is one that allows player A to get the lower price of the game.We have already calculated the value of α, its value is 4 and is achieved when player A chooses strategy A3.

Criterion of Sevage
For this criterion, we compile a risk table (Table 2).

Source: [Compiled by the authors]
where, ri is the maximum risk value from the row (Table 3).

Source: [Compiled by the authors]
According to the Sevage criterion, the optimal strategy is the one in which the risk value takes the least value in the most unfavorable situation, i.e. (determined by formula ( 7)) [30].

Hurwitz criterion
According to this criterion, a strategy is selected at which the maximum value of the quantity is achieved (determined by formula ( 9)) [32]: Where, λ is the coefficient that we set ourselves, and the more we want to be safe, the closer the value is to 1.
We take λ = 0.7 and write down all the intermediate results in the table for convenience of calculations.3.
According to these data, it can be seen that H assumes a maximum value of 5.2 for strategy A3.
Based on the totality of all the criteria, we conclude that the A3 strategy is optimal for the management of an agricultural enterprise, that is, we should deal with the cultivation of spicy crops.
Possible behavior of the company in the presence of large losses In many practical important conflict situations with multi-factor knowledge in agribusiness, the parties involved make their choices step by step.In this way, they make wise use of forward-looking strategies that reflect the dynamics of the conflict and their level of awareness of the real situation in which the conflict develops [19,39,40].
One of the classes of games that describe conflicts, the dynamics of which affect the behavior of participants, are the so-called positional games.
Positional play is a non-collaborative game that mimics sequential, systematic decisionmaking processes by players in a time-changing and incompletely diverse information environment.The game process itself consists of a series (period of uncertainty) transition from one state of the game to another under the influence of many factors, and the players follow the rules of the game when choosing different options.choose one of the possible actions accordingly, or randomly (random action).
Consider the following situation: In the Jizzakh region, there is a monopolist company "Y", which is engaged in the production and processing of vegetables.Our company "X" suffers serious losses, as loans were taken as start-up capital.
The first move is made by company B, i.e. "Y" offers to completely redeem the company with loans (1), or to redeem only the company (2), and the debts will remain.
The second move is made by firm A, i.e.Our company: chooses to agree (1) or refuse (2).
Imagine the sequence of moves in the form of a game tree (Fig. 2): We write the results of the calculations in the form of a payout table for player A and in the form of a game matrix (Table 4).According to the above algorithm, we find the upper and lower prices of the game.As last time, their value coincides: α = β = -10.
Thus, we can conclude that it is more profitable for player B not to buy our company, then we will incur a loss of 10 million UZS. and go broke, thereby "Y" gets rid of a potential competitor and remains a monopolist in this area.Thus, the measures for the intensive development and modernization of agriculture contained in the «Strategy of Action for Five Priority Development Fields of the Republic of Uzbekistan in 2017-2021» [1,41] will ensure the sustainable development of the agricultural sector, strengthen the country's food security, increase export potential and improve the quality of life of the population of the republic.

Conclusion
Using game theory, we found that to create a small business in the agricultural industry, it is better to start with the production of spicy vegetables.We also simulated a situation in which we incur losses, and the possible behavior of our competitors.
The agricultural industry is the most promising in the economy of the Republic of Uzbekistan.Game theory has found wide application in an innovative economy.In recent years, its importance has increased significantly in many areas of 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, which contributes to the choice of the main production area and the adoption of optimal management decisions.Application of game theory contributes to successful business development.Also, as in the problem considered, as the two players-may be a state and subjects of hothouse.With the help of this theory, the country can maintain its food security, which consists in the appropriate provision of citizens with greenhouse crops, reimbursement of the costs of planting certain types of crops.

Fig. 1 .
Fig. 1.A decision-making theorem for agro-industrial facilities (applicable in the Republic of Uzbekistan) the matrix elements were obtained by expert assessment.This matrix is called the payoff matrix or the game matrix.

Table 1 .
The necessary data