Applied Statistical and Optimization Models Topic 10: Decision Making Criteria

Objectives Understand what decision analysis is Understand decision-making criterion Maximax Maximin Maximum Likelihood Equal Likelihood (Laplace) Expected Monetary Value Solve simple decision tree to determine expected monetary value Calculate Expected Value of Perfect Information (EVPI).

Decision Analysis What is decision analysis? States of Nature Better Helps guide decision-making under uncertainty in the presence of alternative decision-making choices for different states of nature whose probabilities and payoffs are a priori (in advance) known. A decision analysis problem can often be set up in a general table like this States of Nature Better As is Worse Some Value to base decision on Project A Payoff $ Project B Project C Probability for States of Nature p1 p2 1-p1-p2

Decision Analysis What is decision analysis? States of Nature Better A decision analysis problem can often be set up in a general table that informs about the possible states of nature, their probabilities, and payoffs. States of Nature Better As is Worse Some Value to base decision on Project A 270 120 30 Project B 210 180 60 Project C 240 150 90 Probability for States of Nature 0.1 0.6 0.3

Decision-Making CriteriA There are five essential decision-making criteria Maximax Maximin Maximum Likelihood Criterion (MLC) Equally likely criterion (ELC) Expected Monetary Value (EMV) The Maximax and Maximin criterion do not involve probabilities, the MLC, ELC, and EMV do.

Decision Analysis Maximax decision (no probabilities involved) – For the hopeless optimist who asks “What is the best of the best that could happen?” A maximax decision taker would choose Project A. States of Nature Better As is Worse Some Value to base decision on Project A 270 120 30 Project B 210 180 60 Project C 240 150 90 Probability for States of Nature 0.1 0.6 0.3

Decision Analysis Maximin decision (no probabilities involved) – For the cautious who asks “What is the best of the worst that can happen?” A Maximin decision taker would assume that the state of nature is “worse” and then choose the best, which is Project C. States of Nature Better As is Worse Some Value to base decision on Project A 270 120 30 Project B 210 180 60 Project C 240 150 90 Probability for States of Nature 0.1 0.6 0.3

Decision Analysis The Maximum Likelihood criterion (probabilities involved) – For the realist who asks “What is the best choice of the most likely outcome?” A maximum likelihood decider would acknowledge that the most likely state of nature is “As is” and then choose the best, which is Project B. States of Nature Better As is Worse Some Value to base decision on Project A 270 120 30 Project B 210 180 60 Project C 240 150 90 Probability for States of Nature 0.1 0.6 0.3

Decision Analysis States of Nature Better As is Worse Some Value to The Equally Likely Criterion (probabilities involved) – For someone who, for some reason, wants to ignore all a priori probabilities and asks “What is the best choice if all outcomes are equally weighted?” An equally likelihood decider (sometimes also called Laplace decider) would choose Project C. States of Nature Better As is Worse Some Value to base decision on Project A 270 120 30 90+40+10=140 Project B 210 180 60 70+60+20=150 Project C 240 150 90 80+50+30=160 Probability for States of Nature 1/3

Decision Analysis States of Nature Better As is Worse Some Value to The Expected Monetary Value (EMV) criterion (probabilities involved) – For someone who takes all a priori probabilities into account and asks “What is the best choice under all a priori probabilities?” An EMV decider would choose Project B. States of Nature Better As is Worse Some Value to base decision on Project A 270 120 30 27+72+9=108 Project B 210 180 60 21+108+18=147 Project C 240 150 90 24+90+27=141 Probability for States of Nature 0.1 0.6 0.3

Decision Analysis The Expected Monetary Value (EMV) criterion can also be illustrated using decision trees and backward induction. In a decision tree, lines that branch off a box are decision alternatives, lines that branch off a circle states of nature (including their probabilities). Backward induction is then used to determine maximum EMV

Decision Analysis Summary Maximax, Maximin, Maximum Likelihood, Equal Likelihood, and Expected Monetary Value Project A Project B Project C Maximax  Maximin Maximum Likelihood Criterion Equal Likelihood Expected Monetary Value

Decision Analysis The Expected Value of Perfect Information (EVPI) Economic actors do naturally have an interest to reduce risk and uncertainty If an economic actor knew with certainty that the state of nature will be “Better” she would surely pursue A, if she knew with certainty that the state of nature will be “As is,” she would pursue B, and if she knew with certainty that the state of nature will be “Worse,” she would pursue C. More certainty would therefore release the economic actor from having to worry about taking into account other states of nature that are less favorable. The EVPI can therefore be written as EVPI = Expected Value of Perfect Information – Max Expected Monetary Value

Decision Analysis The Expected Value of Perfect Information (EVPI) The Expected Value under Certainty is simply the sum of the best expected payoffs under each state of nature. Expected Value under Certainty = 0.1×270+0.6×180+0.3×90 = 27+108+27 = 162 The Maximum Expected Monetary Value we already calculated as Max EMV= 147. The maximum EMV is the expected value under the given imperfect information while the EVPI is the expected value under perfect information. Because EVPI = Expected Value under Certainty – Max EMV = 162-147 = 15, an economic actor has an incentive to pay up to 15 for more information.

What you Should Have Learned Apply to a decision matrix Maximax Maximin Maximum Likelihood Equal Likelihood (Laplace) Expected Monetary Value Solve simple decision trees to determine expected monetary value Calculate Expected Value of Perfect Information (EPVI).