Using the payoff matrix method in production management
1. Payoff matrix method
Although some of the models used in production management are so complex that it is impossible to do without a computer, the concept of modeling is simple.
According to Shannon's definition: "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 shape of the model is less complex, and irrelevant data clouding the problem 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 concrete models of management science is almost as great as the number of problems for which they have been developed.
Almost any decision-making method used in management can technically be considered a form of modeling. In addition to modeling, there are a number of methods that can assist the manager in finding an objectively justified decision to choose from among several alternatives to the one that most contributes to the achievement of goals. These include the payoff matrix.
The essence of each decision taken by management is the choice of the best of several alternatives according to specific criteria established in advance.
The payoff matrix is one of the methods statistical theory decisions, a method that can assist the manager in choosing one of several options. It is especially useful when a manager must determine which strategy will most contribute to the achievement of goals.
In the words of N. Paul Loomba, “A payoff is a monetary reward or utility that results from a specific strategy combined with specific circumstances. If payments are presented in the form of a table (or matrix), we get a payoff matrix, as shown in Table 1.
In the very general view the matrix means that the payment depends on certain events that actually happen. If such an event or state of nature does not actually happen, the payoff will inevitably be different Mescon Michael, Albert Michael, Hedouri Franklin. Fundamentals of management./ Translation from English. - M.: Delo Publishing House, 1997. - http://www.tourlib.columb.net.ua/Lib/meskon.htm.
Table 1. Payoff matrix
In general, a payoff matrix is useful when:
1. There is a reasonably limited number of alternatives or strategies to choose from.
2. What can happen is not known with complete certainty.
3. Results decision depend on which alternative is chosen and what events actually take place.
In addition, the manager must have the ability to objectively assess the likelihood of relevant events and calculate the expected value of such a probability. A leader rarely has complete certainty. But also rarely does he act in conditions of complete uncertainty. In almost all cases of decision-making, the manager has to evaluate the likelihood or possibility of an event. Recall from the foregoing discussion that the probability varies from 1, when the event will definitely occur, to 0, when the event will definitely not happen. Probability can be determined objectively, as a roulette player behaves 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 own experience actions in such situations.
If probability has not been taken into account, the decision will always slide in the direction of the most optimistic outcome.
For example, if we proceed from the fact that investors on a successful film can have 500% of the invested capital, and when investing in a distribution network, 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 said 20% is very significant. To take a simpler example, the payouts for betting long distance races are higher because you are more likely to win nothing at all Mescon Michael, Albert Michael, Hedouri Franklin. Fundamentals of management./ Translation from English. - M.: Delo Publishing House, 1997. - http://www.tourlib.columb.net.ua/Lib/meskon.htm.
Probability directly affects the definition of expected value - the central concept of the payoff matrix. The expected value of an alternative or variant of a strategy is the sum of the possible values multiplied by the respective probabilities.
By determining the expected value of each alternative and arranging the results in the form of a matrix, the manager can easily determine which choice is the most attractive given the criteria. It will of course correspond to the highest expected value (Table 2).
Based on the payoff 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 initial data is determined by the formula
Table 2. Payoff matrix З = ||З ji ||
|
Activity Options |
Combinations of source data |
||||||
The payment risk matrix serves as an information basis for comparison and selection of the final (preferred) option of activity in terms of optimality. To make such a choice, special decision rules are used under conditions of uncertainty and risk. These rules include:
1. Laplace's criterion (minimum arithmetic mean costs Z j).
2. Wald's criterion (minimum cost or maximum utility).
3. Savage's criterion ( minimal risk).
4. Hurwitz criterion.
1. Laplace criterion. By the principle of insufficient reason in conditions when it is impossible to find out the probabilities for the occurrence of a particular state external environment, they are assigned equal probabilities, the average effect is found for each of the considered solutions, and the one where the average effect is maximum is selected:
2. Criterion of Wald (criterion of the greatest caution / pessimist). For each of the considered solutions Xi, the worst situation (the smallest of Wij) is selected and the guaranteed maximum effect is found among them:
3. Hurwitz criterion. Orientation to the worst outcome is a kind of reinsurance, but it is also rash to choose an overly optimistic policy. The Hurwitz criterion offers some compromise:
where the parameter b takes a value from 0 to 1 and acts as an optimism coefficient.
For example, at b = 0 (complete pessimism), the Hurwitz criterion turns into the Wald criterion, at b = 0.5, the chances of success and failure are considered equally probable, at b = 0.2, they are more careful and the probability of success is considered less (0.2) than possible failure.
4. Savage's criterion. Its essence is to find the minimum risk. When choosing a solution according to this criterion:
Dij = Wij-(Wij)
The matrix of the utility function (efficiency) is compared new matrix- a matrix of regrets, the elements of which reflect the losses from an erroneous action, i.e. the profit lost as a result of making the i>-th decision in j-th state;
the matrix D is used to select the solution by the pessimistic Wald criterion, which gives smallest value maximum regret
It is quite logical that different criteria lead to different conclusions regarding the best solution. At the same time, the possibility of choosing a criterion gives freedom to managers who make managerial decisions.
Any criterion must be consistent with the intentions problem solving and correspond to his character, knowledge and beliefs M.A. Tynkevich. Economic-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 an activity option. Therefore, the final choice of the option is the task of experts and specialists.
Selection and implementation of a strategy on the example of culinary production of SM "Elite Center" TS Rainford
For the company's CXE, these figures are, respectively: CXE 1 - 19% and 0.8 CXE 2 - 30% and 1.8 CXE 3 - 13.5% and 1.5 CXE 4 - 10% and 0 Elite Center" Analyzing the matrix, you can determine ...
Using the payoff matrix method in production management
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Virtually any decision-making method used in management can technically be considered a form of modeling. Traditionally, however, the term model usually refers only to the general methods just described above, as well as to many of them. specific varieties. In addition to modeling, there are a number of methods that can assist the manager in finding an objectively justified decision to choose from among several alternatives to the one that most contributes to the achievement of goals. Under the heading of this section are payoff matrix and decision tree, described below. To facilitate the use of these methods and generally improve the quality of decisions made, management uses forecasting. The most common forecasting methods are discussed in the next section. Our goal is to help you understand these tools, not teach you how to use them.
The essence of each decision taken by management is the choice of the best of several alternatives according to specific criteria established in advance. (If you want to revisit the discussion of constraints and decision criteria, see Chapter 6.) The payoff 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 the achievement of goals.
In general, the payoff matrix is useful when
Probability directly affects the definition of expected value - the central concept of the payoff matrix. The expected value of an alternative or variant of a strategy is the sum of the possible values multiplied by the respective probabilities. For example, if you believe that investing (as an action strategy) in an ice cream shop 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 - $3,000, then the expected value is
In table. 12.2 summarizes the results of various possible pricing decisions. When deciding what price to set, two firms play a non-cooperative game - each firm decides for itself how best to proceed, taking into account its competitor. Tab. 12.2 is called the payoff matrix for this game, since it shows the profit of each firm if its solution and the solution of its competitor are known. For example, the top left corner of the payoff matrix tells us that if both firms charge $4, each firm will make a profit of $12. The top right corner shows that if firm 1 charges $4 and firm 2 charges $4 $6, firm 1 makes a profit of $20 and firm 2 a profit of $4.
TABLE 12.2 Payoff matrix for the price prosthetic game
This payoff matrix may clarify the answer to the original question of why the firms do not work together and therefore do not earn higher profits, even if they are able to negotiate. In this case, the contract means that both firms will charge a price of $6 instead of $4 and they will make a profit of $16 instead of $12. The problem is that every firm is always trying to win by charging $4, no matter what its competitor does. As the payoff matrix shows,
Considering the enterprise (P,) and nature (P2) as two players, we get the so-called payoff matrix the following kind(Table 6.11)
It can be seen from the payoff matrix that player P (enterprise) will never receive income less than 6800. But if the weather conditions coincide with the chosen strategy, then the revenue (winning) of the enterprise will be 26000 or 28400. If player P, will constantly apply strategy A, and player P2 - strategy D, then the payoff will decrease to 6800. The same will happen if player P constantly applies strategy B, and player P2 - strategy C. Hence the conclusion that the enterprise will provide the greatest income if it alternately applies that strategy A, then strategy B. Such a strategy is called mixed, and its elements (A and B) are called pure strategies.
Considering AO Silhouette and nature as two players /, and P2, we obtain the so-called payoff matrix of the following form based on the results of the calculations made (p. 53).
According to the payment matrix, player P1 (JSC Silhouette) will never make a profit of less than 136,000 rubles. If the weather conditions coincide with the chosen strategy, then the profit of the JSC (winning) will be 568,000 or 520,000 rubles. If player P constantly adopts strategy A, and player P2 - strategy D, then the profit will decrease to 136,000 rubles. The same will happen if player P constantly adopts strategy B, and player P2 - strategist
Example. The daily demand for a perishable product in tons is expressed by the following distribution (demand/probability) (0.0/0.2) (1.0/0.3) (2.0/0.4) (3.0/0.5 ). Let the cost of a ton be 3,000 rubles, the sale price be 5,000 rubles, and the profit per unit be 2,000 rubles. The store can hold a stock of 0, 1.2, or 3 tons. Suppose that the day's stock cannot be sold tomorrow, and the rest is written off entirely as a loss. The payment matrix is shown in Table. 7.2. Analysis with complete information is given in table. 7.3.
Example. The enterprise plans the production of two products A, B with uncertain demand, the expected level of which is characterized by two states I, P. Depending on these states, the profit of the enterprise is different and is determined by the payoff matrix
It is required to determine the volume of production of each product, at which the enterprise is guaranteed an average value for any state of demand. Solution. Checking the payoff matrix for the presence of a saddle point
Let the payoff matrix of the game be given
The condition of the game is usually written in the form of a payoff matrix, or a game matrix (Table 3.33).
Let the payoff matrix be given in qualitative terms. Data
Analysis of payoff matrices allows us to draw the following conclusions with incomplete information best choice- keep a stock of 2 tons highest value profit 1.90 thousand rubles. This is the best thing you can do with limited information.
In management practice, methods such as the payoff matrix tree of goals or decisions are widely used. The best known of these is the decision tree method for comparing and evaluating alternatives put forward. This method is especially useful in situations where the manager deals with uncertainty. This method gives big picture decisions choices, risks and outcomes that may occur. Moreover, this method helps to discover new alternatives that could previously be omitted for some reason.
The payoff matrix data above reflects an assessment of the consequences different options actions. Additionally, some assumptions are presented regarding the likelihood of fog that will affect the aircraft, but not the train) and clear weather. We see that the probability of clear weather is 10 radars higher than that of cloudy weather. Further, the matrix shows that, under the first strategy option (airplane), if the weather is good (9 out of 10 chances), the valuation salesperson will sell $4,500 worth of goods (this is the result or consequences). Three other possible consequences can be explained in the same way, we omit these considerations.
According to N. Paul Loomba, a Payment is a monetary reward or utility that results from a specific strategy combined with specific circumstances. If payments are presented in the form of a table (or matrix), we get a payoff matrix 24, as shown in Fig. 8.4. The words combined with the specific circumstances are very important to understand when to use the payoff matrix and assess when a decision based on it is likely to be reliable. In its most general form, 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.
By determining the expected value of each alternative and arranging the results in the form of a matrix, the manager can easily determine which choice is the most attractive given the criteria. It will of course match the highest expected value. Studies show when installed exact values probabilities, decision tree and payoff matrix methods provide better decision making than traditional approaches25. saddle point ot = max minay = max (22,21,20) = 22 - lower price
Judgments about the preference of alternatives are made based on the results of their comparison or evaluation. G positive and negative aspects of each of the alternatives and a certain compromise is established, pos] comparison of the alternative with the previously adopted standard, criterion. To do this, use the Kepner-Trego criterion comparison, the payoff matrix, the tree of goals or decisions, as well as i theories of probability, preferences, utility, etc. The most common method in comparison) is the decision tree method, especially in situations of uncertainty, in the presence of uncontrollable
GAME WITH "NATURE" - a game in which there is only one player, and its outcome depends not only on his decisions, but also on the state of "nature", i.e. not from a consciously opposing enemy, but from objective, non-hostile reality. The payment matrix in this case is similar to that shown in Art. "Matrix of the game", but here player X is a person who makes one of m different possible decisions, and player Y is "nature", which accepts also possible states. When choosing a solution, player X can use various criteria, for example.
Lecture 9 The concept of game 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 winning .
For each formalized game, we introduce rules , those. a system of conditions that determines: 1) options for the players' actions; 2) the amount of information each player has about the behavior of partners; 3) the payoff to which each set of actions leads. Typically, gain (or loss) can be quantified; for example, you can evaluate a loss by zero, a win by one, and a draw by 1/2. Quantifying the results of a game is called payment .
The game is called steam room , if there are two players involved, and multiple , if the number of players is more than two. We will consider only paired games. They are played by two players A And IN, whose interests are opposite, and by the game we mean a series of actions on the part of A And IN.
The game is called zero sum game or antagonistic skoy , if the gain of one of the players is equal to the loss of the other, i.e. the sum of the payoffs of both parties is zero. To complete the task of the game, it is enough to indicate the value of one of them . If we designate A- win one of the players, b – the other's payoff, then for a zero-sum game b= –A, so it suffices to consider, for example A.
The choice and implementation of one of the actions provided for by the rules is called move player. Moves can be personal And random . personal move – it 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 the game to be mathematically defined, the rules of the game must specify for each random move probability distribution possible outcomes.
Some games may consist only of random moves (so-called pure games of chance) or only of personal moves (chess, checkers). Most card games belong to the mixed type games, that is, they contain both random and personal moves. In what follows, we will consider only the personal moves of the players.
Games are classified not only by the nature of the moves (personal, random), but also by the nature and amount of information available to each player regarding the actions of another. A special class of games are the so-called "games with complete information». A game with complete information A game is called in which each player at each personal move knows the results of all previous moves, both personal and random. Examples of games with complete information are chess, checkers, and famous game"tic-tac-toe". Most games of practical importance do not belong to the class of games with complete information, since the unknown about the opponent's actions is usually an essential element of conflict situations.
One of the basic concepts of game theory is the concept strategies .
strategy A player is called a set of rules that determine the choice of his action for each personal move, depending on the situation. Usually during the game, at each personal move, the player makes a choice depending on the specific situation. However, in principle it is 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 certain strategy, which can be given in the form of a list of rules or a program. (So 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 decision , it is necessary for each player to choose a strategy that satisfies the condition optimality , those. one of the players must receive maximum win, when the second sticks to his strategy, At the same time, the second player must have minimum loss , if the first adheres to its strategy. Such strategies are called optimal . Optimal strategies must also satisfy the condition sustainability , those. it should be unprofitable for any of the players to abandon their strategy in this game.
If the game is repeated enough times, then the players may not be interested in winning and losing in each particular game, Aaverage win (loss) in all parties.
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
End game in which the player A It has T strategies, and the player B - p strategies is called a game.
Consider the game 
two players A And IN("we" and "opponent").
Let the player A has T personal strategies, which we denote 
. Let the player IN available n personal strategies, we denote them 
.
Let each side choose a particular strategy; for us it will be 
, for the enemy 
. As a result of the players' choice of any pair of strategies 
And 
(
) the outcome of the game is uniquely determined, i.e. win 
player A(positive or negative) and losing 
player IN.
Let's assume that the values 
are known for any pair of strategies ( 
,
).
Matrix 
,
,
whose elements are the payoffs corresponding to the strategies 
And 
,
called payment matrix
or game matrix.
The rows of this matrix correspond to the player's strategies A, and the columns are the player's strategies B. These strategies are called pure.
Game Matrix 
looks like:
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Consider the game 
with matrix 
and determine the best among the strategies 
.
Choosing a strategy 
,
player A should expect the player IN will answer it with one of the strategies 
,
for which the payoff for the player A minimal (player IN seeks to "harm" the player A).
Denote by 
player's lowest payoff A when choosing a strategy 
for all possible player strategies IN(smallest number in i-th row of the payoff matrix), i.e.
(1)
Among all the numbers 
(
) choose the largest: 
.
Let's call 
lower price ngra,
or
maximum win (maxmin).
This is the guaranteed payoff of player A for any strategy of player B.
Hence,
.
(2)
The strategy corresponding to the maximin is called maximin strategy
.
Player IN interested in reducing the player's payoff A, choosing a strategy 
,
it takes into account the maximum possible payoff for A. Denote
.
(3)
Among all the numbers 
choose the smallest 
and call 
top game price
or minimax payoff
(minimax).
Ego guaranteed loss of player B
.
Therefore,
.
(4)
The minimax strategy is called minimax strategy.
The principle that dictates to players the choice of the most "cautious" minimax and maximin strategies is called minimax principle . This principle follows from the reasonable assumption that each player seeks to achieve the opposite goal of the opponent.
Theorem.The lower price of the game never exceeds 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 net price of the game,
or the price of the game.
The minimax strategies corresponding to the price of the game are optimal strategies
,
and their totality optimal solution
or game decision.
In this case the player A receives the maximum guaranteed (independent of the player's behavior) IN) win v, and the player IN achieves the minimum guaranteed (regardless of the player's behavior A) losing v. The solution to the game is said to have sustainability
,
those. if one of the players sticks to his optimal strategy, then it cannot be advantageous 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, this can never be beneficial; such a deviation of the player IN may at best leave the gain unchanged. and in the 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 to 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. Such a situation, if it exists, is called saddle point.
In geometry, a point on a surface that has the property: simultaneous minimum along one coordinate and maximum along the other, is called saddle
dot, by analogy this term is used in game theory.
The game for which
,
called saddle point game.
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, which differs 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 both its lower and upper price.
2) If one of the parties adheres to its optimal strategy, while the other deviates from its own, then the deviating party can only lose from this and in no case can increase its gain.
The class of games with a saddle point is of great interest both from the theoretical and practical points of view.
In game theory, it is proved that, in particular, every game with complete information has a saddle point, and, consequently, every such game has a solution, i.e., there is a pair of optimal strategies for one side and the other, giving an average payoff equal to the price of the game. If a game with complete information consists only of personal moves, then, when each side applies its own optimal strategy, it must always end in a quite definite outcome, namely, a payoff exactly equal to the price of the game.
Service assignment. With the help of the service online mode Can:- determine the price of the matrix game (lower and upper bounds), check for the presence of a saddle point, find a solution to the mixed strategy, find the minimax strategy of the players;
- write a mathematical model of a pair of dual linear programming problems, solve a matrix game using methods: minimax, simplex method, graphical (geometric) method, Brown's method.
Instruction. Select the dimension of the matrix, click Next. In the new dialog box, select a method for solving the matrix game. Fill example. Calculation results are presented in a report in Word format (see the design example).
A game is a mathematical model of a real conflict situation. Conflict situation two players is called a doubles game. It is convenient to study a zero-sum pair game if it is described in the form of a matrix. Such a game is called matrix; the matrix composed of numbers a ij is called payoff. The table shows the options for solving the game given by the payoff matrix A.Description of the algorithm:
- Based on the analysis of the payoff matrix, it is necessary to determine whether there are dominated strategies in it and exclude them.
- Find the upper and lower prices of the game and determine if this game saddle point (the lower price of the game must be equal to top price games).
- If a saddle point exists, then optimal strategies players that are the solution to the game will be their pure strategies corresponding to the saddle point. The price of the game is equal to the upper and lower prices of the game, which are equal to each other.
- If the game has no saddle point, then the solution of the game should be sought in mixed strategies. To determine the optimal mixed strategies in m × n games, the simplex method should be used, after reformulating the game problem into a linear programming problem.
Let us present the algorithm for solving a matrix game graphically.
Figure - Scheme for solving a matrix game.
Methods for solving a matrix game in mixed strategies
So, if there is no saddle point, the solution of the game is carried out in mixed strategies and solved by the following methods:- Solving the game through a system of equations.
If a square matrix nxn (n=m) is given, then the probability vector can be found by solving a system of equations. This method is not always used and is applicable only in some cases (if the matrix is 2x2, then the solution of the game is almost always obtained). If negative probabilities are obtained in the solution, then this system is solved by the simplex method. - The solution of the game by a graphical method.
In cases where n=2 or m=2 , the matrix game can be solved graphically. - Solution of a matrix game by the simplex method.
In this case, the matrix game reduces to
The essence of each decision taken by management is the choice of the best of several alternatives according to specific criteria established in advance. (If you want to revisit the discussion of constraints and decision criteria, see Chapter 6.) Payment matrix- this is one of the methods of statistical decision theory, a method that can help the manager in choosing one of several options. It is especially useful when a manager must determine which strategy will most contribute to the achievement of goals.
According to N. Paul Loomba:<Платеж представляет собой денежное вознаграждение или полезность, являющиеся следствием конкретной стратегии в сочетании с конкретными обстоятельствами. Если платежи представить в форме таблицы (или матрицы), мы получаем платежную матрицу>, as shown in fig. 8.4. Words<в сочетании с конкретными обстоятельствами>are very important to understand when to use the payoff matrix and assess when a decision based on it is likely to be reliable. In its most general form, 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 payoff matrix is useful when:
1. There is a reasonably limited number of alternatives or strategies to choose from.
2. What can happen is not known with complete certainty.
3. The results of the decision taken depend on which alternative is chosen and what events actually take place.
In addition, the manager must have the ability to objectively assess the likelihood of relevant events and calculate the expected value of such a probability. A leader rarely has complete certainty. But also rarely does he 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 foregoing discussion that the probability varies from 1, when the event will definitely occur, to 0, when the event will definitely not happen. Probability can be determined objectively, as a roulette player behaves when betting on odd numbers. The choice of its value can 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 outcome. For example, if we proceed from the fact that investors on a successful film can have 500% of the invested capital, and when investing in a distribution network, in the most favorable case, only 20%, then the decision should always be in favor of film production. However, if you take into account that the probability of a movie being a big success is very low, then investing in stores becomes more attractive, since the probability of obtaining the said 20% is very significant. To take a simpler example, the payouts for long-distance betting on horse races are higher because you are more likely to win nothing at all.
Probability directly affects the definition of expected value - the central concept of the payoff matrix. expected value of an alternative or variant of a strategy is 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 shop 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 dollars, then the expected value will be.
