The Development of Decision Analysis Jason R. W. Merrick Based on Smith and von Winterfeldt (2004). Decision Analysis in Management Science. Management Science 50 (5) 561-574.
Why making decisions can be hard? � There are trade-offs between the alternatives � Consider buying a car, a computer or a phone � There is uncertainty about the outcomes � Consider playing the lottery, investing in the stock market, or choosing health insurance � There is a sequence of decisions to make Th i f d i i t k � Consider choosing a major and then a career � There are disagreements between stakeholders � Consider making any decision with your spouse or significant other � There is a large range of alternatives available confined by constraints � Go see Drs. Brooks, Hardin, and McLay!
Elements of a Decision � Values and Objectives � What you are trying to achieve? � Decisions and Alternatives � What you are choosing between to get what you � What you are choosing between to get what you want? � Uncertainties and Probabilities � The uncertain events that affect you getting what you want?
The Decision Context � Keeney (1992) uses the concept of a decision frame to explain the decisions that people make. � A decision frame consists of a decision maker’s set of alternatives and the objectives that the decision maker is attempting to achieve when choosing. � Suppose you are looking for a car. � What objectives might you have if you wanted a car to get to work, go shopping, and get around town? � Suppose you are looking transportation for the same purpose � How does this change your objectives for just the car choice?
Development of Decision Analysis Bernoulli Bayes Ramsey DeFinetti von Savage Neumann 1738 1763 1931 1937 1954 Morgenstern 1944 • Concerned with the fact that people generally do not follow the expected value model when choosing amongst gambles (e.g. buying insurance). • Proposed the expected utility model with a logarithmic utility function to explain the deviations from the expected value model .
Development of Decision Analysis Bernoulli Bayes Ramsey DeFinetti von Savage Neumann 1738 1763 1931 1937 1954 Morgenstern 1944 • Interested in the revision of probability based on observations and proposed the updating procedure that is now known as Bayes Theorem ( | ) ( ) P B A P A = ( | ) P A B + ( | ) ( ) ( | ) ( ) P B A P A P B A P A
Development of Decision Analysis Bernoulli Bayes Ramsey DeFinetti von Savage Neumann 1738 1763 1931 1937 1954 Morgenstern 1944 • Recognized the notion of probability and utility as intrinsically intertwined and showed that subjective probabilities and utilities can be inferred from preferences among gambles .
Development of Decision Analysis Bernoulli Bayes Ramsey DeFinetti von Savage Neumann 1738 1763 1931 1937 1954 Morgenstern 1944 • Followed a similar path as Ramsey by developing a system of assumptions about preferences among gambles that allowed him to derive subjective probabilities for events. • DeFinetti’s interest was primarily in the representation of beliefs as subjective probabilities , not in the derivation of utilities .
Development of Decision Analysis Bernoulli Bayes Ramsey DeFinetti von Savage Neumann 1738 1763 1931 1937 1954 Morgenstern 1947 • “Theory of Games and Economic Behavior”: Primary purpose was to lay the foundation for the study of games , but also established foundations for decision analysis . • Provided an axiomization of the expected utility model showing that the cardinal utility function could be created from preferences among gambles . • Analysis took the probabilities as a given and their axioms led to the conclusion that decision makers should make decisions to maximize their expected utility . • This is now referred to as the expected utility model .
Development of Decision Analysis Bernoulli Bayes Ramsey DeFinetti von Savage Neumann 1738 1763 1931 1937 1954 Morgenstern 1944 • Extended the work of von Neumann and Morgenstern to consider cases in which the probabilities are not given. • Savage’s goal was to provide a foundation for a “theory of probability based on the personal view of probability derived mainly from the work of DeFinetti.” • Savage proposed a set of axioms about preferences among gambles that enabled him to simultaneously derive the existence of subjective probabilities for events and utilities for outcomes • Combined the ideas of utility theory from economics and subjective probability from statistics in to the subjective expected utility model .
Lotteries � Let’s see what your answers would be ? $30,000 ≈ 1 $0 1-? -$10,000 � What would your answer be? $30,000 $500 ≈ 1 1-? -$10,000 � What would your answer be? � Etc…
How should we decide? � Complete Ordering Axiom ≈ f p or or r r r r r r 1 2 1 2 1 2 ⇒ f f f and r r r r r r 1 2 2 3 1 3 � These are the minimal mathematical conditions for a complete ordering � What does this mean?
How should we decide? � Continuity Axiom ⇒ ∃ > f f and 0 s.t. r r r r c 1 2 2 3 c r 1 ≈ 1 1 r 2 1-c r 3 � This is rather like the mean value theorem in calculus � What does this mean?
How should we decide? � Independence Axiom ≈ ∀ if then and r r r c 1 2 3 c c r r r r 1 2 ≈ 1-c 1-c r r 3 3 � What does this mean?
How should we decide? � Unequal Probability Axiom > f if and then r r p q 1 2 p q r r r r 1 1 f 1-p 1-q r r 2 2 � What does this mean?
How should we decide? � Compound Lottery Axiom p r 2 ≈ 1 r 1 1-p r 3 p r 2 q q r 1 1-p ⇒ ≈ r 3 1-q 1-q r r 4 4 � What does this mean?
Expected Utility Wins � Criteria that don’t satisfy these axioms � Maximin � Maximax � Minimax regret � Minimax regret � They fail the continuity, unequal probability and the compound lottery axioms � Criteria that do satisfy these axioms � Expected value � Expected utility
Three Viewpoints � There are three major angles of study about gambles and decisions Normative : the study of rational choice. � Normative models are built on basic assumptions (axioms) that � people consider as providing logical guidance for their decisions. Examples include the expected utility model and the subjective � expected utility model. t d tilit d l Descriptive : the study of how people actually think and behave. � Descriptive studies may develop mathematical models of behavior, � but such models are judged by the extent to which their predictions correspond to the actual choices people make. Major example is prospect theory. � Prescriptive : focused on helping people make better decisions. � Uses normative models, but with awareness of the limitations and � descriptive realities of human judgment.
Decision Analysis � Focused on the prescriptive power of the subjective expected utility model and Bayesian statistics. � Robert Schlaifer at Harvard wrote “Probability and Statistics for Business Decisions” in 1959. � Howard Raiffa and Schlaifer wrote “Applied Statistical � Howard Raiffa and Schlaifer wrote Applied Statistical Decision Theory” in 1961. � Ron Howard at Stanford first used the term decision analysis. Howard (1966) “Decision Analysis: Applied Decision Theory”. � Howard (1968) “The Foundations of Decision Analysis”. �
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