The payment matrix includes the values ​​of all criteria. Decision making methods

A table showing the payouts to each participant in a two-way game. The rows of the table reflect the results of each choice of strategy by one participant, and the columns reflect the results of the choice of another. There may be one matrix showing the payoff to each player, and alternatively, each square in the multi-dimensional payoff matrix may contain two numbers to show the payoff to both players. In a zero-sum game, the payout to the second player will be equal to the payout to the first; thus, only one row needs to be recorded in detail.

End of work -

This topic belongs to the section:

Limiting risk in a business system is called risk management

Risk is understood as all internal and external prerequisites that can negatively affect the achievement of strategic goals during a precisely defined period of observation time, for example an operational period.

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All topics in this section:

Types of risks. Factors influencing the occurrence of risks
Classification: A) By the nature of the consequences: · Pure (cause only loss-risk of fire or flood); · Speculative (can bring both losses and

Factors influencing the occurrence of risks
All risk factors can be divided into 2 groups: · internal factors arising in the course of the enterprise’s activities; · external factors, essentially

Organization of the risk management process in the organization
The first stage of organizing risk management is determining the risk goal and the goal of risky capital investments. The purpose of the risk is the result that needs to be obtained. They may be

Information Risk Management
The work to minimize information risks involves preventing unauthorized access to data, as well as accidents and equipment failures. To minimize information risks with

Risk map
Risk map - a simple method of risk assessment Representatives from various sectors of the economy often ask, like risk management consultants, the question: are there simple and

Description of the risk map structure
This risk map displays probability or frequency on the vertical axis and impact or significance on the horizontal axis. In this case, the likelihood of risk occurrence increases

Building a risk map
Be carried out as part of the implementation of a risk management system at the level of the entire organization, which is difficult and often impossible to implement internal forces organizations. D

Basic steps of the self-risk mapping process
1. initial training 2. defining the boundaries of analysis 3. forming the team composition 4. analyzing scenarios and ranking 5. defining the boundaries of risk tolerance

Risk management methods
The risk management methods themselves are quite diverse. This is due to the ambiguity of the concept of risk and the presence large number criteria for their classification. In the next section

Parametric method
It is based on the assumption of a normal probability distribution of the risk factors under consideration and requires, in the process of constructing a model for calculating VAR, only an assessment of the parameters of this

Modeling from historical data
The historical simulation method is based on the use of historical data on changes in market risk factors to obtain the distribution of future price fluctuations

Monte Carlo method
from the textbook: The Monte Carlo method involves defining statistical models for portfolio assets and simulating them by generating random trajectories. Z

Scenario Analysis Method
The scenario analysis method examines the effect of changes in portfolio capital depending on changes in the magnitude of risk factors (e.g. interest rate, volatility) or model parameters. Model

Main quantitative characteristics of risks
The risk to which an enterprise is exposed is the probable threat of ruin or incurring such financial losses that could stop the whole business. Because there will be a chance of failure

Selection of projects based on mathematical expectation and standard deviation
The main goal of any investor is to obtain the expected profit from investment results. This profit is expected in the sense that at the stage of investment its value

Normal distribution law (Gauss's law)
Normal distribution(Gaussian distribution) is used to assess the reliability of products that are affected by a number of random factors, each of which has a slight effect on the resulting

Types of Math Games
Cooperative and non-cooperative A game is called cooperative, or coalition, if players can unite into groups, taking on certain obligations to other players and coordinating

Pure strategies in a mathematical game

Mixed strategies in a math game
In game theory, a player's strategy in a game or business situation is a complete plan of action for all possible situations that may arise. Strategy determines the player's actions at any point in the game

Question #24
The main theorem of matrix game theory, or the minimax theorem. If is a matrix

Question No. 25
The graphical method is applicable to those games in which at least one of the players has two strategies. The main stages of finding a solution to the game 2×n or m×2: 1. Construct straight lines, coo

Analytical solution of a mixed game
To find the optimal mixed strategy of player A: and the corresponding game price ν, it is necessary

Methodology for majorizing strategies
Majorization represents the relationship between strategies, the presence of which in many practical cases makes it possible to reduce the size of the original payment matrix games. Consider

Using a decision tree
In practice, the result of one decision forces us to make the next decision, etc. When we need to make several decisions under conditions of uncertainty, when each decision depends on the outcome of

Neumann-Morgenstern utility function
Basic definitions and axioms. The methodology of rational decision making under conditions of uncertainty, based on the individual’s utility function, is based on five axioms that reflect m

VAR Value at Risk Concept
One of the main tasks of financial institutions is to assess market risks that arise due to fluctuations (favorable events) in stock prices, commodities, exchange rates, interest rates.

It is convenient to study a zero-sum pair game if it is described in the form of a matrix.

Let's assume that the player A It has m strategies (let's denote them A 1 , A 2 , …, A m), and the player B(enemy) – n strategies (B 1, B 2, …, B m). Such a game is called a game of dimension m x n. Let player A choose one of his possible strategies A i . Player B without knowing the result of the player's choice A, chose strategy B j . For each pair of strategies (A i, B j), the payment a ij of the second player to the first is determined, i.e. player's winnings A. The player wins B will be accordingly (– a ij). There is no discrimination in relation to the second player here, since the values ​​of a ij can also be negative, then –a ij > 0. For example, a 13 = –2 – winning A, –a 13 = 2 – winning B. This game is called matrix; a matrix composed of numbers a ij is called payment. In example 1, the payment matrix looks like

The rows of this matrix correspond to the strategies of player A, and the columns correspond to the strategies of player B. General view of such a matrix

Example No. 1. Players A and B play the following game. Player A writes down one of the numbers 3, 7, 8, and player B writes down one of the numbers 4, 5. If the sum of the numbers is even, then this is Player A's win. If the sum of the numbers is odd, then this is a win for player B (a loss for player A). Find the payment matrix and the optimal solution.

Solution. If the sum of the numbers is even, player A receives a payoff of +1, otherwise player B receives a payoff of +1 (i.e., A receives a payoff of -1). Payment matrix:

Example No. 2. Two players independently call one number from the range 1-5. If the sum of the numbers is odd, then player 2 pays player 1 an amount equal to the maximum of the numbers; if it is even, then player 1 pays.
Solution. We will write down positive numbers as pay to the first player, negative numbers as pay to the second.
Payment matrix of the game.

Example No. 3. A small private company produces cosmetic products for teenagers. During the month, 15, 16 or 17 packages of goods are sold. From the sale of each package the company receives 75 rubles. arrived. Cosmetics have a short shelf life, so if the packaging is not sold within a month, it must be destroyed. Since the production of one package costs 115 rubles, the firm's loss is 115 rubles if the package is not sold by the end of the month. The probabilities of selling 15, 16 or 17 packages per month are 0.55, respectively; 0.1 and 0.35. How many packages of cosmetics should the company produce monthly? What is the expected cost value of this solution? How many packages could be produced if the shelf life of cosmetic products was significantly extended?

Decide.
Production costs: 115 rub. Income: 115+75 = 190 rub.
Sales of only manufactured products according to the formula: Profit = Price * Sales volume - Cost * Production volume
1125 = 15*190 - 15*115
1010 = 15*190 - 16*115
1200 = 16*190 - 16*115
895 = 15*190 - 17*115
1085 = 16*190 - 17*115
1275 = 17*190 - 17*115

How many packages of cosmetics should the company produce monthly?
It is profitable to produce 16 packages based on average profit and 15 packages based on the probability of sales.

What is the expected cost value of this solution?
1136.7 rub. based on average profit and 1125 rubles, taking into account the probability of sales.

How many packages could be produced if the shelf life of cosmetic products was significantly extended?
With an increase in the shelf life of products, it becomes possible to sell the entire volume of products produced, i.e. As much as they produced, the same amount was sold.
In this case, it is already profitable to produce 17 packages (they are still sold).

A zero-sum game in which each player has a finite set of strategies at his disposal. The rules of the matrix game are determined by the payment matrix, the elements of which are the winnings of the first player, which are also the losses of the second player.

Matrix game is an antagonistic game. The first player receives the maximum guaranteed (independent of the behavior of the second player) winnings, equal to the price of the game; similarly, the second player achieves the minimum guaranteed loss.

Under strategy is understood as a set of rules (principles) that determine the choice of action for each personal move of the player, depending on the current situation.

Now about everything in order and in detail.

Payment matrix, pure strategies, game price

IN matrix game its rules are determined payment matrix .

Consider a game in which there are two participants: the first player and the second player. Let the first player have at his disposal m pure strategies, and at the disposal of the second player - n pure strategies. Since the game is being considered, it is natural that in this game there are wins and there are losses.

IN payment matrix the elements are numbers expressing the players' wins and losses. Wins and losses can be expressed in points, amount of money or other units.

Let's create a payment matrix:

If the first player chooses i-th pure strategy, and the second player - j th pure strategy, then the payoff of the first player will be aij units, and the loss of the second player is also aij units.

Because aij + (- a ij) = 0, then the described game is a zero-sum matrix game.

The simplest example of a matrix game is coin toss. The rules of the game are as follows. The first and second players throw a coin and the result is either heads or tails. If "heads" and "heads" or "tails" or "tails" are thrown at the same time, then the first player will win one unit, and in other cases he will lose one unit (the second player will win one unit). The same two strategies are at the disposal of the second player. The corresponding payment matrix will be as follows:

The task of game theory is to determine the choice of the first player's strategy, which would guarantee him the maximum average win, as well as the choice of the second player's strategy, which would guarantee him the maximum average loss.

How do you choose a strategy in a matrix game?

Let's look at the payment matrix again:

First, let's determine the amount of winnings for the first player if he uses i th pure strategy. If the first player uses i th pure strategy, then it is logical to assume that the second player will use such a pure strategy due to which the first player’s payoff would be minimal. In turn, the first player will use such a pure strategy that would provide him maximum win. Based on these conditions, the winnings of the first player, which we denote as v1 , called maximin winnings or lower price of the game .

At for these values, the first player should proceed as follows. From each line, write down the value of the minimum element and select the maximum one from them. Thus, the first player's winnings will be the maximum of the minimum. Hence the name - maximin winning. The line number of this element will be the number of the pure strategy that the first player chooses.

Now let’s determine the amount of loss for the second player if he uses j th strategy. In this case, the first player uses his own pure strategy in which the loss of the second player would be maximum. The second player must choose a pure strategy in which his loss would be minimal. The loss of the second player, which we denote as v2 , called minimax loss or top price of the game .

At solving problems on the price of the game and determining the strategy To determine these values ​​for the second player, proceed as follows. From each column, write down the value of the maximum element and select the minimum from them. Thus, the loss of the second player will be the minimum of the maximum. Hence the name - minimax win. The column number of this element will be the number of the pure strategy that the second player chooses. If the second player uses "minimax", then regardless of the choice of strategy by the first player, he will lose no more than v2 units.

Example 1.

.

The largest of the smallest elements of the strings is 2, this is lowest price game, the first line corresponds to it, therefore, the maximin strategy of the first player is the first. The smallest of the largest elements of the columns is 5, this is the upper price of the game, the second column corresponds to it, therefore, the minimax strategy of the second player is the second.

Now that we have learned to find the lower and upper price of the game, the maximin and minimax strategies, it’s time to learn how to formally define these concepts.

So, guaranteed win first player:

The first player must choose a pure strategy that would provide him with the maximum of the minimum winnings. This gain (maximin) is denoted as follows:

.

The first player uses his pure strategy so that the loss of the second player is maximum. This loss is indicated as follows:

The second player must choose his pure strategy so that his loss is minimal. This loss (minimax) is indicated as follows:

.

Another example from the same series.

Example 2. Given a matrix game with a payoff matrix

.

Determine the maximin strategy of the first player, the minimax strategy of the second player, the lower and upper price of the game.

Solution. To the right of the payment matrix, we will write out the smallest elements in its rows and note the maximum of them, and below the matrix - the largest elements in the columns and select the minimum of them:

The largest of the smallest elements of the lines is 3, this is the lower price of the game, the second line corresponds to it, therefore, the maximin strategy of the first player is the second. The smallest of the largest elements of the columns is 5, this is the upper price of the game, the first column corresponds to it, therefore, the minimax strategy of the second player is the first.

Saddle point in matrix games

If the upper and lower prices of the game are the same, then the matrix game is considered to have a saddle point. The converse is also true: if a matrix game has a saddle point, then the upper and lower prices of the matrix game are the same. The corresponding element is both the smallest in the row and the largest in the column and is equal to the price of the game.

Thus, if , then is the optimal pure strategy of the first player, and is the optimal pure strategy of the second player. That is, equal lower and upper game prices are achieved using the same pair of strategies.

In this case matrix game has a solution in pure strategies .

Example 3. Given a matrix game with a payoff matrix

.

Solution. To the right of the payment matrix, we will write out the smallest elements in its rows and note the maximum of them, and below the matrix - the largest elements in the columns and select the minimum of them:

The lower price of the game coincides with the upper price of the game. Thus, the price of the game is 5. That is . The price of the game is equal to the value of the saddle point. The first player's maxin strategy is the second pure strategy, and the second player's minimax strategy is the third pure strategy. This matrix game has a solution in pure strategies.

Solve a matrix game problem yourself, and then look at the solution

Example 4. Given a matrix game with a payoff matrix

.

Find the lower and upper price of the game. Does this matrix game have a saddle point?

Matrix games with optimal mixed strategy

In most cases, a matrix game does not have a saddle point, so the corresponding matrix game has no solutions in pure strategies.

But it has a solution in optimal mixed strategies. To find them, you need to assume that the game is repeated a sufficient number of times so that, based on experience, you can guess which strategy is more preferable. Therefore, the decision is associated with the concept of probability and average (mathematical expectation). In the final solution there is also an analogue of the saddle point (that is, the equality of the lower and top price games), and an analogue of their corresponding strategies.

So, in order for the first player to receive the maximum average win and for the second player to have a minimum average loss, pure strategies should be used with a certain probability.

If the first player uses pure strategies with probabilities , then the vector is called a mixed first player strategy. In other words, it is a “mixture” of pure strategies. In this case, the sum of these probabilities is equal to one:

.

If the second player uses pure strategies with probabilities , then the vector is called a second player mixed strategy. In this case, the sum of these probabilities is equal to one:

.

If the first player uses a mixed strategy p, and the second player - a mixed strategy q, then it makes sense expected value the first player's win (the second player's loss). To find it, you need to multiply the first player's mixed strategy vector (which will be a one-row matrix), the payoff matrix and the second player's mixed strategy vector (which will be a one-column matrix):

.

Example 5. Given a matrix game with a payoff matrix

.

Determine the mathematical expectation of the first player's win (the second player's loss), if the first player's mixed strategy is , and the second player's mixed strategy is .

Solution. According to the formula for the mathematical expectation of the first player’s win (the second player’s loss), it is equal to the product of the first player’s mixed strategy vector, the payment matrix and the second player’s mixed strategy vector:

The first player is called such a mixed strategy that would provide him with the maximum average payoff if the game is repeated a sufficient number of times.

Optimal mixed strategy the second player is called such a mixed strategy that would provide him with a minimum average loss if the game is repeated a sufficient number of times.

By analogy with the maximin and minimax notations, in the case of pure strategies the optimal mixed strategies are designated as follows (and are linked to mathematical expectation, that is, the average of the winnings of the first player and the losses of the second player):

,

.

In this case, for the function E there is a saddle point , which means equality.

In order to find optimal mixed strategies and a saddle point, that is, decide matrix game in mixed strategies , we need to reduce the matrix game to the problem linear programming, that is, to an optimization problem, and solve the corresponding linear programming problem.

Reducing a matrix game to a linear programming problem

In order to solve a matrix game in mixed strategies, you need to construct a straight line linear programming problem And dual task. In the dual problem, the extended matrix, which stores the coefficients of the variables in the system of constraints, free terms and coefficients of the variables in the objective function, is transposed. In this case, the minimum of the goal function of the original problem is matched to the maximum in the dual problem.

Goal function in a direct linear programming problem:

.

System of constraints in a direct linear programming problem:

The goal function in the dual problem is:

.

System of restrictions in the dual problem:

The optimal plan for a direct linear programming problem is denoted by

,

and the optimal plan for the dual problem is denoted by

We denote the linear forms for the corresponding optimal plans by and ,

and they need to be found as sums of the corresponding coordinates of optimal plans.

In accordance with the definitions of the previous paragraph and the coordinates of optimal plans, the following mixed strategies of the first and second players are valid:

.

Theoretical mathematicians have proven that game price is expressed in the following way through the linear forms of optimal plans:

,

that is, it is the reciprocal of the sums of coordinates of optimal plans.

We, practitioners, can only use this formula to solve matrix games in mixed strategies. Like formulas for finding optimal mixed strategies the first and second players respectively:

in which the second factors are vectors. Optimal mixed strategies are also, as we already defined in the previous paragraph, vectors. Therefore, multiplying the number (game price) by a vector (with the coordinates of optimal plans) we also obtain a vector.

Example 6. Given a matrix game with a payoff matrix

.

Find the price of the game V and optimal mixed strategies and .

Solution. We create a linear programming problem corresponding to this matrix game:

We obtain a solution to the direct problem:

.

We find the linear form of the optimal plans as the sum of the found coordinates.

The essence of every decision made by management is the choice of the best of several alternatives according to specific criteria established in advance. (In case you want to recall the consideration of constraints and criteria for making decisions, refer to Chapter 6). Payment matrix- this is one of the methods statistical theory decisions, 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.

According to N. Paul Loomba: ʼʼPayment 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 a payment matrix, as shown in Fig. 8.4. The 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 the very general view the matrix means that the 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 are a reasonably limited number of alternatives or strategy options to choose between.

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

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

At the same time, the manager must have the ability 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 cases of decision-making, the manager has to evaluate probability or the 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 in the direction of the most optimistic consequences. 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 benefits of film production. Moreover, 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.

Probability directly influences the determination of expected value, a central concept in the payoff matrix. Expected value alternatives or strategy options are the sum of the possible values ​​multiplied by the corresponding probabilities. For example, if you believe that investing (as an action strategy) in an ice cream kiosk with a probability of 0.5 will provide you with an annual profit of $5,000, with a probability of 0.2 - $10,000, and with a probability of 0, 3 - $3000, then the expected value will be:

5000 (0.5) + 10,000 (0.2) + 3000 (0.3) = $5400

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. Research shows that when accurate probability values ​​are established, decision tree and payoff matrix methods produce better decisions than traditional approaches.

Rice. 8.5.Decision tree.

Payment matrix - concept and types. Classification and features of the “Payment Matrix” category 2017, 2018.

The essence of every decision made by management is the choice of the best of several alternatives according to specific criteria established in advance. Payment matrix- this 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 a payment matrix”, as shown in Fig. 8.4. 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 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 cases, the manager must evaluate probability or the 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.

Fig.4. Payment matrix

Let's imagine the situation of a sales agent who decides whether to fly by plane or go by train out of town where the consumer is located. If the weather is good, he can fly and spend 2 hours on the entire journey from gate to gate, and if he has to go by train - 7 hours. If he goes by train, he will lose a day at his place of work, which, in his estimation, could increase sales by $1,500. According to estimates, an out-of-town consumer should give him an order for $3,000 if he personally visits the client. If he plans to fly to a client, then the plane is forced to land due to fog, he will have to change personal visit by phone call. This will reduce the out-of-town client's order to $500, but the agent will be able to provide orders for $1,500 at home.

The payment matrix data above reflects an assessment of the consequences different options actions. Additionally, some assumptions are presented regarding the likelihood of fog (which would affect the plane, but not the train) and clear weather. We see that the probability of clear weather is 10 times higher than fog. Further, the matrix shows that, acting on the first option of the strategy (airplane), if the weather is good (9 chances out of 10), the sales agent will sell goods worth $4,500 (this is the result or consequences). Three other possible consequences can be explained in the same way; we omit these arguments.

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 you consider that the likelihood of a movie being a big success is very low, investing in stores becomes more attractive, since the probability of achieving the specified 20% is very significant. To take a simpler example, the payout for long-distance bets on horse racing is higher because you are more likely to win nothing at all.

Probability directly influences the determination of expected value, a central concept in the payoff matrix. Expected value alternatives or strategy options are the sum of the possible values ​​multiplied by the corresponding probabilities. For example, if you believe that investing (as an action strategy) in an ice cream kiosk with a probability of 0.5 will provide you with an annual profit of $5,000, with a probability of 0.2 - $10,000, and with a probability of 0.3 - $3000, then the expected value will be:

5000 (0.5) + 10,000 (0.2) + 3000 (0.3) = $5400

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. Research shows that when accurate probability values ​​are established, decision tree and payoff matrix methods produce better decisions than traditional approaches.