Mathematical methods and models in decision making. Mathematical theory of managerial decision making Mathematical model of decision making

Mathematical methods and models in decision making

Introduction!

The purpose of modeling is the process of researching an object at different levels - from qualitative to precise quantitative, as information is collected and the model develops.

In the mathematical field, methods and models are understood as complex categories that include:

methods in decision making;

operations research methods;

economic and mathematical methods;

methods of economic cybernetics;

optimal control methods;

applied mathematics in economics;

applied mathematics in the organization of production.

This list is not complete, indicating a wide range of mathematical methods and models. In various sources, the content of which relates to the topics presented, mathematical models and methods are considered in various combinations.

Practical proof of the indicated thought is possible on the example of the well-known method of "probability theory", which is presented in the framework of mathematical models by a wide class and includes concepts such as "probability", "random event", "random variable", "expectation (average value ) a random variable "," variance (scattering) ", etc. In the late XIX - early XX centuries. a new object is allocated, which is a switched telephone communication system, implying such concepts as "request for a connection", "refusal", "connection timeout", "switching" and the like.

A mathematical probabilistic-theoretical model of processes in switched telephone networks was formed in the 1920s. as a result of combining the presented method and the object. The author of such an operation was A.K. Erlang. As an example of the existing concepts of this model, we can note:

“Flow of applications”;

"Average waiting time";

"Average length of the service queue";

"Latency variance";

"Probability of failure".

The subsequent development of this scientific direction demonstrated the effectiveness of the conceptual categories of the symbiotic model, revealed its large-scale constructive function.

In the course of its development, this model has been transformed into a method for studying complex systems. As an example, we can single out the "queuing theory", the categorical apparatus of which has ceased to be perceived as an integral part of telephone networks. Terminology and conceptual base have acquired a general theoretical character. Thus, the organization of new models can be carried out by applying the theory of queuing to such objects as production processes, operating systems, computers, traffic flows, etc.

As a result, it seems obvious that the method is fully formed in the case of the development of a homogeneous set of models. The degree of research of the object directly depends on the number of developed models of the object. The dual nature of the model forms, in turn, the dualism of the categorical apparatus of modeling, which integrates into itself the concepts of general or specific, formed from the "method" and "object", respectively.

In other words, methods, models, objects organize a continuous sequence, which implies the presence of various groups of models that are formed in accordance with the specifics of their origin and applicability. Among these groups are:

models that imply the interaction of previously developed methods and new objects;

models, first created for the purpose of describing a specific object, while new models can be applied in relation to other objects.

Linear programming- mathematical discipline devoted to the theory and methods of solving extremal problems on sets n-dimensional vector space defined by systems of linear equations and inequalities.

Integer programming- a kind of linear programming, implying that the desired values must be integers.

A branch of mathematical programming that studies methods for finding the extrema of functions in the parameter space, where all or some of the variables are integers.

The simplest method for solving an integer programming problem is to reduce it to a linear programming problem with checking the result for integer values.

Streams in networks

The activities of modern society are closely related to all sorts of networks - take, for example, transport, communications, distribution of goods, and the like. Therefore, the mathematical analysis of such networks has become a subject of fundamental importance.

GEOMETRIC PROGRAMMING- chapter , studies a specific class optimization tasks found mainly in engineering and economic calculations. The main requirement of the method is that all the technical characteristics of the designed objects were expressed quantitatively as dependences on regulated parameters... This type of programming is called geometric because it effectively uses geometric the average and a number of geometric concepts such as vector spaces, vectors, orthogonality and etc.

NONLINEAR PROGRAMMING- chapter mathematical programming studying solution methods extreme tasks with nonlinear target function and / or domain of feasible solutions defined by nonlinear restrictions.

OPTIMUM CONTROL- 1. Basic concept mathematical theory of optimal processes(belonging to the branch of mathematics under the same name - O. u.); means choice such control parameters that would provide the best from the point of view of the given criterion flow process or, otherwise, the best system behavior, its development to goals on optimal trajectory... These control parameters are usually viewed as functions of time, which means the possibility of changing them during the process to select at each stage their best (optimal) values.

MASS SERVICE THEORY- chapter operations research which considers various processes in the economy, as well as in telephone communications, healthcare and other areas, as service processes, i.e., the satisfaction of some requests, orders (for example, servicing ships in the port - their unloading and loading, servicing turners in the toolroom of the workshop - providing them with cutters, customer service in the laundry - washing clothes, etc.).

THEORY OF USEFULNESS- a theoretical direction in economic science, developed by representatives of the Austrian school in the 19th-20th centuries, based on the basic objective concept of "utility", perceived as pleasure, satisfaction received by a person as a result of the consumption of goods. The basic principle of the theory of utility is - law of diminishing marginal utility, according to which the increment of utility received from one added unit of good is continuously decreasing.

Decision theory- an interdisciplinary area of research of interest to practitioners and related to mathematics, statistics, economics, philosophy, management and psychology; examines how real decision-makers make decisions and how optimal decisions can be made.

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 some 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.

Simulation modeling- a method that allows you to build models that describe the processes as they would in reality. Such a model can be “played” in time for both one test and a given set of them. In this case, the results will be determined by the random nature of the processes. Based on these data, one can obtain fairly stable statistics.

Dynamic programming Is a branch of mathematics devoted to the theory and methods of solving multistep optimal control problems.

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Mathematical Methods in Decision Making

From its inception, mathematics as a science is a tool in the search for truth, and therefore it can be considered that any mathematical operations, even the simplest ones, are mathematical methods of decision-making. Currently, decision-making is understood as a special process human activity aimed at choosing the best option (alternative) of action. Decision-making processes underlie any purposeful human activity. For example, when creating new technology(machines, devices, devices), in construction when designing new buildings, when organizing the functioning and development of social processes. In this regard, there is a need for a decision-making guide that would simplify this process and make decisions more reliable. In addition to empirical perception of the situation and intuition, in our time of difficult economic situations and enterprise management processes, managers need some basis and a "proven guarantee" of the decision to be made. Inevitably, formalization of the decision-making process is required. As a rule, important decisions are made by experienced people who are quite far from mathematics, and especially from its new methods, and who are afraid to lose more from formalization than to gain.

Consequently, science is required to advise on optimal decision making. The time has passed when the right decisions were made "by touch", by the "trial and error" method. Today, to work out such a solution requires a scientific approach - the losses associated with errors are too great. Optimal solutions make it possible to provide the enterprise with the most favorable conditions for the production of products (maximum profit with minimum labor costs, material and labor resources).

Currently, the search for optimal solutions can be considered using the sections of classical mathematics. So, for example, in mathematical statistics in the section "decision making" they study the ways of accepting or not accepting some basic hypothesis in the presence of a competing hypothesis, taking into account the loss function. Decision theory develops methods of mathematical statistics - methods for testing hypotheses. Different values of losses when choosing different hypotheses lead to results that differ from those obtained by methods of statistical hypothesis testing. The choice of a less probable hypothesis may turn out to be more preferable if the losses in case of an erroneous choice are less than the losses caused by an erroneous choice of a more probable competing hypothesis. Such problems are called statistical decision problems. To solve these problems, it is necessary to find the minimum value of the risk function on the set of possible outcomes, i.e. solve the problem of finding a conditional extremum. As a rule, for these tasks, you can select a goal and specify conditions, i.e. the constraints under which they must be resolved. Similar problems are dealt with in the "mathematical programming" section of mathematics, which, in turn, is part of the "operations research" section.

The input data is a real task - an arbitrarily formulated set of data about a problem situation. The first step in solving the problem is its formulation - bringing the data to a form convenient for building a model. Model is an approximate (descriptive) representation of reality. Further, according to the constructed model, the search for optimal solutions and the issuance of recommendations are carried out.

The models can be divided into 2 large groups:

Deterministic models:

Linear programming;

Integer programming and combinatorics;

Graph theory;

Streams in networks;

Geometric programming;

Non-linear programming;

Mathematical programming;

Optimal control.

Stochastic models:

Queuing theory;

Utility theory;

Decision making theory;

Game theory and game modeling;

Search theory;

Simulation modeling;

Dynamic simulation.

When making decisions, it is necessary to find the optimum of some functional in a deterministic or stochastic form. Two features should be noted. First, mathematical decision-making methods for problems related to various areas of human activity begin to penetrate each other, for example, optimization control problems in the transition from continuous variables to discrete ones become problems of mathematical (linear) programming, evaluation of the separating function

in statistical methods, decision making can be carried out using linear or quadratic programming procedures, etc. Second, the original numerical data as a result of measurements or observations

in decision-making problems for real situations are not deterministic, but are more often random variables

with known or unknown distribution laws, therefore, subsequent data processing requires the use of methods of mathematical statistics, theory of fuzzy sets or theory of possibilities.

Mathematical methods in economics and decision making can be divided into several groups:

1. Optimization methods.

2. Methods that take into account uncertainty, primarily probabilistic and statistical.

3. Methods for constructing and analyzing simulation models,

4. Methods of analysis of conflict situations (game theory).

Optimization methods

Optimization in mathematics is the operation of finding an extremum (minimum or maximum) of an objective function in a certain region of a vector space bounded by a set of linear or nonlinear equalities (inequalities).

The theory and methods of solving the optimization problem is studied by mathematical programming.

Mathematical programming is a branch of mathematics that develops theory, numerical methods for solving multidimensional constrained problems. Unlike classical mathematics, mathematical programming deals with mathematical methods for solving problems of finding the best possible options.

Optimization problem statement

In the design process, the task is usually to determine the best, in a sense, the structure or values of the parameters of objects. This task is called optimization. If optimization is associated with the calculation of the optimal values of parameters for a given structure of the object, then it is called parametric optimization. The problem of choosing the optimal structure is structural optimization.

A standard mathematical optimization problem is formulated in this way. Among the elements h that form the set H, find an element h * that provides the minimum value f (h *) of a given function f (h). In order to correctly formulate the optimization problem, it is necessary to set:

1. An admissible set is a set

solution math game

2. Objective function - display;

3. Search criterion (max or min).

Then solving the problem means one of:

1. Show that.

2. Show that the objective function is not bounded from below.

If, then find:

If the function to be minimized is not convex, then it is often limited to the search for local minima and maxima: points such that everywhere in some of their neighborhoods for the minimum and for the maximum.

If the set is admissible, then such a problem is called an unconstrained optimization problem, otherwise it is called a conditional optimization problem.

Optimization methods classification

The general record of optimization problems specifies a wide variety of their classes. The selection of the method (the efficiency of its solution) depends on the class of the problem. The classification of problems is determined by: the objective function and the admissible area (set by a system of inequalities and equalities or a more complex algorithm).

Optimization methods are classified according to the optimization tasks:

1. Local methods:

converge to some local extremum of the objective function. In the case of a unimodal objective function, this extremum is unique and will be the global maximum / minimum.

2. Global methods:

deal with multi-extreme target functions. In a global search, the main task is to identify trends in the global behavior of the target function.

The currently existing search methods can be divided into three broad groups:

1.deterministic;

2. random (stochastic);

3. combined.

According to the criterion of the dimension of the admissible set, optimization methods are divided into one-dimensional optimization methods and multidimensional optimization methods.

By the type of the objective function and the feasible set, optimization problems and methods for their solution can be divided into the following classes:

Optimization problems in which the objective function and constraints are linear functions, are resolved by the so-called linear programming methods.

Otherwise, deal with the task nonlinear programming and apply appropriate methods. In turn, two particular tasks are distinguished from them:

if and are convex functions, then such a problem is called a convex programming problem;

if, then we are dealing with an integer (discrete) programming problem.

According to the requirements for smoothness and the presence of partial derivatives in the objective function, they can also be divided into:

· Direct methods requiring only calculations of the objective function at the points of approximation;

· Methods of the first order: they require the calculation of the first partial derivatives of the function;

· Methods of the second order: they require the calculation of the second partial derivatives, that is, the Hessian of the objective function.

In addition, optimization methods are divided into the following groups:

Analytical methods (for example, the Lagrange multiplier method and Karush-Kuhn-Tucker conditions);

Numerical methods;

Graphic methods.

Depending on the nature of the set X, mathematical programming problems are classified as:

· Discrete programming problems (or combinatorial optimization) - if X is finite or countable;

· Problems of integer programming - if X is a subset of the set of integers;

· Problems of nonlinear programming, if the constraints or objective function contain nonlinear functions and X is a subset of a finite-dimensional vector space.

If all the constraints and the objective function contain only linear functions, then this is a linear programming problem.

In addition, sections of mathematical programming are parametric programming, dynamic programming and stochastic programming.

Mathematical programming is used to solve optimization problems of operations research.

The way to find the extremum is completely determined by the class of the problem. But before getting a mathematical model, you need to do 4 stages of modeling:

1. Determination of the boundaries of the optimization system

We discard those connections of the optimization object with the outside world that cannot greatly affect the optimization result, or, more precisely, those without which the solution is simplified

2. Selection of controlled variables

We "freeze" the values of some variables (unmanaged variables). The others are left to take any values from the range of feasible decisions (controlled variables)

3. Determination of constraints on controlled variables (equality and / or inequality).

Choosing a numerical optimization criterion (for example, a performance indicator)

4. Create an objective function.

Probabilistic statistical methods

The essence of probabilistic and statistical methods of decision making

How are the approaches, ideas and results of probability theory and mathematical statistics used in decision making?

The base is a probabilistic model of a real phenomenon or process, i.e. a mathematical model in which objective relationships are expressed in terms of probability theory. Probabilities are used primarily to describe uncertainties that need to be considered when making decisions. This refers to both unwanted opportunities (risks) and attractive ones ("lucky chance"). Sometimes randomness is deliberately introduced into a situation, for example, by drawing lots, randomly selecting units to control, holding lotteries or consumer surveys.

Probability theory allows for some probabilities to calculate others that are of interest to the researcher. For example, based on the probability of a coat of arms falling out, you can calculate the probability that with 10 coin tosses at least 3 coats of arms will fall out. Such a calculation is based on a probabilistic model, according to which coin tosses are described by a scheme of independent tests, in addition, the coat of arms and the lattice are equally possible, and therefore the probability of each of these events is Ѕ. A more complex model is one in which, instead of tossing a coin, checking the quality of a unit of output is considered. The corresponding probabilistic model is based on the assumption that the quality control of various items of production is described by an independent test scheme. In contrast to the coin tossing model, a new parameter must be introduced - the probability P that a unit of production is defective. The model will be fully described if it is assumed that all items have the same probability of being defective. If the latter assumption is incorrect, then the number of model parameters increases. For example, you can assume that each item has its own probability of being defective.

Let us discuss a quality control model with a common defectiveness probability P. To do this, it is necessary to go beyond the probabilistic model and turn to the data obtained during quality control. Mathematical statistics solves the inverse problem in relation to the theory of probability. Its purpose is to draw conclusions about the probabilities that underlie the probabilistic model based on the results of observations (measurements, analyzes, tests, experiments). For example, based on the frequency of occurrence of defective products during inspection, conclusions can be drawn about the probability of defectiveness (see Bernoulli's theorem above). On the basis of Chebyshev's inequality, conclusions were drawn about the correspondence of the frequency of occurrence of defective products to the hypothesis that the probability of defectiveness takes on a certain value.

Thus, the application of mathematical statistics is based on a probabilistic model of a phenomenon or process. Two parallel series of concepts are used - related to theory (probabilistic model) and related to practice (sample of observation results). For example, the theoretical probability corresponds to the frequency found from the sample. The mathematical expectation (theoretical series) corresponds to the sample arithmetic mean (practical series). Typically, sample characteristics are theoretical estimates. At the same time, the values related to the theoretical series “are in the heads of researchers”, refer to the world of ideas (according to the ancient Greek philosopher Plato), and are inaccessible for direct measurement. Researchers have only sample data, with the help of which they try to establish the properties of the theoretical probabilistic model that interest them.

Why is a probabilistic model needed? The fact is that only with its help it is possible to transfer the properties established from the results of the analysis of a particular sample to other samples, as well as to the entire so-called general population. The term “general population” is used when referring to a large but finite population of units of interest. For example, about the aggregate of all residents of Russia or the aggregate of all consumers of instant coffee in Moscow. The purpose of marketing or opinion polls is to transfer statements from a sample of hundreds or thousands of people to populations of several million people. In quality control, a batch of products acts as a general population.

In order to transfer conclusions from a sample to a larger population, one or another assumption about the relationship of the sample characteristics with the characteristics of this larger population is necessary. These assumptions are based on an appropriate probabilistic model.

Of course, it is possible to process sample data without using a particular probabilistic model. For example, you can calculate the sample arithmetic mean, calculate the frequency of the fulfillment of certain conditions, etc. However, the calculation results will relate only to a specific sample; the transfer of the conclusions obtained with their help to any other population is incorrect. This activity is sometimes referred to as “data mining”. Compared to probabilistic-statistical methods, data analysis has limited cognitive value.

So, the use of probabilistic models based on evaluating and testing hypotheses using sample characteristics is the essence of probabilistic-statistical decision-making methods.

We emphasize that the logic of using sample characteristics for making decisions based on theoretical models presupposes the simultaneous use of two parallel series of concepts, one of which corresponds to probabilistic models, and the other to sample data. Unfortunately, in a number of literary sources, usually outdated or written in a recipe spirit, no distinction is made between selective and theoretical characteristics, which leads readers to bewilderment and errors in the practical use of statistical methods.

The application of a specific probabilistic-statistical method consists of three stages:

1. The transition from economic, managerial, technological reality to an abstract mathematical and statistical scheme, that is, building a probabilistic model of a control system, technological process, decision-making procedures, in particular based on the results of statistical control, and the like.

2. Carrying out calculations and obtaining conclusions by purely mathematical means within the framework of a probabilistic model.

3. Interpretation of mathematical and statistical conclusions in relation to a real situation and making an appropriate decision (for example, on the conformity or non-conformity of product quality with established requirements, the need to adjust the technological process), in particular, conclusions (on the proportion of defective product units in a batch, on a specific type of laws distribution of controlled parameters of the technological process and the like).

Mathematical statistics applies concepts, methods and results of the theory of probability. Next, we consider the main issues of constructing probabilistic models in various cases. We emphasize that for the active and correct use of normative-technical and instructive-methodological documents on probabilistic-statistical methods, preliminary knowledge is needed. So, you need to know under what conditions a particular document should be applied, what initial data are needed for its selection and application, what decisions should be made based on the results of data processing, and so on.

Let's consider a few examples when probabilistic-statistical models are a good tool for solving problems.

In the novel by Alexei Nikolaevich Tolstoy, "Walking through the agony" (volume 1), it says: "the workshop gives twenty-three percent of the marriage, and you stick to this figure," Strukov said to Ivan Ilyich. " How are these words to be understood in a conversation between plant managers? A piece cannot be 23% defective. It can be either good or defective. Probably, Strukov thought that a large batch contains about 23% of defective items. Then the question arises: what does "approximately" mean? Let 30 out of 100 tested units of production turn out to be defective, or out of 1,000 - 300, or out of 100,000 - 30,000 ... Should Strukov be accused of lying?

The coin used as a toss must be "symmetrical": on average, half of the tosses should be heads, and half of the cases are tails. But what does “average” mean? If you carry out many series of 10 tosses in each series, then there will often be series in which a coin falls heads 4 times. For a symmetrical coin, this will occur in 20.5% of the series. And if there are 40,000 eagles per 100,000 tosses, can the coin be considered symmetrical? The decision-making procedure is based on the theory of probability and mathematical statistics.

An example may seem frivolous. This is not true. The drawing of lots is widely used in the organization of industrial technical and economic experiments. For example, when processing the results of measuring the quality indicator (frictional moment) of bearings depending on various technological factors (the influence of a conservation medium, methods of preparing bearings before measurement, the effect of bearing load during measurement, and the like). Let's say you want to compare the quality of bearings depending on the results of their storage in different conservation oils. When planning such an experiment, the question arises of which bearings should be placed in oil of one composition, and which in another, but in such a way as to avoid subjectivity and ensure the objectivity of the decision. The answer can be obtained by drawing lots.

A similar example can be given with quality control of any product. To decide whether a controlled batch of products meets or does not meet the established requirements, a representative part is selected from it: the entire batch is judged from this sample. Therefore, it is desirable that every unit in a controlled lot has the same likelihood of being selected. In a production environment, the selection of units of production is usually not done by lot, but according to special tables of random numbers or with the help of computer random number sensors.

Similar problems of ensuring the objectivity of comparison arise when comparing various schemes for organizing production, remuneration, tenders and competitions, and the selection of candidates for vacant positions. Draw lots or similar measures are needed everywhere.

Let it be necessary to identify the strongest and second strongest team when organizing a tournament according to the Olympic system (the loser is eliminated). Let's say that the stronger team always wins the weaker one. It is clear that the strongest team will definitely become the champion. The second strongest team will reach the final only when they have no games with the future champion before the final. If such a game is planned, then the second-strongest team will not make it to the final. Anyone who plans a tournament can either “knock out” the second-strongest team from the tournament ahead of schedule, bringing it together in the first meeting with the leader, or provide it with a second place, ensuring meetings with weaker teams until the final. To avoid subjectivity, a toss is carried out. For a tournament of 8 teams, the probability that the two strongest teams will meet in the final is 4 out of 7. Accordingly, with a probability of 3 out of 7, the second-strongest team will leave the tournament ahead of schedule.

Any measurement of product units (using a caliper, micrometer, ammeter ...) has errors. To find out if there are systematic errors, it is necessary to repeatedly measure a unit of product, the characteristics of which are known (for example, a reference material). It should be remembered that in addition to the systematic error, there is also a random error.

The question arises of how to identify the systematic error by measurements. If we only note whether the error obtained during the next measurement is positive or negative, then this problem can be reduced to the already considered one. Indeed, let us compare the measurement with tossing a coin: a positive error - with the falling heads, negative - tails (zero error with a sufficient number of scale divisions almost never occurs). Then checking the absence of a systematic error is equivalent to checking the symmetry of the coin.

So, the problem of checking for systematic error is reduced to the problem of checking the symmetry of a coin. The above reasoning leads to the so-called "sign criterion" in mathematical statistics.

With the statistical regulation of technological processes on the basis of the methods of mathematical statistics, rules and plans for the statistical control of processes are developed, aimed at timely detection of irregularities in technological processes and taking measures to adjust them and prevent the release of products that do not meet the established requirements. These measures are aimed at reducing production costs and losses from the supply of substandard products. With statistical acceptance control, based on the methods of mathematical statistics, quality control plans are developed by analyzing samples from batches of products. The difficulty lies in being able to correctly build probabilistic and statistical decision-making models. In mathematical statistics, probabilistic models and methods for testing hypotheses have been developed for this, in particular, hypotheses that the proportion of defective units of production is equal to a certain number, for example,.

Game theory

Game theory is a mathematical method for studying optimal strategies in games. A game is understood as a process in which each of the participating parties (two or more) are fighting for their interests. Each side pursues its own goals and uses some strategy, which can, in turn, lead to a win or a loss (the result depends on other players. Game theory provides an opportunity to choose the best strategy, taking into account ideas about other players, their capabilities and 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, jurisprudence 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.

Optimal solutions or strategies in mathematical modeling were proposed as early as the 18th century. The problems of production and pricing in an oligopoly, which later became textbook examples of game theory, were considered in the 19th century. A. Cournot and J. Bertrand. At the beginning of the XX century. E. Lasker, E. Zermelo, E. Borel put forward the idea of a mathematical theory of conflict of interest.

Mathematical game theory has its origins 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 Oscar Morgenstern "The theory of games and economic behavior"(English Theory of Games and Economic Behavior).

This area of mathematics has found some reflection in social culture. In 1998, the American writer and journalist Sylvia Nazar published a book about the fate of John Nash, the 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 NUMBERS, periodically 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 degrees - 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 "managerial dynamics". The first concepts of game theory analyzed antagonistic games, when 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 "noncooperative equilibrium", in a situation the parties use the optimal strategy, which leads to the creation of a stable equilibrium. It is beneficial for the players to maintain this balance, since any change will worsen their situation. These works by J. Nash made a 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 and immediately after World War II, 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 an 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 stated 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 the game 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, that the game is older than the person himself, since animals also play. The concept of play 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. Shchedrovitsky, the founder of the organizational-activity approach. During Perestroika in the USSR G.P. Shchedrovitsky 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 rapidly developing, 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 distribution market power etc. A number of renowned scientists have become Nobel laureates in economics for their contributions 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 tasks that game theory deals with.

Nobel laureates in economics for achievements in the field of game theory and economic theory were: Robert Auman, 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, Jean Tyrol.

Game presentation

Games are strictly defined mathematical objects. The game is formed by the players, a set of strategies for each player and an indication of the winnings, or payments, of the players for each combination of strategies. Most of the cooperative games are characterized by a characteristic function, while for the rest of the species, the normal or extensive form is more often used. Characterizing features of the game as a mathematical model of the situation:

1. The presence of several participants;

2. Uncertainty in the behavior of participants associated with the presence of each of them several options for action;

3. Difference (mismatch) of interests of the participants;

4. The interconnectedness of the behavior of the participants, since the result obtained by each of them depends on the behavior of all participants;

5. Availability of rules of conduct known to all participants.

Extensive form

The game " Ultimatum»In extensive form

Games in extensive or extended form are represented as a directed tree, where each vertex corresponds to a situation where the player chooses his strategy. A whole level of peaks is associated with each player. Payments are recorded at the bottom of the tree, under each leaf vertex.

The picture on the left is a game for two players. Player 1 goes first and chooses strategy F or U. Player 2 analyzes his position and decides whether to choose strategy A or R. Most likely, the first player will choose U, and the second - A (for each of them these are optimal strategies); then they will get 8 and 2 points respectively.

The extensive form is very descriptive and makes it especially convenient to represent games with more than two players and games with consecutive moves. If the participants make simultaneous moves, then the corresponding vertices are either connected by a dotted line, or outlined with a solid line.

Normal form of play

In normal or strategic form, the game is described by a payment matrix. Each side (more precisely, dimension) of the matrix is a player, the rows define the strategies of the first player, and the columns define the strategies of the second. At the intersection of the two strategies, you can see the winnings that the players will receive. In the example on the right, if player 1 chooses the first strategy, and the second player chooses the second strategy, then at the intersection we see (? 1,? 1), which means that as a result of the move, both players lost one point.

The players chose strategies with the maximum result for themselves, but lost, due to ignorance of the move of the other player. Usually games are presented in normal form in which the moves are made at the same time, or at least it is assumed that all players are unaware of what other participants are doing. Such games with incomplete information will be discussed below.

Characteristic function

In cooperative games with transferable utility, that is, the ability to transfer funds from one player to another, it is impossible to apply the concept of individual payments. Instead, a so-called characteristic function is used, which determines the payoff of each coalition of players. In this case, it is assumed that the payoff of the empty coalition is zero.

The foundations of this approach can be found in the book by von Neumann and Morgenstern. Studying the normal form for coalition games, they reasoned that if coalition C is formed in a game with two sides, then coalition N \ C opposes it. It is like a game for two players. But since there are many options for possible coalitions (namely, 2N, where N is the number of players), the payoff for C will be some characteristic value depending on the composition of the coalition. Formally, a game in this form (also called a TU-game) is represented by a pair (N, v), where N is the set of all players and v: 2N> R is the characteristic function.

This form of presentation can be applied to all games, including those without transferable utility. Currently, there are ways to convert any game from normal to characteristic form, but conversion in the opposite direction is not possible in all cases.

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 economics, 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-theoretic analysis was first used to describe animal behavior by Ronald Fisher in the 1930s (although even Charles Darwin used the ideas of game theory without formal justification). The term "game theory" does not appear in 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 does not meet this premise. There are many explanations for this phenomenon - irrationality, modeling discussion, and even various 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, as a result of experiments, it was 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. Another view is that Nash equilibrium is not a prediction of expected behavior; it only explains why populations that are already in Nash equilibrium remain in that state. However, the question of how these populations arrive at a 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 assume 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 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, using the Nash equilibrium concept to select behavior seems to be 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. Second, the famous Prisoner Dilemma provides another counterexample. In Prisoner's Dilemma, pursuing self-interest puts both players in a worse situation than when they would sacrifice self-interest.

Cooperative and non-cooperative

The game is called cooperative, or coalition, if the players can unite in groups, taking on some obligations to other players and coordinating their actions. This differs from non-cooperative games in which everyone is obliged to play for themselves. Entertaining games rarely cooperative, but such mechanisms are not uncommon in everyday life.

It is often assumed that cooperative games are distinguished precisely by the ability of players to communicate with each other. V general case 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 consider the process of the game as a whole. Attempts to combine the two approaches have yielded considerable results. The 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




Asymmetrical play

The game will be symmetrical when the players have equal strategies, that is, they have the same payments. In other words, if the players can change 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 strategy profiles (A, A) and (B, B) will be greater than that of the first.

Zero-sum and non-zero-sum

Zero-sum games are a special kind of fixed-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: in games with a nonzero sum, 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 by introducing a fictitious player who "pockends" the surplus or makes up for the lack of funds.

Another game with a non-zero amount is trading, where each participant benefits. A well-known example of where it is declining is war.

Parallel and sequential

In parallel games, the players move at the same time, or at least they are not aware of others' choices until everyone has made their move. In sequential, or dynamic, 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 not even be completely complete, for example, a player may find out that his opponent out of ten of his strategies did not exactly choose the fifth, without knowing anything about the 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

Games with complete information constitute an important subset of sequential games. In such a game, the participants know all the moves made before of the current moment as well as the possible strategies of 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, the whole "salt" of "Prisoner's Dilemma" or "Coin Comparison" is 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 a similar concept - 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

Games in the real world, or games studied in economics, tend to last a finite 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 an optimal solution, but to find at least a winning strategy. Using the axiom of choice, one can prove that sometimes even for games with complete information and two outcomes - "win" or "lose" - none of the players has such a strategy. The existence of winning strategies for some specially designed games has important role in descriptive set theory.

Discrete and continuous games

Most of the games under study are discrete: they have a finite number of players, moves, events, outcomes, etc. However, these components can be extended to a variety of real numbers. Games that include these elements are often referred to as 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 a target or object game). The purpose of metagames is to increase the usefulness of the set of rules produced. The theory of metagames is related to the theory of optimal mechanisms.

Methods for constructing and analyzing simulation models (simulation).

Simulation modeling ( situational modeling) is a method that allows you to build models that describe the processes as they would in reality. Such a model can be “played” in time for both one test and a given set of them. In this case, the results will be determined by the random nature of the processes. Based on these data, one can obtain fairly stable statistics.

Simulation modeling is a research method in which the system under study is replaced by a model that describes the real system with sufficient accuracy, with which experiments are carried out in order to obtain information about this system. Experimenting with a model is called imitation (imitation is comprehending the essence of a phenomenon without resorting to experiments on a real object).

Simulation modeling is a special case of mathematical modeling. There is a class of objects for which, for various reasons, analytical models have not been developed, or methods for solving the resulting model have not been developed. In this case, the analytical model is replaced by a simulator or simulation model.

Simulation modeling is sometimes called obtaining partial numerical solutions of a formulated problem based on analytical solutions or using numerical methods.

A simulation model is a logical and mathematical description of an object, which can be used for experimenting on a computer in order to design, analyze and evaluate the functioning of an object.

Application of simulation modeling.

Imitation modeling is used when:

· It is expensive or impossible to experiment on a real object;

· It is impossible to build an analytical model: the system has time, causal relationships, consequences, nonlinearities, stochastic (random) variables;

· It is necessary to simulate the behavior of the system in time.

The purpose of simulation is to reproduce the behavior of the system under study based on the results of the analysis of the most significant relationships between its elements or, in other words, to develop a simulation modeling of the studied subject area for various experiments.

Types of simulation

Three Simulation Approaches

Simulation Approaches on the Abstraction Scale

Agent-based modeling is a relatively new (1990-2000) direction in simulation modeling, which is used to study decentralized systems, the dynamics of which are determined not by global rules and laws (as in other modeling paradigms), but vice versa, when these global rules and laws are the result of the individual activity of the members of the group. The goal of agent-based models is to get an idea of these global rules, the general behavior of the system, based on assumptions about the individual, private behavior of its individual active objects and the interaction of these objects in the system. An agent is a certain entity with activity, autonomous behavior, can make decisions in accordance with a certain set of rules, interact with the environment, and also change independently.

· Discrete-event modeling - an approach to modeling that offers to abstract from the continuous nature of events and consider only the main events of the modeled system, such as: "waiting", "order processing", "movement with a load", "unloading" and others. Discrete-event modeling is the most developed and has a huge scope of applications - from logistics and queuing systems to transport and production systems. This type of simulation is most suitable for modeling production processes. Founded by Jeffrey Gordon in the 1960s.

Algorithm of mathematical theory

A feature of the mathematical theory of justifying management decisions is the presence of a certain algorithm in it, which precisely prescribes to perform a certain system of operations in an established sequence to solve a certain class of problems.

The algorithm of the mathematical theory of managerial decision-making must meet a number of requirements:

certainty, i.e. accuracy and unambiguity, leaving no room for arbitrariness;
mass character and universality - applicability for solving a specific class of problems, when the initial data vary within certain limits;
effectiveness, i.e. the ability to solve a specified problem in a limited number of operations.

Mathematical methods for making management decisions

The main methods for solving typical management problems within the framework of mathematical theory are:

The method of mathematical analysis is used in calculations to justify resource requirements, cost accounting, project development, etc.
The method of mathematical statistics is convenient to use when the change in the studied indicators is a random process.
The econometric method involves the use of an economic model - a schematic representation economic process or phenomena.
Linear programming is the solution of a system of equations when there is a strictly functional relationship between the investigated phenomena.
Dynamic programming is used to solve optimization problems where the constraints or objective function have a non-linear relationship.
Queuing theory is used to find the optimal number of service channels for a given level of demand. An example of such a situation is the choice of the optimal option for organizing work with clients, so that the service time is minimal, and the quality is high without additional costs.
Operations research method is the use of mathematical probabilistic models that represent the investigated process, type of activity or system. Optimization comes down to a comparative study of the numerical estimates of those parameters that cannot be estimated by conventional methods.
Situational analysis is a complex technology for making and implementing management decisions, which is based on analyzing a separate management situation. Such an analysis is based on a specific situation, a problem arising in the organization's activities, which requires a management decision.
Game theory methods - modeling a situation in which, when justifying decisions, it is necessary to take into account the conflict or mismatch of interests of different persons.
Break-even points are a method in which total revenues are equalized with total expenses to find the point that brings the enterprise the minimum profit.
Trend projection is a time series analysis based on the assumption that what happened in the past gives a good approximation in the case of an estimate of the future. This method is used to identify past trends and extend them into the future.

Efficiency in general terms is the effectiveness of something (production, labor, management, etc.). In economic theory, there are mainly two types of efficiency - economic and social. Economic efficiency characterizes the ratio of the obtained result to costs, social - the degree of satisfaction of the demand of the population (consumers, customers) for goods and services. They are often combined by a single term - socio-economic efficiency, which is most related to the assessment of managerial decisions, since the latter are aimed at the state and behavior of people and thus have a high social value and their assessment only from the standpoint of the economic effect is not entirely correct. In recent decades, there has been an increasing need to assess many management decisions. environmental efficiency, reflecting both the positive and negative impact of their implementation on the environmental situation. Here, as a rule, the organization's possible costs for eliminating the negative impact on the environment, fines and other related payments or their savings with a positive impact on the environment are reflected.

Quality - from the standpoint of philosophy - expresses a set of essential features, characteristics and properties that distinguish one object or phenomenon from others and give it certainty. The quality of the result of labor (products, services, investment projects, management decisions, etc.) is associated with the concepts of "property" and "utility". Property the result of labor determines the objective aspects without evaluating its importance for the consumer (for example, the technical level of a product, a project); utility - the ability of a given result of labor to be beneficial and meet the requirements of a particular consumer. Hence, quality of management decisions - a set of properties that determine its ability to satisfy certain needs in accordance with the purpose. In the practice of organizations, efficiency and quality are inseparable and mutually condition each other. A solution cannot be highly effective if it is of low quality and, conversely, it cannot be of high quality if it is ineffective, i.e. efficiency – one of the characteristics of quality, and quality is an essential factor in efficiency.

The effectiveness and quality of management decisions are determined by the entire set of management processes that make up it relatively independent and interconnected stages in the technological cycle: development, adoption and implementation of decisions. In accordance with this, it is necessary to consider modifications of the management decision - the effectiveness and quality of the theoretically found, adopted by the decision maker and practically implemented solution.

At the stages of development and adoption of a managerial decision, its quality is the degree to which the parameters of the chosen alternative of the solution correspond to a certain system of characteristics, which satisfies its developers and consumers and ensures the possibility of effective implementation. At the stage of implementation the quality of a managerial decision is expressed in its actual effectiveness, efficiency of implementation.

The main characteristics that determine the quality of decisions are: validity, timeliness, consistency (consistency), reality, completeness of content, authority (authority), efficiency.

Reasonableness of the decision is determined by: the degree of consideration of the regularities of the functioning and development of the control object, the trends in the development of the economy and society as a whole, the competence of its developing specialists and decision makers. It should cover the entire range of issues, the entire completeness of the needs of the managed object. This requires knowledge of the features, ways of development of the controlled system and the external environment. A thorough analysis of resource provision, scientific and technical capabilities, target development functions, economic and social prospects of the company, region, industry, national and world economy is required. Comprehensive validity of decisions requires the search for new forms and ways of processing scientific, technical and socio-economic information, forms and methods of management, theory and practice of development and decision-making, i.e. the formation of advanced professional thinking, the development of its analytical and synthetic functions. Only the decision that is made on the basis of reliable, systematized and scientifically processed information, which is achieved by using scientific methods for developing and optimizing solutions, can be justified.

Thus, the validity of the decision is ensured by the following main factors:

taking into account the requirements of objective economic laws and patterns, current legislation and statutory documents;
knowledge and use of patterns and trends in the development of the control object and its external environment;
availability of complete, reliable, timely information;
availability of special knowledge, education and qualifications of developers and decision makers;
knowledge and application of the decision maker of the main recommendations of management and decision-making theory;
used methods of analysis and synthesis of situations.

The increasing complexity and complexity of the problems to be solved and their consequences require universal knowledge for the development and adoption of well-grounded management decisions, which leads to the increasingly widespread use of collegial forms of decision-making.

The validity of management decisions can be achieved by performing the following actions:

determination of conditions for the formation of acceptable options;
compilation of a list of indicators characterizing the essential properties of the found solutions, and the development of scales for their measurement;
screening out irrational options and determining the range of possible values of each indicator using a variety of mathematical and heuristic methods;
identifying the structure of decision makers' preferences;
formation of a criterion or rules for evaluating decision options;
choosing the best option for management decisions or clarifying the structure of decision makers' preferences.

The implementation of these actions does not always guarantee high quality and efficiency of solutions, since the choice of alternatives is significantly hampered by the following factors.

1. The multidimensional nature of assessments of the effectiveness of alternatives. When determining possible solutions, and even more so when choosing the most appropriate of them, one has to make economic, technical and technological, social, political, and environmental assessments. Moreover, each has several approaches. For example, valuation, according to international, European and Russian standards, uses cost, market (comparative) and income approach s, which use different methods depending on the object and objectives of the assessment. When choosing options for the development of an open joint-stock company, it is necessary to take into account the entire set of stakeholders, since the decisions made can significantly affect various groups of people, which increases the number of possible assessments (both in relation to them and from their side). In many cases, it is necessary to take into account changes in estimates over time. At the same time, more and more often there are problems of taking into account new types of assessments that characterize the consequences of a decision made at different moments of the future.
2. Difficulties in identifying and comparing all aspects of comparing alternatives. The existence of heterogeneous aspects of evaluating alternatives poses difficult problems for developers and decision makers to compare them. It should be borne in mind here that such a comparison is subjective and therefore can be criticized. This is exacerbated many times in collegial decision-making, where each member of the decision-making collective body may have different measures for comparing dissimilar qualities. Some participants in the development and decision-making may be interested mainly in economic criteria, others in political ones, others in environmental ones, etc.
3. The subjective nature of assessments of the effectiveness and quality of alternatives. Many estimates of the effectiveness and quality of alternatives can be obtained either by building special models, or by collecting and processing expert opinions. Both methods involve the use of subjective assessments, either by specialists developing models or experts. When choosing alternatives, it should be borne in mind that the reliability of such subjective assessments cannot be absolute. Even if the experts are completely unanimous, it is possible that their assessments turn out to be incorrect. It is also possible that different models exist or that the assessments of experts do not coincide. Consequently, several alternatives may have different estimates, and the result of the choice depends on which of them will be used by the decision maker.

Timeliness managerial decision means that the decision made should neither lag behind nor outstrip the needs of the development of the situation. Even the most optimal (of those expedient for decision makers) decision, calculated to obtain the greatest socio-economic efficiency, may turn out to be useless if taken late. It can even cause some damage. Premature decisions are just as harmful to the organization as belated decisions. They do not have the conditions necessary for implementation and development, and can give impulses for the development of negative tendencies, do not contribute to the solution of already "overripe" tasks and further aggravate already painful processes.

Consistency (consistency ). Distinguish between internal and external consistency of the solution. Under internal consistency solutions is understood as the correspondence of goals and means of achieving them, the complexity of the problem being solved and methods of developing a solution, individual provisions of the solution to each other and the meaning of the solution as a whole. Under external consistency decisions - their continuity, compliance with the strategy, the goals of the company and previously made decisions (actions necessary to implement one decision should not interfere with the implementation of others). Achieving a combination of these two conditions ensures the consistency and consistency of management decisions. Consistency with previously adopted decisions also means the need to observe a clear cause-and-effect relationship of social development. Previous decisions, if necessary, should be canceled or corrected if they conflict with the new conditions of the controlled system. The emergence of conflicting decisions is a consequence of poor knowledge and understanding of the laws of social development, the manifestation of a low level of management culture.

Reality. The decision should be developed and made taking into account the objective capabilities of the organization, its potential. In other words, the material, financial, informational and other resources, the organization's capabilities must be sufficient for the effective implementation of the chosen alternative.

Completeness of content solutions means that the solution should cover the entire set of parameters of the managed object necessary to ensure the achievement of goals, all areas of its activities, all areas of development. The content of the management decision should reflect:

the goal (set of goals) of the functioning and development of the managed object, to which the decision is directed;
the resources used to achieve these goals;
the main ways and means of achieving goals, the main methods of performing work that determine the implementation of the goals of the solution;
deadlines for achieving goals, the beginning and end of their supporting work;
the order of interaction between departments and individual employees.

So, a managerial decision can be considered to be of high quality if it meets all the requirements listed above. Moreover, we are talking specifically about the system of requirements, since failure to comply with at least one of them leads to a decrease in the quality of the solution and, consequently, to a loss of efficiency, difficulties or even the impossibility of its implementation.

The quality and efficiency of management decisions are determined by many factors acting throughout the entire technological cycle of management or at its individual stages, which have an intra-system or external (environmental influence), objective or subjective nature. The most significant factors include:

the laws of the objective world related to the adoption and implementation of management decisions;
goal formulation; why the management decision is made, what real results can be achieved, how to measure, correlate the goal and the results achieved;
the volume and value of available information - for successful management decision-making, the main thing is not so much the amount of information as its value, determined by the level of professionalism, experience, intuition of staff;
time for developing a management decision - as a rule, a management decision is always made in conditions of time pressure and extraordinary circumstances (lack of resources, activity of competitors, market conditions, inconsistent behavior of politicians);
organizational structure management, defined by organizational documents (formal) and actually existing (informal). In fact, the existing (current) management structure, in almost exceptional cases, coincides with the one determined by the relevant organizational documents, within which all employees of the organization are required to act. The need to take this requirement into account is often a condition for making a not the most optimal decision option;
forms and methods of management activities, including the development and implementation of management decisions;
the state of the control and controlled systems (psychological climate, authority of the leader, professional qualification composition of personnel, etc.);
a system for assessing the level of quality and effectiveness of management decisions;
the degree of risk associated with the consequences of the implementation of the decision. This factor requires the use of various risk assessment techniques (financial, economic, etc.); accordingly, the leader must have the skills to perform such an analysis;
office equipment, including IVS. The use of modern information systems is a powerful factor in enhancing the process of developing, making and implementing decisions. It requires certain knowledge and skills of using modern information technologies in managing the activities of organizations;
subjectivity of the assessment of the option of choosing a solution. The decision-making process, the choice of a specific option is of a creative nature and depends on a particular person, his state at the time of making a decision. Personal assessments of the decision maker act as a compass, showing him the desired direction when he has to choose between alternatives to action. Each person has his own system of values, which determines his actions and influences the decisions made. Personal factors include:
- the psychological state of the decision maker at the moment of making a decision. In a state of irritability, congested with other decisions, the decision maker can make one decision in this situation, and in a good mood, being relatively free - another,
- the measure of responsibility of the decision maker, determined both by the internal sense of responsibility for their actions, and by those regulating it activity by documents,
- the level of knowledge on this issue... The higher the level of knowledge of the decision maker about the object at which the decision is directed, and its external environment, the more likely they are to make a high-quality and effective decision,
- experience, which, as the main resource for the development and implementation of decisions, is a determining factor in the adequate perception of a real assessment and an effective response of a decision maker to what is happening, is a certain bank of approved and adaptable options, from which analogs and prototypes of solutions developed, adopted and implemented are drawn,
- intuition, judgment (common sense) and rationality of the decision maker.

Reference. Intuition manifests itself as some kind of insight or instant understanding of a situation without the use of rational thinking. However, such an insight is usually preceded by a long and painstaking work of consciousness. First, through observation, information is accumulated in a person's memory, systematized and arranged in a certain order. Often in this way they come to an expedient solution to the problem. If this does not happen, intuition and imagination are connected, generating numerous ideas and associations. One of the ideas can evoke an intuitive insight, which, as it were, pushes the corresponding idea from the subconscious into consciousness. Intuition is a powerful decision-making tool that needs constant development and should be actively used in management activities.

When making a decision, the decision maker is often based on his own feeling that his choice is correct. Intuition develops as you gain experience. Judgment decisions are based on the knowledge and meaningful experience of the past. Using them and relying on common sense, with the amendment to date, choose the option that brought the greatest success in a similar situation in the past. However, common sense in people, from the point of view of the author, is rare, so this method of making decisions is not very reliable, although it captivates with its speed and cheapness. With this approach, the decision maker seeks to act mainly in those directions that are familiar to him, as a result of which he risks missing a good result in another area, consciously or unconsciously refusing to invade it;

The criterion of the risk strategy chosen by the decision maker: optimism, pessimism or indifference. The criterion of optimism (maximax) determines the choice of an alternative that maximizes the maximum result for each alternative; pessimism (maximin) - an alternative that maximizes the minimum result for each alternative; indifference - an alternative with the maximum average result (in this case, there is an unspoken assumption that each of the possible states of the controlled system can occur with equal probability: as a result, an alternative is chosen that gives the maximum value of the mathematical expectation).

At the stage of implementation, the effectiveness of decisions is determined by the following factors:

the level of development and state of the controlled system, its technique, technology, personnel (personnel), organization and economy. With a high level of development of all components of a controlled system, when implementing a solution, greater efficiency can be obtained than that provided for by the solution, and vice versa, with a low level, it is rather difficult to ensure the efficiency defined in the solution;
socio-psychological climate in the team implementing the decision. The main criterion for the socio-psychological climate is the level of maturity of the team, which is understood as the degree of coincidence of individual and collective interests. The higher the level of maturity of the team, the more manageable it is, which is a necessary condition for its effective activity;
the credibility of the leaders who ensure the implementation of the solution. The higher the authority of the leaders, the more manageable the team and, accordingly, the higher the level of efficiency of its activities;
the effectiveness of the mechanism for managing the activities of the team, which is expressed in the essence of management as the creation of conditions that induce people to take the actions necessary to achieve the goals;
solution implementation time. A timely, high-quality and effective decision in case of its untimely implementation may turn out to be not only ineffective, but unnecessary;
the correspondence of the number and qualifications (education, skills and experience) of personnel to the volume and complexity of work on the implementation of the solution. When the number of personnel is less than necessary for the implementation of the decision, it is difficult to comply with its deadlines. If the qualifications of employees are below the required level, the quality of work performance decreases and, at the same time, the effectiveness of the solution is implemented;
provision with the necessary material, energy, labor, information and financial resources.

It was shown above that the effectiveness of a solution is determined at the stages of its development and implementation. At the first stage, it is determined according to the well-known methods of calculating the effectiveness of design solutions, at the second - as a rule, but by the methods for calculating the actual profit and profitability of activities. In recent years, to determine the effectiveness of strategic decisions at the stages of their development and implementation, the calculation of the expected and actual changes in the market value of a business is often used, the results of which are the basis for assessing and choosing an organization's strategy.

Evaluation of the effectiveness of management decisions at the stages of their development and adoption can be carried out according to well-known indicators for evaluating investment projects:

net discounted (reduced, current) income (NPV) - NPV (Net Present Value ) - the current value of cash inflows (income) minus the value of cash outflows (investment costs);
internal rate of return (IRR) - IRR (Internal Rate of Return ) Is the discount rate at which there is an equality between the present value of the projected cash inflows (income) and the present value of the projected investment costs (cash outflows), i.e. net current income (NPV) at the same time it is equal to zero;
modified internal rate of return (MIRR) - MIRR (Modified Internal Rate of Return ) Is an indicator characterizing the efficiency of capital investments (investments). If the present value of all investment

to consider investments as initially invested capital, and the future value of all cash inflows - as an accrued amount, then the rate of discount of the accrual coefficient is taken by the Ministry of Internal Affairs;

profitability index (IR) - PI (Profitability Index ) - the amount of net (discounted) cash flow per unit of investment;
payback period - PP (Payback Period ) - the expected period of reimbursement of the invested funds by net cash receipts;
discounted payback period - DPP (Discounted Payback Period ) - the estimated period of reimbursement (equality) of the present value of the invested funds and the present value of the net cash receipts;
cost efficiency ratio - ARR (Accounting Rate of Return ) is equal to the ratio of the predicted average annual net (balance sheet) profit to the average annual investment costs.

These indicators are widely used in practice, and the methods of their calculation are recognized as traditional. In the numerous literature, they are described in detail, examples are given illustrating their calculations for the selection of projects (alternatives) of management decisions with different initial conditions.

These indicators, as well as the corresponding methods, are used in two versions:

to determine the effectiveness of independent (uncontested) management decisions (the so-called absolute effectiveness), when a conclusion is made about whether to accept it or reject it;
to determine the effectiveness of mutually exclusive decision alternatives (comparative effectiveness), when a conclusion is made about which of them to take as a managerial decision.

In assessing the effectiveness of management decisions, like any other activity, the results of its implementation (effect - Er) and the costs of its development, adoption and implementation (Zr) are involved. The effect of managerial decisions is manifested in the final results of the organization's activities. Even in cases where the managerial decision is aimed at changing the technical and economic or socio-economic indicators of the organization's activities (the level of state and development of technology and production technology, the range and range of products, the quality of raw materials, the design characteristics of working premises, social infrastructure, etc. ), the effect of its implementation is ultimately reflected in the change in the level of use of its potential and the satisfaction of social needs in its products and services, i.e.

Er = f (P, Ip, Zr, Up)

at (P - Ip), Zr š min; Up š max,

where P is the organization's potential; Ип - its use; UP - the level of satisfaction of social needs in its products and services.

This approach, called " resource-potential ", to assess the effectiveness of management of the activities of organizations, the product of which are management decisions and the results of their implementation, was proposed by Academician of the USSR Academy of Sciences V. A. Trapeznikov, substantiated and developed by professors F. M. Rusinov and V. I. Busov.

The development of an organization (its potential, referred to a particular goal, expressed in the desire to satisfy a certain type of social needs as much as possible) has limitations determined by the ratio of supply and demand for products and services that it is capable of producing. this organization... The excess of the result for one or another function of the enterprise of the existing needs in it is a negative effect of its activities or an unhelpful result, equivalent to waste and losses of resources spent on it.

The second component of efficiency is the cost of resources for the development, adoption and implementation of management decisions. Increasing the level of return of these costs (their effectiveness) is the most important task of managing the process of developing, making and implementing managerial decisions. A misunderstanding of this task (especially in terms of development and decision-making) often leads in practice to a reduction in these costs, even at the expense of the effectiveness of management decisions. This is due to the fact that the main share of costs is often wages and charges for it, and their reduction is reduced to a reduction in the personnel involved in this process or the level of remuneration for their labor, as a result of which the quality of management decisions and the effect of its implementation, personnel motivation deteriorate. Reducing the cost of developing, making and implementing managerial decisions by a simple voluntaristic decision entails a decrease in the efficiency of the organization, associated with a deterioration in control, an increase in the waiting time for a decision on a particular situation, a deterioration in the quality of preparation, development and decision-making and other factors. affecting the level of resource losses.

An assessment of the effectiveness of the implementation of management decisions can be made for each major management decision or for the totality of those implemented in a certain period of time (for example, a quarter, half a year, a year). It consists of a system of indicators (Figure 3.5), including:

generalizing integral indicator, specifying the criterion of effectiveness;
generalizing indicators reflecting the effectiveness of the implementation of groups of goals for the achievement of which a management decision was made (scientific, technical, economic, social, etc.);
partial indicators reflecting the efficiency of using certain types of resources for individual stages of the reproduction cycle.

When determining the effectiveness of the implementation of a management decision, the value is not used of the potential of the organization's resources in general, but of its potential to perform the functions that this decision covers. To identify such a composition, you can use the matrices shown in table. 1.2-1.5.

The level of potential use is defined as the difference between its value and losses. Moreover, the reserve part of the potential necessary for the sustainable functioning and development of any unit of the organization does not apply to its losses.

Rice. 3.3.

Shown in Fig. 3.5 The scorecard reflects the structure of the "tree" of goals to improve the efficiency of the organization.

The effectiveness of a management decision is defined as

where Ents and Ents, Epts and Epts, ESCs and ESCs, Eeks and Eekts are the effectiveness and effect of management decisions in achieving scientific, technical, production, social and environmental goals, respectively; Ei, is the effect of the implementation of a management decision in t division organization (department workplace); Зр - costs for the development and implementation of management decisions; P - the number of departments involved in the development and implementation of this management solution.

Participation effect i -th subdivision of the organization (workplace) in the development and implementation of management decisions is defined as the sum of the effects of changes in the level of use in the process to which this decision is directed, the existing potential of the subdivision (workplace) - the internal effect (EE) - and the result of the implementation of the goals of the decision - external effect (Ets), i.e.

Ei = Ev + Etz.

The internal effect is determined by intensive (Ei) and extensive factors (Ee), i.e.

Ev = Ei + Ee.

Intensive factors determine the changes in the productive use of potential due to the implementation of this managerial decision, extensive factors - changes in the unproductive use of potential and loss of resources.

The scheme for calculating indicators of the effectiveness of enterprise management is shown in Fig. 3.6.

Since all resources go to the workplaces of the organization and are used here, the level of use of the potential of the resources of the enterprise is determined by the processes at its workplaces. The change in the level of productive use of resources in the workplace is determined by the difference in the use of potential output (or labor productivity) at a given workplace before and after the implementation of this management decision, i.e.

where and Bp - potential production at a given workplace, respectively, before and after the implementation of the management decision; , and Vf - the actual production at a given workplace, respectively, before and after the implementation of the managerial decision.

Actual production (or labor productivity) in any production unit(procurement, mechanical, foundry, assembly, etc.) is determined without much difficulty using generally accepted assessment methods.

Rice. 3.6.

Potential and actual production in the workplace form the basis for determining the potential and actual production by department, function or type of activity of the department. The volume of output at a workplace is influenced by: the performance of equipment with a given technology of work performed at a given workplace; compliance of the employee's qualifications with the level of complexity of work; the timeliness of providing the workplace with the necessary materials, tools, organizational equipment, information and other resources; compliance of the quantity and quality of initial resources with the technology requirements; the rhythm of the employee's activity in the workplace. These factors reduce the actual production compared to the potential.

The potential output of a workplace (Bp (rm)) is determined by the output of the equipment installed on it with a maximum number of hours of one hundred work in a given period, taking into account the time for changeover, repair, adjustment, i.e. according to the formula

Βп (рм) = (Фр - t m) P n ,

where Fr is the operating time of one unit (construction crane, bulldozer, concrete mixer, scraper, etc.) at the workplace per month; t n - standard time for adjustment and repair, changeover of one unit; P - regime (technological) removal of products from a unit of equipment (unit) per unit of time; P - the number of units of the same type at the workplace with multi-station service.

For jobs with low mechanized and manual labor, including engineering and managerial workers, the potential output is calculated based on the maximum shift output of the month, based on the fact that the maximum output in a given shift was achieved due to the maximum use of the resources that make up this workplace, i.e.

Bn (pm) = Sun.max t p,

where Sun.max is the maximum shift production at the workplace in the estimated month, standard hours; m - the number of shifts in the estimated month; R - cost of 1 standard hour, rub.

The initial data for the calculation are taken from the production and wage accounting cards, which must be filled out in the divisions of the enterprise.

A similar approach can be applied to any workplace, but for mechanized and automated workplaces, Bp should be calculated from the performance of the equipment.

Knowing the potential monthly output for all workplaces in a division, it is possible to determine the potential output of a given division. It is calculated according to the technological chain of jobs formed by the system of machines involved in the production of a given type of product, or determined by the sequence of execution of technological operations assigned to the jobs for the production of this type of result of the division's activity.

Extensive use of the economic potential by the internal effect of the processes of the enterprise management system expresses losses and technologically unjustified waste of resources. The change in their value after the implementation of the management decision () in comparison with the baseline (P) reflects the change in the internal effect of management by extensive factors, i.e.

The resources involved in the processes are used productively and unproductively.

The productive use of resources is also divided into two parts. The first part is the consumption of resources, calculated on the basis of unit costs, which are recognized as rational (technologically necessary). The second part is resource expenditures that exceed rational unit costs. Such costs represent a waste of resources.

Unproductive use of resources occurs when products and services are not created. For example, the unproductive use of resources includes the cost of working time of workers, the cost of production capacity of equipment and materials to correct defects, losses - absenteeism, day-long and continuous downtime, unused capacity of installed equipment, irreparable rejects, unused scientific and technical developments, damage to materials in the warehouse. and etc.

The effect of implementing a managerial decision to achieve production goals is determined by an increase in the volume and quality of products and services, compliance with the terms of their provision to the consumer and is expressed in a change in the efficiency of their use by consumers; scientific and technical goals - in the effectiveness of the application of the company's developments in innovative processes; social goals - to save time (increase free time) and increase the social activity of employees of the enterprise and consumers of the products and services of the enterprise; environmental goals - to reduce waste and increase the volume of their disposal, landscaping, etc. The effect on social outcomes is especially important for enterprises providing various services to the population (utilities, transport, household, postal, catering, trade, etc.). Environmental impact - for the fuel, petrochemical and chemical industries.

The costs of developing and implementing a management solution include the entire set of costs for performing work both on our own and for third-party organizations (contractors), as well as for the purchase of the necessary materials, equipment and other necessary resources.

The above approach is applicable only in the presence of the necessary initial data in the organization, provided by an organized system of control and accounting of process parameters at workplaces and in departments, monitoring the needs and consumption of the company's products and services.

In advanced economies, it has long been a textbook cost approach in the management of organizations and, accordingly, in assessing the effectiveness of management decisions.

Reference. In the American capital market, the concept of value is widespread in practice and the only one accepted in the scientific literature. In May 2010, KPMG, in collaboration with the State University - Higher School of Economics (SU-HSE), conducted a study on the application of value-based management practices by Russian companies. It has shown the high relevance of value management for Russian companies in the current market situation and interest for managers, since the growth in business value leads to an increase in the investment attractiveness and competitiveness of the organization.

The main idea of the concept of value management is that the main financial goal of the organization is the growth of its value (value) not only for the owners (shareholders), but also for all legal entities and individuals interested in the company's activities (management of the company's value in the interests of stakeholders). The concept of "value" in this management concept is an internal category that characterizes the value, investment attractiveness of a company for owners, and is expressed in a monetary indicator of future growth opportunities.

Increase in value Is an economic criterion that reflects the integral effect of the influence of management decisions implemented in an organization on all parameters by which its activities are assessed (market share and strength of competitive position, income, investment needs, operational efficiency, tax burden, regulation, cash flows and risk level ), which allows you to rank options in a multiple choice situation.

In the system of value management, the premise was initially laid that the command-and-control style of making managerial decisions "top-down" does not bring the desired results, especially in large diversified corporations. Lower-level managers need to learn how to use cost metrics to make better, more effective management decisions. Cost management requires a reasonable balance of long-term and short-term performance goals. It, in essence, is the development, adoption and implementation of management decisions that ensure continuous reorganization aimed at achieving maximum business value.

An important advantage of the cost management approach is the fact that it offers management a single and understandable criterion for evaluating activities - cost. The parameter of business value growth is a key tool for improving the quality and efficiency of management decisions, allowing you to create a universal coordinate system to determine the vector of business development, as well as create a single scale for changing the results achieved in accordance with the established strategy.

The process of managing the market value of a company uses a profitable approach to the valuation of a company (business) as a base. Under this approach, company value is the sum of the cash flows that will be generated by the company, adjusted for time factors and associated risks, minus all the company's liabilities.

Evaluating the effectiveness of a managerial decision by this method involves comparing two scenarios for the development of an organization "without the development and implementation of a managerial solution to a given situation-problem" and "subject to the development and implementation of a managerial solution to a given situation-problem".

Assessment of the cost of an organization in the first option is reduced to a forecast of cash flows for the enterprise as a whole, provided that nothing in it will fundamentally change in the billing period. This - discounted value business, which is determined by discounting the cash flow at a rate that takes into account the existing risks of the organization as a whole:

where PV 0 - the discounted value of the organization during its development without solving the existing problem situations; CF 0i - expected cash flow in the period г; r - discount rate; P - the number of periods during which the organization will generate cash flows (in years).

The cost of the organization under the scenario of implementing a management decision (strategic value) is determined by discounting the project-adjusted cash flow at an adjusted rate that takes into account both the risk of the organization as a whole and the risks of management decisions. It will be equal to the residual present value of the expected flows of the organization, subject to the implementation of a management decision, i.e. the organization's cash flows in two scenarios of its development are combined:

where PV C is the strategic cost of the organization; CF c - strategic cash flow of the organization; CF pi is the cash flow generated by the implementation of the management decision.

Application capital market and transactions method to assess the increase in the value of an enterprise due to the implementation of a management decision, it is based on information about a similar company that implements a similar solution. In this case, the similarity of decisions is determined by the following factors:

maximum similarity of situations to be solved in compared organizations;
general industry (functional) affiliation of the compared situations;
use of similar resources;
the comparability of the scale of situations and the radicality of changes as a result of the implementation of management decisions.

To determine the increase in value created as a result of the implementation of a management decision, the capital market method uses the market coefficients of an analogous company before and after it implements a solution to a similar situation, i.e.

where Δ CV - an increase in the market value of the appraised company due to the implementation of a management decision; E ok is the current profit of the evaluated company; - the price / profit ratio for a similar company after the implementation of a solution to a similar situation; - the "price / profit" ratio for a similar company before the implementation of a solution to a similar situation.

The transaction method differs from the capital market method in that the price / earnings ratio for the peer company (peer companies) is calculated taking into account only the prices of shares of the peer company (peer companies) that were sale and purchase transactions of large blocks of shares or at the corresponding share quotation. At the same time, large stakes are considered those, the purchase of which makes it possible to acquire at least participation in control over the company by introducing a representative (or oneself) to its board of directors, which allows controlling the company's management. Hence, finding an analogous company that implements a management solution for a similar situation, information on which is publicly available, is an extremely difficult and sometimes simply unrealizable task. In practice, this significantly complicates or makes it impossible to use capital market and transaction methods to assess the effectiveness of management decisions.

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