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IV. Small Worlds And The Grand World

Before constructing my model, some basic ideas on small worlds and the grand world are introduced based on Savage’s notions (1972, pp.8-17). However, I elaborate these ideas more to fit my purposes presented in the following sections. A small world is a confined decision situation derived from the grand world for making one decision, whereas a plan begs coordination of a set of decisions, a set of small worlds. In the small world there exists a mapping from a set of states into consequences through actions. Savage did not define explicitly and mathematically what states are except for a verbal explanation that a state of the world is “a description of the world, leaving no relevant aspect undescribed” (1972, p. 9). I provide a broader definition of states here as follows:

Definition 1: States

States are the realized outcomes of a set of independent random variables.

For example, let X be a set of m independent random variables, X1,X2,X3, . . . , Xm, of which the values are real numbers, representing the grand world. The vector (x1,x2,x3, . . . ,xm) is a state of the grand world. Assume the number of the possible values for each random variable is finite. That is, each random variable can be defined on a particular finite sample space. It follows that the number of the total states are also finite. A state of a small world is a subset of the elementary states in the grand world. For example, the outcomes resulting from the vector of independent random variables (v, Xi,Xi+1, . . . , Xi+k, v)are the states of a particular small world where v denotes the random variables that are irrelevant to the current decision situation and can be ignored, although they do exist. The grand world can thus be represented by a Cartesian product across the set of independent random variables under consideration, i.e., X1 x X2 x X3 x . . . x Xm.

Alternatively, the state in a small world can be any subset of the states in the grand world. Consider throwing a pair of dice. If the 36 outcomes represent all the states in the grand world, the events that the two dice show up even points simultaneously are the states in a particular small world, and are subsets of the states in the grand world. States in a small world are in some sense events in the grand world. The decision maker's degree of belief that a state is realized depends on the joint probability distribution across the independent random variables under consideration. Therefore, for each world, whether a small world or the grand world, there is a probability distribution characteristic of the states in the world. The derivation of such a joint probability distribution is beyond the present scope.

An act is action committed to a decision and taken by the decision maker capable of implementing the action in order to yield expected consequences. Consequences are anything that could happen to the decision maker and he or she is concerned. The notions of acts and consequences introduced here are the same as Savage’s (1972, pp. 3-17). More formally,

Definition 2: Acts

Acts are functions that map states into consequences.

Strictly speaking, there is uncertainty in the mapping between acts and consequences because that mapping is contingent on situations imposed by Nature. For simplicity, I assume that the functions are deterministic, eliminating stochastic factors of such mapping.

According to Savage, a small world is an isolated decision situation in that only subsets of the elementary states in the grand world concern the decision maker, and that only a set of admissible acts are available for the decision maker resulting in various consequences for given states (Savage, 1972, p. 14). Uncertainty plays an important role in deciding an act because no prior knowledge exists about which small world state would obtain. The best the decision maker can do is to select the act yielding the maximum expected utility among those in the available set. Measurements of probability distributions of all possible states and utilities of consequences are central in making such decision. Savage provides a general axiomatic system proving that under strict conditions, such measurements are theoretically attainable and that the decision maker should act accordingly (1972). He coined the term 'microcosm' to represent a small world in which such probability and utility measurements derived from the axiomatic system exist. In my exposition of planning behavior, I treat small worlds as microcosms so that subjective expected utility theory can be applied in my explanations. Savage’s notion also implies that there is a close interplay between a small world and the grand world. More specifically, a small world consequence is an act in the grand world that triggers further consequences in the grand world. Therefore, an act in the small world will lead indirectly to the corresponding consequences in the grand world.

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