Game Theory: An Introduction

What it is and what it is eaten with.

Game theory is a branch of mathematical economics that studies the solution of conflicts between players and the optimality of their strategies. Conflict can relate to different areas of human interest: most often it is economics, sociology, political science, less often biology, cybernetics, and even military affairs. A conflict is any situation in which the interests of two or more participants, traditionally called players, are affected. For each player there is a certain set of strategies that he can apply. Intersecting, the strategies of several players create a certain situation in which each player gets a certain result, called a gain, positive or negative. When choosing a strategy, it is important to consider not only obtaining the maximum profit for yourself, but also the possible steps of the enemy, and their influence on the situation as a whole.


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Brief history of development.

The foundations of game theory originated in the 18th century, with the beginning of the era of enlightenment and the development of economic theory. For the first time, the mathematical aspects and applications of the theory were outlined in the classic 1944 book by John von Neumann and Oscar Morgenstern, Game Theory and Economic Behavior. The first concepts of game theory were analyzed by antagonistic games, when there are losers and winners at their expense. Despite the fact that game theory considered economic models, until the 1950s it was just a mathematical theory. After, as a result of a sharp jump in the US economy after the Second World War, and, as a result, greater funding for science, attempts are being made to put practical use of game theory in economics, biology, cybernetics, technology, anthropology. During World War II and immediately after it, the military became seriously interested in game theory, who saw in it a powerful device for researching strategic decisions. In the early 50s, John Nash (pictured) develops analysis methods in which all participants either win or lose. These situations are called the Nash equilibrium . According to his theory, the parties must use the optimal strategy, which leads to the creation of a stable equilibrium. It is beneficial for players to maintain this balance, since any change will worsen their situation. These works of Nash made a significant contribution to the development of game theory, mathematical tools of economic modeling were revised. John Nash shows that A.Smith’s classic approach to competition, when every man for himself, is not optimal. More optimal strategies when everyone tries to do better for themselves, doing better for others. Over the past 20–30 years, the value of game theory and interest has grown significantly, some areas of modern economic theory cannot be expounded without the use of game theory.

How it works

It seems to me that the meaning of the theory of games is easiest to explain in the Prisoner's Dilemma, the classic wording of which is:

 ,   ,           .   ,     ,  ,     ,       :     ,    ,      ,        (10 ).   ,       ,     6 .      ,     ( 2 ).   ,     .        ,   .  ? 

Representing the game in the form of a matrix, we get:

	Criminal B Silent strategy	Criminal B Strategy "betray"
Criminal A Silent strategy	Half a year to each	10 years to criminal A Release the criminal B
Criminal A Strategy "betray"	10 years to criminal B Release the criminal A	2 years each

And now imagine the development of the situation, putting yourself in the place of prisoner A. If my accomplice is silent, it is better to surrender him and go free. If he says, then it is also better to tell everything, and get only two years, instead of ten. Thus, if each player chooses what is best for him, both will surrender each other, and get two years, which is not the ideal situation for both. If everyone thought about the common good, they would receive only half a year.

Types of games

Cooperative \ non-cooperative game

A cooperative game is a conflict in which players can communicate with each other and form groups to achieve the best result. An example of a cooperative game is the Bridge game, where each player’s points are counted individually, but the pair with the highest amount wins. Of the two types of games, non-cooperative describe the situation in the smallest detail and give more accurate results. Cooperative consider the process of the game as a whole. Despite the fact that these two types are opposite to each other, it is quite possible to combine strategies, which can bring more benefits than following any one.

Zero-sum and non-zero-sum

A zero-sum game is a game in which the winning of one player equals the loss of another. For example, a banal dispute: if you won the sum of N, then someone lost the same amount of N. In a game with a non-zero amount, the total price of the game may change, thus bringing benefits to one player, does not take away its price from another. As an example, chess is perfect here: transforming a pawn into a queen, player A increases the total amount of his pieces, without taking anything from player B. In games with a nonzero amount, losing one of the players is not a prerequisite, although this outcome is not excluded.

Parallel and serial

Parallel is a game in which players make moves simultaneously, or the move of one player is unknown to another, until the overall cycle is completed. In a sequential game, each player has information about the previous course of his opponent before making his choice. And not necessarily the information to be complete, which brings on with the next type.

With complete or incomplete information

These types are subspecies of consecutive games, and their names speak for themselves.

Meta Games

These games are "lemmas" of game theory. They are useful not by themselves, but in the context of any conflict, extending its set of rules.

In any conflict, the types are combined, thus defining the rules of the game, be it a cooperative, successive zero-sum game, or a metagame with incomplete information.

Problems of practical application

Of course, it should be noted that there are certain limits to the use of analytical tools of game theory. In the following cases, it can be used only if additional information is provided.

First, this is the case when the players have different ideas about the game in which they participate, or when they are not sufficiently informed about the capabilities of each other. For example, there may be unclear information about competitor payments (cost structure). If incompleteness is characterized by not too complicated information, then one can apply the experience of such cases, taking into account certain differences.

Secondly, game theory is difficult to apply in a variety of equilibrium situations. This problem can occur even in the course of simple games with simultaneous selection of strategic solutions.

Thirdly, if the situation of making strategic decisions is very complicated, players often cannot choose the best options for themselves. For example, several enterprises may enter the market at different times or the reaction of enterprises already operating there may be more complicated than being aggressive or friendly.

It has been experimentally proven that when a game is expanded to ten or more stages, players are no longer able to use the appropriate algorithms and continue the game with equilibrium strategies.

Unfortunately, real-world situations are often very complex and change so quickly that it is impossible to predict exactly how competitors will react to changing tactics. Nevertheless, game theory is useful when it is necessary to determine the most important factors that need to be taken into account in a decision-making situation in a competitive environment. This information is important because it allows you to take into account additional variables or factors that have the ability to influence the situation, and thereby increase the effectiveness of the decision.

Conclusion

In conclusion, it should be emphasized that game theory is a very complex area of ​​knowledge. When referring to it, it is necessary to observe a certain caution and clearly know the limits of application. Too simple interpretations pose a hidden danger. Analysis and consultations based on game theory because of their complexity are recommended only for particularly important problem areas. Experience shows that the use of appropriate tools is preferable when making one-time, fundamentally important strategic planning decisions, including when preparing large cooperation agreements.

If the topic turns out to be interesting for the community, in the following articles I will try to reveal the types of games and their strategies in more detail.