Recursive Decision Systems: An Existence Analysis

Abstract

In this paper decision systems are structured so that topological concepts can be applied to formulate and help solve existence problems. Existence of stationary states and orbits is established. The analysis is then applied to programs, a special class of decision systems in which the decision operator is a mathematical program. The theory is extended to of many decision makers with rolling schedules of future actions. RECURSIVE DECISION SYSTEMS (RDS's) are dynamic systems based on discrete time that represent the positive behavior of decision makers. They have been put to three basic uses, (1) to describe the behavior of various economic sectors, (2) to show how indirect policies can in some particular way improve the performance of the economic system under investigation, and (3) to formulate and analyze a variety of dynamic economic theories. In this paper we define RDS's so that topological concepts and theorems can be used to study existence questions. Existence theorems are then given for stationary states and compact orbit sets. Special attention is given to the class of RDS's called programs (RP) of which various recursive program- ming models are special cases. The paper concludes with some brief comments on the assumptions used in the analysis. Before proceeding to the formal definitions, we briefly review in a nontechnical manner the basic concepts underlying RDS's and their use in economic research. RDS's as defined here are mathematical of socioeconomic processes having two basic components: (1) a decision operator that describes the manner in which final decisions or actions are derived from a given amount of information about the decision maker's environment ;2 and (2) afeedback operator that describes how decisions once acted on, or once scheduled for the future, interact with the decision maker's environment to produce new information upon which succeeding plans can be based.3 A given decision operator may represent the decision process not only of a single decision maker, but also of a group of decision makers who make their decisions independently-or collusively-during the same time period. Further- more, the decision at a given time may represent not only an immediate choice,