Chapter 19
Chapter 19. Decision Theory. Decision Theory. 19.1 Bayes’ Theorem 19.2 Introduction to Decision Theory 19.3 Decision Making Using Posterior Probabilities 19.4 Introduction to Utility Theory. Bayes’ Theorem.
Chapter 19
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Chapter 19 Decision Theory
Decision Theory 19.1 Bayes’ Theorem 19.2 Introduction to Decision Theory 19.3 Decision Making Using Posterior Probabilities 19.4 Introduction to Utility Theory
Bayes’ Theorem • S1, S2, …, Sk represents k mutually exclusive possible states of nature, one of which must be true • P(S1), P(S2), …, P(Sk) represents the prior probabilities of the k possible states of nature • If E is a particular outcome of an experiment designed to determine which is the true state of nature, then the posterior (or revised) probability of a state Si, given the experimental outcome E, is calculated using the formula on the next slide
Example 19.1: AIDS Testing • Suppose that a person selected randomly for testing, tests positive for AIDS • The test is known to be highly accurate • 99.9% for people who have AIDS, 99% for people who do not • What is the probability that the person actually has AIDS? • Surprisingly, much lower than most of us would guess!
Example 19.1: AIDS Testing #2 • AIDS incidence rate is six cases per 1,0000 Americans • P(AIDS) = 0.006 • P(No AIDS) = 0.994 • Testing accuracy: • P(Positive|AIDS) = 0.999 • P(Positive|No AIDS) = 0.01 • Looking for P(AIDS|Positive)
Introduction to Decision Theory • States of nature: A set of potential future conditions that affects decision results • Alternatives: A set of alternative actions for the decision maker to chose from • Payoffs: A set of payoffs for each alternative under each potential state of nature
Decision Making Under Uncertainty • Maximin: Identify the minimum (or worst) possible payoff for each alternative and select the alternative that maximizes the worst possible payoff (Pessimistic) • Maximax: Identify the maximum (or best) possible payoff for each alternative and select the alternative that maximizes the best possible payoff (Optimistic) • Expected value criterion: Using prior probabilities for the states of nature, compute the expected payoff for each alternative and select the alternative with the largest expected payoff
Example: Condominium ComplexSituation • A developer must decide how large a luxury condominium complex to build • Small, medium, or large • Profitability depends on the level of future demand for luxury condominiums • Low or high • Elements of decision theory • States of nature • Low demand versus high demand • Alternatives • Small, medium, large
Example: Condominium ComplexSituation #2 • Maximin • If a small complex is built, the worst payoff is $8 million • If a medium complex is built, the worst payoff is $5 million • If a large complex is build, the worst payoff is -$11 million • Since $8 million is the maximum of these, choose to build a small complex
Example: Condominium ComplexSituation #3 • Maximax • If a small complex is built, the best payoff is $8 million • If a medium complex is built, the best payoff is $15 million • If a large complex is build, the best payoff is $22 million • Since $22 million is the maximum of these, choose to build a large complex
Example: Condominium ComplexSituation #4 • Expected value • Small: Expected value = 0.3($8 million) + 0.7($8 million) = $8 million • Medium: Expected value = 0.3($5 million) + 0.7($15 million) = $12 million • Large: Expected value = 0.3(-$11 million) + 0.7($22 million) = $12.1 million • Since $12.1 million is the maximum of these, choose to build a large complex
Decision Making Using PosteriorProbabilities • When we use expected value to choose the best alternative, we call this prior decision analysis • Often, sample information can be obtained to help us make a better decision • In this case, we compute expected values by using posterior probabilities • We call this posterior decision analysis
Example 19.3: Decision Tree and Payoff Table for Prior Analysis
Example 19.3: The Oil Drilling Case #2 • The oil company can obtain more information by performing a seismic experiment • There are three outcomes • Low, medium, and high
A Decision Tree for a Posterior Analysis of the Oil Drilling Case
Example 19.3: The Oil Drilling Case #4 • Expected payoff of sampling • Low: Expected payoff is $0, probability is 0.646 • Medium: Expected payoff is $334,061, probability is 0.226 • High: Expected payoff is $1,362,500, probability is 0.128 • Expected payoff of sampling (EPS) is $249,898
Example 19.3: The Oil Drilling Case #5 • Expected payoff of no sampling (EPNS) is $0 • Expected value of sample information (EVSI) is • EPS – EPNS • $249,898 - $0 = $249,898 • Expected net gain of sampling (ENGS) is • EVSI – Cost of sample • $249,898 - $100,000 = $149,898
Introduction to Utility Theory • Utilities are measures of the relative value of varying dollar payoffs for an individual decision maker and thus capture the decision maker’s attitude toward risk • Under certain mild assumptions about rational behavior, decision makers should replace dollar payoffs with their respective utilities and maximize expected utility
