论文修订版英文

The Development History of Game Theory and the Basic Content

Zhe-Zhe Hu, Qian Zhang

Faculty of Science，Xi’an University of Architecture and Technology,Xi’an,China,710055 Abstract Game theory as a mathematical tool to analyze competitive strategy theory．It studies the strategy of interactive behavior and helps them to analyze competitors in response in order to make the best response．

Keywords Game theory prisoner’s dilemma Nash equilibrium

AMS

Introduction After more than half a century of short development，game theory has become an important subject in the field of econamics．And they are widely application field of game theory in economics，political science，military strategy，evolutionary biology，as well as the contemporary computer science has become an important tool to research and analysis．In addition，it also with accounting，statistics，mathematical basic，social psychology and philosophy branches such as epistemology and ethics has important connection．Game theory has been the world’s universal recognition. What is a Game theory，what is the main content of it．This article will discuss briefly.

1．The origin and the development of history of Game theory

Game theory thought is an ancient, sun tzu’s the art in ancient China is not just a military workand is one of the earliest a game theory

monographs. The initial research of game theory is the outcome of chess, bridge, gambling problem. The grasp of the situation in game is only on experience and there is no theoretical development.

Important character of modern game theory research started by Zermelo, Borel and von Neumann.

In 1928, von Neumann proved that basic principle of game theory, which declared the birth of game theory. In 1944, the landmark article 《the game theory and the economic activities》of von Neumann and Morgenstern makes two people game to n game structure and applies game theory system in the economic field, which laid the foundation of the discipline and system of theory.

In 1950-1951,John Forbes Nash proved the existence of the equilibrium by using the fixed point theorem, which laid a solid foundation for the generalization of game theory. Nash’s seminal paper 《n people game equilibrium》(1950),《the non-cooperative game》(1951), and so on, which gives the concept of Nash equilibrium and equilibrium existence theorem. In addition, Seldon, Hassan’s research also play a role in promoting development of game theory.

Previous mathematics is developed on the basis of physics, it describes the objective material behavior. And economics research object is more like a game of the participants, expecting between each other. Therefore, it requires a series of mathematical tools to describe the study

object. They have been named as the game theory.

Now game theory has developed into a more perfect discipline. Game theory. Decision Theory and Operation Research together constitute the theoretical basic of economic of modern enterprise ,military strategy, system management and so on.

2. The model of the prisoner’s dilemma[1]

The model of the prisoner’s delemma is a classic case in game theory, it was suggested by tucker in 1950s.The basic content is:The police caught two criminals who named Jia and Yi. But the police can’t charge them because lack of evidence, but if only one of them admit the crime and they will be accused.

In order to obtain confessions, the police locked up both of them and inquest them. They were locked apart to prevent their gang up. The police told them two choices they were faced and the consequence: First, if both of them admit the crime, they will be sentenced to five years; Second: The person will be released immediately who admit the crime first, and the left one will be sentenced to ten years. The last: if they both don’t admit the crime, because don’t have enough proof, they will be sentenced to one year for crime of interference with public function. The model is:

The game theory researched is Jia and Yi take the strategy in order to maximize their own interests. In this model, Jia and Yi have fully understand the likely profit of each other and have a strategy by the other one’s choice. For Jia, he will be sentenced to five years if he admit the crime when Yi do the same thing, he will be released if he admit the error but Yi don’t. So, Jia always say the truth whatever Yi choice. The same to Yi, so the (admit, admit) is the best strategy.

Besides, we can see from the model, if both of them don’t admit the crime, they will be sentenced to one year. The result is much better than they admit the truth. They want to have the best choice whatever they do, but it can’t be. It means the best choice and the best strategy is not same. It reflect the contradiction between their own interests and collective interests, this very common in life.

3. The definition 、element and classify of game theory

The game theory has another name: strategy theory. It is a theory study some individuals and teams and organizations in a particular environment depend the information they have with some rules, at the

same time or soon or later, one time or some time, take a action from all choices they have, they will get a corresponding result.

3.1 Game and player

The Game and player is mean some people who can have a independent decision and bear the consequence. The game can be individual 、team or organization even a country. According to how much the number of participants, the game can be divided in to single game and multiplayer game. And the multiplayer game can be divided cooperative game and non cooperative game. The non cooperative was applied more widely in the field of economics, it is the focus of the current study of game theory. And the cooperative game was main applied in political and social .

3.2 Strategy

Strategy is also named strategic. It means the player’s choices in game. It decide the time when players should take a action. Every decision has corresponding consequence. According to the partition of the set of strategies, the game is divided to Game and Infinite strategy game. The more the number of the more complex game decision. So the Infinite strategy game is more complex than strategy game.

3.3 Pay off function[2]

The pay off function also call profit, it’s a result by participants take some strategies. It is the embodiment of the benefit level of

participants from the game. The parties in accordance with the revenue division, the game can be divided into zero-sum game, constant-sum game and change-sum game.

3.4 The order of the game

The order of the game is participants select strategies and action sequence. According to the decision problem, the game is divided into static game and dynamic game. The static game is mean the participant can take decisions and actions, don’t need to consider the sequence. The dynamic game is mean the participants in order to take action and the latter can be observed by the former strategies and actions.

3.5 Information

The information refers to the information structure of the game. Especially the level of understanding of the relevant strategies and each game revenue. From the information structure of the participants, the game is divided complete information game and incomplete information game. Complete information refers to the participants must be well know about the strategy space to other people in the game participants and the characteristics of action and pay off function. Otherwise it’s incomplete information.

The general game is made up of at least three factors. In addition there are some other concepts of game theory, for example the action. It means the participants at a certain time point game decision variables.

The result is the set of the most interests to participants. The balance is mean the optimal strategy combination of participants.

Divided according to the combination of game decision and information, the game was divided into: complete information static game, complete information dynamic game and incomplete information static game and incomplete information dynamic game. The same equilibrium can also be divided into Nash equilibrium, subgame perfect Nash equilibrium and Bias Nash equilibrium, perfect Bias Nash equilibrium. 4.Nash equilibrium[3] game theory and Evolutionary game

theory

The basic concept of Nash equilibrium in game theory is Nash equilibrium and refining. Nash equilibrium can be described as: if a game has a strategic combination, any participant to change this strategy may lead to low their level of benefits, thus any participants are not actively to change the strategic combination. This strategy is called the Nash equilibrium of the game.

The Nash equilibrium is completely rational participants and the game information is complete and asked the participants to correct know how other players will choose, that is expected to meet the principle of consistency. The participants in the treatment of incomplete information to assume that the participants knew all the possible state of the world and know the distribution of objective probability in a random sample of

the state and asked the participants to have strong computing ability and reasoning ability.

From the Nash equilibrium of the game theory and evolutionary game theory is a progress and evolutionary game theory is a progressive process, reflects the change from full rationality to bounded rationality, the transition from complete information to incomplete information, from the analysis of behavior change to system analysis, the transition from simple game to repeated game, change from static to dynamic game, study on transformation from individual to group study.

5. The relationship between game theory and reality

Game theory and reality are closely related. The game theory of modern enterprise management concept and mode change has important guiding significance. We often encounter a variety of price wars and alliance in life, But this does not necessarily promote the development of enterprises, while strengthening the cooperation between enterprises and enterprises will be obtained and game is a shortcut to win-win day. In our daily life, about our personal queuing and queue, do not seat and seat, it also reflects the game and so on and take forcible possession of knowledge. The relationship between countries only cooperation to achieve the interests of the two countries.

By virtue of its advantages in the theory of game theory, has developed into the frontiers of knowledge economics. Although it has

been widely recognized, but there are still some problems. Based on game theory and game rules of weak information, a lot of game theory is still in the stage of hypothesis, also need to accept the reality of the inspection. It is because of these problems, game theory have a great development momentum and broad development prospects. We believe that in the near future, the development of game theory will take the last new step. To make more contribution to the society.

6. Convey thanks

I want to thank my teacher, in the topic and writing process, he spent a lot of time and effort during my writing. To adjust the structure of the small word error, he pointed out that one. The rigorous style of the teacher impressed me, once again thank the teacher.

My sincere thanks also go to our team, we encourage each other in the process of writing, to show each other the way forward and let us have more confidence in confidence.

7. References

[1]L-Q Zhang, C-W Feng. Rationality and limited rationality: One the relationship between classical game theory and evolutionary game theory. [J]. World economy, 2001 (8).

[2]L-Guo On the game theory .[J]. The economy of Shandong 1999 (6).

[3]Y-J Zhang The development of game theory and application in practice [J]. Theoretical exploration,2005 (2).