3.4.1. Basic concepts of game theory
Currently, many solutions to problems in production, economic or commercial activities depend on the subjective qualities of the decision maker. When choosing decisions under conditions of uncertainty, an element of arbitrariness, and therefore risk, is always inevitable.
Problems of decision making under conditions of complete or partial uncertainty are dealt with by game theory and statistical solutions. Uncertainty can take the form of opposition from the other party, which pursues opposing goals, interferes with certain actions or states external environment. In such cases, it is necessary to take into account possible options for the behavior of the opposite party.
Possible behavior options for both sides and their outcomes for each combination of alternatives and states can be represented in the form mathematical model which is called a game. Both sides of the conflict cannot accurately predict mutual actions. Despite such uncertainty, each side of the conflict has to make decisions.
Game theory- This mathematical theory conflict situations. The main limitations of this theory are the assumption of the complete (“ideal”) rationality of the enemy and the adoption of the most cautious “reinsurance” decision when resolving the conflict.
The conflicting parties are called players, one implementation of the game – party, outcome of the game - winning or losing.
On the move in game theory is the choice of one of the actions provided for by the rules and its implementation.
Personally called the player's conscious choice of one of possible options actions and their implementation.
Random move called a player's choice that is not made by volitional decision player, but by some mechanism of random selection (tossing a coin, dealing cards, etc.) of one of the possible options for an action and its implementation.
Player strategy is a set of rules that determine the choice of action for each personal move of this player, depending on the situation that arises during the game
Optimal strategy player is a strategy that, when repeated multiple times in a game containing personal and random moves, provides the player with the maximum possible average winnings (or, what is the same, the minimum possible average loss).
Depending on the reasons causing uncertainty of outcomes, games can be divided into the following main groups:
- Combinatorial games in which the rules, in principle, allow each player to analyze everything various options behavior and, comparing these options, choose the best one. The uncertainty here is that there are too many options that need to be analyzed.
- Gambling games in which the outcome is uncertain due to the influence of random factors.
- Strategic games in which the uncertainty of the outcome is caused by the fact that each player, when making a decision, does not know what strategy the other participants in the game will follow, since there is no information about the subsequent actions of the opponent (partner).
- The game is called doubles, if the game involves two players.
- The game is called multiple, if there are more than two players in the game.
- The game is called zero sum, if each player wins at the expense of the others, and the sum of the winnings and losses of one side is equal to the other.
- Zero-sum doubles game called antagonistic game.
- The game is called finite, if each player has only a finite number of strategies. Otherwise it's a game endless.
- One step games when the player chooses one of the strategies and makes one move.
- In multi-step games Players make a series of moves to achieve their goals, which may be limited by the rules of the game or may continue until one of the players has no resources left to continue the game.
- Business games imitate organizational and economic interactions in various organizations and enterprises. The advantages of a game simulation over a real object are:
Visibility of the aftereffects of decisions made;
Variable time scale;
Repetition of existing experience with changes in settings;
Variable coverage of phenomena and objects.
Elements game model are:
- Participants of the game.
- Rules of the game.
- Information array, reflecting the state and movement of the modeled system.
Carrying out classification and grouping of games allows you to find general methods searching for alternatives in decision making, developing recommendations on the most rational course of action during the development of conflict situations in various fields of activity.
3.4.2. Setting game objectives
Consider a finite zero-sum pairs game. Player A has m strategies (A 1 A 2 A m), and player B has n strategies (B 1, B 2 Bn). Such a game is called a game of dimension m x n. Let a ij be the payoff of player A in a situation where player A chose strategy A i, and player B chose strategy B j. The player's payoff in this situation will be denoted by b ij . A zero-sum game, therefore, a ij = - b ij . To carry out the analysis, it is enough to know the payoff of only one of the players, say A.
If the game consists only of personal moves, then the choice of strategy (A i, B j) uniquely determines the outcome of the game. If the game also contains random moves, then the expected win is the average value (mathematical expectation).
Let us assume that the values of a ij are known for each pair of strategies (A i, B j). Let's create a rectangular table, the rows of which correspond to the strategies of player A, and the columns correspond to the strategies of player B. This table is called payment matrix.
Player A's goal is to maximize his winnings, and player B's goal is to minimize his loss.
Thus, the payment matrix looks like:
The task is to determine:
1) The best (optimal) strategy of player A from the strategies A 1 A 2 A m;
2) The best (optimal) strategy of player B from strategies B 1, B 2 Bn.
To solve the problem, the principle is applied according to which the participants in the game are equally intelligent and each of them does everything to achieve their goal.
3.4.3. Methods for solving game problems
Minimax principle
Let us analyze sequentially each strategy of player A. If player A chooses strategy A 1, then player B can choose such strategy B j, in which the payoff of player A will be equal to the smallest of the numbers a 1j. Let's denote it a 1:
that is, a 1 is the minimum value of all the numbers in the first line.
This can be extended to all rows. Therefore, player A must choose the strategy for which the number a i is the maximum.
Value a is a guaranteed win that player a can secure for himself for any behavior of player B. Value a is called the lower price of the game.
Player B is interested in reducing his loss, that is, reducing player A's winnings to a minimum. To choose the optimal strategy, he must find the maximum payoff value in each column and select the smallest among them.
Let's denote by b j the maximum value in each column:
Lowest value b j denote by b.
b = min max a ij
b is called upper limit games. The principle that dictates that players choose appropriate strategies is called the minimax principle.
There are matrix games for which the lower price of the game is equal to the upper price; such games are called saddle point games. In this case, g=a=b is called the net price of the game, and strategies A * i, B * j, allowing to achieve this value are called optimal. The pair (A * i, B * j) is called the saddle point of the matrix, since the element a ij .= g is simultaneously the minimum in the i-row and the maximum in the j-column. Optimal Strategies A * i, B * j, and the net price are the solution to the game in pure strategies, i.e., without involving a random selection mechanism.
Example 1.
Let a payment matrix be given. Find a solution to the game, i.e. determine the lower and upper prices of the game and minimax strategies.
Here a 1 =min a 1 j =min(5,3,8,2) =2
a =max min a ij = max(2,1,4) =4
b = min max a ij =min(9,6,8,7) =6
Thus, lower price of the game (a=4) corresponds to strategy A 3. By choosing this strategy, player A will achieve a payoff of at least 4 for any behavior of player B. The upper price of the game (b=6) corresponds to the strategy of player B. These strategies are minimax. If both sides follow these strategies, the payoff will be 4 (a 33).
Example 2.
The payment matrix is given. Find the lower and upper prices of the game.
a =max min a ij = max(1,2,3) =3
b = min max a ij =min(5,6,3) =3
Therefore, a =b=g=3. The saddle point is the pair (A * 3, B * 3). If a matrix game contains a saddle point, then its solution is found using the minimax principle.
Solving mixed strategy games
If the payment matrix does not contain a saddle point (a
To use mixed strategies, the following conditions are required:
1) There is no saddle point in the game.
2) Players use a random mixture of pure strategies with corresponding probabilities.
3) The game is repeated many times under the same conditions.
4) During each move, the player is not informed about the choice of strategy by the other player.
5) Averaging of game results is allowed.
It is proven in game theory that every zero-sum paired game has at least one mixed strategy solution, which implies that every finite game has a cost g. g- average winnings, per batch, satisfying condition a<=g<=b . Оптимальное решение игры в смешанных стратегиях обладает следующим свойством: каждый из игроков не заинтересован в отходе от своей оптимальной смешанной стратегии.
The players' strategies in their optimal mixed strategies are called active.
Theorem on active strategies.
The application of an optimal mixed strategy provides a player with a maximum average win (or minimum average loss) equal to the cost of the game g, regardless of what actions the other player takes, as long as he does not go beyond the limits of his active strategies.
Let us introduce the following notation:
P 1 P 2 ... P m - the probability of player A using strategies A 1 A 2 ..... A m ;
Q 1 Q 2 …Q n the probability of player B using strategies B 1, B 2….. Bn
We write the mixed strategy of player A in the form:
A 1 A 2…. A m
Р 1 Р 2 … Р m
We write the mixed strategy of player B as:
B 1 B 2…. Bn
Knowing the payment matrix A, you can determine the average winnings (mathematical expectation) M(A,P,Q):
M(A,P,Q)=S Sa ij P i Q j
Player A's average winnings:
a =max minM(A,P,Q)
Player B's average loss:
b = min maxM(A,P,Q)
Let us denote by P A * and Q B * the vectors corresponding to the optimal mixed strategies under which:
max minM(A,P,Q) = min maxM(A,P,Q)= M(A,P A * ,Q B *)
In this case, the following condition is satisfied:
maxM(A,P,Q B *)<=maxМ(А,P А * ,Q В *)<= maxМ(А,P А * ,Q)
Solving a game means finding the price of the game and optimal strategies.
Geometric method for determining game prices and optimal strategies
(For the game 2X2)
A segment of length 1 is plotted on the abscissa axis. The left end of this segment corresponds to strategy A 1, the right end to strategy A 2.
The y-axis shows the winnings a 11 and a 12.
The winnings a 21 and a 22 are plotted along a line parallel to the ordinate axis from point 1.
If player B uses strategy B 1, then we connect points a 11 and a 21, if B 2, then a 12 and a 22.
The average winning is represented by point N, the point of intersection of straight lines B 1 B 1 and B 2 B 2. The abscissa of this point is equal to P 2, and the ordinate of the game price is g.
Compared to the previous technology, the gain is 55%.
Game theory is a mathematical method for studying optimal strategies in games. The term “game” should be understood as the interaction of two or more parties who seek to realize their interests. Each side also has its own strategy, which can lead to victory or defeat, which depends on how the players behave. Thanks to game theory, it becomes possible to find the most effective strategy, taking into account ideas about other players and their potential.
Game theory is a special branch of operations research. In most cases, game theory methods are used in economics, but sometimes also in other social sciences, for example, political science, sociology, ethics and some others. Since the 70s of the 20th century, it also began to be used by biologists to study animal behavior and the theory of evolution. In addition, today game theory is very important in the field of cybernetics and. That's why we want to tell you about it.
History of game theory
Scientists proposed the most optimal strategies in the field of mathematical modeling back in the 18th century. In the 19th century, problems of pricing and production in a market with little competition, which later became classic examples of game theory, were considered by scientists such as Joseph Bertrand and Antoine Cournot. And at the beginning of the 20th century, outstanding mathematicians Emil Borel and Ernst Zermelo put forward the idea of a mathematical theory of conflict of interest.
The origins of mathematical game theory should be sought in neoclassical economics. Initially, the foundations and aspects of this theory were outlined in the work of Oscar Morgenstern and John von Neumann, “The Theory of Games and Economic Behavior” in 1944.
The presented mathematical field also found some reflection in social culture. For example, in 1998, Sylvia Nasar (American journalist and writer) published a book dedicated to John Nash, a Nobel Prize winner in economics and a game theorist. In 2001, based on this work, the film “A Beautiful Mind” was made. And a number of American television shows, such as “NUMB3RS”, “Alias” and “Friend or Foe” also refer to game theory from time to time in their broadcasts.
But a special mention should be made about John Nash.
In 1949, he wrote a dissertation on game theory, and 45 years later he was awarded the Nobel Prize in Economics. In the earliest concepts of game theory, games of the antagonistic type were analyzed, in which there are players who win at the expense of losers. But John Nash developed analytical methods according to which all players either lose or win.
The situations developed by Nash were later called “Nash equilibria.” They differ in that all sides of the game use the most optimal strategies, which creates a stable equilibrium. Maintaining balance is very beneficial for the players, because otherwise one change can negatively affect their position.
Thanks to the work of John Nash, game theory received a powerful impetus in its development. In addition, the mathematical tools of economic modeling were subjected to a major revision. John Nash was able to prove that the classical point of view on the issue of competition, where everyone plays only for themselves, is not optimal, and the most effective strategies are those in which players make themselves better by initially making others better.
Despite the fact that game theory initially included economic models in its field of view, until the 50s of the last century it was only a formal theory limited by the framework of mathematics. However, since the second half of the 20th century, attempts have been made to use it in economics, anthropology, technology, cybernetics, and biology. During the Second World War and after its end, game theory began to be considered by the military, who saw in it a serious apparatus for the development of strategic decisions.
During the 60-70s, interest in this theory faded, despite the fact that it gave good mathematical results. But since the 80s, the active application of game theory in practice began, mainly in management and economics. Over the past few decades, its relevance has grown significantly, and some modern economic trends are completely impossible to imagine without it.
It would also not be superfluous to say that a significant contribution to the development of game theory was made by the 2005 work “Strategy of Conflict” by Nobel Prize laureate in economics Thomas Schelling. In his work, Schelling examined many strategies used by participants in conflict interactions. These strategies coincided with conflict management tactics and analytical principles used in, as well as tactics that are used to manage conflict in organizations.
In psychological science and a number of other disciplines, the concept of “game” has a slightly different meaning than in mathematics. The cultural interpretation of the term “game” was presented in the book “Homo Ludens” by Johan Huizinga, where the author talks about the use of games in ethics, culture and justice, and also points out that the game itself is significantly superior to humans in age, because animals are also inclined play.
Also, the concept of “game” can be found in the concept of Eric Byrne, known from the book “”. Here, however, we are talking about exclusively psychological games, the basis of which is transactional analysis.
Application of game theory
If we talk about mathematical game theory, it is currently at the stage of active development. But the mathematical basis is inherently very expensive, for which reason it is used mainly only if the ends justify the means, namely: in politics, the economics of monopolies and the distribution of market power, etc. Otherwise, game theory is used in studies of human and animal behavior in a huge number of situations.
As already mentioned, game theory first developed within the boundaries of economic science, making it possible to determine and interpret the behavior of economic agents in various situations. But later, the scope of its application expanded significantly and began to include many social sciences, thanks to which game theory today explains human behavior in psychology, sociology and political science.
Experts use game theory not only to explain and predict human behavior - many attempts have been made to use this theory to develop benchmark behavior. In addition, philosophers and economists have long used it to try to understand good or worthy behavior as best as possible.
Thus, we can conclude that game theory has become a real turning point in the development of many sciences, and today it is an integral part of the process of studying various aspects of human behavior.
INSTEAD OF CONCLUSION: As you noticed, game theory is quite closely interconnected with conflictology - a science dedicated to the study of human behavior in the process of conflict interaction. And, in our opinion, this area is one of the most important not only among those in which game theory should be applied, but also among those that a person himself should study, because conflicts, whatever one may say, are part of our lives.
If you have a desire to understand what behavioral strategies exist in general, we suggest you take our self-knowledge course, which will fully provide you with such information. But, in addition, after completing our course, you will be able to conduct a comprehensive assessment of your personality in general. This means that you will know how to behave in case of conflict, and what are your personal advantages and disadvantages, life values and priorities, predispositions to work and creativity, and much more. In general, this is a very useful and necessary tool for anyone who strives for development.
Our course is on - feel free to begin self-knowledge and improve yourself.
We wish you success and the ability to be a winner in any game!
The Game Theory section is represented by three online calculators:
- Solving a matrix game. In such problems, a payment matrix is specified. It is required to find pure or mixed strategies of players and, game price. To solve, you must specify the dimension of the matrix and the solution method.
- Bimatrix game. Usually in such a game two matrices of the same size of payoffs of the first and second players are specified. The rows of these matrices correspond to the strategies of the first player, and the columns of the matrices correspond to the strategies of the second player. In this case, the first matrix represents the winnings of the first player, and the second matrix represents the winnings of the second.
- Games with nature. It is used when it is necessary to select a management decision according to the criteria of Maximax, Bayes, Laplace, Wald, Savage, Hurwitz.
In practice, we often encounter problems in which it is necessary to make decisions under conditions of uncertainty, i.e. situations arise in which two parties pursue different goals and the results of the actions of each party depend on the activities of the enemy (or partner).
A situation in which the effectiveness of a decision made by one party depends on the actions of the other party is called conflict. Conflict is always associated with some kind of disagreement (this is not necessarily an antagonistic contradiction).
The conflict situation is called antagonistic, if an increase in the winnings of one of the parties by a certain amount leads to a decrease in the winnings of the other side by the same amount, and vice versa.
In economics, conflict situations occur very often and are of a diverse nature. For example, the relationship between supplier and consumer, buyer and seller, bank and client. Each of them has their own interests and strives to make optimal decisions that help achieve their goals to the greatest extent. At the same time, everyone has to take into account not only their own goals, but also the goals of their partner and take into account the decisions that these partners will make (they may be unknown in advance). In order to make optimal decisions in conflict situations, a mathematical theory of conflict situations has been created, which is called game theory . The emergence of this theory dates back to 1944, when J. von Neumann’s monograph “Game Theory and Economic Behavior” was published.
The game is a mathematical model of a real conflict situation. The parties involved in the conflict are called players. The outcome of the conflict is called a win. The rules of the game are a system of conditions that determine the players’ options for action; the amount of information each player has about the behavior of their partners; the payoff that each set of actions leads to.
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. Players are designated A And B.
The game is called antagonistic (zero sum), if the gain of one of the players is equal to the loss of the other.
The choice and implementation of one of the options provided for by the rules is called progress player. Moves can be personal and random.
Personal move- this is a conscious choice by a player of one of the options for action (for example, in chess).
Random move is a randomly selected action (for example, throwing a dice). We will only consider personal moves.
Player strategy is a set of rules that determine the player’s behavior during each personal move. Usually during the game at each stage the player chooses a move depending on the specific situation. It is also possible that all decisions were made by the player in advance (i.e. the player chose a certain strategy).
The game is called ultimate, if each player has a finite number of strategies, and endless- otherwise.
Purpose of Game Theory– develop methods to determine the optimal strategy for each player.
The player's strategy is called optimal, if it provides this player with multiple repetitions of the game the maximum possible average win (or the minimum possible average loss regardless of the opponent’s behavior).
Example 1. Each of the players A or B, can write down, independently of the other, the numbers 1, 2 and 3. If the difference between the numbers written down by the players is positive, then A the number of points equal to the difference between the numbers wins. If the difference is less than 0, he wins B. If the difference is 0, it’s a draw.
Player A has three strategies (action options): A 1 = 1 (write 1), A 2 = 2, A 3 = 3, the player also has three strategies: B 1, B 2, B 3.
| B A | B 1 =1 | B2=2 | B 3 =3 |
| A 1 = 1 | 0 | -1 | -2 |
| A 2 = 2 | 1 | 0 | -1 |
| A 3 = 3 | 2 | 1 | 0 |
Player A's task is to maximize his winnings. Player B’s task is to minimize his loss, i.e. minimize the gain A. This zero-sum doubles game.
Preface
The purpose of this article is to familiarize the reader with the basic concepts of game theory. From the article, the reader will learn what game theory is, consider a brief history of game theory, and become familiar with the basic principles of game theory, including the main types of games and forms of their representation. The article will touch upon the classical problem and the fundamental problem of game theory. The final section of the article is devoted to consideration of the problems of using game theory for making management decisions and the practical application of game theory in management.
Introduction.
21 century. The age of information, rapidly developing information technologies, innovations and technological innovations. But why the information age? Why does information play a key role in almost all processes occurring in society? Everything is very simple. Information gives us invaluable time, and in some cases even the opportunity to get ahead of it. After all, it’s no secret that in life you often have to deal with tasks in which you need to make decisions in conditions of uncertainty, in the absence of information about responses to your actions, i.e. situations arise in which two (or more) parties pursue different goals, and the results of any action of each party depend on the activities of the partner. Such situations arise every day. For example, when playing chess, checkers, dominoes, and so on. Despite the fact that games are mainly entertaining in nature, by their nature they relate to conflict situations in which the conflict is already inherent in the goal of the game - the winning of one of the partners. At the same time, the result of each player’s move depends on the opponent’s response move. In economics, conflict situations occur very often and are of a diverse nature, and their number is so large that it is impossible to count all the conflict situations that arise in the market in at least one day. Conflict situations in the economy include, for example, relationships between supplier and consumer, buyer and seller, bank and client. In all of the above examples, the conflict situation is generated by the difference in interests of the partners and the desire of each of them to make optimal decisions that realize their goals to the greatest extent. At the same time, everyone has to take into account not only their own goals, but also the goals of their partner, and take into account the decisions unknown in advance that these partners will make. To competently solve problems in conflict situations, scientifically based methods are needed. Such methods are developed by the mathematical theory of conflict situations, which is called game theory.
What is game theory?
Game theory is a complex, multi-dimensional concept, so it seems impossible to interpret game theory using just one definition. Let's look at three approaches to defining game theory.
1.Game theory is a mathematical method for studying optimal strategies in games. A game is a process in which two or more parties participate, fighting for the realization of their interests. Each side has its own goal and uses some strategy that can lead to winning or losing - depending on the behavior of other players. Game theory helps to choose the best strategies, taking into account ideas about other participants, their resources and their possible actions.
2. Game theory is a branch of applied mathematics, or more precisely, operations research. Most often, game theory methods are used in economics, and a little less often in other social sciences - sociology, political science, psychology, ethics and others. Since the 1970s, it has been adopted by biologists to study animal behavior and the theory of evolution. Game theory is very important for artificial intelligence and cybernetics.
3.One of the most important variables on which the success of an organization depends is competitiveness. Obviously, the ability to predict the actions of competitors means an advantage for any organization. Game theory is a method for modeling the impact of a decision on competitors.
History of game theory
Optimal solutions or strategies in mathematical modeling were proposed back in the 18th century. The problems of production and pricing under oligopoly conditions, which later became textbook examples of game theory, were considered in the 19th century. A. Cournot and J. Bertrand. At the beginning of the 20th century. E. Lasker, E. Zermelo, E. Borel put forward the idea of a mathematical theory of conflict of interest.
Mathematical game theory originates from neoclassical economics. The mathematical aspects and applications of the theory were first outlined in the classic 1944 book by John von Neumann and Oscar Morgenstern, Game Theory and Economic Behavior.
John Nash, after graduating from the Carnegie Polytechnic Institute with two degrees - a bachelor's and a master's degree - entered Princeton University, where he attended lectures by John von Neumann. In his writings, Nash developed the principles of "managerial dynamics". The first concepts of game theory analyzed zero-sum games, where there are losers and winners at their expense. Nash develops methods of analysis in which everyone involved either wins or loses. These situations are called “Nash equilibrium” or “non-cooperative equilibrium”; in the situation, the parties use the optimal strategy, which leads to the creation of a stable equilibrium. It is beneficial for the players to maintain this balance, since any change will worsen their situation. These works of Nash made a serious contribution to the development of game theory, and the mathematical tools of economic modeling were revised. John Nash shows that A. Smith's classic approach to competition, where everyone is for himself, is suboptimal. More optimal strategies are when everyone tries to do better for themselves while doing better for others. In 1949, John Nash wrote a dissertation on game theory, and 45 years later he received the Nobel Prize in Economics.
Although game theory originally dealt with economic models, it remained a formal theory within mathematics until the 1950s. But already since the 1950s. attempts are beginning to apply game theory methods not only in economics, but in biology, cybernetics, technology, and anthropology. During World War II and immediately after it, the military became seriously interested in game theory, who saw in it a powerful tool for studying strategic decisions.
In 1960 - 1970 interest in game theory is fading, despite significant mathematical results obtained by that time. Since the mid-1980s. active practical use of game theory begins, especially in economics and management. Over the past 20 - 30 years, the importance of game theory and interest has been growing significantly; some areas of modern economic theory cannot be presented without the use of game theory.
A major contribution to the application of game theory was the work of Thomas Schelling, Nobel laureate in economics in 2005, “The Strategy of Conflict.” T. Schelling considers various “strategies” of behavior of the participants in the conflict. These strategies coincide with conflict management tactics and principles of conflict analysis in conflictology and organizational conflict management.
Basic principles of game theory
Let's get acquainted with the basic concepts of game theory. The mathematical model of a conflict situation is called game, parties involved in the conflict - players. To describe a game, you must first identify its participants (players). This condition is easily met when it comes to ordinary games such as chess, etc. The situation is different with “market games”. Here it is not always easy to recognize all the players, i.e. current or potential competitors. Practice shows that it is not necessary to identify all players; it is necessary to discover the most important ones. Games typically span several periods during which players take sequential or simultaneous actions. The choice and implementation of one of the actions provided for by the rules is called progress player. Moves can 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). Random move is a randomly selected action (for example, choosing a card from a shuffled deck). Actions may be related to prices, sales volumes, research and development costs, etc. The periods during which players make their moves are called stages games. The moves chosen at each stage ultimately determine "payments"(win or loss) of each player, which can be expressed in material assets or money. Another concept in this theory is player strategy. 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.) In other words, strategy refers to possible actions that allow the player at each stage of the game to choose from a certain number of alternative options the move that seems to him the “best response” to the actions of other players. Regarding the concept of strategy, it should be noted that the player determines his actions not only for the stages that a particular game has actually reached, but also for all situations, including those that may not arise during the course of a given game. The game is called steam room, if it involves two players, and multiple, if the number of players is more than two. For each formalized game, rules are introduced, i.e. 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 ½. A game is called a zero-sum game, or antagonistic, if the gain of one of the players is equal to the loss of the other, i.e., to complete the game, 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 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, you should choose a strategy for each player 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, i.e., it must be disadvantageous for any of the players to abandon their strategy in this game. If the game is repeated quite a few times, then players may be interested not in winning and losing in each specific game, but in average win (loss) in all batches. Purpose game theory is to determine the optimal strategies for each player. When choosing an optimal strategy, it is natural to assume that both players behave reasonably in terms of their interests.
Cooperative and non-cooperative
The game is called cooperative, or coalition, if players can unite in groups, taking on some obligations to other players and coordinating their actions. This differs from non-cooperative games in which everyone must play for themselves. Entertainment games are rarely cooperative, but such mechanisms are not uncommon in everyday life.
It is often assumed that what makes cooperative games different is the ability for players to communicate with each other. In general this is not true. There are games where communication is allowed, but the players pursue personal goals, and vice versa.
Of the two types of games, non-cooperative ones describe situations in great detail and produce more accurate results. Cooperatives consider the game process as a whole.
Hybrid games include elements of cooperative and non-cooperative games. For example, players can form groups, but the game will be played in a non-cooperative style. This means that each player will pursue the interests of his group, while at the same time trying to achieve personal gain.
Symmetrical and asymmetrical
|
Asymmetrical game |
||
The game will be symmetrical when the corresponding strategies of the players are equal, that is, they have the same payments. In other words, if players can change places and their winnings for the same moves will not change. Many two-player games studied are symmetrical. In particular, these are: “Prisoner’s Dilemma”, “Deer Hunt”. In the example on the right, the game at first glance may seem symmetrical due to similar strategies, but this is not so - after all, the payoff of the second player with strategy profiles (A, A) and (B, B) will be greater than that of the first.
Zero-sum and non-zero-sum
Zero-sum games are a special type of constant-sum games, that is, those where players cannot increase or decrease the available resources, or the game fund. In this case, the sum of all wins is equal to the sum of all losses for any move. Look to the right - the numbers represent payments to the players - and their sum in each cell is zero. Examples of such games include poker, where one wins all the others' bets; reversi, where enemy pieces are captured; or banal theft.
Many games studied by mathematicians, including the already mentioned “Prisoner’s Dilemma”, are of a different kind: in non-zero sum games One player's win does not necessarily mean another's loss, and vice versa. The outcome of such a game can be less or more than zero. Such games can be converted to zero sum - this is done by introducing fictitious player, which “appropriates” the surplus or makes up for the lack of funds.
Another game with a non-zero sum is trade, where every participant benefits. This also includes checkers and chess; in the last two, the player can turn his ordinary piece into a stronger one, gaining an advantage. In all these cases, the game amount increases. A well-known example where it decreases is war.
Parallel and serial
In parallel games, players move simultaneously, or at least they are not aware of others' choices until All won't make their move. In sequential, or dynamic In games, participants can make moves in a predetermined or random order, but at the same time they receive some information about the previous actions of others. This information may even be not quite complete, for example, a player can find out that his opponent from ten of his strategies definitely didn't choose fifth, without learning anything about the others.
The differences in the presentation of parallel and sequential games were discussed above. The former are usually presented in normal form, and the latter in extensive form.
With complete or incomplete information
An important subset of sequential games are games with complete information. In such a game, the participants know all the moves made up to the current moment, as well as the possible strategies of their opponents, which allows them to some extent predict the subsequent development of the game. Complete information is not available in parallel games, since the current moves of the opponents are unknown. Most games studied in mathematics involve incomplete information. For example, all the "salt" Prisoner's dilemmas lies in its incompleteness.
Examples of games with complete information: chess, checkers and others.
The concept of complete information is often confused with the similar one - perfect information. For the latter, it is enough just to know all the strategies available to opponents; knowledge of all their moves is not necessary.
Games with an infinite number of steps
Games in the real world, or games studied in economics, tend to last final number of moves. Mathematics is not so limited, and set theory in particular deals with games that can continue indefinitely. Moreover, the winner and his winnings are not determined until the end of all moves.
The task that is usually posed in this case is not to find an optimal solution, but to find at least a winning strategy.
Discrete and continuous games
Most of the games studied discrete: they have a finite number of players, moves, events, outcomes, etc. However, these components can be extended to many real numbers. Games that include such elements are often called differential games. They are associated with some kind of material scale (usually a time scale), although the events occurring in them can be discrete in nature. Differential games find their application in engineering and technology, physics.
Metagames
These are games that result in a set of rules for another game (called target or game-object). The goal of metagames is to increase the usefulness of the given ruleset.
Game presentation form
In game theory, along with the classification of games, the form of presentation of the game plays a huge role. Typically, a normal or matrix form is distinguished and an expanded form, specified in the form of a tree. These forms for a simple game are shown in Fig. 1a and 1b.
To establish a first connection with the realm of control, the game can be described as follows. Two enterprises producing similar products are faced with a choice. In one case, they can gain a foothold in the market by setting a high price, which will provide them with an average cartel profit P K . When entering into fierce competition, both receive a profit P W . If one of the competitors sets a high price, and the second sets a low price, then the latter realizes a monopoly profit P M , while the other incurs losses P G . A similar situation may arise, for example, when both firms must announce their price, which subsequently cannot be revised.
In the absence of strict conditions, it is beneficial for both enterprises to set a low price. The “low price” strategy is the dominant one for any firm: no matter what price a competing firm chooses, it is always preferable to set a low price. But in this case, firms face a dilemma, since profit P K (which for both players is higher than profit P W) is not achieved.
The strategic combination of “low prices/low prices” with corresponding payments represents a Nash equilibrium, in which it is disadvantageous for either player to separately deviate from the chosen strategy. This concept of equilibrium is fundamental in resolving strategic situations, but under certain circumstances it still requires improvement.
As for the above dilemma, its resolution depends, in particular, on the originality of the players' moves. If the enterprise has the opportunity to reconsider its strategic variables (in this case price), then a cooperative solution to the problem can be found even without a rigid agreement between the players. Intuition suggests that with repeated contact between players, opportunities arise to achieve acceptable “compensation”. Thus, under certain circumstances, it is inappropriate to strive for short-term high profits through price dumping if a “price war” may arise in the future.
As noted, both pictures characterize the same game. Presenting the game in normal form in the normal case reflects "synchronicity". However, this does not mean the “simultaneity” of events, but indicates that the player’s choice of strategy is carried out in ignorance of the opponent’s choice of strategy. In an expanded form, this situation is expressed through an oval space (information field). In the absence of this space, the game situation takes on a different character: first, one player would have to make a decision, and the other could do it after him.
Classic problem in game theory
Let's consider a classic problem in game theory. Deer hunting is a cooperative symmetric game from game theory that describes the conflict between personal interests and public interests. The game was first described by Jean-Jacques Rousseau in 1755:
"If they were hunting a deer, then everyone understood that for this he was obliged to remain at his post; but if a hare ran near one of the hunters, then there was no doubt that this hunter, without a twinge of conscience, would set off after him and, having overtaken the prey , very few will lament that in this way he deprived his comrades of prey."
Deer hunting is a classic example of the challenge of providing a public good while tempting man to give in to self-interest. Should the hunter remain with his comrades and bet on a less favorable opportunity to deliver large prey to the whole tribe, or should he leave his comrades and entrust himself to a more reliable opportunity that promises his own family a hare?
Fundamental problem in game theory
Consider a fundamental problem in game theory called the Prisoner's Dilemma.
Prisoner's dilemma A fundamental problem in game theory, players will not always cooperate with each other, even if it is in their best interest to do so. The player (the “prisoner”) is assumed to maximize his own payoff without caring about the gain of others. The essence of the problem was formulated by Meryl Flood and Melvin Drescher in 1950. The name of the dilemma was given by mathematician Albert Tucker.
In the prisoner's dilemma, betrayal strictly dominates over cooperation, so the only possible equilibrium is the betrayal of both participants. Simply put, no matter what the other player does, everyone will win more if they betray. Since in any situation it is more profitable to betray than to cooperate, all rational players will choose betrayal.
While behaving individually rationally, together the participants come to an irrational decision: if both betray, they will receive a smaller payoff in total than if they cooperated (the only equilibrium in this game does not lead to Pareto-optimal decision, i.e. a decision that cannot be improved without worsening the situation of other elements.). Therein lies the dilemma.
In a repeated prisoner's dilemma, the game occurs periodically, and each player can "punish" the other for not cooperating earlier. In such a game, cooperation can become an equilibrium, and the incentive to betray can be outweighed by the threat of punishment.
Classic Prisoner's Dilemma
In all judicial systems, the punishment for banditry (committing crimes as part of an organized group) is much heavier than for the same crimes committed alone (hence the alternative name - “the bandit's dilemma”).
The classic formulation of the prisoner's dilemma is:
Two criminals, A and B, were caught at about the same time for similar crimes. There is reason to believe that they acted in conspiracy, and the police, isolating them from each other, offer them the same deal: if one testifies against the other, and he remains silent, then the first is released for helping the investigation, and the second receives the maximum sentence imprisonment (10 years) (20 years). If both are silent, their act is charged under a lighter article, and they are sentenced to 6 months (1 year). If both testify against each other, they receive a minimum sentence of 2 years (5 years). Each prisoner chooses whether to remain silent or testify against the other. However, neither of them knows exactly what the other will do. What will happen?
The game can be represented in the form of the following table:
The dilemma arises if we assume that both are only concerned with minimizing their own prison term.
Let's imagine the reasoning of one of the prisoners. If your partner is silent, then it is better to betray him and go free (otherwise - six months in prison). If the partner testifies, then it is better to also testify against him in order to get 2 years (otherwise - 10 years). The “testify” strategy strictly dominates the “keep silent” strategy. Similarly, another prisoner comes to the same conclusion.
From the point of view of the group (these two prisoners), it is best to cooperate with each other, remain silent and get six months each, as this will reduce the total prison term. Any other solution will be less profitable.
Generalized form
- The game consists of two players and a banker. Each player holds 2 cards: one says “cooperate”, the other says “defect” (this is the standard terminology of the game). Each player places one card face down in front of the banker (that is, no one knows anyone else's decision, although knowing someone else's decision does not affect dominance analysis). The banker opens the cards and gives out the winnings.
- If both choose to cooperate, both receive C. If one chose “to betray”, the other “to cooperate” - the first receives D, second With. If both chose “betray”, both receive d.
- The values of the variables C, D, c, d can be of any sign (in the example above, all are less than or equal to 0). The inequality D > C > d > c must be satisfied for the game to be a Prisoner's Dilemma (PD).
- If the game is repeated, that is, played more than 1 time in a row, the total payoff from cooperation must be greater than the total payoff in a situation where one betrays and the other does not, that is, 2C > D + c.
These rules were established by Douglas Hofstadter and form the canonical description of the typical prisoner's dilemma.
Similar but different game
Hofstadter suggested that people understand problems like the prisoner's dilemma more easily if they are presented as a separate game or trading process. One example is “ exchange of closed bags»:
Two people meet and exchange closed bags, realizing that one of them contains money, the other contains goods. Each player can respect the deal and put what was agreed upon in the bag, or deceive the partner by giving an empty bag.
In this game, cheating will always be the best solution, which also means that rational players will never play the game and that there will be no market for trading closed bags.
Application of game theory to make strategic management decisions
Examples include decisions regarding the implementation of a principled pricing policy, entry into new markets, cooperation and the creation of joint ventures, identifying leaders and performers in the field of innovation, vertical integration, etc. The principles of game theory can in principle be used for all types of decisions if they are influenced by other actors. These individuals, or players, do not necessarily have to be market competitors; their role may be subsuppliers, leading customers, employees of organizations, as well as work colleagues.
It is especially advisable to use game theory tools when there are important dependencies between the participants in the process in the field of payments. The situation with possible competitors is shown in Fig. 2.
Quadrants 1 And 2 characterize a situation where the reaction of competitors does not have a significant impact on the company's payments. This happens in cases where the competitor has no motivation (field 1 ) or capabilities (field 2 ) strike back. Therefore, there is no need for a detailed analysis of the strategy of motivated actions of competitors.
A similar conclusion follows, although for a different reason, and for the situation reflected by the quadrant 3 . Here, the reaction of competitors could have a significant impact on the company, but since its own actions cannot greatly affect the payments of a competitor, then one should not be afraid of its reaction. An example is decisions to enter a market niche: under certain circumstances, large competitors have no reason to react to such a decision of a small company.
Only the situation shown in the quadrant 4 (the possibility of retaliatory steps by market partners) requires the use of game theory provisions. However, these are only necessary but not sufficient conditions to justify the use of a game theory framework to combat competitors. There are situations when one strategy will undoubtedly dominate all others, regardless of what actions the competitor takes. If we take, for example, the drug market, then it is often important for a company to be the first to introduce a new product on the market: the profit of the “first mover” turns out to be so significant that all other “players” can only quickly intensify their innovation activities.
A trivial example of a “dominant strategy” from the standpoint of game theory is the decision regarding penetration into a new market. Let's take an enterprise that acts as a monopolist in any market (for example, IBM in the personal computer market in the early 80s). Another enterprise, operating, for example, in the market of computer peripheral equipment, is considering the issue of penetrating the personal computer market by reconfiguring its production. An outsider company may decide to enter or not to enter the market. A monopolist company can react aggressively or friendly to the emergence of a new competitor. Both companies enter into a two-stage game in which the outsider company makes the first move. The game situation indicating payments is shown in the form of a tree in Fig. 3.
The same game situation can be presented in normal form (Fig. 4).
There are two states indicated here - “entry/friendly reaction” and “non-entry/aggressive reaction”. Obviously, the second equilibrium is untenable. From the expanded form it follows that for a company that has already established a foothold in the market, it is inappropriate to react aggressively to the emergence of a new competitor: with aggressive behavior, the current monopolist receives 1 (payment), and with friendly behavior - 3. The outsider company also knows that it is not rational for the monopolist begin actions to displace it, and therefore it decides to enter the market. The outsider company will not bear the threatened losses of (-1).
Such rational equilibrium is characteristic of a “partially improved” game, which deliberately excludes absurd moves. In practice, such equilibrium states are, in principle, quite easy to find. Equilibrium configurations can be identified using a special algorithm from the field of operations research for any finite game. The decision maker proceeds as follows: first, the choice of the “best” move at the last stage of the game is made, then the “best” move is selected at the previous stage, taking into account the choice at the last stage, and so on, until the starting node of the tree is reached games.
How can companies benefit from game theory-based analysis? For example, there is a well-known case of conflict of interests between IBM and Telex. In connection with the announcement of the latter's preparatory plans for entering the market, a “crisis” meeting of IBM management was held, at which measures aimed at forcing the new competitor to abandon its intention to penetrate the new market were analyzed. Telex apparently became aware of these events. An analysis based on game theory showed that threats to IBM due to high costs are unfounded. This suggests that it is useful for companies to consider the possible reactions of their gaming partners. Isolated economic calculations, even those based on decision-making theory, are often, as in the situation described, limited in nature. Thus, an outsider company could choose the “non-entry” move if a preliminary analysis convinced it that market penetration would cause an aggressive reaction from the monopolist. In this case, in accordance with the expected value criterion, it is reasonable to choose the “non-intervention” move with a probability of an aggressive response of 0.5.
The following example is related to the rivalry of companies in the field technological leadership. The starting situation is when the enterprise 1 previously had technological superiority, but currently has fewer financial resources for research and development (R&D) than its competitor. Both companies must decide whether to try to achieve global market dominance in their respective technology area through large capital investments. If both competitors invest large amounts of money in the business, then the prospects for success of the enterprise 1 will be better, although it will incur large financial expenses (like the enterprise 2 ). In Fig. 5 this situation is represented by payments with negative values.
For enterprise 1 it would be best if the enterprise 2 refused to compete. His benefit in this case would be 3 (payments). Most likely the enterprise 2 would win the competition when the enterprise 1 would accept a reduced investment program, and the enterprise 2 - wider. This position is reflected in the upper right quadrant of the matrix.
Analysis of the situation shows that equilibrium occurs at high R&D costs of the enterprise 2 and low enterprises 1 . In any other scenario, one of the competitors has a reason to deviate from the strategic combination: for example, for an enterprise 1 a reduced budget is preferable if the enterprise 2 will refuse to participate in competition; at the same time to the enterprise 2 It is known that when a competitor’s costs are low, it is profitable for him to invest in research and development.
An enterprise with a technological advantage can resort to analyzing the situation based on game theory in order to ultimately achieve the optimal result for itself. With the help of a certain signal, it must show that it is ready to make large expenditures on research and development. If such a signal is not received, then for the enterprise 2 it is clear that the enterprise 1 chooses the low cost option.
The reliability of the signal must be evidenced by the enterprise's obligations. In this case, it may be the decision of the enterprise 1 on the purchase of new laboratories or the hiring of additional research personnel.
From the point of view of game theory, such obligations are equivalent to changing the course of the game: the situation of simultaneous decision-making is replaced by a situation of sequential moves. Company 1 firmly demonstrates the intention to make large expenditures, the enterprise 2 registers this step and he no longer has any reason to participate in the rivalry. The new equilibrium follows from the scenario “non-participation of the enterprise 2 " and "high costs of research and development of the enterprise 1 ".
Well-known areas of application of game theory methods also include pricing strategy, creation of joint ventures, timing of new product development.
Important contributions to the use of game theory come from experimental work. Many theoretical calculations are tested in laboratory conditions, and the results obtained serve as an impetus for practitioners. Theoretically, it was clarified under what conditions it is advisable for two selfishly minded partners to cooperate and achieve better results for themselves.
This knowledge can be used in enterprise practice to help two firms achieve a win/win situation. Today, gaming-trained consultants quickly and clearly identify opportunities that businesses can take advantage of to secure stable, long-term contracts with customers, sub-suppliers, development partners, and the like.
Problems of practical application in management
Of course, it should be pointed out that there are certain limits to the application of the analytical tools of game theory. In the following cases, it can only be used if additional information is obtained.
Firstly, this is the case when businesses have different ideas about the game they are playing, or when they are not sufficiently informed about each other's capabilities. For example, there may be unclear information about a competitor's payments (cost structure). If information that is not too complex is characterized by incompleteness, then one can operate by comparing similar cases, taking into account certain differences.
Secondly, Game theory is difficult to apply to many equilibrium situations. This problem can arise even during simple games with simultaneous strategic decisions.
Third, If the strategic decision-making situation is very complex, then players often cannot choose the best options for themselves. It is easy to imagine a more complex market penetration situation than the one discussed above. For example, several enterprises may enter the market at different times, or the reaction of enterprises already operating there may be more complex than being aggressive or friendly.
It has been experimentally proven that when the game expands to ten or more stages, players are no longer able to use the appropriate algorithms and continue the game with equilibrium strategies.
Game theory is not used very often. Unfortunately, real-world situations are often very complex and change so quickly that it is impossible to accurately predict how competitors will react to a firm's changing tactics. However, game theory is useful when it comes to identifying the most important factors to consider in a competitive decision-making situation. This information is important because it allows management to consider additional variables or factors that may affect the situation, thereby increasing the effectiveness of the decision.
In conclusion, it should be especially emphasized that game theory is a very complex field of knowledge. When handling it, you must be careful and clearly know the limits of its use. Too simple interpretations, whether adopted by the firm itself or with the help of consultants, are fraught with hidden dangers. Due to their complexity, game theory analysis and consultation are recommended only for particularly important problem areas. The experience of firms shows that the use of appropriate tools is preferable when making one-time, fundamentally important planned strategic decisions, including when preparing large cooperation agreements.
Bibliography
1. Game theory and economic behavior, von Neumann J., Morgenstern O., Science publishing house, 1970
2. Petrosyan L.A., Zenkevich N.A., Semina E.A. Game theory: Textbook. manual for universities - M.: Higher. school, Book House "University", 1998
3. Dubina I. N. Fundamentals of the theory of economic games: textbook. - M.: KNORUS, 2010
4. Archive of the journal "Problems of Theory and Practice of Management", Rainer Voelker
5. Game theory in the management of organizational systems. 2nd edition., Gubko M.V., Novikov D.A. 2005
- J. J. Rousseau. Reasoning about the origin and foundations of inequality between people // Treatises / Trans. from French A. Khayutina - M.: Nauka, 1969. - P. 75.
In practical activities, it is often necessary to make decisions in the face of opposition from the other party, which may pursue opposing or other goals, or hinder the achievement of the intended goal by certain actions or states of the external environment. Moreover, these influences from the opposite side can be passive or active. In such cases, it is necessary to take into account possible behavior options of the opposite party, retaliatory actions and their possible consequences.
Possible behavior options for both parties and their outcomes for each combination of options and states are often presented in the form of a mathematical model, which is called a game .
If the opposing party is an inactive, passive party that does not consciously oppose the achievement of the intended goal, then this game is called playing with nature. Nature is usually understood as a set of circumstances in which decisions have to be made (uncertainty of weather conditions, unknown behavior of customers in commercial activities, uncertainty of the population’s reaction to new types of goods and services, etc.)
In other situations, the opposite party actively, consciously opposes the achievement of the intended goal. In such cases, there is a clash of opposing interests, opinions, and ideas. Such situations are called conflict , and decision-making in a conflict situation is difficult due to the uncertainty of the enemy’s behavior. It is known that the enemy deliberately seeks to take the least beneficial actions for you in order to ensure the greatest success. It is unknown to what extent the enemy knows how to assess the situation and possible consequences, how he assesses your capabilities and intentions. Both sides cannot predict mutual actions. Despite such uncertainty, each side of the conflict has to make a decision
In economics, conflict situations occur very often and are of a diverse nature. These include, for example, the relationship between supplier and consumer, buyer and seller, bank and client, etc. In all these examples, the conflict situation is generated by the difference in the interests of partners and the desire of each of them to make optimal decisions. At the same time, everyone has to take into account not only their own goals, but also the goals of their partner and take into account his possible actions unknown in advance.
The need to justify optimal decisions in conflict situations has led to the emergence game theory.
Game theory - this is a mathematical theory of conflict situations. The starting points of this theory are the assumption of the complete “ideal” rationality of the enemy and the adoption of the most cautious decision when resolving the conflict.
The conflicting parties are called players , one implementation of the game – party , the outcome of the game is winning or losing . Any possible action for a player (within the given rules of the game) is called his strategy .
The point of the game is that each player, within the given rules of the game, strives to apply the strategy that is optimal for him, that is, the strategy that will lead to the best outcome for him. One of the principles of optimal (expedient) behavior is the achievement of an equilibrium situation, in the violation of which none of the players is interested.
It is the situation of equilibrium that can be the subject of stable agreements between players. In addition, equilibrium situations are beneficial for each player: in an equilibrium situation, each player receives the largest payoff, to the extent that it depends on him.
Mathematical model of a conflict situation called a game , the parties involved in the conflict, are called players.
For each formalized game, rules are introduced. In general, the rules of the game establish the players' options for action; the amount of information each player has about the behavior of their partners; the payoff that each set of actions leads to.
The development of the game over time occurs sequentially, in stages or moves. A move in game theory is called selection of one of the actions provided for by the rules of the game and its implementation. Moves are personal and random. Personally call the player’s conscious choice of one of the possible options for action and its implementation. Random move they call a choice made not by the player’s volitional decision, but by some kind of random selection mechanism (tossing a coin, passing, dealing cards, etc.).
Depending on the reasons causing uncertainty of outcomes, games can be divided into the following main groups:
Combined games, in which the rules provide, in principle, the opportunity for each player to analyze all the various options for his behavior and, having compared these options, choose the one that leads to the best outcome for this player. The uncertainty of the outcome is usually due to the fact that the number of possible behavior options (moves) is too large and the player is practically unable to sort through and analyze them all.
Gambling , in which the outcome is uncertain due to the influence of various random factors. Gambling games consist only of random moves, the analysis of which uses the theory of probability. Mathematical game theory does not deal with gambling.
Strategy games , in which the complete uncertainty of choice is justified by the fact that each of the players, when making a decision on the choice of the upcoming move, does not know what strategy the other participants in the game will follow, and the player’s ignorance of the behavior and intentions of the partners is fundamental, since there is no information about subsequent actions of the enemy (partner).
There are games that combine the properties of combined and gambling games; the strategic nature of games can be combined with combinatoriality, etc.
Depending on the number of participants in the game are divided into paired and multiple. In a doubles game the number of participants is two, in a multiple game the number of participants is more than two. Participants in a multiple game can form coalitions. In this case the games are called coalition . A multiple game becomes a double game if its participants form two permanent coalitions.
One of the basic concepts of game theory is strategy. Player strategy is a set of rules that determine the choice of action for each personal move of this player, depending on the situation that arises during the game.
Optimal strategy A player is called a strategy that, when repeated many times in a game containing personal and random moves, provides the player with the maximum possible average win or minimum possible loss, regardless of the opponent’s behavior.
The game is called ultimate , if the number of player strategies is finite, and endless , if at least one of the players has an infinite number of strategies.
In multi-move game theory problems, the concepts of “strategy” and “option of possible actions” are significantly different from each other. In simple (one-move) game problems, when in each game each player can make one move, these concepts coincide, and, therefore, the set of player strategies covers all possible actions that he can take in any possible situation and under any possible actual situation. information.
Games are also differentiated by the amount of winnings. The game is called game with zero sum th, if each player wins at the expense of the others, and the amount of winning of one side is equal to the amount of loss of the other. In a zero-sum doubles game, the interests of the players are directly opposed. A zero-sum pairs game is called Iantagonistic game .
Games in which one player's gain and another's loss are not equal are callednon-zero sum games .
There are two ways to describe games: positional and normal . The positional method is associated with the expanded form of the game and is reduced to a graph of successive steps (game tree). The normal way is to explicitly represent the set of player strategies and payment function . The payment function in the game determines the winnings of each side for each set of strategies chosen by the players.
