19 Decision theory
Frank Knight influenced decision theory greatly and was one of the first to discuss risk and uncertainty as part of decision making. Knight is best known as the author of the book Risk Uncertainty and Profit, based on his Ph.D. dissertation at Cornell University. In that book, he carefully distinguished between economic risk and uncertainty. Situations with risk were those where the outcomes were unknown but governed by probability distributions known at the outset. He argued that these situations, where decision making rules such as maximising expected utility can be applied, differ in a deep way from “uncertain” ones, in which not only the outcomes, but even the probability models that governed them, were unknown. Knight argued that uncertainty gave rise to economic profits that perfect competition could not eliminate.
The Hurwicz criterion attempts to find a middle ground between the extremes posed by the optimist and pessimist criteria. The Hurwicz criterion incorporates a measure of pessimism and optimism by assigning a certain percentage weight to optimism and a balance to pessimism. A weighted average can be computed for every action alternative with an alpha‐weight α, called the coefficient of realism. The term Realism here means that the unbridled optimism of maximax is replaced by an attenuated optimism as denoted by the α. Note that 0 ≤ α ≤ 1. Thus, a better name for the coefficient of realism is coefficient of optimism. An α=1 implies absolute optimism (maximax) while an α = 0 implies absolute pessimism (maximin).
A utility curve can be constructed by measuring the attitude of the decision maker toward risk. The shape of each curve is a function of the individual´s attitude toward risk (an s-shaped curve describes a person who is risk taker when is poor, but once he has accumulated wealth, will become risk averse).
Decision trees are graphic tools for describing the actions available to the decision maker, the events that can occur, and the relationship between these actions and events. Decision trees are particularly useful for analyzing situations that involve sequential decisions.
After the tree has been drawn, it is analyzed from right to left. The aim of this analysis is to determine the best strategy of the decision maker, that means an optimal sequence of the decisions. To analyze a decision tree, we must know a decision criterion, probabilities that are assigned to each event, and revenues and costs for the decision alternatives and the chance events that occur. A firm is deciding between two alternatives: to introduce a new product or to keep the existing product. Introducing a new product has uncertain outcomes in dependence on the demand. If the demand is high, the resulting profit of the firm will be 140. The low demand will be result in the profit 80. The firm estimates the probabilities of a high and low demand 0.7 and 0.3, respectively. If the firm keeps the existing product, its profit will be 110. The estimated profit is written at the end of the chance branches. The probabilities of a high and a low demand for the new product are written below the branches leaving the chance node. The nodes are numbered. For the chance node 2, we calculate the expected value of the profit (0.7*140 + 0.3*80 = 122) and write this value over the node 2. At the decision node 1, we select the decision alternative with the higher expected profit. Because max (122;110) = 122, introducing the new product is profitable.
The certainty equivalent for an alternative is the certain amount that is equally preferred to the alternative. An equivalent term for certainty equivalent is selling price. If your certainty equivalent for alternatives specified in terms of profits is less than the expected profit for an alternative, you are said to be risk averse with respect to this alternative. If your certainty equivalent is equal to the expected profit for the alternative, then you are said to be risk neutral. Finally, if your certainty equivalent is greater than the expected profit for the alternative, you are said to be risk seeking. These definitions are reversed for an uncertain alternative specified in terms of losses. That is, you are risk averse if your certainty equivalent
is greater than the expected loss and risk seeking if your certainty equivalent is less than the expected loss.
Procedure
- List the possible alternatives (actions/decisions)
- Identify the possible outcomes
- List the payoff or profit or reward
- Select one of the decision theory models
- Apply the model and make your decision