Using the methods of game theory
Let's consider the decision of a concrete problem of definition of cost of the enterprise and an optimum strategy of its management when at the moment of achievement of a stage of a maturity of a product before the enterprise there is a choice presented on fig. 6.15, using the game model with nature.
Fig. 6.15. Enterprise strategy selection
Sales Promotion allows you to extend the life cycle of an existing product. Possible measures to extend the life cycle: advertising, discounts, merchandising actions, etc.
Modification of an already existing product involves the creation of new models, in some way a new product, bypassing the launch phase.
Creating a new product usually requires a lot of capital and time. The optimal situation is when the new product has time to pass the development phase, while the old product is at the stage of maturity. Putting a new product on the market can accelerate the decline of an already existing one.
The choice of strategy depends on the situation prevailing in the market. In the medium and long term, these strategies are mutually exclusive, because the enterprise is not interested in that its products compete with each other.
Drawing up a game matrix
Because in this model, competitors do not act as an aggressive adversary, but only form a market situation, the following points will be adopted for further reasoning.
1. A competitor will be called the aggregate of producers of the goods produced by the company.
2. The choice of the competitor's strategy does not depend on the choice of the firm, this environment is nonaggressive.
3. The existing competitor's product satisfies the same demand as the goods of the company, its new product is comparable with the new product of the company.
We are dealing with a nonantagonistic game, i.e. in this case, the game model with nature is applicable.
Thus, it is necessary to choose one of three strategies in order to maximize the value of the enterprise with six possible states of nature (environment):
1) the stage of developing a competing product;
2) the stage of bringing to market and the growth of a competing product;
3) the maturity stage of the competing product;
4) the stage of decline of a competing product;
5) the stage of developing a new competing product;
6) the stage of bringing to market and the growth of a new competing product.
Thus, in a game with nature, the player A has three possible pure strategies A 1 , A 2, A3, and the nature of P may be in one of the n states of P1, ..., P6.
The game matrix A is the payoff matrix of the player A and has the following appearance.
A = 

Here a ij denotes the value of the company in each option. To fill these values, we will analyze the factors affecting the value of the enterprise as a result of the adopted strategic decision:
• The amount of necessary investment;
• risk of failure to implement the decision;
• risk of premature expulsion of products from the market;
• the length of the period before the profit is generated;
• Duration of profit.
1. Lifecycle extension strategy:
and 11  the company artificially extends the life cycle of the product through advertising and other marketing and merchandising actions. There is still no competing product on the market, but the market is, nevertheless, already full. The volume of necessary capital investments is relatively small, the risk of failure to implement the decision is low. The period between investment and profitmaking is small. Due to the lack of competitors, profits can be obtained for a very long period;
a12  a competing product appears on the market, its sales grow. Higher capital investments are required to convince consumers to make a choice not in favor of a competing product. The risk of low efficiency of the conducted campaign and early replacement of products from the market increases in case of success of a competing product. The time gap between investment and profit is small. Demand for the goods is small, and the duration of the extraction of profits is small;
a13  the competing product is in the same phase of the life cycle (maturity stage) as the goods of the firm. The risk of failure of measures to promote the goods, undertaken by the company, is growing. The risk of early replacement of goods from the market is above average. The time gap between investment and profit is small. Demand for the goods is low, and the duration of profit extraction is small;
a 14  the competing product is already in the decay stage, i.e. the market is already saturated with this product. The time gap between investment and profit is small. Demand for the goods is low, and the duration of profit extraction is small;
and 15  the competitor is developing a new product. The consumer can have this information and limit current consumption while waiting for a new product. The time gap between investment and profit is small. Demand for the goods is low, and the duration of profit extraction is small;
and 16  the competitor has launched a new product on the market, which gradually leads to the consumer's refusal of the goods of the firm. The time gap between investment and profit is small, but demand is low and profit is shortterm.
2. Modification strategy:
a 21  the modification of the product requires new developments, possibly new equipment or personnel retraining. In this regard, the required amount of investment and the period between investment and profit is greater than for the strategy of promoting the old commodity. Research and development may not lead to the creation of new modifications, or the created variations may not find support among consumers, in connection with this the risk of failure of the decision taken is more than average. The absence of a competing product reduces the risk of premature expulsion from the industry;
and 22  a competing product appears on the market. Since there are always consumers among consumers who are conservative, preferring already known goods to its modifications, a competitor will manage to get a part of the market segment. Demand for product modification will decrease slightly;
a 23  a competing product has reached the stage of maturity, it is well known in the market. Accordingly, the demand for product modification will be low. If the competitor adopts an active marketing strategy, the risk of premature expulsion of modifications from the market is high;
and 24  the decline of a competing product speaks of the extreme saturation of the market. It is likely that even a diversified product offered by the enterprise will soon be pushed out of the market by new products. Demand is low;
and 25  the new product is already in the development phase. In the event that this information is publicly available, the rapid ousting of the old product modifications from the market is inevitable;
а 26  there is a competitive new product on the market, whose sales are growing. The period of possible profit for the enterprise that produces the modifications of the old goods is small. Most likely, the modifications of the old goods will have to leave the market in the near future.
3. New Product Strategy:
a 31  the creation of a new product requires extensive scientific development, a drastic change of equipment and training of personnel. This strategy requires a lot of investment, the length of the period from the moment of making a decision to making a profit is great. There is a great risk of failure to implement the decision (scientific research and development may not lead to the creation of a new product). Demand for a new product is great, as the product is oriented to a market not occupied by competitors;
and 32  in the market as an alternative to the new there is an old commodity, which reduces demand. Since a new product may be more expensive than the old because of large investments, consumers who are sensitive to the price will prefer the outofdate products of competitors. The risk of early replacement of the product from the market is low;
and 33  the old competitor's product has reached maturity. The demand for a new product remains at the same level. The risk of premature expulsion from the market is growing due to the fact that competitors are forced to promote their products more actively or to seek replacement;
a 34  a competing product is in a state of decline, therefore, the competitor is forced to make a strategic decision and either modify the old product or create a new one. From its decision and choice depends the position of our product on the market. The achievement of the commodity by the stage of decline speaks of the saturation of the market;
and 35  a competing product is under development. Consequently, there is a high probability that a competing new product will appear on the market earlier, having won a large part of the market. Thus, this situation is characterized by low demand, short duration of profit extraction and high risks;
a 36  because there is a similar competitive product on the market, the demand for the goods offered by the enterprise is small. The duration of profit extraction is declining, as the market will be saturated faster.
Generalize the conclusions drawn in the summary table. 6.15. The numerical estimates of the factors at this stage are subjective. The final values are not the total value of the company, but allow you to evaluate it in the game model.
Table 6.15. Pivot Table
Metric 
The amount of required capital investments (the lower, the higher the score) 
The risk of failure to implement the decision 
Risk of premature expulsion of products from the market 
Shortterm period before profit extraction 
Duration of profit extraction 
Quantity demanded for a product 
Totals 
9 
2 
4 
9 
5 
4 
21 

8 
3 
5 
9 
4 
3 
16 

7 
4 
6 
9 
4 
2 
12 

7 
5 
6 
9 
3 
2 
11 

6 
5 
7 
9 
2 
2 
8 

5 
5 
10 
9 
1 
1 
1 

4 
6 
2 
6 
8 
7 
17 

4 
6 
2 
6 
7 
6 
15 

4 
6 
3 
6 
6 
6 
12 

4 
6 
3 
6 
5 
5 
11 

4 
6 
4 
6 
3 
4 
7 

4 
6 
4 
6 
2 
3 
5 

1 
8 
1 
3 
10 
8 
13 

1 
8 
1 
3 
9 
7 
11 

1 
8 
2 
3 
7 
6 
7 

1 
8 
2 
3 
6 
6 
6 

1 
8 
3 
3 
4 
5 
2 

1 
8 
3 
3 
3 
4 
0 
As a result of the analysis, the matrix of the game with nature acquires the following form.
21 
16 
12 
11 
8 
1 

л = 
17 
15 
12 
11 
7 
5 

13 
11 
7 
6 
2 
0 
Finding the best strategy
We apply the criteria considered above for choosing the optimal enterprise strategy.
1. Bayes criterion for wins. Let us analyze the probabilities with which the nature of P takes its states, and rewrite the game matrix, adding for convenience of calculations the string of probabilities of the state of nature and the effectiveness column of player's strategies A ( column of average wins) E:
21 
16 
12 
11 
8 
1 
10.8 

A = 
17 
15 
12 
11 
7 
5 
10.4 

13 
11 
7 
6 
2 
0 
5.55 

0.05 
0.1 
0.2 
0.3 
0.3 
0.05 
According to this criterion, the strategy with the highest efficiency index is considered optimal. For a given matrix, this strategy is with an efficiency score of 10.8.
2. Bayesian Criteria for Risks. We will apply this criterion to determine the optimal enterprise strategy.
With the a favorable indicator , therefore, ;
With the benefit index , therefore, .
Similarly, we find the values of the remaining cells of the risk matrix A1 at the states of nature ,
Next, we complement the matrix with the column of indicators of the inefficiency of enterprise strategies, R.
0 
0 
0 
0 
0 
4 
10.8 

A1 = 
4 
1 
0 
0 
1 
0 
10.4 

8 
4 
5 
5 
6 
5 
5.55 

0.05 
0.1 
0.2 
0.3 
0.3 
0.05 
According to Bayes criterion for risks, the optimal strategy is the strategy with the lowest inefficiency index, for this matrix it is the A3 strategy.
3. Laplace Criterion for Wins. This criterion is similar to the Bayes criterion for wins. The difference is that in this criterion we allow the possibility of the unknownness of the probabilities of the states of nature. Assuming that the company's management can not reliably determine the probability of occurrence of events, all states of nature are considered to be equally probable.
So, we rewrite the winning matrix by adding a column of strategy performance indicators, E.
A = 
21 
16 
12 
11 
8 
1 
11.5 

17 
15 
12 
11 
7 
5 
11.2 

13 
11 
7 
6 
2 
0 
6.5 
According to this criterion, the strategy with the highest efficiency index is considered optimal. For a given matrix, this is an A1 strategy with an efficiency score of 11.5.
4. Laplace's Criteria for Risks. Let's compose a matrix of risks for the payoff matrix and add the resulting matrix to the column of indicators of strategy inefficiency, calculated by the Laplace formula.
0 
0 
0 
0 
0 
4 
0.7 

4 
X 
0 
0 
1 
0 
1.0 

8 
4 
5 
5 
6 
5 
5.5 
The optimal strategy for Laplace's criterion for risks is also the strategy A x with the lowest inefficiency score of 0.7.
5. Criterion of relative values of probabilistic states of nature (environment) taking into account wins. In a risk situation it is not always possible to determine the probability of occurrence of each of the states. However, you can often say which state is more likely than others.
In this problem, we can conditionally rank the states of nature in descending order of probability of their occurrence as follows: .
Since there are only six possible states of nature,
Therefore,
Add a row of probabilities and an efficiency column to the matrix A.
A = 
11 
8 
12 
16 
1 
21 
15.08 

11 
7 
12 
15 
5 
17 
14.79 

6 
2 
7 
11 
0 
13 
5.55 

0.40 
0.33 
0.27 
0.20 
0.13 
0.07 
According to this criterion, the strategy with the highest efficiency index is considered optimal. For a given matrix, this strategy is with an efficiency score of 15.08.
6. Criterion for the relative values of probability states of nature, taking into account risks. Using the matrices obtained in the analysis within the previous probability criterion, we supplement the risk matrix A1 with a column of indicators of strategy inefficiency and choose the strategy with the smallest indicator, Rcpi.
0 
0 
0 
0 
4 
0 
0.52 

0 
1 
0 
1 
0 
4 
0.81 

5 
6 
5 
4 
5 
8 
7.34 

0.40 
0.33 
0.27 
0.20 
0.13 
0.07 
According to this criterion, the optimal strategy is the one with the lowest inefficiency index. For a given matrix, this strategy is with an inefficiency index of 0.52.
7. Wald criterion, or criterion of extreme pessimism about wins. This criterion is also called maximin. He considers optimal the strategy, in choosing which the minimum winnings are greater than the minimum wins in other strategies.
The minimum winnings for a strategy .
Thus, according to this criterion, the optimal strategy is .
8. Maximax criterion, or criterion of extreme optimism about winnings. According to the extreme optimism criterion, the optimal strategy is that the maximum win is greater than the maximum winnings in other strategies. The maximum payoff for a strategy is .
Thus, according to this criterion, the optimal strategy is
9. Savage criterion, or criterion of extreme pessimism about risks. Optimal among pure strategies by the Savage criterion is that pure strategy, the maximum risk in choosing which is the minimum among the maximum risks of all pure strategies. Therefore, the optimal strategy by the Savage criterion guarantees the player A for any state of nature the risk is not greater than the minimax.
The maximum risks when choosing strategies and are equal, therefore, for this task the Savage criterion is not applicable.
10. A minimum criterion, or criterion of extreme optimism about risks. In accordance with this criterion, the optimal strategy is , at least one of the risks of which is 0. In this problem, this strategy is met both by the strategy , and the strategy . Consequently, this criterion is inapplicable in this case.
11. Generalized criterion of pessimism  of Hurwitz's optimism regarding wins with coefficients Rearrange winnings for each strategy A., arranging them in a nondecreasing order, and denote the elements of the resulting matrix through, and the matrix itself through B:
1 
8 
11 
12 
16 
21 

5 
7 
11 
12 
15 
17 

0 
2 
6 
7 
11 
13 

6 
17 
28 
31 
32 
51 
Now it is necessary to determine the effectiveness of the three strategies of the enterprise. To do this, you need to classify the situation as dangerous or safe.
In the safe situation, the coefficients are determined by the principle of "nondecreasing average wins", then, respectively:
Then
Therefore, according to the Hurwitz criterion for a safe situation, the optimal strategy is .
In the dangerous situation, the coefficients are based on the principle of "nonincrease of average wins":
Then
Thus, according to the Hurwitz criterion for a dangerous situation, the optimal strategy is .
12. The generalized criterion of pessimism  Hurwitz optimism about risks with coefficients . We will rearrange the risks in each line of the risk matrix , so that they stand in a nonincreasing order. Let's call the resulting matrix winnings for each strategy , arranging them in a nondecreasing order, and denote the elements of the resulting matrix through, and the matrix itself through.
4 
0 
0 
0 
0 
0 

4 
1 
1 
0 
0 
0 

8 
6 
5 
5 
5 
4 

16 
7 
6 
5 
5 
4 
Analogous to the analysis within the previous criterion, it is necessary to determine the danger of the situation.
For safe situations, the following formulas apply:
The function P takes the smallest value with the strategy , it is optimal for the safe situation.
>In the dangerous situation, the coefficients are calculated using the following formulas:
The function P takes the least value with the strategy A1, it is optimal for the dangerous situation.
Thus, when applying different criteria, different results are obtained. Different strategies may prove to be optimal depending on the chosen analysis model.
There are several ways to get the final result. You can abandon some of the criteria by determining the answer to the question whether our task is related to a situation of risk or a situation of uncertainty. The remaining criteria can be assigned weight, calculate the weighted average number of optimality cases of each strategy.
There is a simpler and less accurate version of the analysis, when the optimal strategy is considered to be the optimal large number of criteria.
Since this task is of the nature of training and research and information for eliminating the criteria and assigning weight to them is not enough, we will resort to the second method.
To obtain the final result, we reduce the results obtained by applying all the criteria to a single table. 6.16.
Table 6.16. Optimal strategy for different criteria
Criterion 
Optimal strategy 
1. Bayesian winnings 

2. Bayesian About Risks 

3. Laplace relative to winnings 

4. Laplace regarding risks 

5. Criterion for the relative values of probability states of nature with allowance for winnings 

6. A criterion for the relative values of probability states of a nature taking into account risks 

7. The Wald criterion, or the criterion of extreme pessimism about wins 

8. Maximax criterion, or criterion of extreme optimism regarding wins 

9. Criterion Savage, or the criterion of extreme pessimism about the risks 
Not applicable 
10. Miniminny criterion, or criterion of extreme optimism about risks 
Not applicable 
11. The generalized criterion of pessimism is Hurwitz optimism regarding wins with coefficients 
Safely  A 1, it's dangerous  A 2 
12. A generalized criterion for pessimism is Hurwitz's optimism about risks with coefficients 
Safely  A 1, it's dangerous  A 1 
Optimal strategy 
Thus, as an optimal strategy is chosen A 1.
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