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

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?
Decision Analysis

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
Decision Analysis

5. Payoff Table
Decision Analysis

6. An Example
State of Nature
Alternative
Decision Analysis

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
Decision Analysis

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
Decision Analysis

9. Example: Newsboy Problem
Payoff Table
Decision Analysis

10. Example: Newsboy Problem
LaPlace Criterion
Decision Analysis

11. Example: Newsboy Problem
Maximax Criterion
Decision Analysis

12. Example: Newsboy Problem
Maximin Criterion
Decision Analysis

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

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

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

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
Decision Analysis

17. Expected Return & Variance
Example: Newsboy Problem
Expected Return & Variance
Decision Analysis

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 = The probability of state j (Sj)
rij = The payoff if choose Ai and Sj occurs
ERi= Expected return for alternative Ai
Decision Analysis

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)
Decision Analysis

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

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 = The probability of state j (Sj)
OLij = The opportunity loss if choose Ai and Sj occurs
min(EOLi) = EVPI
Decision Analysis

22. Expected Opportunity Loss
Example: Newsboy Problem
Expected Opportunity Loss
EVPI
Decision Analysis

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
Decision Analysis

24. 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 MV
Utility MV
Utility MV
Utility
Decision Analysis

25. Risk-Averse Utility Function
0.910
0.850
0.775
0.680
0.524
100
200
300
400
500
Dollars
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)
Decision Analysis

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
Decision Analysis

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 chance
Then the utility for a particular monetary value x can be found using the above assumed exponential utility function
Decision Analysis

28. Example: Newsboy Problem
Expected Utility
Decision Analysis