DSC 3120 Generalized Modeling Techniques with Applications

1. DSC 3120Generalized Modeling Techniques with Applications Part III. Decision Analysis

2. Decision Analysis • A Rational and Systematic Approach to Decision Making • Decision Making: choose the “best” from several available alternative courses of action • Key Element is Uncertainty of the outcome • We, as decision maker, control the decision • Outcome of the decision is uncertain to and uncontrolled by decision maker (controlled by nature)

3. Example of Decision Analysis You have $10,000 for investing in one of the three options: Stock, Mutual Fund, and CD. What is the best choice? Question: Do you know the choices? Do you know the best choice? What is the uncertainty? How do you make your choice?

4. Components of Decision Problem • Alternative Actions -- Decisions • There are several alternatives from which we want to choose the best • States of Nature -- Outcomes • There are several possible outcomes but which one will occur is uncertain to us • Payoffs • Numerical (monetary) value representing the consequence of a particular alternative action we choose and a state of nature that occurs later on

5. Payoff Table

6. An Example State of Nature Alternative

7. Three Classes of Decision Models • Decision Making Under Certainty • Only one state of nature (or we know with 100% sure what will happen) • Decision Making Under Uncertainty (ignorance) • Several possible states of nature, but we have no idea about the likelihood of each possible state • Decision Making Under Risk • Several possible states of nature, and we have an estimate of the probability for each state

8. Decision Making Under Uncertainty • LaPlace (Assume Equal Likely States of Nature) • Select alternative with best average payoff • Maximax (Assume The Best State of Nature) • Select alternative that will maximize the maximum payoff (expect the best outcome--optimistic) • Maximin (Assume The Worst State of Nature) • Select alternative that will maximize the minimum payoff (expect the worst situation--pessimistic) • Minimax Regret (Don’t Want to Regret Too Much) • Select alternative that will minimize the maximum regret

9. Example: Newsboy Problem Payoff Table

10. Example: Newsboy Problem LaPlace Criterion

11. Example: Newsboy Problem Maximax Criterion

12. Example: Newsboy Problem Maximin Criterion

13. Example: Newsboy Problem Minimax Regret Criterion: Step 1

14. Example: Newsboy Problem Minimax Regret: Step 2 (Regret or Opportunity Loss Table)

15. Decision Making Under Risk • In this situation, we have more information about the uncertainty--probability

16. Decision Making Under Risk • Maximize Expected Return (ER) ERi =  (pjrij)= p1ri1 + p2ri2 +…+ pmrim Where ERi = Expected return if choosing the ith alternative (Ai), (i = 1, 2, …, n) pj=The probability of state j (Sj) rij = The payoff if we choose alternative Ai and Sj state of nature occurs

17. Example: Newsboy Problem Expected Return & Variance

18. Decision Making Under Risk • High return is good, but on the other hand, low risk is also important • Variance -- a measure of the risk Variancei = pj (rij - ERi)2 Where pj=Theprobabilityof state j (Sj) rij = The payoff if choose Ai and Sj occurs ERi= Expected return for alternative Ai

19. Expected Value of Perfect Information • EVPI measures the maximum worth (value) of the “Perfect Information” that we should pay for in order to improve our decisions EVPI = ER w/ perfect info. - ER w/o perfect info. • ER w/ perfect info. = pj max(rij) • ER w/o perfect info. =max(ERi) = max(pjrij)

20. Example: Newsboy Problem Calculate EVPI ER w/o PI ER w/ PI EVPI

21. Expected Opportunity Loss (EOL) • We can also use EOL to choose the best alternative • Minimizing EOL = Maximizing ER • both criteria yield the same best alternative EOLi = pj OLij where pj=Theprobabilityof state j (Sj) OLij = The opportunity loss if choose Ai and Sj occurs • min(EOLi) = EVPI

22. Example: Newsboy Problem Expected Opportunity Loss EVPI

23. Decision Making with Utilities • Problem with Monetary Payoffs • People do not always just look at the highest expected monetary return to make decisions; they often evaluate the risk • Example: A company wants to decide to develop a new product or not

24. Utility Utility Utility 0 0 0 MV MV MV Decision Making with Utilities • Utility -- combines monetary return with people’s attitude toward risk • Utility Function -- a mathematical function that transforms monetary values into utility values • Three general types of utility functions (1) Risk-Averse (2) Risk-Neutral (3) Risk-Seeking

25. Risk-Averse Utility Function Utility 0.910 0.850 0.775 • Properties of Risk-averse Utility Function • non-decreasing: more money is always better • concave: utility increase for unit ($100, e.g.) increase of money is decreasing (extra money is less attractive) 0.680 0.524 0 100 200 300 400 500 Dollars

26. How to Create Utility Function • Method I. Equivalent Lottery • Start with two endpoints A (the worst possible payoff) and B (the best possible payoff) and assign U(A) = 0 and U(B) = 1 • Then to find the utility for a possible payoff z between A and B, select the probability p (=U(z)) such that you are indifferent between the following two alternatives • receive a payoff of z for sure • receive a payoff of B with probability p or a payoff of A with probability 1 - p

27. How to Create Utility Function • Method II. Exponential Utility Function where x is the monetary value, r>0 is an adjustable parameter called risk tolerance • First, the value of r can be estimated such that we are indifferent between the following choices • a payoff of zero • a payoff of r dollars or a loss of r/2 dollars with 50-50 chance • Then the utility for a particular monetary value x can be found using the above assumed exponential utility function

28. Example: Newsboy Problem Expected Utility