Three signs of a good strategy in the face of uncertainty. Open Library - open library of educational information

The above problem from game theory assumed the choice of an optimal strategy under risk conditions. These are situations where the player knows the probabilities of the outcomes and consequences for each decision.

A completely different situation occurs when these probabilities are not known, i.e. there is complete uncertainty about the possibility of realizing the state of the environment. In this case, the game can be represented in such a way that it has one player and a certain reality called nature. The conditions of such a game are usually represented by the same payment matrix as before, in which the rows represent the strategies of the player, and the columns represent the strategies of nature.

In this case, when choosing the best solution, the following criteria are usually used:

1. The maximax criterion, or the criterion of extreme optimism - defines an alternative that maximizes the maximum result for each alternative, i.e. a strategy is selected that corresponds to

2. Wald's maximin criterion, or the criterion of extreme pessimism, defines an alternative that maximizes the minimum result for each alternative, i.e. a strategy is selected that corresponds to

3. Savage's minimax risk criterion. According to this criterion, a strategy is selected in which the magnitude of risk in the worst conditions is minimal, i.e. is equal to

Here risk = () -.

4. The criterion of optimism-pessimism Hurwitz recommends when choosing a solution not to be guided by either extreme pessimism or extreme optimism. According to this criterion, the strategy is selected from the condition

+ (1 – k) }.

The value of the pessimism coefficient k is chosen by the researcher between zero and one for practical reasons.

5. Laplace's criterion of indifference. Under conditions of complete uncertainty, it is assumed that all possible environments (nature) are equally probable. This criterion identifies the alternative with the maximum average result, i.e.

Once the probabilities of realization are known for all environmental states, the expected EMV value can be determined for each alternative. One of the most common criteria for choosing an alternative is the maximum EMV.

For each alternative, the expected EMV value is the sum of all possible gains for that alternative multiplied by the probabilities of realizing those gains:

The maximum EMV in the case of equal probabilities coincides with the Laplace indifference criterion.

Let us illustrate these points with the following example, implemented in the Decision Analysis / Decision Tables module.

In Figure 4.6, almost all the described criteria are calculated, except for the Savage minimax risk criterion, which is calculated in Figure 4.7.

Certain criteria are obvious from the designation of rows and columns. For example, the EMV column (Figure 4.6) below shows the maximum EMV. In addition, at the bottom of this figure, the values ​​of specific criteria are spelled out and it is indicated on which alternatives they are implemented.

Figure 4.6 - Window of the report on the solution of the decision analysis problem

Figure 4.7 – The report window on the calculation of the Savage minimax risk criterion

Figure 4.7 shows the calculations of Savage's minimax risk criterion (it is equal to 4 and is implemented by the second alternative).

Tasks for laboratory work No. 4

You must analyze all the tasks described in this section.

You will take the initial information for the laboratory work from the assignment on the transport problem. The game must be 4x4. The transportation cost matrix is ​​the three strategies of player A. The fourth strategy of this player will be the line of needs (the last row not included in the matrix of transportation costs).

To solve the problem using graphical methods, choose two active strategies of player A with minimum frequencies.

To analyze the game with nature, take the same payment matrix.

Laboratory work No. 5

Queuing systems

General information

There is a wide class of tasks that have to be constantly faced in daily and business activities, where there are processes that lead to delays in service and queues. The systems in which these processes take place are called queuing systems (MO), and the MO theory deals with the mathematical description or the development of mathematical models of the processes occurring in them.

In the process of studying queues, you first need to pay attention to the following main components: incoming flow of requests, service channels, presence of a queue and outgoing flow. These components do not require clarification, except for the discipline of the queue. The latter is just a service rule. In the future, we will consider the rule: first came, first served. ML systems are associated with two types of costs: service costs, which increase as service levels increase, and waiting costs, which decrease as service levels increase. As you know, there is a point of minimum total costs of the MO system.

Determination of the optimal level of service that minimizes the total costs of the ML system is one of the main tasks in the development and operation of ML systems.

In conditions of uncertainty, in the most general case, two approaches to making a strategic decision are possible.

The first approach is when a manager can use available information or experience to identify his assumptions about the probabilities of possible external conditions in which his company will find himself. In the case when the probability of the states of objective conditions is unknown, in accordance with the Bayes-Laplace criterion, one must proceed from their equality. That is, in the absence of grounds for otherwise, it is necessary to assume the equality of the probabilities of the occurrence of conditions leading to each of the possible results. The application of this criterion makes it possible to reduce the problem to a variant with complete information on the state of objective conditions, conditions of uncertainty become similar to conditions of risk.

In the second approach, when the degree of uncertainty is too high, the manager prefers not to make assumptions about the probabilities of various external conditions. Applying this approach to assess the proposed strategies, the following decision criteria are possible:

· Criterion of extreme optimism;

· Wald's criterion, also called maximin;

· Alpha-Hurwitz criterion;

· The Savage criterion, also called the minimax rejection criterion.

The choice of the criterion is predetermined by specific circumstances, as well as subjective psychological characteristics, temperament and general outlook of the company's management (optimistic or pessimistic; conservative or progressive). Let's consider these criteria in the following example.

Example 4 For sale at a price of 50 rubles. for 1 unit a certain amount of perishable product is purchased at a price of 30 rubles. for a unit. It is known from observations that the sale of a product can occur at the level of 1 unit, 2 units, 3 units. and 4 units. If the product is not sold per day, then at the end of the day it is sold at a reduced price of 20 rubles. for a unit. It is necessary to determine how many units the manager should purchase in order for his solution to be optimal in accordance with different criteria of optimality.

With the sale of each unit of the product, a profit of 20 rubles will be made. (50 -30). With the sale of each unit of the purchased product at a reduced price, the loss from the sale will be 10 rubles. (30 - 20).

Let's make a table. 8.5 illustrating the possible trading results.

Table 8.1 Possible income for different scenarios

We will arrange horizontally the possible options for the demand for products per day, vertically - the possible options for the decision maker to purchase products. In each cell of the table, we calculate the profit (with a plus sign) or loss (with a minus sign) from sales operations.

Extreme optimism criterion dictates to the manager a strategy of actions in which he gets the opportunity to earn maximum income. This will happen if the manager purchases the maximum possible amount of products and sells all of them during the day:

4 x (50 - 30) = 80 (rub.).

This strategy is the most risky, because in the case of minimum sales (1 unit), the manager will receive the maximum loss:

1 x (50 - 30) + 3 x (20 - 30) = -10 (rub.).

Wald's solution criterion - criterion extreme pessimism suggests the most cautious strategy of behavior that guarantees the maximization of the minimum income. For example, the minimum income for various purchasing options can be 20, 10, 0 or -10 rubles. If the manager buys 1 unit. products, then the minimum winnings of 20 rubles. guaranteed to him.

This criterion orientates the decision-maker to the worst conditions and recommends choosing the strategy for which the gain is maximized. In other, more favorable conditions, the use of this criterion leads to a loss of efficiency of the system or operation. Because the criterion is conservative, it is especially well suited for small business firms, whose survival depends on the ability to avoid losses.

Savage criterion, also known as the "wild principle", the principle for miscalculations, the criterion of minimax risk, the principle of minimax of the consequences of erroneous decisions, etc. Savij criterion also pessimistic, but when choosing the optimal strategy in accordance with it, one should focus not on income, but on possible losses, taking into account lost profits.

Data on possible losses are presented in table. 8.2.

Table 8.2. Possible losses under various scenarios

As you can see from the table. 8.2, upon purchase of 2 units. and selling 1 unit. at the selling price, the second we sell at a reduced price, losing 10 rubles. When buying 2 units. and the demand of 3 units. 2 units we sell at a regular price, while losing the lost profit from the absence of 1 unit. product in the amount of 20 rubles.

The maximum losses for each of the procurement options will be 60, 40, 20 and 30 rubles. Guided by the Savage criterion, it is necessary to choose from them the minimum value of 20 rubles. and purchase 3 units. products.

In accordance with this criterion, if it is required to avoid a large risk in any conditions, then the optimal solution will be the solution for which the risk, maximum under various conditions, turns out to be minimal. The executive, when using the Savage criterion, clearly abandons attempts to maximize returns, choosing a strategy with a satisfactory return at a lower risk. The Savage criterion is therefore particularly useful for evaluating a series of projects over an extended period.

Hurwitz criterion (pessimism-optimism- this is a compromise way of making decisions in conditions of uncertainty. For each of the possibilities - obtaining the maximum income and the minimum income - the probability of its occurrence is determined. The sum of the probabilities of the two options must be equal to one. Then the value of the objective function is calculated as the sum of the products of each result and the probability of its achievement. The values ​​of the maximum and minimum income are taken from table. 8.1. The calculations are presented in table. 8.3.

Table 8.3. Calculating Objective Function Values

Suppose it is determined by an expert that the probability of obtaining the maximum income is 0.7, the minimum income is 0.3. Then for the decision to buy 2 units. production, the value of the objective function will be equal to

0.7 x 40 + 0.3 x 10 = 31 (rub.)

The maximum value of the objective function is 53 rubles. achieved when choosing a solution to purchase 4 units. products. This solution will be optimal according to the Hurwitz criterion. The relativity of this choice is determined by the degree of objectivity in assessing the probabilities of different outcomes.

Decision tree. In management practice, situations often arise when the adoption of one decision puts the manager or the owner of the company before the next choice. When it is necessary to make several decisions under conditions of uncertainty, and each subsequent decision depends on the previous one, a scheme called a decision tree is used to solve such a problem.

Decision tree is a graphical representation of the decision-making process, which reflects alternative decisions and states of the environment, corresponding to the probabilities, and "gains" for any combination of decisions and states of the environment.

The construction and analysis of a "decision tree" is acceptable in any case, if a sequential series of contingent decisions is made in conditions of risk. A contingent decision is a decision that depends on circumstances or options that arise later. The construction of a "decision tree" begins with the very first, or original, decision and moves forward in time through a series of successive events and decisions. With every decision or event, this "tree" has branches that show every possible direction of action until, finally, all the logical sequences and the resulting returns are drawn.

Maximax criterion(extreme, "pink" optimism) is based on the optimistic principle of L. Hurwitz, according to which the option is selected that provides the greatest effect in the most favorable situation.

If the consequence matrix (3.1) is considered as the effect matrix E,

This criterion corresponds to strategy 1 (see Figure 3.6), it is advisable to apply it in cases where it is possible to influence the opposite side in order to make the uncontrolled external environment more favorable, and to realize the possibilities of optimal use of controlled internal factors.

Example 3.3. Taking the matrix of consequences in example 3.2 as the matrix of effects, choose a solution option according to the maximax criterion.

1. The initial data is entered into Excel (Fig. 3.9). Then, using the MAX function for cells (B4: F4;…; B7: F7), the maximum values ​​for each solution are sequentially found: a 1 = 8, a 2 = 12, a 3 = 10, a 4 = 8.

Rice. 3.9. Results of choosing the optimal solution according to the maximax criterion

2. From the sequence of found maximum values a i(G4: G7) The MAX function (cell G8) selects the largest value: a 2 = 12, with this in mind, it is recommended to make a second decision.

If the elements of the matrix A (3.1) are the costs Z, then they can be considered as losses, and then the solution that provides the lowest costs is selected from the conditions for minimizing costs:

. (3.10)

Minimin criterion(pessimism) is based on the pessimistic principle, according to which, in an unfavorable external environment, controlled factors can be used in an unfavorable way. Then, if the consequences matrix is ​​the effect matrix E, then the effective solution is selected from the conditions for ensuring the maximum:

. (3.11)

In real conditions, it is not always possible to control the uncontrollable factors of the external environment, especially when it is necessary to take into account the time factor. For example, with long-term forecasting and planning; design of complex objects, etc. Or, for example, production costs are controllable factors in short time intervals and uncontrollable in the long term, since the cost of electricity, the cost of materials and purchased products, etc. are unknown in advance. Another example is the determination of the volume of production of the company's products (controlled factor), which depend on various factors associated with the production process. These factors relate to the internal environment of the enterprise: the level of design and technological preparation of production, the type of equipment used, the qualifications of workers, etc.

This criterion corresponds to strategy 2 (see Figure 3.6).

Example 3.4. Taking the matrix of consequences in example 3.2 as the matrix of effects, choose a solution option according to the minimin criterion.

1. Initial data are entered into Excel (Fig. 3.10). Then, using the MIN function for cells (B4: F4;…; B7: F7), the minimum values ​​for each i-th solution: .

Rice. 3.10. Results of choosing the optimal solution according to the minimin criterion

3. From the sequence of found minimum values a i(G4: G7) Use the MIN function (cell G8) to select the smallest value: a 4 = 1, with this in mind, it is recommended to make a fourth decision.

When analyzing the cost matrix, the pessimism criterion takes the following form

(3.12)

Maximin criterion (extreme pessimism) based on the pessimistic principle of A. Wald, according to which the option is chosen, the result of which is the most favorable among the least favorable.

If the expected situation is unfavorable, i.e. will bring the smallest income: a i= min a i j, then a solution is chosen for which the minimum (guaranteed) income will be the greatest

. (3.13)

This criterion is conservative, since it offers a choice with a cautious line of behavior, so it is advisable to use it in cases where it is necessary to ensure success under any possible conditions. In the decision matrix (Figure 3.6), Wald's criterion corresponds to strategy 3.

Example 3.5. For the consequences matrix in example 3.2, choose a solution option according to the maximin criterion.

1. For each i–Th alternative solution, using the MIN function, the minimum values ​​are found: a 1 = 2, a 2 = 2, a 3 = 3, a 4 = 1(see figure 3.11, cells G4: G7)

Rice. 3.11. Results of choosing the optimal solution according to the maximin criterion

2.Using the MAX function from the sequence of found minimum values a i(G4: G7) the maximum is selected a 3= 3 (cell G8).

3. According to Wald's rule (3.11), preference should be given to the third solution ( i = 3), with the maximum guaranteed result (gain) regardless of the variant of the situation (external conditions).

Minimax criterion (minimax risk, expectation of losses) based on L. Savage's frustration principle. According to this principle, an option is selected, in the implementation of which the maximum possible disappointment (the difference between the maximum possible result and the results that can be obtained for each of the remaining options) turns out to be the smallest.

Here they are guided by the worst situation, which is associated with the greatest risk. When choosing a solution, a risk matrix is ​​used R(3.5). The best solution is the one in which the maximum risk value will be the smallest:

. (3.14)

When making investment decisions under conditions of uncertainty with a focus on the worst outcomes, the pessimistic criterion (maximin) and the criterion of disappointment (minimax) are applied.

This criterion is used in cases where it is required to avoid a large risk in any conditions; it corresponds to strategy 4 (Fig. 3.6).

Example 3.6. Using the consequences matrix in example 3.2, choose a solution option based on the minimax criterion.

1. Preliminarily, according to the matrix of consequences of example 2, using expression (3.4), the elements of the risk matrix are calculated in Fig. 3.12.

2. In each row of the risk matrix, using the MAX function, its maximum element is selected (cells G4: G7): r i = : r 1 = 8, r 2 = 6, r 3 = 5, r 4 = 7.

Rice. 3.12. Results of choosing the optimal solution according to the minimax criterion

3. According to Savage's rule, the smallest of these values ​​is selected (MIN function in cell G8): r 3 = 5, i.e. the 3rd decision should be made ( i = 3). The choice of this option means that the maximum losses in various situations will be minimal and will not exceed 5 units.

Hurwitz criterion for generalized maximin(pessimism-optimism) presupposes a choice mixed strategy, when, in a certain proportion, pessimism (caution) and optimism (propensity to take significant risks) are combined, i.e. an intermediate solution is chosen between the line of behavior in the expectation of the worst and the line of behavior in the expectation of the best.

According to this criterion, a solution option is selected at which the maximum indicator is achieved G defined from the expression:

G i = max[a min a i j + (1 - a) max a i j]. (3.15)

where and ij- the payoff for i-th solution for j-th variant of the situation,

a- coefficient reflecting the degree of optimism ( 0 ≤ a ≤ 1): at a = 0 a line of behavior is chosen based on the best, i.e. focus on marginal risk is made (we get the maximax criterion); at a = 1 focus on the worst, then we get the Wald criterion - a guideline for cautious behavior. Intermediate values a between 0 and 1 and are chosen depending on the specific situation and the risk appetite of the decision maker.

Example 3.7. The enterprise is preparing to release new types of products, with four possible solutions Q 1 , Q 2 , Q 3 , Q 4 , each of which corresponds to a certain type of product or their combination. The structure of demand for products is characterized by three options for the situation S 1 , S 2 , S 3 ... Efficiency of the release of new types of products a i j for each pair of solutions Q i (i = 1,2, ..., m) and setting S j (j = 1,2, ..., n) are given in the table in Figure 3.12. It is necessary to find the most profitable solution according to the Hurwitz criterion Q i and to evaluate the influence of the optimism coefficient on the choice of the solution.

1. Let us set the sequence of coefficients k with a step of 0.25: 0; 0.25; 0.50; 0.75; 1.00 and enter the initial data on the Excel worksheet, fig. 3.12.

2. The results of calculating the indicator G by expression (3.13) for various solutions depending on the value of the coefficient k are shown in the lower table in Figure 3.13.

Rice. 3.13. Initial data, calculation formulas and calculation results of the Hurwitz criterion (arrows show effective solutions)

As can be seen from the figure (cells B18: F18), the change in the coefficient k influences the choice of a solution that should be preferred.

The choice of this or that criterion depends on a number of factors:

The nature of the problem being solved;

The set goals,

Sets of restrictions

Risk appetite of decision-makers.

It should be noted that the considered methods and techniques for solving problems in conditions of risk and uncertainty are not limited to the listed methods. Depending on the specific situation, other methods can be used in the analysis process, for example, using the standard deviation and the coefficient of variation as a measure of risk.

Decision making in conditionsuncertainties

1. Wald's maximal criterion.

2. Criterion Savage (minimax risk).

3. The Hurwitz criterion (pessimism-optimism).

1. Maximin Wald criterion (criterion of extreme pessimism)

("Count on the worst")

In the group of criteria for choosing the optimal strategy, statistics used when unknown prior probabilities states of nature , includes criteria Walda, Savage and Hurwitz... They use either a Payments Matrix or a Risk Matrix Analysis.

If the distribution probabilities of future states of nature are unknown, then all information about nature is reduced to the list of its possible states.

Maximin Wald test- this is criterion of extreme pessimism, or the criterion of the careful observer. It can be formulated for both pure and mixed strategies.

Wald's criterion is criterion of extreme pessimism, since the statistician assumes that nature implements such states in which the magnitude of his payoff takes the smallest value.

The criterion is identical maximin (pessimistic) criterion, used in solving matrix games in pure strategies.

Of each strings are selected minimum elements, i.e. which correspond to the worst result of the decision maker for the known states of "nature". Is then selected strategy Decision maker, corresponding maximum element from the selected minimum:

. (1)

The options chosen in this way completely eliminate the risk, since the decision maker cannot face a worse result than the one on which he is guided.

Application of this criterion justified if the situation in which the decision is made is characterized by the following features:

the probabilities of states of "nature" are unknown;

the solution is implemented only once or a small number of times;

complete inadmissibility of risk.

Thus, the optimal strategy according to Wald's criterion is considered to be a pure strategy, which in the worst conditions guarantees the maximum profit. Means, the optimal will be the maximin clean strategy, and the maximum payoff is the net play price in a zero-sum doubles game.

Example 1.

Supplier game.

The production output of the company significantly depends on perishable material, for example, milk or berries, supplied in batches of 100 units.

If the delivery does not arrive on time, the company loses 400 units. from under-production.

The company can send its own transport to the supplier (costs 50 units), but experience shows that in half of the cases, the transport is returned with nothing.

You can increase the probability of receiving the material up to 80% if you first send your representative, but the costs will increase by another 50 units.

It is possible to purchase a more expensive (50%) substitute material from another, completely reliable supplier, however, in addition to transport costs (50 units), additional costs of storing material in the amount of 30 units are possible if its quantity in the warehouse exceeds the permissible norm equal to one batch.

What strategy should the plant adhere to in this situation?

Solution

Nature has two states: the supplier is reliable and the supplier is unreliable. The company has four strategies: 1) do not carry out any additional actions, 2) send its own transport to the supplier, 3) send a representative and transport to the supplier, 4) buy and bring a substitute material from another supplier.

Let's make a calculation table:

Solution

Based on the calculated results, you can compose a payment matrix:

Answer. It is necessary to adhere to the third strategy and the costs will not exceed 260 units if you send a representative and transport to the supplier.

1 ... The considered way of finding the optimal solution is criterion Wald ( maximin test decision making). A solution is chosen that guarantees a gain not less than maxmin:

units

Applying this criterion, we imagine an active and malicious adversary in place of nature. it pessimistic an approach .

2. Maksimaxny criterion... The most favorable case:

units

If the firm does nothing, it will spend no more than 100 units. This is the criterion absolute optimism.

Wald criterion for mixed strategies

The mixed strategy of statistics is considered optimal. , at which the minimum average payoff will be the maximum: . (2)

Wald's criterion orients statistics on the most unfavorable conditions of nature, that is, they express a pessimistic assessment of the situation.

2. Savage criterion (minimax risk )

In practice, when choosing one of the possible solutions, they often stop at the implementation of which will lead to the least serious consequences if the choice turns out to be wrong. This approach to choosing a solution was mathematically formulated by the American statistician Savage in 1954 and was named the Savage principle... It is especially convenient for economic problems and is often used to choose solutions in the games between man and nature.

According to the Savage principle each solution is characterized by the amount of additional losses that arise during the implementation of this solution, in comparison with the implementation of a solution that is correct for a given state of nature. Naturally, the correct decision does not entail any additional losses, and their value is equal to zero.

When choosing a solution that best suits various states of nature, only these additional losses should be taken into account, which, in essence, will be the result of selection errors.

To solve the problem, the so-called " risk matrix», The elements of which show what loss the player (decision maker) will incur as a result of choosing a non-optimal solution.

Recall that At risk the player when choosing a strategy in the conditions (states) of nature is called the difference between the maximum winnings, which you can get it under these conditions, and the gain, which will receive the player is in the same conditions applying the strategy.

The Savage criterion is a criterion of minimax risk, minimization of "regrets". This criterion, like Wald's criterion, is the most cautious and pessimistic.

In Savage's criterion, pessimism manifests itself in a different way: the worst is not the minimum gain, but the maximum loss of gain in comparison with what could be achieved under the given conditions (maximum risk).

Savage criterion focuses not on the result, but on the risk(losses or fines).

The strategy is chosen as the optimal one, in which the value of losses in the worst conditions is minimal. The Savage criterion recommends choosing as optimal that a strategy that minimizes the maximum risk:

. (3)

Requirements presented to the situation in which the decision is made according to the Savage criterion coincide with the requirement to use the Wald criterion. Savage's criterion, like Wald's criterion, orients statistics on the most unfavorable states of nature.

Example 2. For the Supplier problem, the minimum risk is achieved at once with two strategies А 2 and А 3:

Find the optimal solution to the game by applying the Savage criterion.

Solution.

We focus on the most unfavorable conditions of "nature". Let's calculate the risks statistics.

For the first column:

For the second column:

For the third column:

Let's write down risk matrix.

We define in each line the largest number - the greatest risk statistician, if he applies the strategy, and nature changes its states , , ... Let's supplement the risk matrix with the last column, "the greatest risks".

Risk Matrix and Greatest Risks

Let's find the smallest risk:.

Hence, the optimal strategy according to the Savage criterion is strategy .

4.3. Hurwitz criterion (pessimism-optimism)

The Hurwitz criterion is a criterion for a generalized maximum, or pessimism-optimism.

It seems logical that when choosing a solution, instead of two extremes in assessing the situation, adhere to some intermediate position, taking into account the possibility of both the worst and the best, favorable behavior of nature.

This compromise option was proposed by Hurwitz. According to this approach, for each solution, it is necessary to determine a linear combination of min and max gains and take the strategy for which this value will be the largest.

This criterion provides intermediate solution between extreme optimism and extreme pessimism, which is determined according to the principle:

. (4)

Number () - degree of optimism , satisfies the condition and is selected on the basis of subjective considerations, environmental characteristics, common sense, based on the experience of the decision maker, his attitude to risk, etc. The choice of the value of the degree of optimism is influenced by the measure of responsibility: the more serious the consequences of erroneous decisions, the greater the desire of the decision maker to insure, that is, the degree of optimism is closer to zero.

For each strings calculated weighted average(taking into account the selected value) of the smallest and largest results, after which the string with maximum value.

For we have extreme optimism criterion, i.e. reflects the position of a gambler who expects the most favorable state of the environment.

For, the Hurwitz criterion becomes criterion of extreme pessimism Wald.

If 0 is the intermediate ratio of the decision maker to possible risks. If you want to insure yourself in this situation, take it close to one.

The choice of meaning is subjective, and, therefore, the choice of a solution is also subjective, which is absolutely inevitable in conditions of uncertainty.

The more dangerous the situation, the more decision maker seeks to insure himself against possible risks, the closer to 0. And the less reckless he is, the closer to 1.

A Hurwitz-optimal strategy should guarantee the statistician a greater gain than the statistician's gain intuitively or from experience.

Application of the Hurwitz criterion is justified if the situation in which the decision is made is characterized by signs:

the probabilities of states of nature are unknown;

the solution is implemented by a small number of solutions;

some risk is allowed.

Example 3. Find the optimal solution to a statistical game given by a payoff matrix using the Hurwitz criterion.

Solution.

For application Hurwitz criterion you need to know the value of the probability. For example, let. This means that we want to make the event “the smallest possible gain for a statistician” more plausible (close to one), that is, we insure against unfavorable situations in the game. Then

.

Let's write down all the intermediate results in the table.

From the last column of the table, it can be seen that the maximum value is (-7.2) and corresponds to the net strategy ; it will be optimal according to the Hurwitz criterion.

The analysis of practical situations is carried out according to several criteria. simultaneously, which allows a deeper study of the essence of the phenomenon and choose the most informed management decision... As optimal based on cumulative research the strategy is taken that is most often called optimal by all criteria.

The choice of a criterion (as well as the choice of the principle of optimality) is the most difficult and responsible task in the theory of decision-making. However, a specific situation is never so uncertain that it would be impossible to obtain at least partial information on the probabilistic distribution of the states of nature. In this case, having estimated the probability distribution of the states of nature, the Bayes-Laplace method is applied, or an experiment is carried out to clarify the behavior of nature.

Control questions

What is meant by playing with nature?

What criteria does the statistician use to determine his optimal strategy in the face of uncertainty?

What is a player's risk?

Explain the principles of using game theory models in economic problems under conditions of uncertainty (playing with nature).

1. conditions uncertainties using a fuzzy apparatus ...

2. Adoption decisions v conditions uncertainties (5)

   Abstract >> State and law

   A risk situation, and for another - uncertainties... Risk adoption worst solutions v conditions, when all the initial ones are known ... because in the process adoption decisions have to make a choice in conditions uncertainties.. Procedures and methods of system ...

3. Adoption managerial decisions v conditions risk and uncertainty

   Abstract >> Management

   ... Adoption managerial decisions v conditions risk and uncertainty. Plan: Introduction. Sources and types of uncertainty. Adoption decisions v conditions uncertainties... and types of uncertainty. Adoption

One of the most important conditions for making an effective decision aimed at achieving a goal in the time perspective is the availability of an appropriate amount of relevant information. Incomplete information, the inability to reliably predict future events and factors that could affect the result that the decision is made to are signs of uncertainty. A fairly large part of management decisions is made under conditions of uncertainty. Uncertainty potential is the external environment of the organization.

Decision-making under conditions of uncertainty is associated with the concept of risk and is carried out using the methods of operations research and the theory of statistical decisions. In general, the problem of making a decision under conditions of uncertainty is presented in the form of an efficiency table (Table 1).

Table 1.

where O n - conditions of the situation, which are not known exactly, but about which n-proposals can be made (demand, number of suppliers, satisfaction with materials);

P m - possible strategies, decision lines of behavior.

Each pair of strategy and environment corresponds to the payoffs -A mn.

The payoffs shown in the table are calculated indicators of the effectiveness of the strategy (solution) in various settings.

The presented task is aimed at making decisions when developing plans for the development of enterprises, developing production programs, plans for the release of new types of products, focusing on innovations, choosing insurance strategies, investments, funds, etc.

In the theory of statistical decisions, a special risk indicator is used, which shows the profitability of the adopted strategy in a given situation, taking into account its uncertainty. Risk is calculated as the difference between the expected outcome of an action given the availability of accurate situational data and the outcome that can be achieved if this data is uncertain. Based on this difference, a table of risks of releasing a new type of product is calculated. The risk table makes it possible to assess the quality of various solutions and establish the completeness of the implementation of opportunities in the presence of risk. The choice of the best solution depends on the degree of uncertainty.

Depending on the degree of uncertainty of the situation, there are 3 options for making decisions:

1. Choosing the optimal solution when the probabilities of possible scenarios are known. The optimal solution is determined by the max sums of the products of the probabilities of various scenarios P ​​(O 1) by the corresponding values ​​of the gains A (efficiency table 6) for each solution.

2. Choosing the optimal solution when the probabilities of possible scenarios are unknown.

3. The choice of the optimal solution according to the principles of the approach to assessing the result of actions.

In conditions of an unknown probability of the situation, the following decisions can be made:

a) max-min or “count on the worst” - the choice of a solution that guarantees a win in any conditions, not less than the greatest possible in the worst conditions;

b) min max risk in any conditions. For the optimal decision is made, for which the risk, max under various scenarios, seems to be minimal.

For the optimal solution, depending on the line of orientation of the decision maker, a decision is made for which the indicator G (the criterion of pessimism - Hurwitz's optimism) turns out to be maximum:

where is the minimum payoff corresponding to solution m;

Maximum payoff corresponding to solution m;

k - coefficient characterizing the line of behavior (orientation) of the decision maker,.

Graphically meaning k in relation to the line of behavior can be interpreted as follows:

k value

0 0,25 0,5 0,75 1

Orientation line in calculation

for the best for the worst

Task:

There are 3 options for investment:

1) Invest all available funds in the shares of the company "Oil-AG", which guarantees a high income under the appropriate conditions;

2) Invest all funds in GKOs with a guarantee of low and stable income;

3) Invest part of the funds in the shares of "Oil-AG", part in GKOs - that is, diversify the portfolio of funds.

The perspective is indicated by three scenarios (outcome of events).

Make a decision on the problem of investing, having as the initial data the payoff table (Table 2).

Table 2.

P i - solution option;

O i - a variant of the situation;

O 1 - the company "Oil-AG" - has gone bankrupt, GKO - brings a stable income.

O 2 - the company "Oil-AG" - is flourishing;

O 3 - crisis in the economy.

Let us determine the optimal solution for which the gain in any conditions will be no less than the maximum possible in the worst conditions (max-min).

From table. 2 for the solution P 1 the smallest payoff is 0, for P 2 - 0.3, for P 3 - 0.25.

The greatest possible gain in the worst possible combination of circumstances will be 0.3, which corresponds to the adoption of the decision P 2, i.e. in all situations, the solution P 2 will not be the worst.

The optimal solution, provided that the risk turns out to be the minimum of its maximum values ​​for various solutions, is determined from Table 7. The matrix of markets is pre-calculated. At the same time, the maximum risk when making a decision is P 1 - 0.5; at P 2 - 0.49; at P 3 - 0.29. From a number of maximum risks, the optimal solution is P 3, which has a minimum risk level of 0.29.

Let us calculate the criterion of pessimism - Hurwitz optimism for various solutions depending on the value of the adopted coefficient k.

To solve P 1

Solution:

Let's calculate the matrix of investment risks (Table 3).

Table3.

Provided that the situations are equally probable, their probabilities are equal and amount to:

P (O 1) = P (O 2) = P (O 3) = 0.33

Mathematically, the expectations of winnings, provided that the situations are equally probable, are determined from the expression:

W i = P (O i) * A ij,

where P (O i) is the probability of a future situation;

A ij - the payoff corresponding to the i-th solution in the j-th situation.

W 1 = 0.33 * 0 + 0.33 * 0.99 + 0.33 * 0.1 = 0.3597

W 2 = 0.33 * 0.5 + 0.33 * 0.5 + 0.33 * 0.3 = 0.329

W 3 = 0.33 * 0.25 + 0.33 * 0.7 + 0.33 * 0.4 = 0.445

In conditions of equiprobability of future situations, the most optimal solution is P 3.

For other values ​​of the probabilities of situations, the solution may be different.

Choosing a solution according to the Hurwitz criterion:

to solve P 1: G 1 = 0.495;

to solve P 2: G 2 = 0.5 * 0.3 + (1-0.5) * 0.5 = 0.4;

to solve P 3: G 3 = 0.5 * 0.25 + (1-0.5) * 0.7 = 0.475.

For k = 0.5, the optimal solution is P 1.

The values ​​of G i are calculated similarly for other values ​​of the coefficient.

The obtained values ​​of G i are summarized in table 4.

Table4.

The person making the decision in accordance with the selected k i for the optimal makes the decision having the maximum value of G i. With k i = 0.75 - G max = 0.362. The decision P 1 or P 3 is taken as the optimal one.