Research On The Grey Game Theory And Its Application In Economy

In Game theory, there are many problems, such as incomplete information, bounded rationality and uncertainty of future (or poor information, Grey System and its Application, Sifeng Liu, 1999). However nowadays in the view of information in game theory, the complete and incomplete information means respectively symmetry and asymmetry information of players, and to a certain extent, there exits a serious defect of omitting the uncertainty of information such as lack of information. In fact, because of all sorts of stochastic and non-stochastic factors,the pay-offs of random twice games could not be consistent though it is on strict restricted condition. In other words, the pay-off matrix of game can not be clear and exact, so that the analysis condition of classical game theory is hard to be satisfied and lack of information and finite knowledge is ubiquitous [1-3]The understanding of people to system is impossible to be absolutely complete for that the system people face usually is not"white"but"grey". Here use"white"to represent completely known information, and"grey"for those information which are is partially known and partially unknown. Many problems discussed in Game Theory almost are grey. But in classical game theory, the grey systems of real life are always dealt with by white system by devilishly predigestion, with the resultant that the guidance of forecast by game theory largely decreases.The project of my dissertation, Research on the Grey Game Theory and Its Application in Economy, aims to study the problems of bounded rationality and incomplete information of Game Theory by the abundant theory and correlative methods of Grey System. The game models of grey matrix, grey bi-matrix and grey evolutionary game, which could describe economic problems better and more truly, are put forward, and concise and effective conceptions and structure of solution are worked out. By some foundation works of system of grey matrix, grey bi-matrix and grey evolutionary game, this dissertation presents a powerful tool to know and settle real economic problem.Above all, the dissertation discovers the game problem with incomplete information, which is defined as grey game problem by grey system theory, and advances the Grey Matrix Game Model based on pure strategy and mixed strategy. The main works are setting up the conceptions and structure of grey saddle and grey mixed strategy, and settling the difficulty of finding grey mixed strategy of grey matrix game due to the original systematic defect of grey number computation. It has been found that, mixed strategy solution is easy to be magnified to useless black number by present method. The definition and form of grey saddle of pure strategy of game problem are provided in the dissertation, and also arithmetic of solution is researched. And it proves that random grey matrix game must have a solution of grey form, which is given in maximum minimize theorem.In this dissertation, a series of important conceptions are advanced, such as conceptions of the most optimistic and pessimistic value matrix, superiority and inferior strategy, long row grey matrix, long column grey matrix, grey full rank extended square matrix and its grey inverse matrix. Moreover a series of theorems are proved, including theorem of virtual added strategy of player being the most inferior strategy of original game, theorem of constitution of virtual grey pay-off square matrix, and also including theorem of the optimal strategy and grey value of player, the sufficient and necessary condition of grey full rank extended square matrix, theorem of full rank process of grey singular extended square matrix and theorem that full rank process does not change the optimal solution of original game. On the basis of these works, author found the solution to non-full rank grey pay-off matrix game, and gave the simple, practical and effective solution to mixed strategy of grey matrix game. For the first time, the dissertation puts forward and solutes the denotation, measure and control problems of overrated and underrated risks of optimal mixed strategy, which are due to the incomplete information of grey matrix game. Furthermore the theoretic system of measure and control of risk of grey matrix game is set up and developed. And here the main structure and theory of grey matrix game is mainly completed on the whole.For the first time, the dissertation advanced the equilibrium analysis problem and existence problem of equilibrium point in the condition of incomplete information. And the static game structure with revenue of symmetric information loss is built up, and a series of conceptions are developed, including superiority, inferior and equipollence position degrees of income in interval grey number, grey position-pure strategy Nash equilibrium, and grey position-pure dominant strategy equilibrium, also decision rules of grey positions of grey interval numbers are presented. It is proved that, if the grey position-pure strategy Nash equilibrium exists in n -players static game with revenue of symmetric information loss, it could be easily found according to different conditions by the analysis methods of grey position-pure dominant strategy, grey position-pure dominated strategy and grey position-underline-arrow, therefore the result of this category game could be forecasted quite reliably. Taking the example of price competition of color TV set, whose revenue is in symmetric information loss in real life, thesis discusses its grey position-pure strategy Nash equilibrium point and its grey position- dominant strategy, which explain real life effectively.The core study of dynamic game is perfect Nash equilibrium analysis of subgame, the key method of which is converse inductive method[8,9]. For a long time, the situation of paradox of converse inductive method always puzzled academia. This thesis discovers the root of paradox of converse inductive method, and it has two aspects. On one hand, during using this method, the microscopically logical illation makes the macroscopically logical viewpoint omitted, and in other words, emphasizing short-term interest but neglecting long-term interest. On the other hand, the classical model structure of multi-phase dynamic game could not be used to analyze the whole and long-term interest. This thesis builds up a new model structure of dynamic game based on pilot value of future income, designs the arithmetic of regularization of grey number and the conception system of the end and the pilot of Nash equilibrium solution, and also provides the effective equilibrium analysis method. All these works help solute the Paradox of Centipede Game well.The dissertation overcomes lacuna of evolutionary game model that the model can not forecast results of one-off game and short-term economic equilibrium, and designs evolutionary game chain model based on symmetry and asymmetry cases. Furthermore, it discovers profoundly relationships that players in the game are in dependence and conversion each another and establishes transfer formulas of player quantities and expectation average payments in per step of the game. On the basis of that, taking the eagle-pigeon game as an example, author imitates and analyzes copy dynamics and evolutionarily stable strategies (ESS). Imitated experiment shows the conclusion that there is exclusive equilibrium point of ESS x on symmetry case of the game, the left area of which is copy evolutionary area of the eagle that quantity of the eagle increase and quantity of the pigeon decrease in process of the game, and the right area of which is copy evolutionary area of the pigeon that quantity of the pigeon increase and quantity of the eagle decrease in process of the game. On asymmetry case, the equilibrium point, which is decided by copy evolutionary areas of the eagle and pigeon, curves of initial critical value and evolutionarily stable strategies, could be gotten by imitation. This thesis firstly discovers the biology evolutionary phenomenon of trial and error.It is found that there are some defects in the classical first-price sealed auction model, whose conditions are restricted too much to fit the real situation, by thoughts of grey system. The dissertation designs grey correction factor of experiential ideal quotation, and builds optimal grey quotation model based on accurate evaluation of value and experiential ideal quotation based on finite reasonability. However, bidder's menace reflection coefficient is gotten by the way of first standard grey transform. It is discovered that bidder's optimal grey quotation depends not only on values the bidder itself estimates but on values the rival estimates, and also on menace reflection grey coefficient. The optimal grey quotation of bidder is not the half of values of goods at auction, but higher than the half of the values generally. Furthermore, with simulation of the model, the dissertation proposes significantly some conclusions which are different from the classical model, and the optimal patterns of bidding.