Decision Analysis

Decision Analysis Chapter 15: Hillier and Lieberman Dr. Hurley’s AGB 328 Course

Terms to Know • Alternative, State of Nature, Payoff, Payoff Table, Prior Distribution Prior Probabilities, Maximin Payoff Criterion, Maximum Likelihood Criterion, Bayes’ Decision Rule, Crossover Point, Posterior Probabilities, Probability Tree Diagram, Expected Value of Perfect Information, Expected Value of Experimentation, Nodes,Branches, Decision Node, Event Node

Terms to Know Cont. • Backward Induction Procedure, Spider Chart, Tornado Chart, Utility Function for Money, Decreasing Marginal Utility for Money, Risk Averse, Increasing Marginal Utility of Money, Risk Neutral, Risk Seekers, Exponential Utility Function

Goferbroke Company Example • Trying to maximize payoff from land that may have oil given then could drill or sell the land • If the company drills for oil and oil exists, they expect a payoff of $700K • If the company drills for oil and oil does not exist, they expect a payoff of -$100K • If the company sells the land it receives $90K whether the oil exists or not • There is a 1 in 4 chance that oil exists

Payoff Table for Goferbroke

Maximin Payoff Criterion • This criterion identifies the worst payoff for each decision that you could make and maximizes the highest of these amounts • For Goferberoke this would be sell the land • This criterion is for the very cautious

Maximum Likelihood Criterion • This criterion requires you to select the best payoff from the highest likelihood state of nature • For Goferbroke, the best decision based on this criterion is to sell the land

Bayes’ Decision Rule • This criterion calculates the expected value of each decision and then chooses the maximum of these expected values • For Goferbroke, the expected payoff for drilling is 100K while for selling it is 90K • A nice attribute about Bayes decision rule is that you can conduct a sensitivity analysis to find what probability would cause you to change your decision from the given prior probabilities • You can do this by finding the probability that will cause one decisions expected payoff to equal nother decisions expected payoff

Bayes Theorem • Let Ai represent the true state is i where i = 1,2,…,n • Let Bj represent the finding/event j occurring where j = 1,2,…,m • Let P(•) represent the probability operator and P(•|•) represent the conditional probability operator • Then Bayes Theorem states:

Bayes Theorem Using a Tree Diagram P(B1 |A1) P(B1|A1)P(A1) P(A1) P(B2 |A1) P(B2|A1)P(A1) P(B1 |A2) P(B1|A2)P(A2) P(A2) P(B2 |A2) P(B2|A2)P(A2)

Using Bayes Theorem Using a Tree Diagram for Goferrbroke • Let the probability of finding oil be 25% and 75% for not finding oil given no prior information • Let the probability of finding oil be 60% given information that is favorable to finding oil • Let the probability of finding oil be 40% given information that is not favorable to finding oil

Using Bayes Theorem Using a Tree Diagram for Goferrbroke Cont. • Let the probability of not finding oil be 20% given information that is favorable to finding oil • Let the probability of not finding oil be 80% given information that is not favorable to finding oil

In-Class Activity (Not Graded) • What are: • P(Oil|Unfavorable) • P(No Oil|Favorable) • P(No Oil|Unfavorable)

Calculating Expected Payoffs of the Alternatives Given Information from Seismic Study • Expected payoff if you drill given that the findings were unfavorable: • Expected payoff if you sell given that the findings were unfavorable: • Expected payoff if you drill given that the findings were favorable: • Expected payoff if you sell given that the findings were favorable:

Expected Payoff with Perfect Information (EPPI) • EPPI calculates the expected value of the decisions made given perfect information • This measures assumes that you will have chosen the best alternative given the state of nature that occurs • Hence you will multiply the probability of the state of nature by the best payoff achievable in that state • For the Goferbroke example, if oil exists you would choose to drill receiving 700 and ifoil does not exists you would choose to sell receiving 90 • Goferbroke’s EPPI = 0.25*700+0.75*90 = 242.5

Expected Value of Perfect Information (EVPI) • EVPI = Expected Payoff with Perfect Information – Expected Payoff without Perfect Information • Expected Payoff without Perfect Information is just the value you get by using Bayes Decision Rule of maximizing expected payoff • Goferbroke’s EVPI = 242.5-100=142.5 • If the seismic survey was a perfect indicator, you would choose to do it because the EVPI is greater than the cost of the survey

Expected Payoff with Experimentation (EPE) • EPE = • Where: • P(Bj) is the probability that finding j occurs • E(payoff|Bj) represents the expected payoff that you get if finding j occurs • Note that this payoff does not factor in the cost of colleing the needed information • Goferbroke’s EPE = P(Favorable)*E(payoff|Favorable) + P(Unfavorable)*E(payoff|Unfavorable) = 0.3*300 + 0.7*90 = 153

Expected Value of Experimentation (EVE) • EVE = expected payoff with information – expected payoff without experimentation • Goferbroke’s EVE = 153 – 100 = 53 • Since this exceeds the cost od the information, Goferbroke would proceed with undergoing the survey

Decision Trees • Decision trees can be a useful tool when examining how to make the optimal decisions when there is multiple alternatives to choose from • In the trees, you have decision nodes which are represented as squares and event/chance nodes that are represented by circles • You also have the payoffs that occur due to a sequence of decision and event nodes occurring • To solve these decision trees you work your way from the end of the tree to the beginning of the tree

Goferbroke’s Decision Tree Example • Discussed in class

In-Class Activity (Not Graded) • Do problem 15.2-7 • Do Problem 15.4-3

Decision Analysis