Studying On Group Decision-making Dynamics Based On The Hamilton-Jacobi Equation

Human behaviors are usually determined by some social and/or economic trend. In the past few years, many attempts have been made, in the field of complex scientific systems, to describe the dynamics of these behaviors quantitatively and have obtained some accurate understandings of the corresponding mechanisms. These researches have made many progresses in exploring spatial-temporal dynamics of human mobility, dynamics of opinion, social networks and some related transportation, and so forth. In this thesis, we focus on the first two branches in complexity study.Firstly, a generalized potential, that is, a migration desire function defined by the age of the migrating people, the migrating distance, and the so-called economic-population density of the emigration area, is proposed. It can be transformed into Hamilton-Jacobi equation by using a random dynamical method, from Langevin equation to Fokker-Planck equation. So the decision-making behavior can be captured based on a statistic framework during a group migration process. By taking use of the multi-dimensional steepest descent method, the Hamilton-Jacobi equation is solved. The solution shows that the information entropy of the system varies, leading by a single peak, as the age of the migrating people increases. It also demonstrates that the second derivative of the migrating distance to the information entropy has a change of zero-crossing (which actually means a phase change).The third characteristic of the solution is that the information entropy follows another single peak as the so-call economic-population density increases. These relationships incarnate the correlations of the migration with the concrete characteristics of migrants or visitors, age, distance and economic-population density.The deeper analysis reveal that the significance behind these findings and the corresponding mechanisms. So some new understandings of the group behaviors can be obtained, and some worthy references can be provided for some related administrative offices.Secondly, a similar theoretical framework is suggested for investigating the dynamics of public opinion so as to reveal the physical mechanism of opinion formation. The investigation mainly focuses on the heterogeneities such as age, connection circle and overall quality of the participants involved in the opinion exchange process. The evolutionary patterns of opinion can be described by solution of the Hamilton-Jacobi equation, information entropy. In order to exhibit more detailed dependences of opinion formation on the three determining factors of the participants, the results are presented in three phase-diagrams by the contours of information entropy. The phase-diagrams presented in spaces, age-connection circle, overall quality-connection circle and overall quality-age, incarnate different aspects of the correlations of opinion evolution with the determining factors. In general, the results indicate that the overall qualities of the participants play critical roles in forming an opinion. The higher the overall quality is, the easier the consensus can reach. The solution also demonstrates that the age and the connection circle of the agents play equally important roles in forming an opinion. The essence of the age, overall quality, and connection circle corresponds to the maturity of thought (opinion inertia), reason and intelligence, influence strength of the environment, respectively.Specifically, information entropy distribute regularly in age-connection circle space. Its typical characteristics are that the entropy varies with connection circle extending following a single peak and changes with age increasing following a "U-type" valley. In overall quality-connection circle space, information entropy distributes irregularly in right-middle local region, and distributes symmetrically in the top region and bottom region. One of the typical features is that there exist a series of island chains formed by the contours of entropy with different heights. These island chains correspond to the irregular variation of information entropy. Information entropy also distribute regularly in overall quality-age space. Its typical characteristic is that the entropy is of negative value in the zonal area being parallel to the leading diagonal of the phase plane. And that the information entropy distributing in the phase planes can exhibit, one may say, the evolution rules of an opinion in detail.So, one may draw a conclusion that the theoretical method proposed here can captured some nature of human behaviors, migration and opinion formation.