Create a payment matrix. See pages where the term payment matrix is ​​mentioned

Payment matrix- one of the methods statistical theory decisions that assist the manager in choosing one of several options. It is especially useful in a situation where a manager must determine which strategy will most contribute to achieving goals. In the very general view matrix means that payment depends on certain events that actually occur. If the event or state of nature does not actually occur, the payment will invariably be different.

In general, a payment matrix is ​​useful when:

• · there is a reasonably limited number of alternatives or strategy options to choose between.
• · what may happen is not known with complete certainty.

results decision taken depend on which alternative is chosen and what events actually take place.

In addition, the manager must be able to objectively assess the probability of relevant events and calculate the expected value of such probability.

Probability directly affects the definition of expected value - the basic concept payment matrix. The expected value of an alternative or option is the sum of the possible values ​​multiplied by the corresponding probabilities.

By determining the expected value of each alternative and arranging the results in the form of a matrix, the manager can easily choose the most optimal option. payment matrix relevant

In the words of N. Paul Loomba: "A payoff represents the monetary reward or utility resulting from a specific strategy combined with specific circumstances. When payments are represented in table (or matrix) form, we get a payoff matrix," Words "combined with specific circumstances " are very important to understand when a payoff matrix can be used and to evaluate when a decision made based on it is likely to be reliable. In its most general form, a matrix means that payment depends on certain events that actually occur. If such an event or state of nature does not actually occur, the payment will inevitably be different.

A leader rarely has complete certainty. But it is also rare for him to act in conditions of complete uncertainty. In almost all decision-making situations, a manager must evaluate the likelihood or possibility of an event.

A payment matrix presents payments in combination with specific circumstances in the form of a table or matrix.

To apply this method, the manager needs to determine the probability of the event, which varies from 1 to 0. The choice of its value can be based on past trends or the subjective assessment of the manager, who comes from his own experience of acting in similar situations.

Probability directly influences the determination of expected value, a central concept in the payoff matrix. The expected value of an alternative or strategy option is the sum of the possible values ​​multiplied by the corresponding probabilities.

By determining the expected value of each alternative and arranging the results in the form of a matrix, a manager can easily determine which choice is most preferable given the given criteria. It will, of course, correspond to the highest expected value.

Probability can be determined objectively, just like a roulette player does when betting on odd numbers. The choice of its value may be based on past trends or the subjective assessment of the manager, who proceeds from his own experience of acting in similar situations.

If probability has not been taken into account, the decision will always slide towards the most optimistic outcome.

Factors influencing the adoption process management decisions are important. The management process is the activity of those integrated into a certain system subjects of management, aimed at achieving the goals of the company by implementing certain functions using management methods.

Decision-making methods are varied. When making a decision, regardless of the models used, there are some decision rules. A decision rule is a criterion by which a judgment is made about the optimality of a given specific outcome. There are two types of rules. One does not use numerical values ​​of probable outcomes, the second uses given values.

To the first type The following decision rules apply:

1. A maximax decision is a decision in which a decision is made to maximize the maximum possible income. This method is very optimistic, that is, it does not take into account possible losses and, therefore, is the most risky.

2. A maximizing solution is a solution that maximizes the minimum possible income. This method in to a greater extent takes into account the downsides of various outcomes and is a more cautious approach to decision making.

3. A minimax solution is a solution that minimizes maximum losses. This is the most cautious approach to decision making and the most risk-aware one. Losses here take into account not only real losses, but also missed opportunities.

4. Gurvich criterion. This criterion is a compromise between the maximin and maximax solutions and is one of the most optimal.

To the second type decision-making refers to decisions in which, in addition to the possible gains and losses themselves, the probabilities of the occurrence of each outcome are taken into account. TO this type decision making include, for example, the maximum likelihood rule and the mathematical expectation optimization rule. With these methods, an income table is usually compiled, which indicates all possible income options and the likelihood of their occurrence. When using the maximum likelihood rule, one of the outcomes with the maximum probability is selected according to one of the rules of the first type.

When using the rule for optimizing mathematical expectations, we calculate mathematical expectations for gains or losses and then the optimal option is selected.

Since probability values ​​change over time, the application of rules of the second type usually involves testing the rules for sensitivity to changes in the probabilities of outcomes.

In addition, the concept of utility is used to determine risk attitudes. That is, for each possible outcome, in addition to the probability, the utility of this outcome is calculated, which is also taken into account when making decisions.

In addition to modeling, there are a number of methods that can assist a manager in finding an objectively justified decision to select from several alternatives the one that most contributes to achieving goals.

To make optimal decisions, the following methods are used:

ü payment matrix;

ü decision tree;

ü forecasting methods.

Payment matrix. The essence of every decision made by management is the choice of the best of several alternatives according to specific criteria established in advance. The payment matrix is ​​one of the methods of statistical decision theory, a method that can help a manager choose one of several options. It is especially useful when a manager must determine which strategy will most contribute to achieving goals. A payoff represents a monetary reward or utility resulting from a specific strategy in combination with specific circumstances. If payments are presented in the form of a table (or matrix), we obtain a payment matrix. The words “in combination with the particular circumstances” are very important to understand when a payment matrix can be used and to assess when a decision made based on it is likely to be reliable. In its most general form, a matrix means that payment depends on certain events that actually occur. If such an event or state of nature does not actually occur, the payment will inevitably be different. In general, a payment matrix is ​​useful when:

1) there is a reasonably limited number of alternatives or strategy options to choose between;

2) what can happen is not known with complete certainty;

3) the results of the decision made depend on which alternative is chosen and what events actually take place.

In addition, the manager must be able to objectively assess the probability of relevant events and calculate the expected value of such probability. A leader rarely has complete certainty, but he also rarely acts in conditions of complete uncertainty. In almost all decision-making situations, a manager must evaluate the likelihood or possibility of an event. Probability can be determined objectively, just like a roulette player does when betting on odd numbers. The choice of its value may be based on past trends or the subjective assessment of the manager, who proceeds from his own experience of acting in similar situations.

Many of the assumptions a manager makes relate to future conditions over which the manager has little or no control. However, these types of assumptions are necessary for many planning operations. It is clear that what better leader will be able to predict external and internal conditions in relation to the future, the higher the chances of drawing up feasible plans.

Using a decision tree, a manager can calculate the outcome of each alternative and choose the best sequence of actions. The result of an alternative is calculated by multiplying the expected result by the probability and then summing the same products located to the right on the decision tree.

Decision tree is a schematic representation of a decision-making problem. Like the payment matrix, the decision tree gives the manager the opportunity to take into account various directions of action and correlate them with financial results, adjust them according to the probability assigned to them, and then compare the alternatives. The concept of expected value is an integral part of the decision tree method (Figure 3.1).

Rice. 3.1. Decision Tree

A decision tree can be built under difficult situations when the results of one decision influence subsequent decisions. So the decision tree is useful tool to make consistent decisions.

• General Fundamentals of Management
  • DIDACTIC PLAN
  • LITERATURE
  • List of skills
  • Definition of organization. The need for management
  • The essence of management activities. The role of the leader and management functions. Levels of Management
  • The essence and purpose of the main management functions. Definition of management and its main goals
  • The evolution of management as a scientific discipline. Approaches to management based on the identification of various schools: systemic, process and situational approaches. External environment of the organization
  • Communications in management: concept and process of communication
  • Group dynamics and leadership: groups and their importance; effectiveness of teamwork. Leadership, power and influence: the relationship between leadership and power. Communication process and management effectiveness
  • The concepts of “incentive” and “reward” refer to ways to motivate staff. Modern technology in communication processes using the concepts of “incentive” and “reward” involves the use of a traditional base and modern methods,
  • Labor organization at communication enterprises: formation of labor resources; personnel management in the context of a reduction in the number of employees. Labor rationing
  • Types of solutions. Decision making: models and process of making management decisions

Using the payment matrix method in production management

1. Payment matrix method

Although some models used in production management are so complex that they cannot be done without a computer, the concept of modeling is simple.

As Shannon defines it: “A MODEL is a representation of an object, system, or idea in some form other than the whole itself.” An organization chart, for example, is a model that represents its structure.

The main characteristic of the model can be considered a simplification of the real life situation, to which it applies. Because the form of the model is less complex, rather than irrelevant data clouding the issue in real life, are eliminated, the model often increases the manager’s ability to understand and resolve the problems facing him.

The number of possible specific models of management science is almost as large as the number of problems for which they were developed.

Almost any decision-making method used in management can technically be considered a form of simulation. In addition to modeling, there are a number of methods that can assist a manager in finding an objectively justified decision to select from several alternatives the one that most contributes to achieving goals. These include the Payment Matrix.

The essence of every decision made by management is the choice of the best of several alternatives according to specific criteria established in advance.

The payment matrix is ​​one of the methods of statistical decision theory, a method that can assist a manager in choosing one of several options. It is especially useful when a manager must determine which strategy will most contribute to achieving goals.

According to N. Paul Loomba: “Payment represents the monetary reward or utility resulting from a specific strategy in combination with specific circumstances. If payments are presented in the form of a table (or matrix), we obtain the payment matrix”, as shown in Table 1.

In its most general form, a matrix means that payment depends on certain events that actually occur. If such an event or state of nature does not actually occur, the payment will inevitably be different Meskon Michael, Albert Michael, Khedoury Franklin. Fundamentals of Management./ Translation from English. - M.: Publishing house "Delo", 1997. - http://www.tourlib.columb.net.ua/Lib/meskon.htm.

Table 1. Payment matrix

In general, a payment matrix is ​​useful when:

1. There are a reasonably limited number of alternatives or strategy options to choose between.

2. What may happen is not known with complete certainty.

3. The results of the decision made depend on which alternative is chosen and what events actually take place.

In addition, the manager must be able to objectively assess the probability of relevant events and calculate the expected value of such probability. A leader rarely has complete certainty. But it is also rare for him to act in conditions of complete uncertainty. In almost all decision-making situations, a manager must evaluate the likelihood or possibility of an event. Recall from the previous discussion that probability ranges from 1, when an event will definitely occur, to 0, when an event will definitely not occur. Probability can be determined objectively, just like a roulette player does when betting on odd numbers. The choice of its value may be based on past trends or the subjective assessment of the manager, who proceeds from his own experience of acting in similar situations.

If probability has not been taken into account, the decision will always slide towards the most optimistic outcome.

For example, if we assume that investors in a successful film can have 500% of the invested capital, and when investing in a retail chain - in the most favorable case - only 20%, then the decision should always be in favor of film production. However, if we take into account that the probability great success movie is very low, investment in stores becomes more attractive, since the probability of obtaining the specified 20% is very significant. To take a simpler example, the payouts for long-distance bets on horse racing are higher because you are more likely to win nothing at all Mescon Michael, Albert Michael, Hedowrie Franklin. Fundamentals of Management./ Translation from English. - M.: Publishing house "Delo", 1997. - http://www.tourlib.columb.net.ua/Lib/meskon.htm.

Probability directly influences the determination of expected value, a central concept in the payoff matrix. The expected value of an alternative or strategy option is the sum of the possible values ​​multiplied by the corresponding probabilities.

By determining the expected value of each alternative and arranging the results in the form of a matrix, a manager can easily determine which choice is most attractive given the given criteria. It will, of course, correspond to the highest expected value (Table 2).

Based on the payment matrix З = ||З ji || the risk matrix is ​​calculated - =|| ji || . In this case, the risk ji for the activity option x j and the combination of input data is determined by the formula

Table 2. Payment matrix З = ||З ji ||

The payment risk matrix serves as the information basis for comparison and selection of the final (preferred) activity option from the point of view of optimality. To make such a choice, special decision-making rules are used under conditions of uncertainty and risk. These rules include:

1. Laplace criterion (minimum arithmetic mean costs Z j).

2. Wald criterion (minimum costs or maximum utility).

3. Savage criterion (minimal risk).

4. Hurwitz criterion.

1. Laplace criterion. According to the principle of insufficient reason in conditions where it is impossible to determine the probabilities for the occurrence of a particular state external environment, they are compared equal probabilities, find the average effect for each of the considered solution options and select the one where the average effect is maximum:

2. Wald criterion (criterion of the greatest caution/pessimist). For each of the considered solution options Xi, the worst situation (the smallest of Wij) is selected and the guaranteed maximum effect is found among them:

3. Hurwitz criterion. Focusing on the worst outcome is a kind of reinsurance, but it is reckless to choose an overly optimistic policy. The Hurwitz criterion offers some compromise:

where parameter b takes a value from 0 to 1 and acts as an optimism coefficient.

For example, when b = 0 (complete pessimism), the Hurwitz criterion turns into the Wald criterion, when b = 0.5 the chances of success and failure are considered equally likely, when b = 0.2 they are more cautious and the probability of success is considered lower (0.2) than possible failure.

4. Savage criterion. Its essence is to find the minimum risk. When choosing a solution based on this criterion:

Dij = Wij- (Wij)

· the utility (efficiency) function matrix is ​​compared new matrix- a matrix of regrets, the elements of which reflect losses from an erroneous action, i.e. benefit lost as a result of making the i>th decision in j-th state;

· using matrix D, a solution is selected using the pessimistic Wald criterion, giving smallest value maximum regret

It is logical that different criteria lead to different conclusions regarding the best solution. At the same time, the ability to choose a criterion gives freedom to managers making management decisions.

Any criterion must be consistent with intentions problem solver and correspond to his character, knowledge and beliefs M.A. Tynkevich. Economic and mathematical methods (operations research). - Kemerovo: KuzGTU, 2000. .

There are other generalized criteria, which are essentially combinations of the above criteria). However, none of them is free from conventions and does not provide an unambiguous choice of activity option. Therefore, the final choice of option is the task of experts and specialists.

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Using the payment matrix method in production management

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Lecture 9. The concept of gaming models. Payment matrix.

§ 6 ELEMENTS OF GAME THEORY

6.1 The concept of game models.

The mathematical model of a conflict situation is called game , parties involved in the conflict - players, and the outcome of the conflict is win .

For each formalized game, rules , those. a system of conditions that determines: 1) options for players’ actions; 2) the amount of information each player has about the behavior of their partners; 3) the gain that each set of actions leads to. Typically, winning (or losing) can be quantified; for example, you can value a loss as zero, a win as one, and a draw as 1/2. Quantifying the results of a game is called payment .

The game is called steam room , if it involves two players, and multiple , if the number of players is more than two. We will only consider doubles games. They involve two players A And IN, whose interests are opposite, and by game we mean a series of actions on the part of A And IN.

The game is called zero sum game or antagonistic sky , if the gain of one of the players is equal to the loss of the other, i.e. the sum of the winnings of both sides is zero. To complete the game task, it is enough to indicate the value of one of them . If we designate A– winnings of one of the players, b – the other's winnings, then for a zero-sum game b = –A, therefore it is enough to consider, for example A.

The choice and implementation of one of the actions provided for by the rules is called progress player. The moves may be personal And random . Personal move – this is a conscious choice by the player of one of the possible actions (for example, a move in a chess game). The set of possible options for each personal move is regulated by the rules of the game and depends on the totality of previous moves on both sides.

Random move – it is a randomly chosen action (for example, choosing a card from a shuffled deck). For a game to be mathematically defined, the rules of the game must indicate for each random move probability distribution possible outcomes.

Some games may consist only of random moves (so-called pure gambling) or only of personal moves (chess, checkers). Most card games belong to mixed type games, that is, they contain both random and personal moves. In the future, we will consider only the personal moves of the players.

Games are classified not only by the nature of moves (personal, random), but also by the nature and amount of information available to each player regarding the actions of the other. A special class of games consists of the so-called “games with complete information». A game with complete information is a game in which each player, with each personal move, knows the results of all previous moves, both personal and random. Examples of games with complete information include chess, checkers, and famous game"crosses and toes". Most games of practical importance do not belong to the class of games with complete information, since uncertainty about the actions of the enemy is usually an essential element of conflict situations.

One of the main concepts of game theory is the concept strategies .

Strategy A player is a set of rules that determine the choice of his action at each personal move, depending on the current situation. Usually during the game, with each personal move, the player makes a choice depending on the specific situation. However, it is in principle possible that all decisions are made by the player in advance (in response to any given situation). This means that the player has chosen a specific strategy, which can be specified as a list of rules or a program. (This way you can play the game using a computer.) The game is called ultimate , if each player has a finite number of strategies, and endless .– otherwise.

In order to decide game , or find game solution , for each player we should choose a strategy that satisfies the condition optimality , those. one of the players must receive maximum win, when the second one sticks to his strategy, At the same time the second player must have minimum loss , if the first one sticks to his strategy. Such strategies are called optimal . Optimal strategies must also satisfy the condition sustainability , those. It must be disadvantageous for either player to abandon their strategy in this game.

If the game is repeated quite a few times, then players may not be interested in winning and losing in each specific game, but Aaverage win (loss) in all batches.

The goal of game theory is to determine the optimal strategy for each player.

6.2. Payment matrix. Lower and upper price of the game

The ultimate game in which the player A It has T strategies, and the player V – p strategies is called a game.

Consider the game
two players A And IN(“we” and “enemy”).

Let the player A has T personal strategies, which we denote
. Let the player IN available n personal strategies, let's designate them
.

Let each side choose a specific strategy; for us it will be , for the enemy . As a result of the players choosing any pair of strategies And (
) the outcome of the game is uniquely determined, i.e. winnings player A(positive or negative) and loss
player IN.

Let's assume that the values known for any pair of strategies ( ,). Matrix
,
, the elements of which are winnings corresponding to the strategies And , called payment matrix or matrix of the game. The rows of this matrix correspond to the player's strategies A, and the columns – the player’s strategies B. These strategies are called pure.

Game Matrix
has the form:

Consider the game
with matrix

and determine the best among strategies
. Choosing a strategy , player A must expect that the player IN will answer it using one of the strategies , for which the payoff for the player A minimal (player IN seeks to "harm" the player A).

Let us denote by player's smallest winnings A when choosing a strategy for all possible player strategies IN(smallest number in i th row of the payment matrix), i.e.

(1)

Among all the numbers (
) choose the largest:
.

Let's call
lowest price ngra, or maximum winnings (maxmin). This is a guaranteed win for player A for any strategy of player B. Hence,

. (2)

The strategy corresponding to maximin is called maximin strategy . Player IN interested in reducing the player's winnings A, choosing a strategy , it takes into account the maximum possible gain for A. Let's denote

. (3)

Among all the numbers let's choose the smallest

and let's call top price of the game or minimax win (minimax). Ego guaranteed loss of player B . Therefore,

. (4)

The strategy corresponding to minimax is called minimax strategy.

The principle that dictates players to choose the most “cautious” minimax and maximin strategies is called minimax principle . This principle follows from the reasonable assumption that each player strives to achieve a goal opposite to that of his opponent.

Theorem.The lower price of the game always does not exceed the upper price of the game
.

If the upper and lower prices of the game are the same, then the total value of the upper and lower prices of the game
called the pure price of the game, or at the cost of the game. Minimax strategies corresponding to the price of the game are optimal strategies , and their totality - optimal solution or solution of the game. In this case the player A receives the maximum guaranteed (independent of the player’s behavior) IN) winnings v, and the player IN achieves the minimum guaranteed (regardless of the player’s behavior A) losing v. They say that the solution to the game has stability , those. if one of the players sticks to his optimal strategy, then it cannot be profitable for the other to deviate from his optimal strategy.

If one of the players (for example A) sticks to his optimal strategy, and the other player (IN) will deviate from its optimal strategy in any way, then for the player who made the deviation, it can never be profitable; such player deviation IN can at best leave the winnings unchanged. and in worst case– increase it.

On the contrary, if IN adheres to its optimal strategy, and A deviates from its own, then this can in no way be beneficial for A.

A couple of pure strategies And gives an optimal solution to the game if and only if the corresponding element is both the largest in its column and the smallest in its row. This situation, if it exists, is called saddle point. In geometry, a point on a surface that has the property of having a simultaneous minimum in one coordinate and a maximum in another is called saddle point, by analogy this term is used in game theory.

The game for which
, called playing with saddle point. Element , which has this property, is a saddle point of the matrix.

So, for every game with a saddle point, there is a solution that determines a pair of optimal strategies for both sides, differing in the following properties.

1) If both sides stick to their optimal strategies, then the average payoff is equal to the net cost of the game v, which is simultaneously its lower and upper price.

2) If one of the parties adheres to its optimal strategy, and the other deviates from its own, then the deviating party can only lose and in no case can increase its winnings.

The class of games that have a saddle point is of great interest from both theoretical and practical points of view.

In game theory, it is proven that, in particular, every game with complete information has a saddle point, and, therefore, every such game has a solution, i.e., there is a pair of optimal strategies of both sides, giving an average payoff equal to the cost of the game. If a game with complete information consists only of personal moves, then when each side applies its optimal strategy, it should always end in a well-defined outcome, namely, a win exactly equal to the cost of the game.

2.2 Examples of matrix games in pure and mixed strategies Reducing the order of the payoff matrix

The order of the payoff matrix (number of rows and columns) can be reduced by eliminating dominated and duplicate strategies.

Strategy K* is called a dominated strategy K** if, for any variant of behavior of the opposing player, the following relation is satisfied:

where and are the payoff values ​​when the player chooses strategies K* and K**, respectively.

If the relation is satisfied

strategy K* is called duplicate with respect to strategy K**.

For example, in the matrix

Payment matrix with dominated and overlapping strategies. Strategy A1 is dominant to strategy A2, strategy B6 is dominant to strategies B3, B4 and B5, and strategy B5 is duplicate to strategy B4. These strategies will not be chosen by players, since they are obviously losing and removing these strategies from the payment matrix will not affect the determination of the lower and upper prices of the game described by this matrix.

The set of non-dominated strategies obtained after reducing the dimension of the payment matrix is ​​also called the Pareto set (named after the Italian economist Vilfredo Pareto, who was involved in research in this area)

An example of solving a matrix game in pure strategies

Let us consider an example of solving a matrix game in pure strategies, under the conditions real economy, in a situation of struggle between two enterprises for the market for the region’s products.

Two enterprises produce products and supply them to the regional market. They are the only suppliers of products to the region, therefore they completely determine the market for these products in the region.

Each of the enterprises has the ability to produce products using one of three different technologies. Depending on the quality of the products produced by each technology, enterprises can set the unit price at 10, 6 and 2 monetary units, respectively. At the same time, enterprises have different costs per unit of production.

Costs per unit of products produced at regional enterprises (units).

As a result of marketing research of the regional product market, the demand function for products was determined:

Y = 6 – 0.5×X,

where Y is the quantity of products that the population of the region will purchase (thousand units), and X is the average price of enterprises’ products, unit units.

Data on demand for products depending on sales prices are shown in the table.

Demand for products in the region, thousand units.

The values ​​of the shares of the products of enterprise 1 purchased by the population depend on the ratio of prices for the products of enterprise 1 and enterprise 2. As a result of marketing research, this dependence was established and the values ​​were calculated.

The share of enterprise 1 products purchased by the population depending on the ratio of product prices (Table 1.1)

According to the problem, there are only 2 enterprises operating in the regional market. Therefore, the share of the second enterprise’s products purchased by the population, depending on the ratio of product prices, can be defined as one minus the share of the first enterprise.

The strategies of enterprises in this problem are their decisions regarding production technologies. These decisions determine the cost and selling price per unit of production. In the task it is necessary to determine:

1. Is there an equilibrium situation in this problem when choosing production technologies for both enterprises?

2. Are there technologies that enterprises obviously will not choose due to unprofitability?

3. How many products will be sold in an equilibrium situation? Which company will be in an advantageous position?

The solution of the problem

1. Let us determine the economic meaning of the winning coefficients in the payment matrix of the problem. Every enterprise strives to maximize profits from production. But besides that, in in this case enterprises are fighting for the product market in the region. In this case, the gain of one enterprise means the loss of another. Such a problem can be reduced to a zero-sum matrix game. In this case, the winning coefficients will be the difference between the profits of enterprise 1 and enterprise 2 from production. If this difference is positive, enterprise 1 wins, and if it is negative, enterprise 2 wins.

2. Let's calculate the winning coefficients of the payment matrix. To do this, it is necessary to determine the profit values ​​of enterprise 1 and enterprise 2 from production. The profit of the enterprise in this problem depends on:

From the price and cost of production;

On the amount of products purchased by the population of the region;

From the share of products purchased by the population from the enterprise.

Thus, the values ​​of the difference in the profit of enterprises corresponding to the coefficients of the payment matrix must be determined using formula (1):

D = p×(S×R1-S×C1) – (1-p) ×(S×R2-S×C2) (1),

where D is the difference in profit from the production of products of enterprise 1 and enterprise 2;

p is the share of enterprise 1’s products purchased by the population of the region;

S – the amount of products purchased by the population of the region;

R1 and R2 - sales prices per unit of production by enterprises 1 and 2;

C1 and C2 – the total cost of a unit of production produced at enterprises 1 and 2.

Let's calculate one of the coefficients of the payment matrix.

Let, for example, enterprise 1 decide to produce products in accordance with technology III, and enterprise 2 - in accordance with technology II. Then the selling price per unit. products for enterprise 1 will amount to 2 units. at unit cost. products 1.5 units For enterprise 2, the selling price per unit. products will amount to 6 units. at a cost of 4.00. (Table 1.1).

The amount of products that the population of the region will purchase when average price 4 units equals 4 thousand units. (Table 1.2). The share of products that the population will purchase from enterprise 1 will be 0.85, and from enterprise 2 – 0.15 (Table 1.3). Let's calculate the coefficient of the payment matrix a 32 using formula (1): a 32 = 0.85×(4×2-4×1.5) – 0.15×(4×6-4×4) = 0.5 thousand. units

where i=3 is the technology number of the first enterprise, and j=2 is the technology number of the second enterprise.

Similarly, we calculate all the coefficients of the payment matrix. In the payment matrix, strategies A1 - A3 - represent decisions about production technologies for enterprise 1, strategies B1 - B3 - decisions about production technologies for enterprise 2, winning coefficients - the difference in profit between enterprise 1 and enterprise 2. Payment matrix in the game “Struggle of two enterprises” for the region's product market."

There are no dominant or overlapping strategies in this matrix. This means that for both enterprises there are no obviously unprofitable production technologies. Let us determine the minimum elements of the matrix rows. For enterprise 1, each of these elements has a minimal value guaranteed win when choosing an appropriate strategy. The minimum elements of the matrix by row have the following values: 0.17, -1.5, 0.4.

Let us determine the maximum elements of the matrix columns. For enterprise 2, each of these elements also has the value of the minimum guaranteed gain when choosing the appropriate strategy. The maximum matrix elements by column have the following values: 3, 0.62, 0.4.

The lower price of the game in the matrix is ​​0.4. The top price of the game is also 0.4. Thus, the lower and upper price of the game in the matrix are the same. This means that there is a technology for producing products that is optimal for both enterprises under the conditions of a given task. This is technology III, which corresponds to strategies A3 of enterprise 1 and B3 of enterprise 2. Strategies A3 and B3 are pure optimal strategies in this problem.

The difference between the profits of enterprise 1 and enterprise 2 when choosing a pure optimal strategy is positive. This means that enterprise 1 will win this game. The profit of enterprise 1 will be 0.4 thousand. At the same time, 5 thousand units will be sold on the market. products (sales equal to demand for products, table 1.2). Both enterprises will set the price per unit of production at 2.00. In this case, for the first enterprise the total cost per unit of production will be 1.5 units, and for the second - 1 unit (Table 1.1). Enterprise 1 will benefit only due to the high share of products that the population will purchase from it.

Mixed strategies in matrix games

The concept of matrix games with mixed extension

Research in matrix games begins with finding it net price. If a matrix game has a solution in pure strategies, then the study of the game ends with finding the pure price. If the game does not have a solution in pure strategies, then one can find the lower and upper prices of this game, which indicate that player 1 should not hope to win more than the upper price of the game, and can be sure of receiving a win no less than the lower price of the game . Improvement of solutions to matrix games should be sought in the use of secrecy in the use of pure strategies and the possibility of repeated games in the form of a game. This result is achieved by applying pure strategies randomly, with a certain probability.

Definition. A player's mixed strategy is a complete set of pure strategies applied in accordance with a specified probability distribution. A matrix game solved using mixed strategies is called a mixed expansion game.

Strategies applied with a probability other than zero are called active strategies.

It has been proven that for all games with mixed expansion there is an optimal mixed strategy, the payoff value of which is in the interval between the lower and upper price of the game:

Vн £ V £ Vв.

Under this condition, the value V is called the price of the game.

In addition, it has been proven that if one of the players sticks to his optimal mixed strategy, then the payoff remains unchanged and equal to the cost of the game V, regardless of what strategies the other player follows, unless he goes beyond his active strategies. Therefore, to achieve the largest guaranteed win, the second player also needs to adhere to his optimal mixed strategy.

Solving matrix games with mixed expansion methods linear programming

Solving a matrix game with mixed expansion is the determination of optimal mixed strategies, that is, finding such values ​​of the probabilities of choosing pure strategies for both players at which they achieve biggest win.

For a matrix game, the payoff matrix of which is shown in Fig. 1.1, V n ¹ V in, we determine such values ​​of the probabilities of choosing strategies for player 1 (p 1, p 2,..., p m) and for player 2 (q 1, q 2,..., q n), at which the players would achieve their maximum guaranteed winnings.

If one of the players adheres to his optimal strategy, then, according to the conditions of the problem, his payoff cannot be less than the game price V. Therefore this task can be presented to players in the form of the following systems of linear inequalities:

For the first player:

For the second player:

To determine the value of V, we divide both sides of each equation by V. We denote the value p i /V by x i, and q j /V by y j.

For player 1 we obtain the following system of inequalities, from which we find the value 1/v:

For player 1, it is necessary to find the maximum game price (V). Therefore, the value of 1/V should tend to a minimum.

min Z = min 1/V = min (x 1 + x 2 + … + x m)

For player 2 we obtain the following system of inequalities, from which we find the value 1/v:

For player 2 it is necessary to find the minimum price of the game (V). Therefore, the value of 1/V should tend to the maximum.

The objective function of the problem will have the following form:

All variables in these systems of linear inequalities must be non-negative: x i = p i /V, and y i = q j /V. The values ​​p i and q j cannot be negative, since they are the probabilities of choosing the players’ strategies. Therefore, it is necessary that the value of the game price V is not negative. The price of the game is calculated based on the winning coefficients of the payment matrix. Therefore, in order to guarantee the non-negativity condition for all variables, it is necessary that all matrix coefficients be non-negative. This can be achieved by adding, before starting to solve the problem, to each matrix coefficient the number K, corresponding to the modulus of the smallest negative coefficient of the matrix. Then, in the course of solving the problem, it will not be the price of the game that will be determined, but the value

To solve linear programming problems, the simplex method is used. .

As a result of the solution, the values ​​of the objective functions are determined (for both players these values ​​are the same), as well as the values ​​of the variables x i and y j.

The value of V* is determined by the formula: V* = 1/z

The probabilities of choosing strategies are determined: for player 1: P i = x i ×V*: for player 2: q i = y i ×V*.

To determine the price of the game V, it is necessary to subtract the number K from the value V*.

An example of solving a matrix game with mixed expansion

Let's consider an example of solving a matrix game with mixed extension. We will compile the payment matrix of the game based on the initial data, replacing only the values ​​of the shares of enterprise 1’s products purchased by the population depending on the price ratios (Table 2.1).

Table 2.1 - Share of enterprise 1 products purchased by the population depending on the ratio of product prices

Applying formula (1) for determining the difference in profit from production to the initial data of the problem, we obtain the following payment matrix

Payment matrix in the game “The struggle of two enterprises for the market for regional products”

There are no dominant or overlapping strategies in this matrix. The low game price is 0.175 and the high game price is 0.24. The lower price of the game is not equal to the upper price. Therefore, there is no solution in pure strategies and for each player it is necessary to find an optimal mixed strategy.

The solution of the problem

1. This matrix has negative coefficients. To comply with the non-negativity condition in linear programming problems, we add to each matrix coefficient the modulus of the minimum negative coefficient. In this problem, to each matrix coefficient you need to add the number 1.5 - the modulus value of the smallest negative element of the matrix. We obtain the payment matrix transformed to satisfy the non-negativity condition

Payment matrix transformed to satisfy the non-negativity condition