The Banzhaf Of Interval Cooperative Games And Its Expansion

Game Theory is a basic theory and approach of the interactions among the people when involving in the conflict of interest. Due to the uncertainty and variability of the real life as well as the high complication of strategies and the decision-making behaviors, people become more focused on the extension of the interval cooperative game theory. And in recent years, the interval cooperative game, as one of the mainstream research areas of the game theory, aroused the interest of many scholars who have made some great achievements.The paper is an extension of the model of the interval cooperative game from such three aspects as the Banzhaf Index of the players’ profit, the coalition structure and the coalition profit with the purpose of further improving the theory of the interval cooperative game as a powerful research tool to deal with the uncertainties in the real life.The research content of the paper is as follows:Firstly, it imposes restrictions on the formation of the alliance in the interval cooperative game by establishing a model in the augmented system-a special coalition structure; based on the features of the augmented system, it presents an axiom system of the interval Banzhaf Index of the model to prove the existence and the uniqueness of the index; It presents a method to identify the income distribution scheme of the players with the interval Banzhaf Index when the profits of the grand alliance is relatively accurate.Secondly, based on the establishment of the axiom system and by extending the crisp case, a definition similar to the Banzhaf Index-a cooperative game with fuzzy profits is proposed. Then the fuzzy Banzhaf Index is redefined according to it; It explains the relevant characteristics of the fuzzy Banzhaf Index and the conditions for its existence based on the existence conditions of the interval Banzhaf Index.Thirdly, based on the theory on the arrangement of the all-in-one machine, it defines the interval cooperative game in the arrangement. Besides, it explores the features of the cooperative game of the interval arrangement based on the corresponding axiom system and the operation of the interval numbers. What is more, it expands the analysis on the arrangement of the all-in-one machine and the relevant interval cooperative game in order to decide what the optimal arrangement is and how to make the agents take the optimal sequence.As a conculusion, it defines and illustrates the Banzhaf Index of the uncertain interval game involving limited players. The game value can be expressed with the characteristic functions and the payoff functions in the cooperative game takes the form of the interval numbers which would promote the Banzhaf Index of the TU cooperative game to the uncertain interval game with numerous applications.