ABSTRACT
In Chapter 2, we present an introduction to business analytics of decision theory using stochastic modeling. Expected value is a main component, so we provide a definition and many examples. We discuss decision under uncertainty and risk. A decision tree can be used as a model for a sequential decision problem under uncertainty. A decision tree describes graphically the decisions to be made, the events that may occur, and the outcomes associated with combinations of decisions and events. Probabilities are assigned to the events, and values are determined for each outcome. A major goal of the analysis is to determine the best decisions.
We present the process of using decision trees to draw out the problem and then use the expected value to solve for the best decision. We have a method for determining certain equivalents (expected values for a risk neutral decision-maker); we do not need to examine every possible strategy explicitly. Instead, the method known as rollback determines the single best strategy. The rollback algorithm, sometimes called backward induction or average out and fold back, starts at the terminal nodes of the tree and works backward to the initial decision node, determining certain equivalent rollback values for each node. Rollback values are determined as follows:
At a terminal node, the rollback value equals the terminal value.
At an event node, the rollback value for a risk neutral decision-maker is determined using an expected value (probability-weighted average); the branch probability is multiplied times the successor rollback value, and the products are summed.
At a decision node, the rollback value is set equal to the highest rollback value of the immediate successor nodes. We present examples of the forward and fold-back method for decision trees.
