Purpose of the service. This type of problem refers to decision making problems under conditions of uncertainty. Using the service, you can choose the optimal strategy using:

• minimax criterion, maximax criterion, Bayes criterion, Wald criterion, Savage criterion, Laplace criterion, Hodge-Lehman criterion, see Typical tasks;
• Hurwitz criterion, generalized Hurwitz criterion with efficiency calculation.

Planning of the ideal experiment is also carried out. The results of online calculations are presented in a report in Word format (see sample format).

Instructions. To select the optimal strategy online, you need to set the matrix dimension. Then, in a new dialog box, select the necessary criteria and coefficients. You can also paste data from Excel.

Dimension of the payment matrix(target function of the ZPR under conditions of uncertainty)
2 3 4 5 6 7 8 9 10 x 2 3 4 5 6 7 8 9 10 ",0);">

Note: First, if possible, simplify the matrix by crossing out unprofitable strategies A. Strategies of nature cannot be crossed out, since each of the states of nature can occur randomly, regardless of the actions of A.

Any human economic activity can be considered as a game with nature. In a broad sense, “nature” refers to the totality of uncertain factors; influencing the effectiveness of decisions made. The indifference of nature to the game (winning) to the possibility of the economist (statistician) obtaining additional information about its state distinguishes the game of the economist with nature from an ordinary matrix game in which two conscious players take part.

Example. An enterprise can produce 3 types of products A 1, A 2 and A 3, while receiving a profit depending on demand, which can be in one of 4 states (B 1, B 2, B 3, B 4). The elements of the payment matrix characterize the profit that will be received upon issue i-th products at j-th state demand. The game of enterprise A against demand B is given by the payment matrix:

	IN 1	AT 2	AT 3	AT 4
A 1	2	7	8	6
A 2	2	8	7	3
A 3	4	3	4	2

Determine the optimal proportions in the output that guarantee the maximization of the average profit under any state of demand, considering it certain. The problem comes down to a game model in which.

Solution.
Maximax criterion.

Select from (8; 8; 4) the maximum element max=8

Laplace criterion.

Select from (5.75; 5; 3.25) the maximum element max=5.75
Conclusion: choose strategy N=1.

Wald criterion.

Select from (2; 2; 2) the maximum element max=2
Conclusion: choose strategy N=1.

Savage criterion.
We find the risk matrix.
Risk- a measure of the discrepancy between different possible outcomes of adopting certain strategies. The maximum gain in the jth column b j = max(a ij) characterizes the favorable state of nature.
1. Calculate the 1st column of the risk matrix.
r 11 = 4 - 2 = 2; r 21 = 4 - 2 = 2; r 31 = 4 - 4 = 0;
2. Calculate the 2nd column of the risk matrix.
r 12 = 8 - 7 = 1; r 22 = 8 - 8 = 0; r 32 = 8 - 3 = 5;
3. Calculate the 3rd column of the risk matrix.
r 13 = 8 - 8 = 0; r 23 = 8 - 7 = 1; r 33 = 8 - 4 = 4;
4. Calculate the 4th column of the risk matrix.
r 14 = 6 - 6 = 0; r 24 = 6 - 3 = 3; r 34 = 6 - 2 = 4;

The calculation results will be presented in the form of a table.
Select from (2; 3; 5) the minimum element min=2
Conclusion: choose strategy N=1.

Thus, as a result of solving a statistical game according to various criteria Strategy A 1 was recommended most often.

Close in ideas and methods to game theory is the theory of statistical decisions. It differs from game theory in that the situation of uncertainty does not have a conflict overtones - no one opposes anyone, but there is an element of uncertainty. In problems of the theory of statistical decisions, the unknown conditions of an operation depend not on a consciously acting enemy, but on objective reality, which in the theory of statistical decisions is usually called “nature.” The corresponding situations are often called games with nature (statistical games).

Often these situations are generally referred to as game theory, stipulating in the definition of the game that one of the participants may be the environment (nature), acting as the sum of disorganizing circumstances, the entire complex of external conditions in which the player has to make a decision. Let's call this player a statistician.

Nature is indifferent to winning and does not seek to turn the mistakes of statistics to its advantage. Let the statistician havemstrategies, and nature can implementntheir states. If a statistician has the ability to numerically evaluate the consequences of each of his pure strategies for any state of nature, then the game can be specified by a payoff matrix. When simplifying the payment matrix, there is a specificity: one cannot discard certain strategies of “nature,” since it can implement them regardless of whether they are beneficial to the statistician or not.

When solving such games, there can be 2 situations:

· player A does not know the probabilitiespj, with which nature realizes its states;

· probabilities pj known.

Various criteria are used to make decisions in such games.

If the probabilitiespj states of nature are unknown, then you can use the criteria of Wald, Laplace, Savage, Hurwitz, etc. The main difference between these criteria is determined by the strategy of behavior of the decision maker under conditions of uncertainty. For example, the Laplace criterion is based on more optimistic assumptions than the Wald criterion. The Hurwitz criterion can be used in different approaches: from the most optimistic to the most pessimistic. Thus, the listed criteria, despite their quantitative nature, reflect a subjective assessment of the situation in which the statistician has to make a decision. Unfortunately it doesn't exist general rules assessing the applicability of a particular criterion, since the behavior of the decision maker is likely to be the most important factor in choosing the appropriate criterion. Let us formulate these criteria.

1. Laplace criterion

This criterion is based on the principle insufficient justification, according to which it is believed that the occurrence of all states of nature is equally probable, that isp 1 = p 2 =...= p n =1/ n, and the optimal strategy is considered Ai , providing

. (5.1)

2. Wald criterion (minimax or maxmin criterion )

This criterion is the most cautious, since it is based on choosing the best of worst opportunities:

– if a win is found;

– if losses are found.

These are pessimistic criteria.

3. Savage criterion (minimax risk)

The Wald criterion is so pessimistic that it can lead to illogical conclusions. Consider the following loss matrix, which is usually cited as a classic example to justify the “less pessimistic” Savage criterion.

	11000
	10000	10000

Applying the minimax criterion leads to the choice of strategy A2, although intuitively one can choose A1, since with this choice one can hope to lose 90, while choosing A2 always leads to a loss of 10,000 units in any weather condition.

Savage’s criterion “corrects” the situation by introducing a new loss matrix, in which are replaced by font-size:14.0pt;line-height: 150%">, defined as follows:

It means thatis the difference between the best value in the columnj and meaning.

Essentially expresses the decision maker's regret for not choosing best action regarding the conditionj . Matrix R =() ê called the regret matrix or risk matrix.

Let's find the optimal strategy for the previous problem using this criterion:

.

Let’s apply it to the “regret” matrix R minimax criterion. We find that the optimal strategy is A1.

Note that regardless of– income or losses,– always losses. Therefore, the minimax criterion is always applied to the “regret” matrix.

4. Hurwitz criterion (pessimism-optimism)

This criterion covers a range of different decision-making approaches, from the most optimistic to the most pessimistic.

With an optimistic approach, a strategy is chosen that gives :

, if is a win, and

, if – losses.

Similarly, under the most pessimistic assumptions, the chosen solution matches: , if is a win, and

font-size:14.0pt;line-height: 150%">, if – losses.

The Hurwitz criterion establishes a balance between cases of extreme optimism and pessimism by weighing both behaviors with appropriate weights a and 1- a , where 0 £ a £ 1.

If – profit, then the strategy is selected according to the rule:

If – costs, the criterion selects a strategy that gives

Parameter a interpreted as optimism indicator; at a =1 criterion is too optimistic, when a =0 he is too pessimistic. Meaning a between 0 and 1 can be determined depending on the decision maker's tendency towards pessimism or optimism. a =0.5 seems most reasonable.

Analysis of practical situations is usually carried out on the basis of several criteria, which allows for a deeper exploration of the essence of the phenomenon.

Example.

One of the enterprises must determine the level of service offerings to satisfy customer needs. The exact number of clients is not known, but it is expected that it can take one of the following values: 200, 250, 300, 350. For each of these possible values there is a best supply level (in terms of possible costs). Deviations from these levels lead to additional costs either due to supply exceeding demand or due to incomplete satisfaction of demand.

The losses depending on the situation are shown in the following table:

Clients Proposed.
a 1
a 2
a 3
a 4

· Wald criterion. Because – losses, we apply the minimax criterion.

The optimal strategy would be A3.

· Laplace criterion. Let the strategies of the 2nd player be equally probable. Hence. Then:

EN-US">EN-US">EN-US">font-size:14.0pt;line-height:150%">Thus, best level proposals in accordance with the Laplace criterion will be strategy A2.

· Savage criterion . Let's build a risk matrix:

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The best strategy is A2.

· Hurwitz criterion. Let a =1 / 2.

			5/2+25/2=15
			7/2+23/2=15
			12/2+21/2=16,5
			15/2+30/2=22,5

The best strategies are A1 and A2.

If we find a solution using game theory methods, we first look for the presence of a saddle point:

This game has a saddle point and the optimal strategy is A3.

5. Bayes criterion

If the probabilities of states of nature– pj are known, then we can use the Bayes criterion, according to which:

A pure strategy corresponding to the maximum average payoff is considered optimal: , If – winnings and minimum average losses: , If -losses.

If in the previous example the demand probabilities are known font-size:14.0pt;line-height: 150%">, then to find the optimal strategy it is necessary to find the average losses for each pure strategy of the enterprise and select the one that provides a minimum of average losses: font-size:14.0pt;line-height : 150%;font-family:Symbol">® strategy A2.

It can be shown that the strategy that maximizes the average gain also minimizes the average risk.

All considered criteria were formulated for pure strategies, but each of them can be extended to mixed strategies, just as is done in game theory. In statistical decision theory, mixed strategies make sense when the game is repeated many times.

But by repeating the game many times, you can determine the frequency of repetitions of a particular situation and subsequently apply a stochastic approach to the decision-making problem.

If you use mixed strategies, then Wald test is formulated as follows: will be optimal mixed strategy , providing , i.e. maximizing average winnings(If -winning)

Savage criterion for mixed strategies : The optimal mixed strategy is considered to be the one with the maximum average risk statistics is minimal, that is, the strategy , found from the condition .

The optimal mixed strategies in this case are found in the same way as in a regular matrix game.

Many of us do not like to find ourselves in a situation where there is very little or no information about external factors, and at the same time we urgently need to make an important choice. Most likely, this is why most people prefer to avoid responsibility at work and are content with a modest, but at the same time relatively calm official position. If they knew about game theory and how the criteria of Wald, Savage, and Hurwitz could be useful, the careers of the smartest among them would probably skyrocket.

Expect the worst

This is exactly how the first of these principles can be characterized. The Wald criterion is often called the criterion of extreme pessimism or the rule of minimal evil. In conditions of a shaky, unstable situation, it seems quite logical to take a reinsurance position, which is designed for the worst case. The Wald maximin criterion focuses on maximizing the payoff under the most unfavorable circumstances. An example of its use would be maximizing the minimum income, maximizing the minimum amount of cash, etc. This strategy justifies itself in cases where the decision maker is not so much interested in great luck as wants to insure himself against sudden losses. In other words, the Wald criterion minimizes risk and allows you to make the safest decisions. This approach makes it possible to get a guaranteed minimum, although the actual result may not be so bad.

Wald criterion: example of use

Suppose a certain enterprise is going to produce new types of goods. In this case, you should make a choice between one of four options B 1, B 2, B 3, B 4, each of which involves a certain type of release or a combination of them. The decision will ultimately determine how much profit the company will make. It is unknown exactly how market conditions will develop in the future, but analysts predict three main scenarios for the development of events: C 1, C 2, C 3. The data obtained allows you to create a table possible options winnings that correspond to each pair possible solution and probable situation.

Types of products	Market Scenarios			Worst result

Using the Wald criterion, you should choose the one that will be the most optimal for the enterprise in question. In our case, the efficiency indicator

E = max (25;22;15;20) = 25.

We obtained it by selecting the minimum result for each of the options and identifying among them the one that will bring the greatest income. This means that solution B 1 will be the most optimal for the company, according to this criterion. Even in the most unfavorable circumstances, a result of 25 (C 1) will be obtained, while at the same time it is possible that it will reach 45 (C 3).

Let us note once again that the Wald criterion guides a person towards the most cautious line of behavior. In other circumstances, it is quite possible to be guided by other considerations. For example, option B 3 could bring a win of 90 with a guaranteed result of 15. However, this case is beyond the scope of this article, and therefore we will not consider it for now.

In a situation of uncertainty, it is impossible to determine the likelihood of the occurrence of certain consequences of decisions made. Therefore, the criterion of mathematical expectation, which is widely used in risk situations, and which necessarily requires the mentioned probabilities, is not applicable here. Instead, other criteria are used.

To select the optimal strategy in a situation of uncertainty, there are two main criteria: maximin and minimax. Maximin is also called the pessimist criterion or Wald criterion, and minimax - Savage criterion.

The Wald (Wald) criterion is maximin. This criterion is based on the principle of greatest caution - the criterion of extreme pessimism, which is based on the choice of "from the worst - the best." In fact, this is the minimax criterion - the main one in game theory. According to this criterion, nature (environment) behaves like an intelligent aggressive adversary, doing everything to prevent us from achieving success. The optimal strategy is considered to be the one that guarantees the greatest (max) payoff out of all the worst (min) possible outcomes of the action for each strategy - the security level:

The strategy chosen in this way, optimal according to the Wald criterion, is called maximin, and the value of W is called maximin.

Savage criterion – minimax risk. The criterion presupposes the preliminary compilation of a so-called matrix of “risks” (losses, regrets). In the theory of statistical decisions, the risk rij when using strategy Qi under conditions Gj is the difference between the payoff that could be obtained if the conditions Gj were known and the payoff that would be obtained without knowing them and choosing the strategy Qi:

Minimax is focused not so much on minimizing losses as on minimizing regrets about lost profits. He allows reasonable risk in order to obtain additional profit. This criterion can be used to choose a behavior strategy in a situation of uncertainty only when there is confidence that an accidental loss will not lead the company to complete collapse

Works criterion

High-yielding stocks are rarely reliable enough, and the most reliable are high-yielding. Therefore, when buying shares, you always have to choose between their profitability and their reliability and somehow link them together. This problem always arises whenever investment project. If the probabilities of maintaining or losing an investment are known, then this problem is solved using the mathematical expectation criterion. If they are not there, you have to turn to other criteria. One of these criteria is the criterion of works. It allows you to choose a project that would be the most profitable and at the same time the least risky. The product criterion is calculated using the formula:

The product criterion is capable of working with a minimum of information.

Example of problem solution

Suppose that in conditions of fluctuations in demand G j = (3000, 6000, 9000, 12000), a trading enterprise has three strategies for selling any product: Q p (1) = 6000 pcs; Q p (2) = 9000 pcs; Q p (3) = 12000 pcs. at selling price C p = 70 rub. at the purchase price C p = 30 rub. and average costs I = 10 rubles/piece.

In accordance with the resource capabilities of a trading enterprise, we will calculate the options for average annual profit using formula (1), and summarize the results in Table 3.

Table 3 - Gain (profit) matrix of commercial strategies under uncertain market conditions

1. Wald criterion. To determine the optimal strategy based on the criterion of greatest caution, let us add to the table. 3, column 6 on the right, we indicate for each row the minimum profit and select the strategy for which the minimum of the row is maximum (see Table 4).

Table 4 - Summary profit matrix

2. Hurwitz criterion. Let the pessimism index λ be defined as λ = 0.4.

To calculate the values ​​of strategies according to the criterion of weighted (reasonable) caution in additional column 7 of Table. 4 we'll find maximum values for each line. Then:

The maximum value corresponds to two purchasing strategies Q p (1) and Q p (2).

3. Laplace criterion. Based on the principle of equal probability of states of nature, we will find the average values ​​of “payoffs” – profits for each strategy:

According to the Laplace gain averaging criterion, the best purchasing strategy is Q p (2)

4. Bayes-Laplace criterion. To determine the optimal strategy based on the criterion of weighted average evaluation of winnings, it is necessary to know the probability distribution of demand. Let such probabilities be determined from past experience or by expert analysis (bottom line in Table 4).

Then the assessments by criterion for each strategy will be:

The maximum value corresponds to the strategy Q p (2).

Table 5 - Risk matrix of commercial strategies

5. Savage criterion. Let's move from the winning matrix to the risk matrix (Table 5). To do this, we first indicate in an additional row of the table the maximum possible gains for each state of nature (penultimate row) and then calculate the corresponding risks r i j = max P i j – P i j

to fill out the risk matrix (Table 5). Based on the principle of greatest caution, we find the maximum risk values ​​by row and from them we select the strategies Q p (1) and Q p (2) with the minimum value of the maximum possible risk. Let's transfer the obtained values ​​to the table. 4 to summarize the selection.

So, the strategies Q p (1) and Q p (2) turned out to be competing (the choice of strategy Q p (3) according to the Hurwitz criterion was most likely caused by excessive optimism when choosing the λ indicator). Strategy Q p (1) was selected according to the Wald, Laplace and Savage criteria, strategy Q p (2) - according to the Laplace, Bayes-Laplace and Savage criteria.

The preference for one or another strategy is chosen as the best according to most criteria. But in our case, the two strategies Q p (1) and Q p (2) are equivalent in this sense.

Problem with options:

Table 6 - Profit matrix of commercial strategies under uncertain market conditions

Option 1

Option 2

Option 3

Option 4

Option 5

Option 6

Option 7

Option 8

Option 9