Research On A Type Of Repeated Intertemporal Choice Problems With Markov Properties

Decision-making is a process of selecting a preferred solution of an individual or a group from a set of alternative solutions. When faced with a complex decision-making problems, it's difficult for an individual to make the best decision or scientific decision just depending on its own experience and wisdom, There exists many deficiencies. Assembling preferences of several individuals and gathering more information can effectively reduce the risks of individual decisions or prejudice. Therefore, compared with individual decision-making, group decision-making sometimes is of more rationality and scientificity and group decision-making is so far one of the major areas of decision-making theory. Intertemporal choice is a multi-stage dynamic group decision making problem. With the development of the decision-making stage, decision-making individuals of groups, set of alternative options, and group's preference options may have some change, making the whole process more complicated.Although the individual's preferences may change with the decision-making stage, and the choice of each stage is closely concerned with the group's long-term social welfare, individual's long term profit and the result of each stage, such seemingly complex intertemporal choice sometimes show some Markov properties. This article establishes a Markov Decision dynamic programming model, and aims to provide a tool for the analysis and solution of some intertemporal choice or complex group decision-making problems. However, finding the optimal strategy to general Markov decision model directly often face difficulties of too large state space and difficulties of operating and calculating. In this paper, considering a typical voting problem, we utilize the symmetry existing in the change of individual voting preferences to reduce the time complexity of solving the model, and explain the feasibility and practicability of this simplified approach.Condorcet cycles may occur in the social choice problem with three voters having preferences over three alternatives. For the repeated intertemporal to max the reward of voters, voters can vote to avoid the cycles by appropriate strategies and select a unique winner. In this paper, the repeated voting is regarded as a Markov process. Assuming that the preferences are completed or incompleted revelatory, with the regard of long-term utility, we discuss the establishment of infinite-stage discount Markov decision model based on the maximization of the participants' expected utility, study the existence of Condorcet winner and Condorcet cycles when rational voters vote by mixed strategies(equilibrium strategies), analyze and work out the mixed strategies; and then analyze the possibilities and the results of voters revealing their preferences under different information structures based on the above-mentioned works.Finally, the paper cited the example of a real political campaign, analyse and interpret some of the game phenomenon combining theoretical research findings. And therefore the theory and application of our research results is proved.