**Game Theory** . It consists of circular reasoning, which cannot be avoided when considering strategic issues. By nature, humans are not very good at thinking about the problems of strategic relationships, since the solution is generally the reverse logic .

## Summary

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- 1 Importance of the Game
- 2 History of Theory
- 3 Applications
- 4 Symmetrical and asymmetric games
- 1 Symmetrical play
- 2 Asymmetric Play

- 5 Economy and business
- 1 Descriptive
- 2 Regulations

- 6 Biology
- 7 Computing and Logic
- 8 Political science
- 9 Philosophy
- 10 Game Theory and Statistics
- 11 References
- 12 External links

## Importance of the Game

Psychologists emphasize the importance of play in childhood as a means of forming personality and learning experimentally to relate in society, to solve problems and conflict situations.

All games, for children and adults, board games or sports games, are models of conflictive and cooperative situations in which we can recognize situations and patterns that are frequently repeated in the real world.

## History of the Theory

It was developed in its beginnings as a tool to understand the behavior of the Economy . Game Theory is currently used in many fields, such as Biology , Sociology , Psychology, and Philosophy .

It experienced substantial growth and was first formalized from the works of Johnn Neumann and Oskar Morgenstern , before and during the Cold War , mainly due to its application to the Military Strategy in particular because of the concept of mutual guaranteed destruction. Since the 1970s, game theory has been applied to animal behavior, including the development of species by natural selection.

As a result of games like the prisoner’s dilemma, in which generalized selfishness hurts players, Game Theory has also attracted the attention of researchers in Computer Science , being used in Artificial Intelligence and Cybernetics .

Games theory. Area of applied mathematics that uses models to study interactions in formalized incentive structures (the so-called games) and carry out decision processes. Its researchers study optimal strategies as well as the predicted and observed behavior of individuals in games. Apparently different types of interaction can, in fact, present similar incentive structures and, therefore, the same game can be represented a thousand times together.

Initially developed as a tool to understand the behavior of economics, game theory is currently used in many fields, from biology to philosophy. It experienced substantial growth and was formalized for the first time from the work of John von Neumann and Oskar Morgenstern, before and during the Cold War, mainly due to its application to military strategy – in particular because of the concept of mutual destruction. guaranteed.

Since the 1970s, game theory has been applied to animal behavior, including the development of species by natural selection. Following games like the prisoner’s dilemma, in which widespread selfishness hurts players, game theory has been used in economics, political science, ethics, and philosophy. Finally, it has also attracted the attention of computer researchers, being used in artificial intelligence and cybernetics.

Although it has some points in common with decision theory, game theory studies decisions made in environments where they interact. In other words, study the choice of optimal behavior when the costs and benefits of each option are not fixed in advance, but depend on the choices of other individuals.

A well-known example of the application of game theory to real life is the Prisoner’s Dilemma, popularized by mathematician Albert W. Tucker, which has many implications for understanding the nature of human cooperation. The psychological theory of games, which is rooted in the psychoanalytic school of transactional analysis, is entirely different.

Game analysts assiduously use other areas of mathematics, in particular Probabilities, Statistics, and Linear Programming, in conjunction with game theory. In addition to its academic interest, game theory has received the attention of popular culture. The life of the theoretical mathematician John Forbes Nash , developer of the Nash Balance and who received a Nobel Prize, was the subject of the biography written by Sylvia Nasar, A brilliant mind ( 1998 ), and of the film of the same name ( 2001). Various television programs have explored game theory situations, such as the Catalan television competition (TV3) Sis a traïció (six to treason), the American television program Friend or foe? (Friend or Foe?) And, to some extent, the Survivors contest.

The first known discussion of game theory appears in a letter written by James Waldegrave in 1713 . In this card, Waldegrave provides a mixed strategy minimax solution to a two-person version of the Her card game. However, a theoretical analysis of game theory in general was not published until the publication of Antoine Augustin Cournot’s Recherches sur les princes mathématiques de la théorie des richesses in 1838 . In this paper, Cournot considers a duopoly and presents a solution that is a restricted version of the Nash Equilibrium.

Although Cournot’s analysis is more general than Waldegrave’s, game theory did not really exist as a separate field of study until John von Neumann published a series of articles in 1928 . These results were later expanded on in his 1944 book , The Theory of Games and Economic Behavior, written together with Oskar Morgenstern.

This work contains a method to find optimal solutions for two-person zero-sum games. During this period, work on game theory focused primarily on cooperative game theory. This type of game theory analyzes the optimal strategies for groups of individuals, assuming that they can establish agreements among themselves about the most appropriate strategies.

In 1950 , the first discussions of the prisoner’s dilemma appeared, and an experiment was started on this game at the RAND Corporation. Around this same time, John Nash developed a definition of an optimal strategy for multiplayer games where the optimal was not previously defined, known as Nash equilibrium. This balance is general enough, allowing analysis of non-cooperative games in addition to cooperative games.

Game theory underwent notable activity in the 1950s, at which time the core concepts, extensive play, fictional play, repetitive play, and Shapley value were developed. In addition, at that time, the first applications of game theory appeared in philosophy and political science.

In 1965 , Reinhard Selten introduced his solution concept of subgame perfect equilibria, which further refined the Nash equilibrium. In 1967 John Harsanyi developed the concepts of complete information and Bayesian games. He, along with John Nash and Reinhard Selten, won the Nobel Prize for Economics in 1994 .

In the 1970s game theory was applied extensively to biology, largely as a result of the work of John Maynard Smith and his concept of stable evolutionary strategy. Furthermore, the concepts of correlated balance, perfection of hand tremor, and common knowledge were introduced and analyzed.

In 2005 , game theorists Thomas Schelling and Robert Aumann won the Nobel Prize in Economics. Schelling worked on dynamic models, the earliest examples of evolutionary game theory. For his part, Aumann contributed more to the school of balance.

In 2007 Roger Myerson, along with Leonid Hurwicz and Eric Maskin, received the Nobel Prize in Economics for “laying the foundation for mechanism design theory.”

## Applications

Although it has some points in common with Decision Theory, game theory studies decisions made in environments where they interact. In other words, study the choice of optimal behavior when the costs and benefits of each option are not fixed in advance, but depend on the choices of other individuals.

A well-known example of the application of game theory to real life is the Prisoner’s Dilemma, popularized by mathematician Albert W. Tucker , which has many implications for understanding the nature of human cooperation.

Game theory has the characteristic of being an area in which the underlying substance is mainly a category of applied mathematics , but most of the fundamental research is carried out by specialists in other areas. In some universities it is taught and researched almost exclusively outside the mathematics department.

This theory has applications in numerous areas, among which it is worth highlighting the economic sciences, evolutionary biology, psychology, political science, operational research, computer science, and military strategy.

The psychological theory of games , which is rooted in the psychoanalytic school of transactional analysis, is entirely different.

## Symmetric and asymmetric games

### Symmetrical play

A Symmetric Game is a game in which the rewards for playing a particular strategy depend only on the strategies that other players employ and not on who plays them.

If player identities can be changed without changing strategy rewards, then the game is symmetrical. Many of the most studied 2 × 2 Games are symmetrical.

The standard depictions of the Game of the Hen , the Prisoner’s Dilemma, and the Deer Hunt are symmetrical games.

### Asymmetric Play

The most studied asymmetric games are games where there are no identical sets of strategies for both players. For example, the Ultimatum Game and the Dictator Game have different strategies for each player.

However, there may be asymmetric games with identical strategies for each player. For example, the game shown on the right is asymmetric despite having identical Strategy Sets for both players.

## Economy and business

Economists have used game theory to analyze a wide range of economic problems, including auctions, duopolies, oligopolies, the formation of social networks, and voting systems. These investigations are normally focused on particular sets of strategies known as solution concepts.

These solution concepts are normally based on what is required by the standards of perfect rationality. The most famous is the Nash equilibrium. A set of strategies is a Nash equilibrium if each represents the best response to other strategies. In this way, if all the players are applying the strategies in a Nash equilibrium, they have no incentive to change their behavior, because their strategy is the best one that they can apply given the strategies of the others.

Game rewards typically represent the usefulness of individual players. Often rewards represent money, presumed to correspond to an individual’s utility. This assumption, however, may not be correct.

A game theory paper in economics begins by presenting a game that is an abstraction of a particular economic situation. One or more solutions are chosen, and the author demonstrates which set of strategies correspond to the balance in the game presented. Economists and business school teachers suggest two main uses.

### Descriptive

The main use is to inform about the behavior of current human populations. Some researchers believe that finding the balance of games can predict how human populations would behave if they faced situations analogous to the game studied. This particular view of game theory has been criticized today. First, it is criticized because the assumptions of theorists are frequently violated. Game theorists may assume players who always behave rationally and act to maximize their benefits (the homo oeconomicus model), but real humans often act irrationally or rationally but seeking the benefit of a larger group (altruism).

Game theorists respond by comparing their assumptions with those used in physics. Thus, although their assumptions are not always maintained, they can treat game theory as a reasonable idealization, in the same way as the models used by physicists. However, this use of game theory has continued to be criticized because some experiments have shown that individuals do not behave according to equilibrium strategies. For example, in the centipede game, the 2/3 mean guessing game, and the dictator game, people often do not behave according to the Nash equilibrium. This controversy is currently being resolved.

On the other hand, some authors argue that Nash equilibria do not provide predictions for human populations, but rather provide an explanation of why populations that behave according to the Nash equilibrium remain in that behavior. However, the question of how many people behave like this remains open.

Some game theorists have put hopes in evolutionary game theory to resolve those concerns. Such models presuppose or not rationality or a limited rationality in the players. Despite the name, evolutionary game theory does not necessarily presuppose natural selection in the biological sense. Evolutionary game theory includes biological and cultural evolution and also models individual learning.

### Normative

On the other hand, some mathematicians do not see game theory as a tool that predicts the behavior of human beings, but as a suggestion on how they should behave. Since the Nash equilibrium constitutes the best response to the actions of other players, following a strategy that is part of the Nash equilibrium seems the most appropriate. However, this use of game theory has also received criticism. First, in some cases it is appropriate to play according to a non-equilibrium strategy if one expects that others will also play according to equilibrium. For example, in the game guess 2/3 of the mean.

The prisoner’s dilemma presents another potential counterexample. In this game, if each player pursues his own benefit, both players get a worse result than not having done so. Some mathematicians believe that this demonstrates the failure of game theory as a recommendation of what to do.

## biology

Unlike the use of game theory in economics, the rewards of games in biology are often interpreted as adaptation. Furthermore, his study has focused less on the equilibrium that corresponds to the notion of rationality, focusing on the equilibrium maintained by evolutionary forces.

The best-known balance in biology is known as an evolutionarily stable strategy, and was first introduced by John Maynard Smith. Although his initial motivation did not carry the mental requirements of the Nash Equilibrium, any evolutionarily stable strategy is a Nash equilibrium.

In biology, game theory is used to understand many different problems. It was first used to explain the evolution (and stability) of the 1: 1 sex ratios (same number of males as females). Ronald Fisher suggested in 1930 that the 1: 1 ratio is the result of the action of individuals trying to maximize the number of their grandchildren subject to the restriction of evolutionary forces.

In addition, biologists have used evolutionary game theory and the concept of evolutionarily stable strategy to explain the rise of animal communication (John Maynard Smith and Harper in 2003 ). Analysis of signal games and other communication games has provided new interpretations of the evolution of communication in animals.

Finally, biologists have used the hawk-dove problem (also known as the chicken problem) to analyze combative behavior and territoriality.

## Computing and Logic

Game theory has begun to play an important role in logic and computing. Many logical theories are based on the semantics of games. Additionally, computer researchers have used games to model programs that interact with each other.

## Political Sciences

Research in political science has also used results from game theory. One explanation for democratic peace theory is that open and public debate in democracy sends clear and reliable information about governments’ intentions towards other states.

Furthermore, it is difficult to know the interests of undemocratic leaders, what privileges they will grant and what promises they will keep. According to this reasoning, there will be mistrust and little cooperation if at least one of the participants in a dispute is not a democracy.

## Philosophy

Game theory has been shown to have many uses in philosophy. From two WVO Quine works published in 1960 and 1967 , David Lewis ( 1969 ) used game theory to develop the philosophical concept of convention.

In this way, he provided the first analysis of common knowledge and used it to analyze coordination games. Furthermore, he was the first to suggest that the meaning could be understood in terms of signal sets. This suggestion has been followed by many philosophers since the work of Lewis (Skyrms 1996 , Grim et al. 2004 ).

Leon Henkin, Paul Lorenzen and Jaakko Hintikka began an approach to the semantics of formal languages that explains with concepts of game theory the concepts of logical truth, validity and the like. In this approach the “players” compete by proposing quantifications and instances of open sentences; the rules of the game are the rules of interpretation of sentences in a model, and the strategies of each player have properties that semantic theory deals with – being dominant if and only if the sentences that are played meet certain conditions, etc.

In ethics, some authors have attempted to continue Thomas Hobbes’s idea of deriving morality from self-interest. Since games like the prisoner’s dilemma present an apparent conflict between morality and self-interest, explaining why cooperation is necessary for self-interest is an important component of this project. This general strategy is a component of the idea of social contract in political philosophy (examples in Gauthier 1987 and Kavka 1986 ).

Finally, other authors have attempted to use evolutionary game theory to explain the birth of human attitudes towards morality and corresponding animal behaviors. These authors have looked for examples in many games, including the prisoner’s dilemma, the deer hunt, and the Nash deal game to explain the reason for the emergence of attitudes about morality (see Skyrms 1996 , 2004 ; Sober and Wilson 1999 ).

## Game Theory and Statistics

The study of games has inspired scientists of all time to develop mathematical models and theories . The Statistics is a branch of mathematics that emerged precisely from the calculations to design strategies winning in gambling.

Concepts such as probability, weighted average and distribution or standard deviation, are terms coined by mathematical statistics and that have application in the analysis of games of chance or in the frequent social and economic situations in which decisions must be made and risks taken before random components.

But game theory has a very distant relationship with statistics. Its objective is not the analysis of chance or random elements but of the strategic behavior of the players.

In the real world, both in economic and political or social relations, situations are very frequent in which, as in games, their result depends on the conjunction of decisions of different agents or players. It is said of a behavior that is strategic when it is adopted taking into account the joint influence on the own and external result of the own and other people’s decisions.

The technique for analyzing these situations was developed by a mathematician, John von Neumann . In the early 1940s he worked with economist Oskar Morgenstern on the economic applications of that theory. The book they published in 1944 , “Theory of Games and Economic Behavior”, opened an unexpectedly wide field of study in which thousands of specialists from all over the world currently work.

Game Theory has reached a high degree of mathematical sophistication and has shown great versatility in problem solving. Many fields of the General Equilibrium Economy , cost distribution, have benefited from the contributions of this method of analysis.

In the half century since its first formulation, the number of scientists dedicated to its development has not stopped growing. And they are not only economists and mathematicians but Sociologists , Political Scientists , Biologists or Psychologists . There are also legal applications: assignment of responsibilities, adoption of litigation or conciliation decisions, etc.