Three signs of a good strategy in the face of uncertainty. Strategy under uncertainty

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

Decision-making under 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 task of making a decision under uncertainty is presented in the form of an efficiency table (Table 1).

Table 1.

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

P m - possible strategies, lines of behavior of the solution.

For each pair of strategy and environment, there are payoffs -A mn .

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

The presented task is aimed at making decisions in the development of enterprise development plans, the development of production programs, plans for the release of new types of products, the direction of innovation, the choice of insurance strategies, investments, funds, etc.

In the theory of statistical decisions, a special indicator of risk is used, which shows the profitability of the adopted strategy in a given situation, taking into account its uncertainty. The risk is calculated as the difference between the expected result of actions in the presence of accurate data of the situation and the result that can be achieved if these data are uncertain. Based on this difference, a table of risks for the release of 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. Choice 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) and the corresponding values ​​of the payoffs A (efficiency table 6) for each solution.

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

3. Choosing the optimal solution according to the principles of the approach to evaluating the result of actions.

Under 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, no less than the largest possible in the worst conditions;

b) min max risk in any conditions. The optimal solution is the one for which the risk, max under different scenarios, seems to be minimal.

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

where is the minimum gain corresponding to the solution m;

The maximum payoff corresponding to the solution m;

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

Graphically meaning k in relation to the line of conduct 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

A task:

There are 3 investment options available:

1) Invest all available funds in the shares of Neft-AG, which guarantees a high income under the right conditions;

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

3) Invest part of the funds in Neft-AG shares, part in GKOs - i.e. to diversify the portfolio of funds.

The prospect is indicated by three options for the situation (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 - variant of the situation;

O 1 - the Neft-AG company - went bankrupt, GKO - brings a stable income.

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

O 3 - crisis in the economy.

Let's determine the optimal solution, in which the gain in any conditions will be no less than the largest possible one in the worst conditions (max-min).

From Table. 2 for solution P 1 the smallest gain will be 0, for P 2 - 0.3, for P 3 - 0.25.

The largest possible gain under the worst set of circumstances is 0.3, which corresponds to the decision P 2 , i.e. under any scenarios, 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 preliminarily 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 decision P 3 with a minimum risk level of 0.29 is taken as the optimal one.

Let's calculate the criterion of pessimism - Hurwitz's optimism for various solutions depending on the value of the accepted coefficient k.

For solution P 1

Solution:

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

Table3.

Under the condition of equiprobability of situations, their probabilities are equal and are:

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

Mathematically, the expectation of winnings under the condition of equiprobability of situations is determined from the expression:

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

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

A ij is the payoff corresponding to the i-th decision under the j-th situation.

W 1 \u003d 0.33 * 0 + 0.33 * 0.99 + 0.33 * 0.1 \u003d 0.3597

W 2 \u003d 0.33 * 0.5 + 0.33 * 0.5 + 0.33 * 0.3 \u003d 0.329

W 3 \u003d 0.33 * 0.25 + 0.33 * 0.7 + 0.33 * 0.4 \u003d 0.445

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

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

Choosing a solution by the Hurwitz criterion:

for solution P 1: G 1 =0.495;

for solution P 2: G 2 =0.5*0.3+(1-0.5)*0.5=0.4;

for solution P 3: G 3 \u003d 0.5 * 0.25 + (1-0.5) * 0.7 \u003d 0.475.

When k=0.5, the decision P 1 is taken as the optimal one.

G i values ​​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 chosen k i takes the optimal decision with the maximum value G i . When k i =0.75 - G max =0.362. The decision P 1 or P 3 is taken as the optimal one.

See P.N. Brusov, p. 3.8., A.N. Garmash, p. 3.3.2.

Uncertainty will be considered as such a state of knowledge of a decision maker (DM), in which one or more alternative decisions lead to a block possible outcomes, corresponding to various states of the external environment ("nature"), the probabilities of which are unknown. This is usually because there is no reliable data from which probabilities can be calculated a posteriori, and also because there is no way to derive probabilities a priori. Under these conditions, elements of game theory, in particular, games with nature, can be used to determine the best, so-called rational, solutions. In them, one player (man) tries to act prudently, and the second player (nature) acts randomly.

Games with nature- these are games in which uncertainty is caused not by the conscious opposition of the opponent, but by insufficient awareness of the conditions in which the parties operate. For example, the weather in a certain region or the consumer demand for certain products is not known in advance.

The conditions for such a game are usually presented decision table, in which rows A 1 , A 2 , ..., A m correspond to the strategies of the decision maker (decision maker), and columns B 1 , B 2 , ... B n - strategies of nature; and ij is the decision maker's payoff corresponding to each pair of strategies А i , В j .

In the situation under consideration, when choosing from the set ( a 1 , a 2 ,..., a m ) best solution usually use the following criteria.

1. Wald criterion. Based on the principle pessimism(most careful). When choosing a solution, one must rely on the worst case scenario on the part of nature. It is recommended to use the maximin strategy. She is selected from the condition

and coincides with the lower price of the game.

2. Maximum criterion. It is selected from the condition

The maximum criterion is optimistic: it is believed that nature will be the most favorable for man.

where - the degree of optimism (an indicator of pessimism-optimism) - varies in the range .

The Hurwitz criterion adheres to some intermediate position, taking into account the possibility of both the worst and the best behavior of nature. At = 1, the criterion turns into the Wald criterion, at = 0, into the maximum criterion. It is influenced by the degree of responsibility of the person making the decision on the choice of strategy. The greater the consequences of erroneous decisions, the greater the desire to insure, the closer to one.

4. Savage's criterion. The essence of the criterion is to choose such a strategy in order to prevent excessively high losses to which it can lead. Located risk matrix, the elements of which show what kind of loss a person (firm) will suffer if for each state of nature he does not choose the best strategy:

R=

The elements of the risk matrix are found by the formula

,

where is the maximum element in the column of the original matrix.

When making decisions under conditions of uncertainty, different options should be evaluated in terms of several criteria. If the recommendations match, you can choose the best solution with more confidence; if recommendations contradict each other, the final decision should be made taking into account the results of additional studies.

Example. As the planting season approaches, the farmer has four alternatives: A 1 - grow corn, A 2 - wheat, A 3 - vegetables, or A 4 - use the land for pasture. Payments associated with these opportunities depend on the amount of precipitation, which can be conditionally divided into four categories: B 1 - heavy precipitation, B 2 - moderate, B 3 - slight, B 4 - dry season.

The payoff matrix is ​​estimated as follows:

Which managerial decision should the farmer take?

Solution.

The land should be used for pasture.

2. Maximum Criteria:

Max(80,90,150,35)=150.

This is in line with strategy A 3 – grow vegetables.

2. Let's use Savage's criterion. Let's make a risk matrix, the elements of which are found by the formula

The optimal strategy is determined by the expression

Wheat should be sown according to this criterion.

3. Let's use Hurwitz criterion. The optimal strategy is determined by the formula

Assume that the degree of optimism Then

those. decide to grow vegetables.

4. The rule for maximizing the average expected return. Assuming what is known probability distribution for different states of nature, for example, these states are equally probable (Laplace's rule of equal opportunity) then to make a decision, one should find the mathematical expectations of the payoff:

Since M 2 has the maximum value, wheat should be sown.

Conclusion: two criteria simultaneously recommend the choice of management strategy A 2 (sow wheat), two criteria recommend strategy A 3 (grow vegetables).

The table shows that the optimal behavior largely depends on the accepted criterion for choosing the best solution, so the choice of criterion is the least simple and most responsible issue in game theory.

Decision making under conditions of partial uncertainty (see P.N. Brusov, p. 3.9).

Pareto optimal financial transaction. Consider the consequences matrix , i=1,2,…,m, j=1,2,…,n. Alternative dominates Pareto alternative if , j=1,2,…,n, and at least for one index j this inequality is strict. The dominated alternative cannot be the optimal solution, because it is by all measures not “better” than the dominant alternative. The alternative is called Pareto optimal(or Pareto optimal) if it is not diminated by any other alternative.

All Pareto optimal solutions form Pareto optimality set.

Example. For the consequence matrix, find a set of Pareto optimal alternatives.

In the table - possible alternatives (strategies) of the decision maker, - one of the states of an uncertain real situation.

Solution.

The strategy dominates the strategies , and . Therefore, we exclude the 4th, 5th and 6th rows of the matrix.

There are no more dominated strategies. We get the Pareto optimality set consisting of three alternatives: , , .

The choice of the optimal strategy under risk and uncertainty involves consideration of various optimality criteria, developed in the framework of the so-called "playing with nature". This model assumes the conscious action of only one participant - the so-called "player", which in the investment analysis is an investor, within the limits of his uncontrolled objective reality. At the same time, the term "nature" describes a set of objective factors that change regardless of the desire of the player-investor, but have a decisive influence on his investment decision. In investment analysis, this is the state of the investment market.

The investor has a predictive estimate of possible combinations of these factors (investment market conditions (P.)), which occur randomly, regardless of his actions. In some cases, forecasts may contain an estimate of the probabilities of occurrence of these states (p), the sum of which for all possible options for the development of the investment situation is equal to 1.

The investor develops options for possible investment strategies (A) and evaluates the possible return on investment for each strategy and for each variant of the state of the investment market

Based on this information, the so-called payoff matrix can be formed (table 11.1).

Table 11.1

Payoff Matrix

The difference between the maximum payoff of a player in a given state of nature (max (u])) and the payoff of a certain strategy of the player's behavior that can be implemented in this state of nature is called the risk of strategy A. in a state of nature P:

mu = maxC ^) _ ar]. (11.1)

Thus, the risk is part of the largest investment income in a given state of the investment market, the investor does not receive in case of using an imperfect investment strategy.

For risks, you can build a risk matrix similar in form to the payoff matrix.

The investor is faced with the task of choosing the optimal one among the many possible investment strategies.

To select the optimal investment strategy in a situation of uncertainty (when the probabilities are not known), the following criteria are used:

Maximax criterion - a criterion of extreme optimism, according to which an investment strategy is chosen that provides the maximum gain (income) among all the maximum gains allocated for each of the possible states of the investment market;

Wald's criterion - the so-called "pessimist's criterion", according to which it is assumed that any decision should be expected to have the worst consequences, and, therefore, it is necessary to find such an option in which the worst result will be relatively better than others bad results. That is, the worst result is found for each state of the investment market, and then the investment strategy with the best result among them is selected from them;

The Savage criterion is a minimax risk criterion, similar to the Wald criterion, but it provides for the choice analysis according to the risk matrix data;

The Hurwitz criterion is a maximin-maximum criterion, according to which, when choosing an investment strategy, it recommends choosing an alternative with the maximum average result (in this case, there is an unspoken assumption about the same probability of occurrence for all possible states of the investment market).

The following criteria are used to select the optimal strategy under risk conditions:

Mathematical expectation criterion - provides for the selection of an investment strategy for which the average probability-weighted gain (expectation of the gain, M) is maximum:

mg = Xa, o Pj-> max; (11.2)

The Laplace criterion is a criterion for maximizing the weighted average of the optimality of the strategy, according to which, with approximately the same probability of occurrence of events, the optimal strategy is the one for which the total gain over all possible states of the investment environment is maximum. It is this criterion that underlies the comparative evaluation of the effectiveness of projects according to the criterion of net present value.

The final choice of the optimal investment strategy is carried out on the basis of a generalization of the results of the evaluation according to the above criteria. At the same time, it is advisable to accept for implementation a strategy that is optimal according to most criteria.

Choice of Alternatives Under Uncertainty

The choice of the best solution under conditions of uncertainty essentially depends on its degree, i.e. what information the decision maker has. The choice of alternatives under conditions of uncertainty, when the probabilities of their possible options are unknown, but there are principles for the approach to assessing the results of actions, ensures the use of various criteria.

Given the dependence on this, the consequences of decisions can be assessed through a system of criteria that provide for a different degree of risk.

1. Wald's maximin criterion (criterion of extreme pessimism) - "count on the worst". In accordance with it, if a guarantee is required that the payoff in any conditions is not less than the largest possible under the worst conditions, then the optimal solution will be the one for which the payoff will be the maximum of all the minimum under various conditions.

This criterion orients the decision maker to the worst conditions and recommends choosing the strategy for which the payoff is maximum. In other, more favorable conditions, the use of this criterion leads to a loss in the efficiency of the system or operation.

2. Savage's Minimax Criterion (minimization of high risk) - "count on the best." When used, it provides smallest value the maximum amount of risk. The Savage criterion, like the Wald criterion, is a criterion of extreme pessimism, but pessimism is manifested in the fact that the maximum loss in gain is minimized compared to what could be achieved under given conditions.

3. Laplace or Bayes criterion - Focus on the average.

According to this criterion, if the probability of the state of the environment is unknown, the variants of the conditions must be taken as equal. In this case, the alternative is chosen, characterized by the most estimated cost, subject to equal probabilities. The Laplace criterion allows the uncertainty condition to be reduced to risk conditions. It is called the criterion of rationality, it is suitable for strategic long-term decisions, like the criteria described above.

4. Criterion of extreme optimism - "believe in luck."

The maximum criterion assumes that the state of the environment will be the most prosperous, in this regard, it is extremely important to choose a solution that provides the maximum gain among the maximum possible.

5 . Criterion of pessimism - Hurwitz's optimism - "compromise".

According to this criterion, when choosing a solution under conditions of uncertainty, one should not be guided by either extreme pessimism (always count on the worst) or optimism (everything will be the best). Some middle solution is recommended. That is, it is extremely important to choose between two lines of behavior. The optimal solution will be the one for which the indicator G will be the maximum. This criterion has the form:

G =max[ h min a0+ (1 -h)max aij], (6)

where h- a coefficient selected by an expert from the interval between 0 and 1. The use of this coefficient introduces additional subjectivity in decision-making.

6. Expectation criterion designed to select the optimal strategy of behavior, ᴛ.ᴇ. to make a series of decisions:

7. Generalized Hurwitz criterion.

Let us consider in more detail the ways of choosing solutions in the financial and economic area at risk,ᴛ.ᴇ. under environmental conditions. A mathematical model of situations of this type is usually called a game with the external environment (nature). The game is played by two players - the decision maker and nature. At the same time, the player acts consciously, trying to choose the most satisfactory solution for himself, while nature randomly manifests its states objectively, without consciously opposing the player, without taking into account the player’s possible choice of his strategies and is absolutely indifferent to the result of the game. The following is a risk matrix.

A risk situation is understood to mean when it is possible to indicate not only possible consequences(payoff) of each alternative, but also the probabilities of their occurrence. The main criterion here is the mathematical expectation. The rest are of subordinate importance.

If none of the states of the "environment" can be called more probable than others, ᴛ.ᴇ. if all of them are approximately equally probable, then the decision can be made using the Laplace criterion. In this case, the optimal solution should be considered the one that corresponds to the largest amount of payments.

When two different criteria prescribe to make the same decision, this is considered an additional confirmation of its optimality. If they point to different solutions, then preference in a risk situation should be given to one of them, ĸᴏᴛᴏᴩᴏᴇ indicates the criterion of mathematical expectation. It is he who is the main one for this situation.

Additional information may help you make a better choice. The question arises, what is the maximum price you can pay for it in order to benefit from it. Decision theory to answer this question proposes to find the mathematical expectation of the payout corresponding to ideal information, and then compare it with the mathematical expectation, ĸᴏᴛᴏᴩᴏᴇ can be obtained with ordinary information. The difference between them is proposed to be considered the upper limit of the price of any information.

Projects should provide for specific mechanisms

stabilization, ensuring the protection of the interests of participants in the event of an unfavorable change in the conditions for the implementation of the project (even if the goals of the project are not fully achieved or not achieved at all) and preventing possible actions of the participants that jeopardize its successful implementation. It is possible to reduce the degree of risk or redistribute it among the participants.

1. GENERAL METHODOLOGY FOR FORMING CRITERIA

The essence of the proposed methodology for the formation of criteria is to implement the following points.

1) From the payoffs aij, i=1,…,m; j=1,…,n, player A, we compose the matrix A, assuming that it satisfies the above conditions: m³2, n³2 and it does not contain dominated (in particular, duplicated) rows.

The payoffs aij of player A, presented in the form of a matrix A, provide an opportunity for a better overview of the results of choosing strategies Аi, i=1,…,m, by player A for each state of nature Пj, j=1,…,n.

2) We fix the distribution of the probabilities qj=p(Пj), j=1,…,n, of the states of nature Пj, j=1,…n, satisfying the condition (1), of course, if they are known. Thus, point 2 is involved in the method of forming a criterion in case of making a decision under risk.

3) Based on points 1 and 2, we choose natural number l, 1£l£n, and in a certain way we construct the matrix

Let's call them the coefficients of the criterion being formed. They are intended to play the role of quantitative assessments of some subjective manifestations of player A (decision maker), namely the degree of confidence in the probability distribution of the states of nature and the degree of his pessimism (optimism) when making decisions.

5) Using the matrix B and the coefficients l1,…, ll, each strategy Аi, i=1,…,m, of player A, we assign the number

7) Let's define the optimal strategy.

An optimal strategy is a strategy Ak with the maximum efficiency indicator, in other words, a strategy whose efficiency indicator Gk coincides with the price of the game G:

It is clear that such a definition of the optimal strategy does not imply its uniqueness.

Note that, according to the logic of this paragraph, player A, choosing the optimal strategy, maximizes the index Gi (see (5)). This circumstance justifies the fact that we called this indicator (in paragraph 5) an indicator of efficiency.

2. FORMATION OF SOME KNOWN CRITERIA - SPECIAL CASES OF THE GENERAL METHOD

Bayes criterion (, , , ).

1) Let A be the payoff matrix of player A.

2) Known probabilities qj=p(Пj), j=1,…,n, states of nature Пj, j=1,…,n, satisfying condition (1). Therefore, we are talking about decision-making under conditions of risk.

3) We assume l=n and choose matrix B equal to matrix A, i.e.

bij=aij for all i=1,…,m and j=1,…,n.

4) The coefficients l1,…,ln, are chosen equal to the corresponding probabilities q1,…,qn, i.e. ll=qi, i=1,…,n. By this, player A expresses full confidence in the truth of the probability distribution q1,…,qn, states of nature.

From (1) it follows that the coefficients lj, j=1,…,n satisfy condition (3).

5) The indicator of the effectiveness of the strategy Аi according to the Bayes criterion will be denoted by Вi and we find it according to the formula (3):

Obviously, Вi is the weighted average payoff for the strategy Аi with weights q1,…,qn.

If the strategy Аi is interpreted as a discrete random variable that takes the values ​​of the payoffs for each state of nature, then the probabilities of these payoffs will be equal to the probabilities of the states of nature, and then Вi is the expectation of this random variable(see (6)).

6) The price of the game according to the Bayes criterion, denoted by us as B, is determined by the formula (4):

7) Optimal among pure strategies according to the Bayes criterion is the strategy Ak, for which the efficiency indicator is maximum:

Laplace criterion (, , , ).

2) Based on theoretical or practical considerations, it is stated that none of the possible states of nature Пj, j=1,…,n, can be given preference. Therefore, all states of nature are considered equally probable, i.e. qj=n-1, j=1,…,n. This principle is called Laplace's principle of "insufficient reason". Probabilities qj=n-1, j=1,…,n, satisfy condition (1).

Since the probabilities of the states of nature are known: qj=n-1, j=1,…,n, then we are in a decision-making situation under risk.

3) Let l=n, and as a matrix B, you can take the matrix obtained from the matrix A, if each row of the latter is replaced by an arbitrary permutation of its elements. In particular, we can put B=A. In the general case, the elements of the matrix B have the form bij=aikj(i), i=1,…, m; j=1,…,n, where aik1(i), aik2(i),…,aikn(i) is some permutation of elements ai1, ai2,…,ain i-th line matrices a.

4) Let the coefficients lj=n-1, j=1,…,n. Obviously, they satisfy condition (2).

The choice of the coefficients lj, j=1,…,n, thus confirms the full confidence of player A in the Laplace principle of insufficient reason.

5) According to the formula (3), the indicator of the effectiveness of the strategy Аi according to the Laplace criterion, denoted by us as Li, is equal to:

7) The optimal strategy Ak according to the Laplace criterion is the strategy with the maximum efficiency indicator:

Note that, as follows from (7) and (8), the efficiency indicator Li will be maximum if and only if the sum is maximum, and therefore the number can be considered as an indicator of the effectiveness of the strategy Аi, and the number as the price of the game.

Then the optimal strategy is the one with the maximum payoff.

Wald criterion ( - ).

1) Assume that A is the payoff matrix of player A.

2) The probabilities of the states of nature are unknown and there is no way to obtain any statistical information about them. Therefore, player A is in a decision-making situation under conditions of uncertainty.

3) Let l=1 and

4) Let the coefficient l1=1. Obviously, condition (2) is satisfied.

5) Let's designate the efficiency indicator of the strategy Аi according to the Wald criterion as Wi. By virtue of (9) and the value of the coefficient l1=1, according to formula (3) we have:

Thus, the indicator of the effectiveness of the strategy Аi according to the Wald criterion is the minimum payoff of player A when he applies this strategy.

6) The price of the game according to the Wald criterion, denoted by W, is found by formula (4):

7) The optimal strategy among pure strategies according to the Wald criterion is the strategy Ak with the maximum efficiency indicator:

In other words, according to the Wald criterion, the optimal pure strategy among pure strategies is the pure strategy for which the minimum payoff is the maximum among the minimum payoffs of all pure strategies. Thus, the optimal strategy according to the Wald criterion guarantees a payoff no less than the maximin for any state of nature:

By virtue of (10), the Wald criterion is the criterion of extreme pessimism of player A, and the quantitative expression of this extreme pessimism is the value of the coefficient l1, equal to 1. Player A, when making a decision, acts according to the principle of the greatest caution.

Although the Arabic proverb says: “He who is afraid of his own shadow, there is no place for him under the sun,” nevertheless, this criterion is relevant in cases where player A does not so much want to win as he does not want to lose. The use of the Wald principle in everyday life is confirmed by such sayings as “Measure seven times - cut once”, “God protects the safe”, “Better a titmouse in the hands than a crane in the sky”.

Hodge-Lehmann criterion.

1) Assume that player A's payoff matrix is ​​matrix A.

2) Known probabilities qi=p(Пj), j=1,…,n, states of nature Пj, j=1,…,n, satisfying condition (1).

Thus, player A must make a decision under risk.

3) Let l=2,

· indicator of the effectiveness of the strategy Аi according to the Bayes criterion.

Matrix B will take the form

Obviously, these coefficients satisfy condition (2).

5) According to formula (3), taking into account (11), (12), and (13), the strategy efficiency indicator Аi according to the Hodge-Lehman criterion is equal to:

On the right side of formula (14), the coefficient lО is a quantitative indicator of the degree of trust of player A to the given probability distribution qi=p(Пj), j=1,…,n, states of nature Пj, j=1,…,n, and the coefficient (1 -l) quantitatively characterizes the degree of pessimism of player A. The more confidence player A has in the given probability distribution of states of nature, the less pessimism and vice versa.

6) The price of the game according to the Hodge-Lehman criterion is found by the formula (4):

7) The optimal strategy according to the Hodge-Lehman criterion is the strategy Ak with the highest efficiency indicator:

Note that the Hodge-Lehman criterion is, as it were, an intermediate criterion between the Bayes and Wald criteria. When l=1, from (14) we have: Gi=Bi and therefore the Hodge-Lehman criterion turns into the Bayesian criterion. And when l=0, from (14): Gi=Wi and, therefore, from the Hodge-Lehman criterion, we obtain the Wald criterion.

Germeier's criterion.

1) Let matrix A be the payoff matrix of player A.

2) Probabilities qi=p(Пj), j=1,…,n, of states of nature Пj, j=1,…,n, satisfying condition (1) are given.

That. Player A is in a decision-making situation at risk

size m x 1.

4) We set l1=1. Condition (2) is obviously satisfied.

5) The indicator of the effectiveness of the strategy Аi according to the Germeier criterion is determined by the formula (3) taking into account (15) and the fact that l1=1:

If player A adheres to the strategy Аi, then the probability of winning aij with this strategy and under the state of nature Пj is obviously equal to the probability qj of this state of nature. Therefore, formula (16) shows that the indicator of the effectiveness of the strategy Аi according to the Germeier criterion is the minimum gain for this strategy, taking into account its probability.

6) The price of the game according to the Germeier criterion is determined by the formula (4):

7) The optimal strategy according to the Germeier criterion is the strategy Ak with the highest efficiency indicator:

Note that the Germeier criterion can be interpreted as the Wald criterion applicable to the game with the matrix

The Germeier criterion, like the Wald criterion, is a criterion for the extreme pessimism of player A, but, unlike the Wald criterion, player A, making a decision with maximum discretion, takes into account the probabilities of states of nature.

In the case of a uniform distribution of the probabilities of the states of nature: qj=n-1, j=1,…,n, the strategy efficiency indicator Аi, due to formula (16), will be equal to Gi=n-1aij and, therefore, the Germeier criterion is equivalent to the Wald criterion , i.e. a strategy that is optimal according to the Germeier criterion is also optimal according to the Wald criterion, and vice versa.

Criteria of works.

1) Let the payoff matrix of player A be the matrix A, all elements of which are positive:

aij>0, i=1,…,m; j=1,…,n.

2) The probabilities qj=p(Пj), j=1,…,n, of the states of nature Пj, j=1,…,n, are known and satisfy condition (1).

3) Let l=1 and

size m x 1.

4) Let l1=1. Condition (2) is satisfied.

5) The indicator of the effectiveness of the strategy Аi according to the criterion of products in accordance with formulas (3) and (17) is equal to

.

6) The price of the game according to the criterion of works is calculated by the formula (4):

7) The optimal strategy according to the product criterion is the strategy Аk with the highest efficiency indicator:

Note that for the criterion of products, it is essential that all states of probabilities of states of nature and all payoffs of player A be positive.

Maxmax criterion (.-).

2) The probability of the states is unknown. The decision is made under conditions of uncertainty.

3) Let l=1 and

size m x 1.

4) Coefficient l1 is chosen equal to 1: l1=1. In this case, condition (2) is obviously satisfied.

5) The indicator of the effectiveness of the strategy Аi according to the maximum-maximum criterion will be denoted by Мi and determined by formula (3) taking into account (18) and the fact that l1=1:

Thus, the indicator of the effectiveness of the strategy Аi according to the maximmax criterion is the greatest gain for this strategy.

6) The price of the game according to the maximum criterion, denoted by us as M, is determined by the formula (4):

Obviously, this is the largest element of the matrix A.

7) The optimal strategy according to the maximax criterion is the strategy Ak with the highest efficiency indicator:

From formula (19) we conclude that the maxmax criterion is the criterion of extreme optimism of player A. Quantitatively, this is expressed by the fact that l1=1. This criterion is opposite to the Wald criterion. Player A, using the maximum-maximum criterion, assumes that the nature of P will be in the most favorable state for him, and, as a result, he behaves very frivolously, with a "hat-captive" mood, because he is sure of the greatest gain. However, in some cases, this criterion is used consciously, for example, when player A is faced with a dilemma: either get the biggest win or go bankrupt. The everyday reflection of such situations is illustrated by the sayings: "Pan or lost", "Who does not risk, he does not win", etc.

The optimal strategy by the maximum criterion guarantees player A the possibility of winning equal to the maximum maximum.

.

Hurwitz pessimism-optimism criterion with optimism indicator lО ( – ).

1) Let A be the payoff matrix of player A.

2) The probabilities of the states of nature are unknown and there is no way to obtain any reliable statistical information about them.

Thus, the decision to choose the optimal strategy will be made under conditions of uncertainty.

3) Let l=2. Matrix B elements

4) The coefficients l1 and l2 are chosen as follows:

In formula (22) l is an indicator of optimism, and (1-l) is an indicator of player A's pessimism when choosing the optimal strategy. The closer the optimism indicator is to one, the closer the pessimism indicator is to zero, and the more optimism and less pessimism. And vice versa. If l=0.5, then 1-l=0.5, i.e. indicators of optimism and pessimism are the same. This means that player A behaves neutrally when choosing a strategy.

Thus, the number l is chosen in the range from 0 to 1, depending on the propensity of player A to be optimistic or pessimistic.

6) The price of the game according to the Hurwitz criterion N is determined from the formula (5):

7) The optimal strategy Ak according to the Hurwitz criterion corresponds to the efficiency indicator

The Hurwitz criterion is intermediate between the Wald criterion and the maximum-maximum criterion and turns into the Wald criterion at l=0 and - into the maximum-maximum criterion at l=1.

Generalized Hurwitz test with coefficients l1,…, ln (, ).

1) Let A be the payoff matrix of player A.

2) The probabilities of the states of nature are unknown. So the decision is made under conditions of uncertainty.

3) Matrix B is obtained from matrix A by permuting the elements of each of its rows in non-decreasing order:

bi1£bi2£…£bin, i=1,…,m.

Thus, the 1st column of the matrix B contains the minimum, and the nth column contains the maximum payoffs of the strategies. In other words, in the 1st column of matrix B there are indicators of the effectiveness of strategies according to the Wald criterion, and in the nth column - indicators of the effectiveness of strategies according to the maxmax criterion.

4) The coefficients l1,…, ln are chosen to satisfy conditions (2) according to the different degree of player A's inclination to optimism. In this case, the indicator of player A's pessimism is the number

where is the integer part of the number , and the indicator of the optimism of player A is the number

Obviously, lр+l0=1.

5) The indicator of the effectiveness of the strategy Аi according to the generalized Hurwitz criterion is determined by the formula (3):

6) The value of the game according to the generalized Hurwitz criterion is determined by the formula (4):

7) Optimal strategies are found in the standard way: Аk is the optimal strategy if Gk=G.

Note that the generalized Hurwitz criterion takes into account all payoffs for each strategy, which is necessary for a more complete picture of the effectiveness of strategies. We also note that some of the above criteria are special cases of the generalized Hurwitz criterion.

Note that if B=A, then the coefficients lj, j=1,…,n, can be formally interpreted as the probabilities of the states of nature, and in this case, the generalized Hurwitz criterion coincides with the Bayes criterion.

If lj=n-1, j=1,…,n, then the generalized Hurwitz criterion turns into the Laplace criterion.

If l1=1, l2=…=ln=0, then the generalized Hurwitz criterion is the Wald criterion.

When l1=…=ln-1=0, ln=1, from the generalized Hurwitz criterion we obtain a maximax criterion.

If l1=1-l, l2=…=ln-1=0, ln=l, where lн, then the generalized Hurwitz criterion is the Hurwitz criterion.

If В=А and qi=p(Пj), j=1,…,n – probabilities of states of nature that satisfy conditions (1), then choosing the coefficients lj, j=1,…,n, as follows: l1=1- l+lq1, lj=lqj, j=2,…,n, where lн, we obtain the Hodge Lehman criterion from the generalized Hurwitz criterion.

3. PROBLEM UNDER COMPLETE UNCERTAINTY

Suppose an investor decides to build a certain type of housing in a certain location. The investor operates in conditions of uncertainty (information opacity) in the housing market. To form an idea of ​​the situation in the housing market at the time of completion of construction, he needs to take into account real estate prices, competition in the housing market, the ratio of supply and demand, exchange rates and much more. Statistics show that one of the main components of the cost of housing is its location.

Consider a mathematical model of this situation. We have a game with nature, where player A is an investor, nature P is a set of possible situations in the housing market at the time of construction completion, from which, for example, five states P1, P2, P3, P4, P5 of nature can be formed. The approximate probabilities of these states are known q1=p(П1)»0.30; q2=p(P2)»0.20; q3=p(P3)»0.15; q4=p(P4)»0.10; q5=p(P5)»0.25. Suppose that player A has four (pure) strategies A1, A2, A3, A4, representing the choice of a specific place to build housing. Many of these places are limited by urban planning decisions, the cost of land, etc. The investment attractiveness of the project is defined as the percentage of income growth in relation to the amount of capital investments, the assessment of which is known for each strategy and each state of nature. This data is presented in the following payoff matrix for Player A:

size 4 x 5, in the last, additional line of which the probabilities of the states of nature are indicated. Matrix (24) does not contain dominated (in particular, duplicated) rows and all its elements are positive.

The investor will have to choose a plot of land in such a way as to make the most efficient use of capital investments.

Calculate the performance indicators of strategies

By the Bayesian, Germeier and product criteria, provided that investor A trusts the given probability distribution of the states of nature,

according to the Laplace criterion, if investor A does not trust the given probability distribution of states of nature and cannot give preference to any of the considered states of nature,

· according to the Hodge-Lehman criterion with a confidence factor in the probabilities of the states of nature, for example, l=0.4,

· according to Wald's criterion, maximax criterion, Hurwitz's pessimism-optimism criterion with optimism indicator, for example, l=0.6, and according to generalized Hurwitz's criterion with coefficients, for example, l1=0.35; l2=0.24; l3=0.19; l4=0.13; l5=0.09.

The results of the calculation of performance indicators and optimal strategies are presented in the following table:

Table of performance indicators and optimal strategies

Note that since, in the Hodge-Lehman criterion, the indicator of player A's confidence in the probability distribution of the states indicated in the last row of matrix (24) is l=0.4, then the indicator of player A's pessimism is 1-l=0.6.

In the Hurwitz criterion, the indicator of optimism of player A is l=0.4 and, therefore, the indicator of his pessimism is also equal to 1-l=0.6.

In the generalized Hurwitz criterion according to formula (23), the pessimism indicator

= 0.35+0.24+0.5×0.19=0.685

and, consequently, the indicator of optimism l0=1-0.685=0.315.

Thus, in all the applied criteria, taking into account the individual manifestations of player A to pessimism and optimism, player A is more inclined to a pessimistic assessment of the situation than to an optimistic one, with approximately the same indicators.

As a result of the application of nine criteria, we see that the optimal strategy A1 is 3 times, strategy A3 - 6 times and strategy A4 - 1 time. Therefore, if investor A does not have any reasonable serious objections, then strategy A3 can be considered as optimal.